Study of Debt Market
Study of Debt Market
Study of Debt Market
dicp
dflc
c AI . . . ( 13. 3)
Where df lc represent s days f rom t he last coupon and dicp represent s t he days in t he
coupon period and is t he coupon payment . We know t hat bot h t hese values depend on
t he day count convent ion and can be f ound wit h t he help of t he coupon f unct ions in
Excel.
Let us consider t he bonds in Table 13. 5. We can comput e t he accrued int erest f or t hese
bonds using t he dat a f or coupons ( provided in column 1) , given t he set t lement dat e of
February 5, 2001. The accrued int erest is t he amount of coupons t hat are due t o t he
seller, having held t he bond f rom t he previous coupon dat e unt il t he set t lement dat e.
I f t he price of t he bond includes accrued int erest , it is called as t he dirt y price or full
price of t he bond. Price t hat excludes accrued int erest is called clean price. I n most
market s t he convent ion is t o quot e t he clean price, t hough t he buyers always pay t he
seller t he clean price and t he accrued int erest , t hat is t he dirt y price. I t is import ant t o
remember t hat t he price f unct ion in Excel provides t he clean price of t he bond.
Tabl e 13. 5: Accr u ed I n t er est on Set t l emen t Dat e Febr u ar y 5, 2001
Secu r i t y Semi - an n u al
Cou pon ( Rs. )
Mat u r i t y Days si n ce l ast
cou pon / Days
i n t h e cou pon
per i od
Accr u ed
I n t er est
( Rs. )
CG 12. 5 %2004 6. 25 23- Mar- 04 0. 7333 4. 5833
CG 11. 68%
2006
5. 84 10- Apr- 06 0. 6389 3. 7311
CG 11. 5% 2008 5. 75 23- May- 08 0. 4000 2. 3000
CG 11. 3% 2010 5. 65 28- Jul- 10 0. 0389 0. 2197
CG 11. 03 %
2012
5. 515 18- Jul- 12 0. 0944 0. 5209
13. 5 Yi el d
The ret urns t o an invest or in bond is made up of t hree component s: coupon, int erest
f rom re- invest ment of coupons and capit al gains/ loss f rom selling or redeeming t he
bond. When we are able t o compare t he cash inf lows f rom t hese sources wit h t he
invest ment ( cash out f lows) of t he invest or, we can comput e yield t o t he invest or.
Depending on t he manner in which we t reat t he t ime value of cash f lows and re-
invest ment of coupons, we are able t o get various int erpret at ions of t he yield on an
invest ment in bonds.
13. 5. 1 Cu r r en t Yi el d
One of t he earlier measures on yield on a bond, current yield was a very popular
measure of bond ret urns in t he I ndian market s, unt il t he early 1990s. Current yield is
measured as
= Annual coupon receipt s/ Market price of t he bond
133
This measure of yield does not consider t he t ime value of money, or t he complet e series
of expect ed f ut ure cash f lows. I t inst ead compares t he coupon, as pre- specif ied, wit h
t he market price at a point in t ime, t o arrive at a measure of yield. Since it compares a
pre- specif ied coupon wit h t he current market price, it is called as current yield.
For example, if a 12. 5% bond sells in t he market f or Rs. 104. 50, current yield will be
comput ed as
= ( 12. 5/ 104. 5) * 100
= 11. 96%
Current yield is no longer used as a st andard yield measure, because it f ails t o capt ure
t he f ut ure cash f lows, re- invest ment income and capit al gains/ losses on invest ment
ret urn. Current yield is considered a very simplist ic and erroneous measure of yield.
13. 5. 2 Yi el d t o Mat u r i t y ( YTM)
I n t he previous sect ion on bond valuat ion, we used equat ion 13. 1 t o show t hat t he value
of a bond is t he discount ed present value of t he expect ed f ut ure cash f lows of t he bond.
We solved t he equat ion t o det ermine a value, given an assumed required rat e. I f we
inst ead solve t he equat ion f or t he required rat e, given t he price of t he bond, we would
get an yield measure, which is knows as t he YTM or yield t o mat urit y of a bond. That is,
given a pre- specif ied set of cash f lows and a price, t he YTM of a bond is t hat rat e which
equat es t he discount ed value of t he f ut ure cash f lows t o t he present price of t he bond.
I t is t he int ernal rat e of ret urn of t he valuat ion equat ion.
For example, if we f ind t hat a 11. 99% 2009 bond is being issued at a price of Rs. 108,
( f or t he sake of simplicit y we will begin wit h t he valuat ion on a cash f low dat e) , we can
st at e t hat ,
18 2
) 1 (
995 . 105
.. ..........
) 1 (
995 . 5
) 1 (
995 . 5
108
r r r +
+
+
+
+
This equat ion only st at es t he well known bond valuat ion principle t hat t he value of a
bond will have t o be equal t o t he discount ed value of t he expect ed f ut ure cash f lows,
which are t he 18 semi- annual coupons of Rs. 5. 995 each and t he redempt ion of t he
principal of Rs. 100, at t he end of t he 9
t h
year. That value of r which solves t his
equat ion is t he YTM of t he bond. We can f ind t he value of r in t he above equat ion using
t he I RR f unct ion in Excel
16
. The value of r t hat solves t he above equat ion can be f ound
t o be 5. 29%, which is t he semi - annual rat e. The YTM of t he bond is 10. 58%.
However, as we have already not ed in t he sect ion on valuat ion, we should be able t o
comput e price and yield f or a bond, at any given point of t ime. We t heref ore have t o be
able t o comput e t he yield by plot t ing t he cash f lows accurat ely on t he t ime line ( using
t he appropriat e day count convent ion) and calculat e YTM, given t he price at any point on
16
I RR can be comput ed by list ing t he cashf lows in a single column, wit h init ial out f low
st at ed as a negat ive number, say b2: b20 and using f ormula = I RR ( b2: b20) .
134
t he t ime line. We have t o adopt a procedure very similar t o t he one we used f or bond
valuat ion
17
and we can use t he yield f unct ion in Excel t o comput e t he YTM f or a bond.
Yield t o mat urit y represent s t he yield on t he bond, provided t he bond is held t o mat urit y
and t he int ermit t ent coupons are re- invest ed at t he same YTM rat e. I n ot her words,
when we comput e YTM as t he rat e t hat discount s all t he cash f lows f rom t he bond, at
t he same YTM rat e, what we are assuming in ef f ect is t hat each of t hese cash f lows can
be re- invest ed at t he YTM rat e f or t he period unt il mat urit y.
Let us illust rat e t his limit at ion of YTM wit h an example. Suppose an invest or buys t he
11. 75% 2006 bond at Rs. 106. 84. The YTM of t he bond on t his dat e is 10. 013%.
Consider t he inf ormat ion about t he cash f lows of t he 11. 75% 2006 bond in Table 13. 6.
I t is seen t hat cash f lows f rom coupon and redempt ion are Rs. 164. 625, if t he bond is
held t o mat urit y. However, t he act ual yield on t he bond depends on t he rat es at which
t he coupons can be re- invest ed. The YTM of 10. 02 is also t he act ual ret urn on t he bond,
at mat urit y, only if all coupons can be re- invest ed at 10. 02%. I f t he act ual rat es of re-
invest ment of t he bond are dif f erent , as in columns 5 and 7 in Table 13. 6, as is most ly
t he case, t he act ual yield on t he bond could be dif f erent .
Box 13. 3: Usi n g t h e Yi el d Fu n ct i on
The yield f unct ion in Excel will comput e t he yield of a bond, given t he f ollowing:
Set t lement : t he dat e on which t he yield is sought t o be comput ed
Mat urit y: t he dat e on which t he bond mat ures
Rat e: t he rat e at which coupon is paid
Price: t he market price of t he bond
Redempt ion: t he redempt ion value of t he bond
Frequency: number of coupons per year
Basis: Day count convent ion t o be used ( represent ed by numbers 0- 4)
On providing t hese input s, Excel comput es t he cash f lows f rom t he coupon rat e and
redempt ion values, t he t ime as t he dist ance bet ween set t lement dat e and each of t he
cash f lows, given t he day count convent ion specif ied in t he basis and f ind by t rial and
error, t he rat e t hat equat es t he f ut ure t he cash f lows t o t he price on t he set t lement
dat e.
Use t he f unct ion as = yield( set t lement , mat urit y, rat e, price, redempt ion, f requency,
basis)
For example, in order t o value t he 11. 75% 2006 bond, mat uring on April 16, 2006, on
February 2, 2001, using t he day count convent ion of 30/ 360, at price of Rs. 106. 84, we
shall st at e t he f ollowing:
= yield( 16/ 04/ 2006, 02/ 02/ 2001, 0. 1175, 106. 84, 100, 2, 0)
Excel will ret urn a yield of 10. 002%, which is t he YTM of t he bond.
13. 5. 3 Yi el d t o Mat u r i t y of a Zer o Cou pon Bon d
I n t he case of a zero coupon bond, since t here are no int ermit t ent cash f lows in t he f orm
of coupon payment s, t he YTM is t he rat e t hat equat es t he present value of t he mat urit y
17
Readers who have skipped t he earlier discussion are ref erred t o sect ion 13. 3 on
valuat ion of bonds.
135
or redempt ion value of t he bond t o t he current market price, over t he dist ance in t ime
equal t o t he set t lement and mat urit y dat es. For example, if a zero coupon bond sells at
Rs. 93. 76 on February 5, 2001 and mat ures on January 1, 2002, it s YTM is comput ed as
) 365 / 330 (
) 1 (
100
76 . 93
ytm +
= 7. 39%
I n t he case of zero coupon bond, int erest is accrued on an everyday basis unt il mat urit y,
at t his discount ing rat e.
136
Tabl e 13. 6: Wh y YTM i s n ot ear n ed even i f a Bon d i s h el d t o Mat u r i t y
Case- I Case- I I Days t o
mat u r i t y
Cash
f l ow
dat e
Cash
f l ow
Fu t u r e
val u e i f
r e-
i n vest e
d at
YTM of
10. 02%
Assu me
d
r e-
i n vest
men t
r at es
Re-
i n vest
men t
r et u r n s
Assu me
d r e-
i n vest
men t
r at es
Re-
i n vest
men t
r et u r n
s
1800 16- Apr- 01 5. 875 7. 5022 10. 25 9. 5698 9. 25 9. 1436
1620 16- Oct - 01 5. 875 7. 3210 10. 00 9. 0214 9. 00 8. 6582
1440 16- Apr- 02 5. 875 7. 1442 9. 75 8. 5237 8. 75 8. 2172
1260 16- Oct - 02 5. 875 6. 9717 9. 50 8. 0716 8. 50 7. 8165
1080 16- Apr- 03 5. 875 6. 8033 9. 25 7. 6608 8. 25 7. 4523
900 16- Oct - 03 5. 875 6. 6389 9. 00 7. 2874 8. 00 7. 1214
720 16- Apr- 04 5. 875 6. 4786 8. 75 6. 9481 7. 75 6. 8209
540 16- Oct - 04 5. 875 6. 3221 8. 50 6. 6398 7. 50 6. 5482
360 16- Apr- 05 5. 875 6. 1694 8. 25 6. 3597 7. 25 6. 3009
180 16- Oct - 05 5. 875 6. 0204 8. 00 6. 1055 7. 00 6. 0771
0 16- Apr- 06 105. 875 105. 8750 7. 75 105. 8750 6. 75 105. 8750
Al t er n at e
Val u es 164. 625 173. 2469 182. 0626
180. 031
4
13. 5. 4 Usi n g t h e Zer o- Cou pon Yi el d f or Bon d Val u at i on
I f int erest rat es are a f unct ion of t ime t o mat urit y, t hen valuat ion of a bond, using t he
same YTM rat e, can lead t o erroneous result s, as we saw in t he pervious sect ion. I n
ot her words, t he YTM of a zero coupon bond is a pure int erest rat e f or t he t enor of t he
bond. I n all t he ot her cases, if we used a YTM rat e f or valuat ion, we have assumed t hat
a single rat e, equivalent t o t he YTM, exist s f or all t he t ime periods f or which coupons
have t o be invest ed. Theref ore, t he appropriat e rat es f or various t enors will have t o be
used t o value cash f lows f or t hat t enor. We call such a valuat ion as t he zero coupon
yield based valuat ion. I n t he next chapt er, we shall discuss t he met hodology used f or
est imat ing t he zero coupon yield curve ( ZCYC) . I n t his sect ion, we shall see how t he
valuat ion of a bond changes if we use t he ZCYC f or valuat ion. The equat ion we use will
be
Consider t he 12. 5% 2004 bond, whose cash f lows are in Table 13. 7.
The valuat ion in Table 13. 7 uses a dif f erent rat e f or each of t he cash f lows. I n t he next
chapt er on yield, we shall see how t he appropriat e ZCYC rat e is est imat ed. The NSE
est imat es t he ZCYC f rom market prices and enables t he comput at ion of appropriat e
discount rat es, used in t he t able.
) 4 . 13 .....( .......... ..........
) 1 (
......
) 1 ( ) 1 (
2
2 1
m
m
r
R C
r
C
r
C
PV
+
+
+ +
+
+
+
137
Tabl e 13. 7: Usi n g t h e ZCYC f or val u at i on of bon ds
Cou pon
dat es
Cash
f l ow s
( Rs. )
Di st an ce i n
year s f r om
set t l emen t dat e
Appr opr i at e
ZCYC r at e
Pr esen t val u e
of t h e
cash f l ow ( Rs. )
23- Mar- 01 6. 25 0. 13611 9. 6148 6. 168741
23- Sep- 01 6. 25 0. 64722 9. 5108 5. 876878
23- Mar- 02 6. 25 1. 15000 9. 4519 5. 606264
23- Sep- 02 6. 25 1. 66111 9. 4272 5. 344058
23- Mar- 03 6. 25 2. 16389 9. 4302 5. 096338
23- Sep- 03 6. 25 2. 67500 9. 4548 4. 853332
23- Mar- 04 106. 25 3. 18056 9. 4956 78. 55363
Val u e of t h e bon d 111. 4992
13. 5. 5 Bon d Equ i val en t Yi el d
I n all t he examples which we have seen so f ar, we have det ermined t he semi- annual
coupon f rom t he annual coupon, by simply dividing t he annual coupon by 2. We have
comput ed t he semi- annual yield f or t he purpose of det ermining t he price, by similarly
dividing t he annual yield by 2. I f cash f lows are compounded mult iple t imes during a
year, t he ef f ect ive rat es are not t he annual rat e divided by t he number of compounding
periods. This is because, int ermit t ent cash f lows can be re- deployed, at prevailing rat es,
t o arrive at an ef f ect ive annual rat e.
For example, if annual yield is 11. 75%, t he semi- annual yield is simply t aken as
11. 75/ 2, which is 5. 875%. However, if t he six mont hly coupon is re- invest ed at
5. 875%, t he ef f ect ive annual yield will be higher t han 11. 75%, at 12. 095%. I n ot her
words, semi- annual yields should be annualised, by incorporat ing t he ef f ect of t he re-
invest ment , as f ollows:
Ef f ect ive Annual yield = ( 1+ Periodic int erest rat e)
k
1
where k is t he number of payment s in a year. This f ormula can be used t o comput e
ef f ect ive yields f or any number of compounding periods in a year.
I n t he above example,
Ef f ect ive annual yield = { ( 1+ 0. 05875)
2
} - 1
= 12. 095%
Though it is well known t hat semi- annual yields are t heref ore not half t he annual yields,
in most bond market s, t he convent ion is t o simply divide t he annual yield by 2, t o get
t he semi- annual yield. The semi- annual yield t hus simplist ically comput ed is called t he
Bond Equivalent Yield ( BEY) . Given t he f ormula above, bond equivalent yield is
= ( 1+ ef f ect ive yield)
1/ k
- 1
Using t he numbers f rom t he same example,
BEY = ( 1+ . 12095)
1/ 2
1
138
= 5. 875%
I n t he yield calculat ions f or most f ixed income securit ies, unless ot herwise st at ed, it is
t he bond- equivalent - yield t hat is used.
13. 6 Wei gh t ed Yi el d
When bonds are t raded at dif f erent prices during a day, t he yield f or t he day is usually
report ed as t he weight ed yields, t he weight s being t he market value of t he t rades ( price
t imes quant it ies t raded) . For example, assume t hat t he t rades in CG11. 3% 2010 are as
in Table 13. 8. The weight ed yield is comput ed using market values f or each t rade as
t he weight age.
Tabl e 13. 8: Wei gh t ed Yi el d
Qu an t i t y Pr i ce
( Rs. )
Mar k et Val u e
( Rs. )
YTM ( % ) YTM as Pr opor t i on of
mar k et val u e
10000 105. 23 1052300 10. 4177 1. 4925
2500 105. 45 263625 10. 3820 0. 3726
4000 105. 47 421880 10. 3787 0. 5961
6500 105. 50 685750 10. 3739 0. 9685
9000 105. 63 950670 10. 3528 1. 3399
8500 105. 71 898535 10. 3399 1. 2649
12000 105. 8 1269600 10. 3253 1. 7847
6000 105. 95 635700 10. 3011 0. 8915
5500 106. 00 583000 10. 2931 0. 8170
3500 106. 20 371700 10. 2609 0. 5192
2000 106. 25 212500 10. 2528 0. 2966
Tot al
Val u e 7345260 Wei gh t ed Yi el d 10. 3435
13. 7 YTM of a Por t f ol i o
YTM of a port f olio is not comput ed as t he average or weight ed average of t he YTMs of
t he bonds in t he port f olio. We are able t o comput e weight ed yields only when t he cash
f lows of t he bonds under quest ion are t he same, as was t he case in weight ed yields. I n
a port f olio of bonds, each bond would have a dif f erent cash f low composit ion and
t heref ore, using a weight ed yield would provide erroneous result s. We t heref ore f ind
t he YTM of t he port f olio as t hat rat e which equat es t he expect ed cash f lows of t he bonds
in t he port f olio, wit h t he market value of t he port f olio. Consider f or example, a
port f olio of bonds as in Table 13. 9.
139
Tabl e 13. 9: YTM of a por t f ol i o: Sampl e Bon ds
Bon d Mat u r i t y
Dat e
Nu mber of
Bon ds
Pr i ce as
on Feb 5,
2001 ( Rs. )
Mar k et Val u e
( Rs. )
CG 11. 75 2001 25/ 08/ 01 20000 101. 1 2022000
CG 11. 68 2002 6/ 08/ 02 25000 102. 915 2572875
CG 12. 5 2004 23/ 03/ 04 32000 107. 48 3439360
Tot al 8034235
Box 13. 4: XI RR Fu n ct i on
The XI RR f unct ion comput es t he I RR ( equivalent t o YTM in our case) f or a series of cash
f lows, occurring at dif f erent point s in t ime, when we provide t he dat es and t he cash
f lows. The f unct ion requires { values, dat es, guess} . The values have t o be in a column,
wit h t he init ial cash out f low shown as a negat ive number. I n t he above example, t he
market value on February 05, Rs. 80, 34, 235 is t o be shown as a negat ive value. The
dat es on which t he cash inf lows occur are shown in a corresponding column.
When we use t he f unct ion as, f or inst ance,
= XI RR ( b2: b14, c2: c14) we get t he result 0. 13145, which is 13. 145%. We have t o
remember however, t hat t he XI RR f unct ion support s only t he act ual/ 365 day count
convent ion. We use t his f unct ion as an approximat ion, because f inding t he YTM is an
it erat ive t rial and error process, which can be complex ot herwise.
The cash f lows f rom t hese bonds accrue on dif f erent dat es, as t hese bonds have
dif f erent dat es t o mat urit y. Table 13. 10 shows t he dat es and t he cash f lows f or t hese
bonds and given t he quant it y of bonds held, t he t ot al cash f lows f rom t his port f olio, on
t he given dat es. The yield t o mat urit y of t his port f olio is t hat rat e which equat es t his
series of cash f lows in column 3 of t able 13. 9, t o t he market value on t able 13. 8, as on
February 5, 2001. We can f ind t he YTM by using t he XI RR f unct ion in Excel.
Tabl e 13. 10: Por t f ol i o Cash Fl ow s
Dat e
Cash f l ow
per bon d
Tot al
cash f l ow s
25- Feb- 01 5. 875 117500
25- Aug- 01 105. 875 2117500
6- Feb- 01 5. 84 146000
6- Aug- 01 5. 84 146000
6- Feb- 02 5. 84 146000
6- Aug- 02 105. 84 2646000
23- Mar- 01 6. 25 200000
23- Sep- 01 6. 25 200000
23- Mar- 02 6. 25 200000
23- Sep- 02 6. 25 200000
23- Mar- 03 6. 25 200000
23- Sep- 03 6. 25 200000
23- Mar- 04 106. 25 3400000
YTM 13. 145%
140
13. 8 Real i sed Yi el d
The act ual yield realised by t he invest or in a bond, over a given holding period, is called
realised yield. Realised yield represent s t he horizon ret urn t o t he invest or, f rom all t he
t hree component s of bond ret urn, namely, coupon, ret urn f rom re- invest ment of coupon
and capit al gain/ loss f rom selling t he bond at t he end of t he holding period. The realised
yield t o t he invest or is t he rat e which equat es cash f lows f rom all t hese t hree sources, t o
t he init ial cash out f low. Realised yield is also called t ot al ret urn f rom a bond.
Depending upon t he reinvest ment rat es available t o t he invest or and t he yields which
prevail at t he end of t he holding period, t he invest ors realized yield f rom holding a
bond can vary. For example, consider t he 12. 5% 2004 bond. The realized yield on a 1-
year horizon based on a set of assumpt ions about re- invest ment rat es and YTM at t he
end of t he holding period, are as f ollows:
Purchase price of t he bond on 23 March 2001 Rs. 107. 42 ( YTM 9. 6%)
Coupons received: 2 Semi- annual Rs. 12. 50
Reinvest ment of 1
st
coupon f or 1 year @ 7. 5% Rs. 6. 7185
Reinvest ment of 2
nd
coupon f or 6 mont hs @ 7% Rs. 6. 4651
Sale of bond at t he end of 1 year @ 7. 8% yield Rs. 108. 5
Coupon income f rom t he bond f or 1 year Rs. 12. 5
I ncome f rom coupon re- invest ment Re. 0. 6838
Capit al gain on sale Rs. 1. 08
Tot al cash f lows f rom t he bond Rs. 14. 26
Tot al r et u r n f or 1 year h ol di n g per i od 14. 26/ 107. 42
= 13. 2785%
The t ot al ret urn t o t he invest or is at t ribut able t o all t he t hree sources of income and
depends on t he re- invest ment rat e and t he sale price. An increase in int erest rat es will
enhance t he reinvest ment income of t he invest or, while reducing t he capit al gains; a
decrease in int erest rat es will generat e capit al gains, while reducing t he re- invest ment
income of t he invest or. The invest ment horizon will also impact t he percent age
composit ion of each of t hese component s t o t he t ot al ret urn of t he invest or. Holding
t he bond over a longer t ime will enhance coupon component of t he ret urn and
reinvest ment , if rat es are increasing. However, t he capit al gains will drop, due t o a f all
in yield, as well as due t o t he t ime pat h ef f ect , leading t o t he bond t ending t o
redempt ion value, as it nears mat urit y.
Realised yield, or t ot al ret urn t heref ore provides t he invest or t he t ool t o analyse impact
of int erest rat es and holding period, on t he act ual ret urns earned f rom a bond.
13. 9 Yi el dPr i ce Rel at i on sh i ps of Bon ds
The basic bond valuat ion equat ion shows t hat t he yield and price are inversely relat ed.
This relat ionship is however, not unif orm f or all bonds, nor is it symmet rical f or
increases and decreases in yield, by t he same quant um.
141
Consider Figure 13. 1 which plot s t he price- yield relat ionship f or a set of bonds:
Fi gu r e 13. 1 Pr i ce Yi el d Rel at i on sh i p of Bon ds
0
20
40
60
80
100
120
140
160
180
200
0 0.05 0.1 0.15 0.2 0.25
YTM (%)
P
r
i
c
e
(
R
s
.
)
CG2001 CG2002 CG2005 CG2009 CG2013
13. 9. 1 Pr i ce Yi el d Rel at i on sh i p: Some Pr i n ci pl es
a. Price- yield relat ionship bet ween bonds is not a st raight line, but is convex. This
means t hat price changes f or yield changes are not symmet rical, f or increase
and decrease in yield.
b. The sensit ivit y of price t o changes in yield in not unif orm across bonds.
Theref ore f or a same change in yield, depending on t he kind of bond one holds,
t he changes in price will be dif f erent .
c. Higher t he t erm t o mat urit y of t he bond, great er t he price sensit ivit y. We
not ice in Figure 13. 1, t hat CG2013 has t he st eepest slope, while 2001 and 2002
are virt ually f lat . Price sensit ivit ies are higher f or longer t enor bonds, while in
t he short - t erm bond, one can expect relat ive price st abilit y f or a wide range of
changes in yield.
d. Lower t he coupon, higher t he price sensit ivit y. Ot her t hings remaining t he
same, bonds wit h higher coupon exhibit lower price sensit ivit y t han bonds wit h
higher coupons.
I n t he bond market s t heref ore, we are int erest ed in t wo key quest ions: What is t he yield
at which reinvest ment and valuat ion happens and how t he change in t his yield impact s
t he value of t he bonds held. These are t he quest ions we address in t he next t wo
chapt ers.
142
Model Quest i ons
1. A GOI secu r i t y w i t h cou pon of 11. 68% , mat u r i n g on 6- Au g- 2002, i s t o be
set t l ed on
143
1- Feb- 01. Wh at ar e t h e n u mber of days f r om t h e pr evi ou s cou pon dat e?
a. 179
b. 176
c. 178
d. 175
An sw er : d.
We use t he cou pdaybs f unct ion in Excel and specif y t he f ollowing:
Set t lement dat e: February 1, 2001
Mat urit y Dat e: August 6, 2002
Frequency: 2
Basis: 4
The answer is: 175 days
2. Wh at i s t h e accr u ed i n t er est on a 11. 68% GOI secu r i t y, mat u r i n g on 6- Au g-
2002, t r adi n g on 1- Ju n - 200 1 at a YTM of 7. 7395% ?
a. Rs. 3. 6901
b. Rs. 3. 7311
c. Rs. 3. 7105
d. Rs. 3. 7520
An sw er : b
Accrued int erest is comput ed as
Coupon payment * ( number of days f rom previous coupon/ days in t he coupon period)
We use t he cou pbs and cou pdays f unct ions t o ascert ain days f rom previous coupon
and days in t he coupon period.
The amount of coupon is Rs. 11. 68/ 2.
Theref ore, t he accrued int erest is
= 5. 84 * ( 115/ 180)
= Rs. 3. 7311
3. A 11. 68% GOI secu r i t y mat u r i n g on 6- Au g- 2002, i s bei n g pr i ced i n t h e
mar k et on 11- Ju l - 01 at Rs. 104. 34. Th e YTM of t h e bon d i s
a. 7. 3728%
b. 7. 3814%
c. 7. 3940%
d. 7. 3628%
We use t he Yi el d f unct ion in Excel, specif ying set t lement ( 11 July 2001) and mat urit y
dat es ( 6 Aug 2002) , coupon ( 0. 1168) , price of t he securit y ( 104. 34) , redempt ion ( 100)
f requency ( 2) , basis ( 4) .
The answer obt ained is 7. 3728%
An sw er : a
144
4. Th e f ol l ow i n g i s t h e descr i pt i on of bon ds h el d i n a por t f ol i o. Wh at i s t h e
por t f ol i o yi el d, u si n g t h e w ei gh t ed yi el d met h od?
Cou pon
( % p. a. )
Mat u r i t y dat e Mar k et pr i ce on Ju l y 11,
2001 ( Rs. )
Nu mber of
bon ds
11. 68 6- Aug- 2002 104. 34 5400
11. 15 1- Sep- 2002 104. 03 5560
13. 82 30- May- 2002 105. 5 5720
12. 69 10- May- 2002 104. 9 5880
11. 00 23- May- 2003 105. 74 6040
An sw er :
The yield of each of t he bonds can be comput ed using t he yield f unct ion ( see solved
example 3 above) . The market value of each bond can be comput ed as t he product of
number of bonds and market price as on July 11, 2001.
Cou pon
( % p. a. )
Mat u r i t y dat e Pr i ce ( Rs. ) Yi el d ( % ) Nu mber of
bon ds
Mar k et Val u e
( Rs. )
11. 68 6- Aug- 2002 104. 34 7. 3728% 5400 563436
11. 15 1- Sep- 2002 104. 03 7. 3770% 5560 578406. 8
13. 82 30- May- 2002 105. 5 7. 2731% 5720 603460
12. 69 10- May- 2002 104. 9 6. 5056% 5880 616812
11. 00 23- May- 2003 105. 74 7. 6309% 6040 638669. 6
The yield of t he port f olio can be f ound by weight ing each bonds yield by t he market
value of t he bond in t he port f olio. This is done as:
{ ( 7. 3728* 563436) + ( 7. 3770* 578406. 8) + ( 7. 2731* 603460) + ( 6. 5056* 616812) + ( 7. 6309
* 638669. 6) } / ( 563436+ 578406. 8+ 603460+ 616812+ 638669. 6)
We can do t he same in Excel, using t he f ormula
= sumproduct ( yield array, market value array) / sum( market value array)
The answer in bot h cases is 7. 2302%, which is t he port f olio yield.
5. On Apr i l 12, 2001, a deal er pu r ch ases a 11. 68% GOI bon d mat u r i n g on 6-
Au g- 2002 f or Rs. 104. 34. He h ol ds t h e bon d f or 1 year , an d sel l s i t on Apr i l 11,
2001, f or Rs. 100. 90. I f t h e cou pon s r ecei ved du r i n g t h e h ol di n g per i od ar e r e-
i n vest ed at 8. 2405% ( 1st cou pon ) an d 6. 7525% ( 2n d cou pon ) , w h at i s t h e
r eal i sed yi el d on t h e i n vest men t ?
An sw er :
The component s of realized yield are:
Coupon income, re- invest ment of coupons and capit al gains/ losses.
Cou pon i n come:
The number of coupons bet ween t he acquisit ion dat e and dat e of sale of t he bond can be
f ound wit h t he coupnum f unct ion. I n t his case t here are t wo coupons. Theref ore t he
coupon received is:
145
Rs. 11. 68
Re- i n vest men t I n come:
We can f ind t he f irst coupon dat e, by using t he coupncd f unct ion in Excel. We t hen
use t he Days360 f unct ion t o know t he number of days f or which t he f irst coupon will be
invest ed.
The f irst coupon is due on August 6, 2001. Since t he bond will be sold on April 11,
2002, t he number of days f or which t he coupon will be re- invest ed will be 245 days.
The int erest rat e applicable t o t his coupon, as given in t he quest ion, is 8. 2405%.
Tehref ore t he re- invest ment income can be comput ed as:
= ( 11. 68/ 2) * ( 245/ 360) * 0. 082405
= 0. 327514
Similarly, t he second coupon is due on 6
t h
Feb 2002. I t will be reinvest ed f or 65 days, at
6. 7525%. The reinvest ment income will be
= ( 11. 68/ 2) * ( 65/ 360) * 0. 067525
= 0. 071201
Capit al gain/ loss:
Rs. 100. 90 104. 34
= - 3. 44
The t ot al rupee ret urn f rom holding t he bond f or a year is
= 11. 68 + 0. 327514+ 0. 071201 3. 44
= 8. 6347
The released yield t heref ore is
= ( 8. 6347/ 104. 34) * 100
= 8. 2756%
146
Ch apt er 14
Yi el d Cu r ve an d Ter m St r u ct u r e of
I nt er est Rat es
I nt erest rat es are pure prices of t ime, and are t he discount ing f act ors used in t he
valuat ion equat ion f or bonds. I t is crucial t hat we are able t o derive t hese discount
f act ors f rom t he market such t hat t he valuat ions we do are current and accurat e. The
process of det ermining t he discount f act ors, ( which we know as t he yields or int erest
rat es) will have t o t heref ore draw f rom t he current market prices of bonds. The broad
pict ure of t he debt market can be discerned in t erms of a f unct ional relat ionship bet ween
t wo variables: t ime and int erest rat es. The f ocus of t his chapt er is t he underst anding of
t his relat ionship bet ween t ime and int erest rat es. This relat ionship not only provides
t ools f or valuat ion of bonds, but also enables ident if icat ion of arbit rage opport unit ies in
t he market and assessment market expect at ions of f ut ure int erest rat es.
14. 1 Yi el d Cu r ve: A Si mpl e Appr oach
The simplest approach t o observing t he int erest rat es in t he market , is t o draw t he yield
curve f rom t he YTM of t raded bonds. The YTMs of t raded bonds is used as an
approximat ion of t he int erest rat e f or t he given t erm t o mat urit y of t he bond. When we
obt ain a plot of t hese relat ionships bet ween YTMs and t erm t o mat urit y of a set of
t raded bonds, we can ident if y t he f unct ional relat ionship bet ween t ime and yield, by
f it t ing a curve t hrough t he plot of point s. Alt ernat ively, we can use t hese YTMs t o
est imat e yields f or any t enor, by met hods of int erpolat ion.
14. 1. 1 Yi el d Cu r ve f r om a Sampl e of Tr aded Bon ds
Consider f or example, bonds t raded on March 29, 2001 ( Table 14. 1) . From t he
observed market prices in column 5, we can comput e t he YTM of t hese bonds, using t he
yield f unct ion in Excel. The t erm t o mat urit y of t he bonds is t he dist ance in t ime
bet ween t he mat urit y dat e of t he bonds ( column 3) and t he set t lement dat e ( March 29,
2001) . The t erm t o mat urit y is shown in column 4. We can see t hat bonds wit h varying
t erms t o mat urit y have t raded at dif f erent yields, and t he general t endency is f or yields
t o increase as t he t erm increases.
I n order t o be able t o model t his relat ionship int o a f unct ion, t hat can be used f or
valuing bonds, we need t o est imat e t he relat ionship as an equat ion, so t hat given values
of t enor ( x) , we can est imat e values of yield ( y) .
This can be done by plot t ing t he t erm t o mat urit y and t he yield t o mat urit y, and f it t ing a
3
rd
degree polynomial t o describe t he f unct ional relat ionship. A t hird degree polynomial
is specif ied as f ollows:
147
it it it it it it
e x b x b x b a y + + + +
3
3
2
2 1
. . ( 14. 1)
where b1, b2 and b3 are est imat ed co- ef f icient s, given values of t erm t o mat urit y ( x) and
yield t o mat urit y ( y) .
Tabl e 14. 1: Sampl e Bon ds f or Yi el d Cu r ve
Name Cou pon
( % )
Mat u r i t y
Dat e
Ter m t o Mat u r i t y
( year s)
Pr i ce
( Rs. )
YTM
( % )
CG2001 11. 75 25- Aug- 01 0. 41 101 0. 090924
CG2002 11. 15 9- Jan- 02 0. 79 102. 75 0. 074125
CG2003 11. 1 7- Apr- 03 2. 05 103. 515 0. 091537
CG2004 12. 5 23- Mar- 04 3. 03 108. 31 0. 092473
CG2005 11. 19 12- Aug- 05 4. 44 106. 19 0. 094220
CG2006 11. 68 10- Apr- 06 5. 11 107. 58 0. 097364
CG2007 11. 9 28- May- 07 6. 25 109. 31 0. 098426
CG2008 11. 4 31- Aug- 08 7. 53 107. 6 0. 099240
CG2009 11. 99 7- Apr- 09 8. 14 109. 18 0. 102808
CG2010 11. 3 28- Jul- 10 9. 47 106. 6 0. 101823
CG 2011 12. 32 29- Jan- 11 9. 98 110. 97 0. 104987
CG2013 12. 4 20- Aug- 13 12. 58 111. 2 0. 107401
Fi g 14. 1: Yi el d Cu r ve as on Mar ch 29, 2001
y = 8E-06x
3
- 0.0003x
2
+ 0.004x + 0.0817
0
0.02
0.04
0.06
0.08
0.1
0.12
0 2 4 6 8 10 12 14
Term To Maturity ( in Years)
Y
i
e
l
d
t
o
M
a
t
u
r
i
t
y
(
%
)
148
The equat ion in Fig 14. 1 provides t he generalised relat ionship bet ween t erm t o mat urit y
and yield t o mat urit y. By f it t ing int o t he equat ion, t he t erm t o mat urit y of any given
bond, ( by subst it ut ing t he value of x) , t he corresponding YTM can be est imat ed. The
given bond can be valued by f it t ing int o it s cash f low f eat ures, t he YTM t hus derived, so
t hat value of t he bond can be comput ed.
Box 14. 1: Usi n g Ex cel t o dr aw t h e Yi el d Cu r ve
The f ollowing are t he st eps t o drawing t he yield curve using Excel:
1. Comput e yield and t erm t o mat urit y f or a set of bonds, using t he yield f unct ion,
and f inding t he dif f erence in years, bet ween set t lement dat e and mat urit y dat e
of t he bond. ( ( mat urit y dat e set t lement dat e) / 360) ) .
2. Draw an XY graph ( XY scat t er) of t hese point s, using t erm f or x values and
Yield f or Y values.
3. Choose Chart / Add t rend line/ t ype: Choose polynomial, and order 3.
4. Choose Chart / Add t rend line/ opt ions: display equat ion on t he chart .
5. Excel plot s t he graph and est imat es t he 3
rd
degree polynomial, displaying t he
equat ion of t he yield curve.
For example, we can use t he equat ion in t he yield curve above t o value t he 12. 5% 2004
bond on March 29, 2001. The bond has 3. 03 years t o mat urit y on t hat dat e, t heref ore
we plug int o t he yield curve equat ion, t his value in t he place of x, as f ollows:
Y = ( ( - 0. 000008* ( 3. 03) ^ 3) - ( 0. 0003* ( 3. 03) ^ 2) + ( 0. 004* 3. 03) + 0. 0817)
We obt ain t he value 0. 09843 as t he Y value. Since we know t he cash f lows of t his bond,
we can use t he Price f unct ion t o est imat e t he value of t his bond, plugging in 0. 09843
as t he value f or yield. The result ing price of t he bond is Rs. 108. 7497. ( The last t raded
price of t his bond on t hat dat e was Rs. 108. 31) . We can t hus use t he yield curve t o
mark a port f olio t o market , or value a given bond, which may not be t raded.
14. 1. 2 Li mi t at i on s of t h e Si mpl e Yi el d Cu r ve
The yield curve which we have drawn f rom t he market prices above, is a summary of
t he YTMs f or various t raded bonds, on a given dat e. They, however, may not t ruly
represent t he yields or int erest rat es f or various t enors. The YTM of a bond represent s a
single rat e, at which all t he cash f lows of a bond are discount ed. This act ually t ranslat es
int o a valuat ion proposit ion where, cash f lows accruing at varying point s in t ime are all
discount ed at t he same rat e, i. e t he YTM of t he bond. I n realit y, such a discount ing
process represent s a scenario where cash f lows accruing at any point in t he lif e of t he
bond, can be deployed at a single rat e. This would t hen t ranslat e int o a sit uat ion where
int erest rat es f or all t enors f or a given bond are equal, and hence a f lat yield curve.
What we see when we plot t he YTMs of t raded bonds, is a t endency f or YTMs of bonds
wit h varying t enors t o be dif f erent . This means t hat rat es f or varying t enors are not
unif orm, but dif f erent . I f t his were t rue, we can not use t he same YTM f or valuing all
t he cash f lows of a bond. The t rue int erest rat es, which are implicit in t he prices of
149
t raded bonds, are t heref ore not observed. The YTM is a simplif icat ion, wit h an
erroneous assumpt ion on t he re- invest ment of int ermit t ent coupons. I f we know t hat
dif f erent rat es exist across t enors, t he valuat ion equat ion will have t o be recast as
f ollows:
n
n
n
r
c
r
c
r
c
P
) 1 (
. ..........
) 1 ( 1
2
2
2
1
1
0
+
+
+
+
+
. . ( 14. 2)
Where r1, r2 . r n represent t he rat es f or t he respect ive t enors. These rat es are pure
spot rat es, in t hat t here are no assumpt ions on reinvest ment of coupons. I n ot her
words t hey are rat es t hat would be implicit in a bond t hat has a single cash f low at t he
end of t he t erm, i. e a zero coupon bond. These rat es are also called as t he zero coupon
rat es, and t he yield curve t hat is drawn f rom t hese rat es is called t he zero coupon yield
curve ( ZCYC) .
We can t hus look upon every coupon paying bond, as a bundle of zero coupon bonds,
wit h each cash f low accruing at t he end of a t erm r1, r2, r3 rn, being valued as if t hey
were zero coupon bonds of t hat t enor. The est imat ion problem t heref ore is one of
ident if ying t hese unique rat es, and modeling t heir relat ionship wit h one anot her, which
in t urn is t he basis f or t he valuat ion of t he bond.
The act ual modeling of t he t rue rat es across t enor, and t heir relat ionship across t erm is
called t he modeling of t he t erm st ruct ure of int erest rat es, which at t empt s t he
est imat ion of t he t heoret ical spot rat es, f rom a set of market prices of bonds, based on a
t heoret ical f ramework t hat explains t he relat ionship bet ween t he rat es across various
t enors. There are a number of met hods t o do t his, and we shall discuss one of t hem in
a subsequent sect ion in t his chapt er.
14. 2 Boot st r appi n g
The error caused by t he reinvest ment assumpt ion in t he yield curve derived f rom t he
YTMs of t raded bonds can be eliminat ed, if we are able t o observe t he rat es of bonds
wit hout int ermit t ent coupons, i. e zero coupon bonds. However, in most market s, zero
coupon bonds across varying t enors do not exist , and even if t hey do, are not as act ively
t raded as t he coupon paying bonds. However, in most market s, t reasury bills which are
discount ed securit ies, wit h no int ermit t ent coupons, exist at t he short end of t he market .
Theref ore, we could boot st rap f rom t he zero coupon t reasuries, and derive t he r1, r2 r n
of t he coupon paying bonds.
For example, if a t reasury bill wit h 6 mont hs t o mat urit y, is t raded in t he market at
Rs. 96. 5 and mat ures t o t he par value of Rs. 100, t he 6 mont h zero coupon rat e can be
comput ed by solving t he equat ion:
5 . 0
5 . 0
) 1 (
100
5 . 96
r +
The 6 mont h rat e t hat solves t his equat ion is 7. 492%. We can now look f or a coupon
paying bond wit h 1 year t o mat urit y, whose valuat ion equat ion, in zero coupon t erms
can be st at ed as
150
1
1
2
5 . 0
5 . 0
1
0
) 1 ( ) 1 ( r
c
r
c
P
+
+
+
. ( 14. 3)
I n t his equat ion, we know t he periodic coupons as well as t he market price. From t he
earlier equat ion, we can subst it ut e t he value of r0.5. Then t he only unknown in t his
equat ion would be r1, f or which we can solve. The process of t hus discovering t he zer o
rat es f rom prices of coupon bonds, by subst it ut ing zero rat es est imat ed f or short er
durat ions is called boot st rapping. The yield curve is t hen drawn f rom t he plot of t hese
derived zero rat es, in t he similar manner as we drew t he par yield curve.
Boot st rapping is a very popular met hod wit h bond market dealers, f or est imat ing t he
t erm st ruct ure f rom market prices. Some of t he pract ical considerat ions in est imat ing
t he zero curve in t his manner are t he f ollowing:
1. The choice of bonds f or varying mat urit ies has t o ref lect market act ivit y.
Depending on t he bonds chosen f or est imat ing t he rat es, t he derived zero rat es
can vary. I t is, however, possible t o obt ain a plot of all implied zero rat es f or
all t raded bonds, and adopt t he curve f it t ing procedure, t o overcome t his
problem.
2. I t may not be possible t o obt ain zero rat es f or t he f irst cash f low of a bond, if a
zero coupon t reasury bill wit h mat ching mat urit y is not f ound. For example,
t here could be a bond, wit h t he f irst coupon 42 days away. We, t heref ore,
need t he r1 f or 42 days, in order t o value t his bond. A t reasury bill wit h exact ly
42 days t o mat urit y may not be t raded in t he market . Dealers most ly use a
linear int erpolat ion t o sort t his problem. Traded t reasury bills f or available
mat urit ies are picked up. Assume f or inst ance we have t he rat es f or 2 bills, one
wit h 40 days t o mat urit y, and anot her wit h 52 days t o mat urit y. The zero rat e
f or t he 42- day bond is comput ed by linearly int erpolat ing bet ween t hese t wo
rat es.
Example of linear int erpolat ion:
I f t he rat e f or t he 40- day bond is 6. 542%, and t he rat e f or t he 52- day bond is
6. 675%, t he rat e f or t he 42- day bond can be f ound as
= 6. 542 + [ ( 6. 675- 6. 542) ] x [ ( 42- 40) / ( 52- 40) ]
= 6. 56416%
3. The boot st rapping t echnique is sensit ive t o t he liquidit y and dept h in t he
market . I n a market wit h f ew t rades, and limit ed liquidit y, boot st rapping is
only an approximat ion of t he t rue t erm st ruct ure, due t o simple assumpt ions
( like linear int erpolat ion) made f or linking up rat es f or one t enor and t he rat es
f or anot her. I t is not uncommon f or some t o use more sophist icat ed non- linear
int erpolat ions.
151
14. 3 Al t er n at e Met h odol ogi es t o Est i mat e t h e Yi el d
Cu r ve
I n t he est imat ion of t he yield curve f rom a set of observed market prices, t he f ollowing
are import ant considerat ions:
a. The spot rat es and t he yield curve t hat is est imat ed should have a close f it wit h
market prices. That is, t he prices est imat ed by t he model and t he prices
act ually prevalent in t he market should have a close f it .
b. The model must apply equally well t o bonds which are not part of t he sample
used f or est imat ion. That is, if a very close f it is sought t o be achieved, it may
come at t he cost of t he model not being able t o value out - of - sample bonds.
The model would have incorporat ed noise in t he est imat ion.
c. The est imat ed yield curve should be smoot h, such t hat t he spot and f orward
rat es derived f rom t hem do not show excessive volat ilit y.
A number of mat hemat ical t echniques are used t o generat e a f it t ed yield curve f rom a
set of observed int erest rat e point s. They involve an opt imalit y crit eria consist ent wit h
t he assumpt ions regarding t he t erm st ruct ure of int erest rat es.
14. 3. 1 NSE ZCYC ( Nel son Sei gel Model )
I n t he I ndian market s, t erm st ruct ure est imat ion has been done, and is disseminat ed
every day by t he Nat ional St ock Exchange. The Zero Coupon Yield Curve ( ZCYC)
published by t he NSE, uses t he Nelson- Seigel met hodology.
18
The Nelson- Siegel
f ormulat ion specif ies a parsimonious represent at ion of t he f orward rat e f unct ion given by
)] / exp( ) / [( ) / exp( ) , (
2 1 0
m m m b m f + + . . ( 14. 4)
where m denot es mat urit y and b= [ 0, 1, 2 and ] are paramet ers t o be est imat ed.
Since t he model is based on t he expect at ions hypot hesis, it develops t he t erm st ruct ure
f rom t he no- arbit rage relat ionship bet ween spot and f orward rat es. The f orward rat e
f unct ion can be mat hemat ically manipulat ed ( int egrat ed) t o obt ain t he relevant spot rat e
f unct ion, t he t erm st ruct ure:
) / exp( ) / /( )] / exp( 1 [ ) ( ) , (
2 2 1 0
m m m b m r + +
( 14. 5)
I n t he spot rat e f unct ion, t he limit ing value of r( m, b) as mat ur it y get s large is 0 which
t heref ore depict s t he long t erm component ( which is a non- zero const ant ) . The limit ing
value as mat urit y t ends t o zero is 0 + 1, which t heref ore gives t he implied short - t erm
rat e of int erest .
Wit h t he above specif icat ion of t he spot rat e f unct ion, t he PV relat ion can now be
specif ied using t he discount f unct ion given by
18
The paper ( Gangadhar Darbha, et al, 2000) t hat describes t he met hodology can be
downloaded f rom www. nse. co. in \ ` product s\ ` zcyc. The f ollowing sect ion is ext ract ed
f rom t his paper.
152
,
_
100
) , (
exp ) , (
m b m r
b m d . . ( 14. 6)
The present value arrived at is t he est imat ed/ model price ( p_est ) f or each
bond. I t is common t o observe secondary market prices ( pmk t ) t hat deviat e
f rom t his value. For t he purpose of empirical est imat ion of t he unknown
paramet ers in t erm st ruct ure equat ion above, we post ulat e t hat t he observed
market price of a bond deviat es f rom it s underlying valuat ion by an error t erm
ei, which gives us t he est imable relat ion:
i i i
e est p pmkt + _ ( 14. 7)
This equat ion is est imat ed by minimising t he sum of squared price errors. The
st eps f ollowed in t he est imat ion procedure are as f ollows:
i. A vect or of st art ing paramet ers ( 0, 1, 2 and ) is select ed,
ii. The discount f act or f unct ion is det ermined using t hese st art ing paramet ers,
iii. This is used t o det ermine t he present value of t he bond cash f lows and t hereby t o
det ermine a vect or of st art ing model bond prices,
iv. Numerical opt imisat ion procedures are used t o est imat e a set of paramet ers
( under a given set of const raint s viz. non- negat ivit y of long run and short run
int erest rat es) t hat minimise t he sum of squared price errors,
v. The est imat ed set of paramet ers are used t o det ermine t he spot rat e f unct ion and
t heref rom t he model prices ( t his is t he f irst set of result s we comput e f or each
day) ,
vi. These model prices are used t o comput e associat ed model YTMs f or each bond
( t his is t he second set of result s) .
Plot s of t he est imat ed t erm st ruct ure f or any part icular day can be obt ained by f ollowing
t he
153
procedure below:
i. Creat e a series of mat urit y values; f or inst ance 1 t o 25 years, wit h st ep lengt hs of
( say) 0. 5 years
ii. For each mat urit y, use t he est imat ed paramet ers f or t he required day t o derive
corresponding spot rat es
iii. Wit h mat urit y values on t he X- axis, plot t he spot rat es against t he mat urit y
values,
iv. Spot rat e associat ed wit h any desired mat urit y ( eg. 8. 2 years) can be similarly
derived by subst it ut ing t he est imat ed paramet ers and m= 8. 2 in t he t erm st ruct ure
equat ion.
14.4 Th eor i es of t h e Ter m St r u ct u r e of I n t er est
Rat es
The t erm st ruct ure represent s t he dif f erent rat es of market int erest rat es f or dif f erent
periods of t ime. The shape of t he curve t heref ore cont ains crucial inf ormat ion on t he
f unct ional relat ionship bet ween price and t ime. The normally observed shapes of t he
yield curve are t he f ollowing:
a. upward sloping
b. downward sloping
c. humped
d. invert ed
The most commonly known t heories t hat at t empt an int erpret at ion of t he shape of t he
yield curve are:
The pure expect at ion hypot hesis
The liquidit y pref erence hypot hesis
The pref erred habit at hypot hesis
14. 4. 1 Pu r e Ex pect at i on Hypot h esi s
This int erpret at ion explains t he yield curve as a f unct ion of a series of expect ed f orward
rat es. Pioneered by I rving Fisher in 1896, t his is t he oldest t heory of t he t erm st ruct ure,
and is t he easiest t o quant if y and apply. The t radit ional f orm of t he pure expect at ions
t heory implies t hat t he expect ed average annual ret urn on a long t erm bond is t he
geomet ric mean of t he expect ed short t erm rat es. For example, t he one year spot rat e
can be t hought of as t he product of t he six- mont h spot and t he six mont h rat e six
mont hs f rom now ( six mont h f orward) . A risk neut ral invest or would t heref ore be
indif f erent bet ween t he one year spot rat e, and a one year posit ion f ormed by a
combinat ion of a six mont h spot and a six mont h f orward. Theref ore shape of t he yield
curve is driven by t he expect at ions about t he int erest rat es. Based on t he expect at ions
hypot hesis, we can calculat e a series of short t erm rat es, which over any given period
will, in aggregat e, reproduce t he market rat es expressed in t he yield curve.
14. 4. 2 Li qu i di t y Pr ef er en ce Hypot h esi s
This hypot hesis is a modif icat ion of t he expect at ion hypot hesis, incorporat ing risk.
Ot her t hings remaining t he same, invest ors would pref er short t erm bonds t o long t erm
154
bonds, because pricing of short t erm bonds is made easier by t he lower price risk of
t hese bonds and t he short er t erm t o mat urit y. Theref ore short t erm inst rument s will
enj oy a higher liquidit y t han long t erm inst rument s. I f invest ors pref er short t erm rat es
t o long t erm rat es, int erest rat es at t he lower end of t he yield curve would be lower, and
t he yield curve would slope upwards. The liquidit y pref erence hypot hesis posit s t hat
t he long t erm rat es are not only composed of expect ed short t erm rat es, but also
cont ain a liquidit y premium. The liquidit y premium is t he addit ional yield demanded by
invest ors t o ext end t he mat urit y of inst rument s, over longer periods of t ime. Theref ore
liquidit y premium can be expect ed t o increase wit h t ime t o mat urit y.
14. 4. 3 Pr ef er r ed Habi t at Hypot h esi s
Pref erred habit at hypot hesis recognizes t hat t he market is segment ed and t hat
expect at ions of invest ors is not unif orm across various t enors. This hypot hesis posit s
t hat dist inct cat egories of invest ors exist , and t hat each of t hese cat egories pref ers t o
invest at cert ain segment s of t he yield curve. For example, corporat es wit h short t erm
surplus f unds, would pref er t o deploy t he same in short t erm inst rument s, and may be
unwilling t o t ake price risks associat ed wit h invest ing in long t erm inst rument s. On t he
ot her hand, pension and insurance companies would pref er t o invest in long t erm bonds,
t o mat ch t he liabilit y prof ile of t heir port f olios. Since t he port f olios and t he asset
requirement s of invest ors vary, t hey would pref er some t enors over t he ot her, and
t heref ore f ocus on segment s of t he yield curve. The pref erred habit at t heory t heref ore
posit s t hat depending on demand and supply at varying t enors of t he yield curve,
invest ors will have t o be receive( pay) premiums( discount s) t o shif t away f rom a
pref erred habit at . The shape of t he yield curve t heref ore is a f unct ion of demand and
supply, and does not have any f ormal relat ionship t o int erest rat e expect at ions.
We can summarise t he int erpret at ion of t he alt ernat e shapes of t he yield curve under
t hese t hree hypot heses, as f ollows:
Sh ape of t h e yi el d cu r ve Ter m
st r u ct u r e
h ypot h eses
Fl at Upw ar d
sl opi n g
Dow n w ar d
sl opi n g
Hu mped
Ex pect at i on
s
Hypot h esi s
Short t erm
int erest rat es are
not expect ed t o
change.
Short t erm
int erest rat es
are expect ed
t o increase.
Short t erm
rat es are
expect ed t o
decrease.
Short t erm
rat es are
expect ed t o
init ially
increase, and
t hen decrease.
Li qu i di t y
Pr emi u m
Hypot h esi s
There is no
liquidit y premia
on long t erm
rat es, over short
t erm rat es.
Liquidit y
premia are
posit ive wit h
increases in
t erm.
Liquidit y
premia are
negat ive wit h
increases in
t erm.
Liquidit y premia
are posit ive
upt o a cert ain
t erm, af t er
which t hey t urn
negat ive.
Pr ef er r ed
Habi t at
Hypot h esi s
Demand and
supply are
mat ched at all
Excess of
supply over
demand in
Excess of
supply over
demand in
Excess of
supply over
demand in t he
155
mat urit ies. short er
mat urit ies.
longer
mat urit ies.
int ermediat e
t erm.
The t erm st ruct ure of int erest rat es becomes very import ant in a market in which
f orwards and derivat ives t rade, as t he valuat ion and t rading of t hese inst rument s is not
possible wit hout a dependable model of t erm st ruct ure. The NSE- ZCYC is an import ant
development in t his cont ext . I n t he I ndian market s, pending t he development of t he
f orward and derivat ive market s in int erest rat e product s, and limit ed liquidit y in t he spot
market s, yield curve est imat ions are yet t o gain import ance. However, t he increasing
f ocus on valuat ion and marking t o market of port f olios, has creat ed t he need f or t he
market yield curve, f or banks, PDs, inst it ut ions and mut ual f unds. The RBI used t o
publish t he yield curves f or valuat ion of bank port f olios. Af t er t he RBI discont inued t his
pract ice nearly 2 years ago, t he FI MMDA has creat ed a st andard yield curve, based on
polled yields at t he end of every t rading day, t o enable valuat ion of port f olios on t he
basis of a st andard yield curve. This has enabled st andard indust ry pract ice on
valuat ion. SEBI has mandat ed a st andard valuat ion model f or bonds in mut ual f und
port f olios, f rom December 1, 2000, based on a durat ion- based valuat ion model
developed by CRI SI L.
Model Quest i ons
1. Th e NSE ZCYC est i mat es f or Ju l y 11, 2001 ar e as f ol l ow s:
Bet a 0 = 11. 4652
Bet a 1 = - 2. 2510
Bet a 2 = - 10. 7202
Tau = 1. 4197
Wh at i s t h e spot r at e f or a t er m t o mat u r i t y of 3. 5 year s?
An sw er :
We use t he ZCYC valuat ion equat ion ( 14. 5)
) / exp( ) / /( )] / exp( 1 [ ) ( ) , (
2 2 1 0
m m m b m r + +
We can t ake t he values provided by NSE t o an Excel Spreadsheet , and key in t he
f ormula above, subst it ut ing 3. 5 f or m in t he equat ion, and subst it ut ing t he NSE
est imat es f or B0, B1 and B2 and Tau.
We t hen get
= 11. 4652 + ( ( - 2. 2510- 10. 7202) * ( 1- exp( - 3. 5/ 1. 4197) ) / ( 3. 5/ 1. 4197) - ( - 10. 7202* exp( -
3. 5/ 1. 4197) )
= 7. 56185%
2. I f t h er e ar e 2 bon ds t r adi n g i n t h e mar k et as f ol l ow s, on Ju l y 11, 2001 as
det ai l ed bel ow :
i . 11. 98% 2004 ( Mat u r i t y 8- Sep- 2004) : Rs. 111. 8
i i . 11. 19% 2005 ( Mat u r i t y 12 Au g 2005) : Rs. 111. 83
Wh at i s t h e l i n ear l y i n t er pol at ed r at e f or 3. 5 year s, u si n g t h e above dat a?
An sw er :
156
Using t he Yi el d f unct ion, we can f ind out t he YTM of t he above bonds as 7. 6917% and
7. 7524% respect ively. Using t he year f r ac f unct ion, we can f ind t he t erm t o mat urit y of
t hese bonds as 3. 1583 years and 4. 0861 years respect ively.
To f ind t he YTM f or a 3. 5 year bond, we can do a linear int erpolat ion, as f ollows:
= 7. 6917 + ( 7. 7524- 7. 6917) * ( ( 3. 5- 3. 1583) / ( 4. 0861- 3. 1583) )
= 7. 7141%
3. I f t h e yi el d cu r ve i s u p w ar d sl opi n g, w h i ch of t h e f ol l ow i n g i s f al se?
a. The market expect s short t erm int erest rat es t o increase.
b. The liquidit y premium is increasing wit h increase in t enor.
c. There is an excess of demand over supply in short er mat urit ies.
d. The int erest rat es are posit ively relat ed t o t erm, along t he yield curve.
An sw er : c
157
4. Th e f ol l ow i n g t er m st r u ct u r e of i n t er est r at es i s gi ven t o you :
Ten or
( i n
year s)
Yi el d
( % p. a. )
0. 30 7. 0257
0. 35 7. 0487
0. 40 7. 0847
0. 45 7. 1589
0. 50 7. 1905
0. 55 7. 2025
0. 60 7. 2368
0. 65 7. 2604
0. 70 7. 2928
0. 75 7. 3138
0. 80 7. 3388
0. 85 7. 3704
0. 90 7. 3939
0. 95 7. 4181
1. 00 7. 4379
On 15t h June 2001, you are required t o value a bond wit h a coupon of 11. 04%,
mat uring on 10- Apr- 2002. The f ace value of t he bond is Rs. 100. Given t he yield curve
inf ormat ion in t he t able above, what is t he value of t he bond? ( Use linear int erpolat ion
t o f ind discount ing rat es f or each of t he component cash f lows) .
An sw er :
We have t o f irst f ind t he cash f lows of t he bond upt o t he dat e of mat urit y, and t he
dist ance in years of each of t he cash f lows t o t he set t lement dat e.
We use t he coupncd f unct ion and f ind t hat t here are 115 days t o t he f irst coupon and
295 days t o t he next coupon, which t ranslat e int o 0. 319444 years and 0. 819444 years
respect ively.
The discount rat e f or t hese t wo t enors can be f ound wit h by int erpolat ion f rom t he t erm
st ruct ure inf ormat ion t hat is given in t he t able above.
The rat e f or t he t enor of 0. 319444 years can be f ound by linear int erpolat ion bet ween
t he t enors 0. 3 and 0. 35 years, as f ollows:
= 7. 0257+ ( 7. 0487- 7. 0257) * ( 0. 31944 0. 3) / ( 0. 35- 0. 3)
= 7. 0346%
Similarly t he rat e f or t he t enor of 0. 819444 can be f ound by int erpolat ion bet ween t he
t enors 0. 8 and 0. 85 years, as f ollows:
= 7. 3388 + ( 7. 7304 - 7. 7304) * ( 0. 81944 0. 8) / ( 0. 85- 0. 8)
= 7. 3511%
We can now value t he bond by discount ing t he cash f lows using t hese rat es, as f ollows:
81944 . 0 31944 . 0
) 073511 . 1 (
525 . 105
) 070346 . 1 (
525 . 5
+
= Rs. 104. 9627
158
This is t he value of t he bond, comput ed by discount ing each cash f low by t he
int erpolat ed yield f rom t he t erm st ruct ure of int erest rat es.
5. Th e NSE- ZCYC est i mat e of t h e spot r at e f or t h e t er m 7. 2876 year s i s
9. 1648% . Wh at i s t h e di scou n t ed val u e of a cash f l ow of Rs. 100, r ecei vabl e at
t h e en d of t h at t er m?
An sw er :
We can use t he ZCYC est imat es t o arrive at t he discount ed value of any cash f low, by
using t he f ormula:
}
100
* ) , (
exp{ ) , (
m b m r
b m d
Theref ore t he discount f act or t o be applied t o t he cash f low of Rs. 100, receivable at t he
end of 7. 2876 years is ( Excel recognises t he t erm exp in t he f ormula)
= exp ( ( - 9. 1684 * 7. 2876) / 100)
= 0. 5128
Theref ore, t he discount ed value of Rs. 100 will be
= 100* 0. 5128
= Rs. 51. 28
159
Ch apt er 15
Du r at i on
Durat ion, as t he name suggest s is, in a simple f ramework, a measure of t ime, t hough it s
applicat ions in underst anding t he price- yield relat ionship are more int ense. We shall
begin wit h t he simple def init ion, and lat er illust rat e t he alt ernat e applicat ions, including
modif ied durat ion and PV01.
15. 1 I n t r odu ct i on an d Def i n i t i on
I n t he case of bonds wit h a f ixed t erm t o mat urit y, t he t enor of t he bond is a simple
measure of t he t ime unt il t he bond' s mat urit y. However, if t he bond is coupon paying,
t he invest or receives some cash f lows prior t o t he mat urit y of t he bond. Theref ore it
may be usef ul t o underst and what t he average mat urit y of a bond, wit h int ermit t ent
cash f lows is. I n t his case we would f ind out what t he cont ribut ion of each of t hese cash
f lows is, t o t he t enor of t he bond. I f we can comput e t he weight ed average mat urit y of
t he bond, using t he cash f lows as weight s, we would have a bet t er est imat e of t he t enor
of t he bond. Since t he coupons accrue at various point s in t ime, it would be appropriat e
t o use t he present value of t he cash f lows as weight s, so t hat t hey are comparable.
Theref ore we can arrive at an alt ernat e measure of t he t enor of a bond, account ing f or
all t he int ermit t ent cash f lows, by f inding out t he weight ed average mat urit y of t he
bond, t he present value of cash f lows being t he weight age used. This t echnical measure
of t he t enor of a bond is called durat ion of t he bond.
Let s us at t empt an int uit ive underst anding of durat ion, wit h t he help of an example.
Suppose one had t wo opt ions:
Buy bond A selling at Rs. 100. 25 wit h 1 year t o mat urit y. The redempt ion value of
t he bond is Rs. 110. 275.
Buy bond B, also selling at Rs. 100. 25, and 1 year t o mat urit y. However, t he bond
pays Rs. 50. 5 at t he end of 6 mont hs, and Rs. 57. 5 at t he end of 1 year, on
mat urit y.
Bot h t hese bonds have t he same t enor of 1 year, and are priced at t he same yield 10%.
Would one t heref ore be indif f erent bet ween t he t wo opt ions? Why not ?
I nt uit ively, we seem t o pref er opt ion ( b) t o opt ion ( a) , because we receive some cash
f lows earlier, in t he second case. I n ot her words, t hough t he t wo opt ions are f or 1
years t enor, we int uit ively underst and t hat t he second opt ion places some f unds earlier
t han a year wit h us, and t heref ore must have an average mat urit y of less t han 1 year.
I f we are able t o comput e what percent age of f unds, in present value t erms is available
t o us, in t he case of bond B, we can underst and what t he average mat urit y of bond B is.
We at t empt doing t hat in Table 15. 1.
The 2 cash f lows accruing at t he end of 6 mont hs and 1 year have dif f erent present
values. At a discount ing rat e of 5% ( bond equivalent yield of 10% f or half year) , t he
cash f lows present values are Rs. 48. 1 and Rs. 52. 15 respect ively.
This present value cash f low st ream act ually means t hat 48% of t he bonds cash f lows
accrue at t he end of 6 mont hs, and 52% of cash f lows accrue at t he end of 1 year.
160
( Not e t hat t he sum of t he cash f lows is t he current value of t he bond, i. e. Rs. 100. 25;
and t he sum of t he weight s of t he cash f lows adds up t o 1) . I f we apply t hese weight s
t o t he period associat ed wit h t he cash f low, we know t hat t he weight ed mat urit y of t he
bond is 1. 52 half years, or 0. 76 years.
This is why we seem t o pref er bond B, whose average mat urit y is act ually less t han a
year. The durat ion of t his bond is 0. 76 years. I n t he case of bond A, all t he cash f lows
accrue at t he end of t he year. Theref ore, t he durat ion of t he bond is also 1 year.
I n any bond wit h int ermit t ent cash f lows accruing prior t o mat urit y, t he average
mat urit y will be lesser, and durat ion is a measure of t his average mat urit y of a bond.
Tabl e 15. 1: Wei gh t ed Pr esen t Val u es an d Du r at i on
Per i od Cash
f l ow
( Rs. )
Pr esen t val u e
of cash f l ow
( Rs. )
Wei gh t of
t h e pr esen t
val u e
Wei gh t ed
t en or of t h e
bon d ( Year )
1 50. 5 48. 10 0. 48 0. 48
2 57. 5 52. 15 0. 52 1. 04
Tot al 100. 25 1. 000 1. 52
Du r at i on 1. 52/ 2 = 0. 76
yr s
15. 2 Cal cu l at i n g Du r at i on of a Cou pon Payi n g Bon d
Fredrick Macaulay, in 1938, f irst propounded t he idea of durat ion, and we call his
measure as Macaulays durat ion.
Macaulay durat ion in years
n
i
t
pvtcf k
pvcf t
1
. ( 15. 1)
Where k = number of payment s per year ( in t he case of semiannual coupon paying
bonds, k = 2)
n = number of periods unt il mat urit y ( years t o mat urit y x k)
t = period in which cash f low is expect ed t o be received ( t = 1, 2, n)
pvcf t = present value of t he cash f low in period t discount ed at t he yield t o mat urit y
pvt cf = Tot al present value of t he cash f lows of t he bond, discount ed at t he bonds yield
t o mat urit y ( t his would act ually be t he price of t he bond) .
The above equat ion can also be st at ed as
( 1xPVCF1 + 2xPVCF2 + 3xPVCF3 . + nxPVCFn) / ( k x PVTCF) ( 15. 2)
161
Let us consider an example. See Table 15. 2. Column 1 list s t he period in which t he
cash f lows accrue. Column 2 is t he list of cash f lows, which in t his case are t he coupons
f or all t he periods, except t he last one, when t he coupon and redempt ion amount are
due. Column 3 is t he present value of each of t he cash f lows, discount ed f or t he
appropriat e period, at t he YTM rat e of 9%. ( 4. 5% on a semi - annual basis) . For example,
Rs. 5. 26 is t he discount ed value of Rs. 5. 5 receivable in six mont hs, discount ed at t he
rat e of 4. 5%.
The sum of t he present values is Rs. 107. 91 which is t he value of t he bond at a YTM of
9%. Column 4 provides t he weight ed value of t he present values, by comput ing t he
product of t he present values and t he period in column 1. Durat ion of t he bond is t he
sum of t hese weight ed values divided by t he sum of t he present value of t he cash f lows.
8. 039 is t he durat ion in half - years. Theref ore durat ion in years is 8. 039/ 2, which is 4. 02
years.
Tabl e 15. 2: Du r at i on of a 5 year 11% bon d, at a YTM of 9%
Per i od Cash f l ow s
( Rs. )
Pr esen t
Val u e of
Cash Fl ow s ( Rs. )
Wei gh t ed
Pr esen t
Val u es
( a)
Wei gh t ed
Cash
Fl ow s
( b)
Du r at i on
( c)
1 5. 5 5. 26 5. 263 0. 049 0. 049
2 5. 5 5. 04 10. 073 0. 047 0. 093
3 5. 5 4. 82 14. 459 0. 045 0. 134
4 5. 5 4. 61 18. 448 0. 043 0. 171
5 5. 5 4. 41 22. 067 0. 041 0. 204
6 5. 5 4. 22 25. 341 0. 039 0. 235
7 5. 5 4. 04 28. 291 0. 037 0. 262
8 5. 5 3. 87 30. 940 0. 036 0. 287
9 5. 5 3. 70 33. 309 0. 034 0. 309
10 105. 5 67. 93 679. 344 0. 630 6. 295
Tot al 107. 91 867. 535 1. 00 8. 04
( a)
Present Value in column ( 3) t imes period in column ( 1) .
( b)
Present Value in column ( 3) as f ract ion of Tot al present value.
( c)
Weight ed Cash f lows in column ( 5) t imes period in column ( 1) .
We can arrive at t he same result by f inding out t he weight of each of t he discount ed
cash f lows t o t he t ot al, and applying t his weight t o t he periods in which cash f lows
accrue. I n column 5 we f ind t he proport ion of cash f lows accruing in each of t he
periods, t o t he t ot al cash f lows. Durat ion is t he sum product of t hese weight s, mult iplied
by t he period in column 1, and summed up. We arrive at t he same value of 4. 02 years.
We also not ice what proport ion of t he cash f lows of t he bond accrue in each of t he
periods, in column 5. Only 63% of t he bonds cash f lows accrue in 5 years.
15. 3 Compu t i n g Du r at i on on Dat es ot h er t h an
Cou pon Dat es
162
I n t he example above, we had comput ed durat ion, discount ing t he cash f lows f or whole
periods, as we had assumed t hat t he calculat ions are made at t he beginning of t he cash
f low st ream. I n realit y, we should be able t o comput e durat ion on any day when a bond
is out st anding. I n order t o do t his, t he f ract ional periods represent ing t he dist ance of
each of t he cash f low f rom t he dat e of mat urit y will have t o be calculat ed, and t he
discount ing of cash f lows done f or t hese f ract ional periods. As in t he case of yield and
price calculat ions, t he day count convent ion in t he market should be known, apart f rom
t he set t lement and t he mat urit y dat es. We could t hen use t he Excel f unct ion Durat ion.
163
Box 15. 1: Fu n ct i on Du r at i on
I n order t o use Excel t o comput e t he durat ion of a bond on any given set t lement dat e,
we provide t he f ollowing values:
Set t lement dat e: t he dat e on which we want t o comput e t he durat ion, in dat e f ormat
Mat urit y dat e: t he dat e on which t he bond would redeem, in dat e f ormat
Coupon: Coupon of t he bond, as a rat e
Yield: YTM of t he bond, as a rat e
Frequency: Frequency of payment of coupons per year, 2 f or semi annual bonds
Basis: Day count convent ion in t he market . 4 f or European 30/ 360 convent ion.
Excel will ret urn t he durat ion of t he bond in years.
Tabl e 15. 3 Du r at i on of Sel ect G- Secs on Mar ch 29, 2001
Name Cou pon
( % )
Mat u r i t y
Dat e
Ter m t o
Mat u r i t y ( yr s)
Pr i ce
( Rs. )
Yi el d t o
Mat u r i t y
Du r at i on
( yr s)
CG2001 11. 75
25- Aug-
01 0. 41 101. 00 0. 09092 0. 406
CG2002 11. 15 9- Jan- 02 0. 79 102. 75 0. 07413 0. 752
CG2003 11. 10 7- Apr- 03 2. 05 103. 52 0. 09154 1. 779
CG2004 12. 50
23- Mar-
04 3. 03 108. 31 0. 09247 2. 593
CG2005 11. 19
12- Aug-
05 4. 44 106. 19 0. 09422 3. 554
CG2006 11. 68 10- Apr- 06 5. 11 107. 58 0. 09736 3. 794
CG2007 11. 90
28- May-
07 6. 25 109. 31 0. 09843 4. 457
CG2008 11. 40
31- Aug-
08 7. 53 107. 60 0. 09924 5. 239
CG2009 11. 99 7- Apr- 09 8. 14 109. 18 0. 10281 5. 217
CG2010 11. 30 28- Jul- 10 9. 47 106. 60 0. 10182 6. 006
CG2011 12. 32 29- Jan- 11 9. 98 110. 97 0. 10499 6. 054
CG2013 12. 40
20- Aug-
13 12. 58 111. 00 0. 10740 6. 849
Not i ce t h at t h e du r at i on of t h e 2013 12. 4 secu r i t y i s on l y 6. 85 year s, w h i l e i t s
t er m t o mat u r i t y i s 12. 58 year s.
The basic relat i onship bet ween coupon, t erm t o mat urit y and t he yield and durat ion can
be int uit ively underst ood, by viewing durat ion as t he f ulcrum t hat balances t he present
value of cash f lows of a bond. I f we view t he present values of t he cash f lows f rom a
bond, as weight s placed on a scale, durat ion represent s t he f ulcrum which would balance
t hese weight s on t he t ime scale. We have diagrammat ically represent ed t his in Figure
15. 3. which present s t he cash f lows of a 11%, 5- year bond, semi annual coupons,
selling at YTM of 11%. The durat ion of t his bond is 4. 02 years. I f we can imagine t hat
t here is a f ulcrum at 4. 02 on t he graph, we could begin t o see how t he f ulcrum would
behave f or changes in t he f act ors inf luencing durat ion. An increase in t he t erm would
mean more number of bars on t he chart . The f ulcrum would move t o t he right . Higher
t he t erm, great er t he durat ion. I f t he coupon rat es were higher, t he size of each of t he
bars would be higher. The f ulcrum would t hen move lef t . Durat ion and coupon are
inversely relat ed. Higher t he coupon, lower t he durat ion. I f t he yield at which we
discount t he cash f low increases, t he size of t he bars would decrease. The f ulcrum
would move t o t he lef t . Yield and durat ion are inversely relat ed.
164
Fi gu r e 15. 3: Pr esen t val u e of Cash Fl ow s on t h e Ti me Scal e
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
1 2 3 4 5 6 7 8 9 10
Years
P
r
e
s
e
n
t
V
a
l
u
e
o
f
C
a
s
h
F
l
o
w
s
Apart f rom t hese f act ors, durat ion is also impact ed by t he st ruct ure of t he bond. A bond
wit h sinking f und provisions would have a lower durat ion, as a higher percent age of t he
cash f lows of t he bond would accrue bef ore mat urit y. Similarly, callable bonds have
short er durat ion t han ot herwise comparable non- callable bonds. Call opt ions reduce t he
out st anding mat urit y period of a bond. Est imat ing t he durat ion of a callable bond is,
however, complicat ed by t he need t o est imat e t he probabilit y t hat t he opt ion will be
exercised.
15. 4 Modi f i ed Du r at i on
Though int uit ively we have known durat ion as t he weight ed average t erm t o mat urit y of
a bond, an alt ernat e explanat ion which looks at durat ion as t he approximat ion of t he
slope of t he price- yield relat ionship, is signif icant , and has import ant applicat ions. We
have known t hat a bonds realized yield is impact ed by coupon, t erm t o mat urit y and
yield. Durat ion is a single measure approximat ion of t he impact of all of t hese t hree
f act ors, on t he price of a bond, f or a given change in yield. Theref ore, durat ion is an
import ant measure of sensit ivit y of a bond t o changes in underlying yield, and hence t he
int erest rat e risk of a bond.
The price of a bond is t he present value of t he cash f lows associat ed wit h t he bond, and
can be represent ed as
n
n
n
t
t
t
r
C
r
C
r
C
r
C
P
) 1 (
.........
) 1 ( ) 1 ( ) 1 (
2
2 1
1
+ +
+
+
. . ( 15. 3)
165
I n order t o underst and how price changes f or a small change in yield, we can t ake t he
f irst derivat ive
166
of t he above equat ion wit h respect t o r, t o get t he f ollowing equat ion:
1
]
1
+
+
+
+
+ +
n
n
r
nC
r
C
r
C
r dr
dP
) 1 (
.......
) 1 (
2
) 1 ( ) 1 (
1
2
2
1
1
. ( 15. 4)
This equat ion comput es t he absolut e change in t he price of a bond f or a given change in
yield. I n order t o convert t he same int o a percent age change in price f or a percent age
change in yield, we divide bot h sides of equat ion by t he bond price, as f ollows:
P r
nC
r
C
r
C
r P dr
dP
n
n
1
.
) 1 (
.......
) 1 (
2
) 1 ( ) 1 (
1 1
.
2
2
1
1
1
]
1
+
+
+
+
+ +
. . ( 15. 5)
Th e t er m on t h e r i gh t h an d si de of t h e above equ at i on can be r ew r i t t en as
=
r
Duration
+
1
( 15. 6)
This f ormula represent s t he percent age change in price of a bond, f or small changes in
yield. This measure is known as t he modif ied durat ion of a bond. We can st at e t his
relat ionship in a generalized f orm as
% change in price of a bond = Modif ied durat ion * % change in yield
For example, f or a bond wit h a modif ied durat ion of 7. 5, a 50 basis point change in yield
will result in a 7. 5 * 50/ 100 = 3. 75% change in price, in t he opposit e direct ion ( not ice
t he minus sign in t he equat ion, signif ying t hat t he change in price is in t he opposit e
direct ion of t he change in yield - yield and price are inversely relat ed) . I f t he yield is
semi- annual, we use half t he yield in t he equat ion.
Modif ied durat ion is t he slope of t he line in t he priceyield f unct ion, and f or small
changes in t he yield of a bond, modif ied durat ion indicat es t he percent age change in
price t hat can be expect ed. Modif ied durat ion is, t heref ore, a direct measure of t he
int erest rat e sensit ivit y of a bond. Higher t he modif ied durat ion of a bond, great er t he
percent age change in price f or a given change in yield.
Box 15. 2: Modi f i ed Du r at i on Fu n ct i on
I n order t o use Excel t o comput e t he modif ied durat ion of a bond on any given
set t lement dat e, we use t he f unct ion Mdurat ion, and provide t he f ollowing values:
Set t lement dat e: t he dat e on which we want t o comput e t he modif ied durat ion, in dat e
f ormat
Mat urit y dat e: t he dat e on which t he bond would redeem, in dat e f ormat
Coupon: Coupon of t he bond, as a rat e
Yield: YTM of t he bond, as a rat e
167
Frequency: Frequency of payment of coupons per year, 2 f or semi annual bonds
Basis: Day count convent ion in t he market . 4 f or European 30/ 360 convent ion.
Excel will ret urn t he modif ied durat ion of t he bond in years.
I t has t o be remembered t hat modif ied durat ion will provide close approximat ion of t he
act ual change in prices, f or small changes in yield. For large changes in yield, however,
t he f irst order derivat ive, which is what modif ied durat ion is, is inadequat e.
Tabl e 15. 4: Modi f i ed Du r at i on of a Set of Bon ds
Name Cou pon
( % )
Mat u r i t y
Dat e
Ter m t o
mat u r i t y
( yr s)
Pr i ce on
Mar ch 29,
2001 ( Rs. )
YTM ( % ) Modi f i ed
Du r at i on
( Yr s)
CG2001 11. 75 25- Aug- 01 0. 41 101 0. 09092 0. 388
CG2002 11. 15 9- Jan- 02 0. 79 102. 75 0. 07413 0. 725
CG2003 11. 1 7- Apr- 03 2. 05 103. 515 0. 09154 1. 701
CG2004 12. 5 23- Mar- 04 3. 03 108. 31 0. 09247 2. 479
CG2005 11. 19 12- Aug- 05 4. 44 106. 19 0. 09422 3. 394
CG2006 11. 68 10- Apr- 06 5. 11 107. 58 0. 09736 3. 618
CG2007 11. 9 28- May- 07 6. 25 109. 31 0. 09843 4. 248
CG2008 11. 4 31- Aug- 08 7. 53 107. 6 0. 09924 4. 991
CG2009 11. 99 7- Apr- 09 8. 14 109. 18 0. 10281 4. 962
CG2010 11. 3 28- Jul- 10 9. 47 106. 6 0. 10182 5. 715
CG 2011 12. 32 29- Jan- 11 9. 98 110. 97 0. 10499 5. 752
CG2013 12. 4 20- Aug- 13 12. 58 111. 2 0. 10740 6. 500
Not ice t hat modif ied durat ion is lower t han durat ion of t he same set of bonds, comput ed
in t he beginning of t his chapt er. Tables 15. 4 and 15. 5 illust rat e t he applicat ion of
modif ied durat ion t o measuring t he int erest rat e sensit ivit y of bonds.
The modif ied durat ion of t hese bonds will provide input s f or underst anding t he int erest
rat e sensit ivit y of t he bonds. We change t he yield of t hese bonds by 50bps, and re-
comput e t he value of t he bonds, in Table 15. 5.
We can use t he same set of bonds t o illust rat e how modif ied durat ion helps est imat e
changes in t he price of bonds f or a given change in yield. We have used t he same set of
bonds in t he Table 15. 3, but changed t he yield by 50 basis point s ( column 3 in t he t able
15. 4) . The new price f or t he now changed yield is comput ed and post ed in column 4.
The act ual percent age change in price, f or a 50 bp change in yield is in column 5. The
percent age change in price, comput ed wit h modif ied durat ion ( Mdurat ion * basis point
change in yield/ 100) is in t he last column. Not ice t hat t he numbers in t he last 2
columns are f airly comparable.
Tabl e 15. 5: I n t er est Rat e Sen si t i vi t y of a Set of Bon ds - Usi n g Modi f i ed
Du r at i on
Ol d pr i ce
of t h e
bon d ( Rs. )
Modi f i ed
Du r at i on
( Yr s)
New yi el d
( + 50 bp)
( % )
New Pr i ce
( Rs. )
Act u al %
ch an ge i n
pr i ce
% Ch an ge i n
pr i ce
compu t ed w i t h
mdu r at i on
101 0. 388 0. 09592 100. 80 0. 1973 0. 1940
102. 75 0. 725 0. 07913 102. 37 0. 3701 0. 3625
103. 515 1. 701 0. 09654 102. 60 0. 8887 0. 8504
108. 31 2. 479 0. 09747 106. 98 1. 2320 1. 2394
106. 19 3. 394 0. 09922 104. 38 1. 7019 1. 6971
107. 58 3. 618 0. 10236 105. 56 1. 8785 1. 8091
109. 31 4. 248 0. 10343 106. 94 2. 1708 2. 1241
168
107. 6 4. 991 0. 10424 104. 93 2. 4773 2. 4957
109. 18 4. 962 0. 10781 106. 38 2. 5660 2. 4809
106. 6 5. 715 0. 10682 103. 56 2. 8521 2. 8575
110. 97 5. 752 0. 10999 107. 78 2. 8707 2. 8761
111. 2 6. 500 0. 11240 107. 63 3. 2116 3. 2498
15. 5 Ru pee Du r at i on
Modif ied durat ion provides a measure of percent age change in price, f or a percent age
change in yield. However t wo bonds wit h t he same measure of modif ied durat ion will
change in value, in rupee t erms, in much dif f erent manner, depending on t he price at
which t hey are t rading. Consider Table 15. 5. The rupee change in value of t he bond is
dif f erent across t he bonds, and is a f unct ion of bot h modif ied durat ion and t he price.
Theref ore rupee price change can be calculat ed as:
Modif ied durat ion * yield change ( in basis point s) * rupee price of t he bond.
We can st andardize t he expect ed price change in rupee t erms, f or a 100 basis point
change in yield as
Modif ied durat ion * 0. 01( 100 basis point s) * rupee price of t he bond.
This value is called t he dollar ( rupee) durat ion of a bond, and is comparable across
bonds selling at various prices. Table 15. 6 shows t he rupee durat ion of a set of bonds.
Rupee durat ion represent s t he change in price f or a 100 basis point change in yield.
( From t he dat a on t he bonds t hat is available, calculat e act ual change in price f or a
100bp change in yield, and compare t he same wit h value in t he last column in t he
t able15. 6) .
Tabl e 15. 6: Ru pee Du r at i on of Bon ds
Name Cou pon ( % ) Pr i ce ( Rs. ) Mdu r at i on
( Yr s)
Ru pee
Du r at i on
CG2001 11. 75 101 0. 388 0. 391799
CG2002 11. 15 102. 75 0. 725 0. 744885
CG2003 11. 1 103. 515 1. 701 1. 760554
CG2004 12. 5 108. 31 2. 479 2. 684794
CG2005 11. 19 106. 19 3. 394 3. 604203
CG2006 11. 68 107. 58 3. 618 3. 892417
CG2007 11. 9 109. 31 4. 248 4. 643677
CG2008 11. 4 107. 6 4. 991 5. 370745
CG2009 11. 99 109. 18 4. 962 5. 417223
CG2010 11. 3 106. 6 5. 715 6. 09217
CG 2011 12. 32 110. 97 5. 752 6. 383322
CG2013 12. 4 111. 2 6. 500 7. 227553
Pr i ce Val u e of a Basi s Poi n t ( PV01)
Anot her import ant variat ion t o t he rupee durat ion, which is used ext ensively in pract ice,
is t he price value of a basis point ( known commonly as PVBP or PV01) . The PV01 of a
bond is t he rupee value of change in price of a bond, f or a 1 basis point change in yield.
PV01 is calculat ed as
169
Modif ied durat ion * Price of t he bond* 0. 01/ 100.
PV01 is also = Rupee Durat ion of a bond/ 100.
PV01 of a bond is a number t hat can be applied f or any ant icipat ed change in yield, t o
ascert ain t he change in price value. I n t able 15. 6, t he last column has t o be divided by
100, t o obt ain t he PV01 of each of t he bonds. I n pract ice, PV01 is ext ensively used in
ascert aining t he price sensit ivit y of a port f olio. PV01 of a port f olio is t he port f olios
modif ied durat ion t imes t he market value of t he port f olio, mult iplied by t he value
0. 0001.
PV01 is a usef ul number in buying hedge product s f or a port f olio. The payof f f rom a
hedge has t o mat ch t he PV01 of t he port f olio, t o enable ef f ect ive hedging.
15. 7 Por t f ol i o Du r at i on
The durat ion of a port f olio of bonds can be comput ed in t wo ways:
( a) Map t he cash f lows of t he bond int o various t erm bucket s, when t hey are due, and
using yield of t he port f olio, discount t he t ot al cash f lows of t he port f olio. Comput e
durat ion wit h t he usual f ormula, t reat ing t he aggregat e cash f lows as if t hey were a
single bond.
( b) Comput e t he weight ed durat ion of a port f olio, using t he market value of t he bond as
t he weight age.
Though ( a) is concept ually sound, it is a comput at ionally int ensive procedure. Theref ore
in pract ice ( b) is a more commonly used approach t o det ermine t he durat ion of a
port f olio.
Consider t he set of bonds we have been using in t his chapt er. Table 15. 7 shows t he
durat ion of a port f olio t hat holds one each of all t he bonds. ( The weight ages can be
changed f or any quant it y holding in each of t he bond. What we require f or comput at ion
is t he market value of t he port f olios exposure t o a given bond, as a proport ion of t he
t ot al market value of t he port f olio) .
) (
1
i
N
i
i P
W D Dur
. . ( 15. 7)
Durat ion of a port f olio is t he sum product of durat ion of each securit y in t he port f olio
( Di) t imes t he proport ion of t he securit y t o t ot al port f olio value ( as a decimal) ( Wi) .
I n t he t able 15. 7, t he proport ional weight of each bond is comput ed as t he price of t he
bond divided by t he value of t he port f olio ( in t his case t he sum of t he prices of all t he
bonds, as we assume t hat we hold one bond each) .
For example, t he value 0. 079 = 101/ 1284. 205. The last column applies t hese
proport ions t o t he durat ion of each bond. The durat ion of t he port f olio is t he sum of t he
last column, which is t he weight ed durat ion of all t he bonds in t he port f olio.
170
Tabl e 15. 7: Du r at i on of a Por t f ol i o of Bon ds
Name Cou pon ( % ) Pr i ce ( Rs. ) Du r at i on
( Yr s)
Wei gh t s Wei gh t ed
Du r at i on
CG2001 11. 75 101. 00 0. 4056 0. 079 0. 032
CG2002 11. 15 102. 75 0. 7518 0. 080 0. 060
CG2003 11. 10 103. 52 1. 7786 0. 081 0. 143
CG2004 12. 50 108. 31 2. 5934 0. 084 0. 219
CG2005 11. 19 106. 19 3. 5540 0. 083 0. 294
CG2006 11. 68 107. 58 3. 7943 0. 084 0. 318
CG2007 11. 90 109. 31 4. 4572 0. 085 0. 379
CG2008 11. 40 107. 60 5. 2391 0. 084 0. 439
CG2009 11. 99 109. 18 5. 2168 0. 085 0. 444
CG2010 11. 30 106. 60 6. 0059 0. 083 0. 499
CG 2011 12. 32 110. 97 6. 0543 0. 086 0. 523
CG2013 12. 40 111. 20 6. 8486 0. 087 0. 593
Por t f ol i o val u e 1284. 205 Por t f ol i o Du r at i on 3. 942
We can comput e modif ied durat ion also in a similar manner, using market values of t he
bonds as weight s. We can t hen est imat e t he int erest rat e sensit ivit y of t he port f olio.
The modif ied durat ion of t his port f olio of bonds can be comput ed, using t he mdurat ion
f unct ion, and using t he value weight s as in t he case of port f olio durat ion.
Table 15. 8 shows t he modif ied durat ion of t his port f olio, which is 3. 754. We have also
t aken f rom our earlier illust rat ion of price sensit ivit y, t he new prices of bonds, when
int erest rat es increase by 50 basis point s.
We see t hat t he value of t he port f olio has f allen t o Rs. 1259. 906 due t o t his change in
rat es. I n percent age t erms, t his change is 1. 89%. Given t he port f olios modif ied
durat ion of 3. 754, we can expect f or a 50 basis point change in yield, a price change of
3. 754 * 0. 5 = 1. 877%
171
Modif ied durat ion of t he port f olio t hus provides a close approximat ion of t his change in
price.
Tabl e 15. 8: Modi f i ed Du r at i on of a Por t f ol i o
Name Cou pon
( % )
Pr i ce
( Rs. )
Modi f i ed
Du r at i on
Wei gh t
Wei gh t ed
Mdu r at i on
New Pr i ce
( a)
CG200
1 11. 75 101. 00 0. 388 0. 079 0. 031 100. 801
CG200
2 11. 15 102. 75 0. 725 0. 080 0. 058 102. 370
CG200
3 11. 10 103. 52 1. 701 0. 081 0. 137 102. 595
CG200
4 12. 50 108. 31 2. 479 0. 084 0. 209 106. 976
CG200
5 11. 19 106. 19 3. 394 0. 083 0. 281 104. 383
CG200
6 11. 68 107. 58 3. 618 0. 084 0. 303 105. 559
CG200
7 11. 90 109. 31 4. 248 0. 085 0. 362 106. 937
CG200
8 11. 40 107. 60 4. 991 0. 084 0. 418 104. 934
CG200
9 11. 99 109. 18 4. 962 0. 085 0. 422 106. 378
CG201
0 11. 30 106. 60 5. 715 0. 083 0. 474 103. 560
CG
2011 12. 32 110. 97 5. 752 0. 086 0. 497 107. 784
CG201
3 12. 40 111. 20 6. 500 0. 087 0. 563 107. 629
Tot al 1284. 205
Por t f ol i o Modi f i ed
Du r at i on 3. 754 1259. 906
( a)
Price assuming 50 basis point increase in yield.
15. 8 Li mi t at i on s of Du r at i on
Durat ion is not a st at ic propert y of a bond. Durat ion of a bond changes over t ime, and
wit h changes in market yields. Any st rat egy based on durat ion values of a bonds will,
t heref ore, require dynamic t uning.
Comput ing durat ion involves t he discount ing of cash f lows of a bond. I t is common t o
use t he YTM of t he bond, as t he rat e at which cash f lows are discount ed. Theref ore, t he
limit at ions of YTM ext end t o t he comput at ion of durat ion.
We use durat ion based on t he view t hat equal changes in int erest rat es occur across
various t erms. I n ot her words, when we measure change in yield and use durat ion t o
est imat e change in price , we assume t hat t he given change in yield occurs across t he
t enor spect rum. This act ually t ranslat es int o an assumpt ion of parallel shif t s in t he yield
curve, which is not a very realist ic assumpt ion t o make.
Durat ion is t he f irst derivat ive of t he price- yield f unct ion. The result s obt ained by using
durat ion t o measure price change is only an approximat ion of t he act ual price yield
relat ionship, which is not linear, but convex.
Model Quest i ons
172
1. Th e du r at i on of a cou pon payi n g bon d i s al w ays l ow er t h an i t s t er m t o
mat u r i t y, becau se:
a. Since durat ion is t he measure of average mat urit y, it has t o be lower t han t he
t enor.
b. Durat ion measures t he weight ed mat urit y, and t heref ore cannot be compared t o
t enor of a bond.
c. As long as some cash f lows are received prior t o mat urit y, t he weight age of t he
t erminal cash f low cannot be 1.
An sw er : c
2. On Ju l y 11, 2001, t h e f ol l ow i n g i s t h e mar k et val u e of t h e bon ds i n you r
por t f ol i o. ( Assu me equ al h ol di n gs i n al l t h e bon ds) . Wh at i s t h e du r at i on of
t h e por t f ol i o?
Cou pon
( % )
Mat u r i t y
dat e
Pr i ce on
11- Ju l - 2001 ( Rs. )
11. 68 6- Aug- 2002 104. 34
11. 00 23- May- 2003 105. 74
12. 50 23- Mar- 2004 111. 63
11. 98 8- Sep- 2004 111. 8
11. 19 12- Aug- 2005 111. 83
11. 68 10- Apr- 2006 114. 4
11. 90 28- May- 2007 116. 6
An sw er :
We can use t he Yield f unct ion t o f ind t he YTM and t he Durat ion Funct ion t o comput e
durat ion, as f ollows:
Cou po
n ( % )
Mat u r i t y
dat e
Mar k et Pr i ce
on 11- Ju l - 2001 ( Rs. )
YTM
( % )
Du r at i on
( Yr s)
11. 68 6- Aug- 2002 104. 34 7. 3728% 0. 990695
11. 00 23- May- 2003 105. 74 7. 6309% 1. 720562
12. 50 23- Mar- 2004 111. 63 7. 6399% 2. 318881
11. 98 8- Sep- 2004 111. 8 7. 6917% 2. 653983
11. 19 12- Aug- 2005 111. 83 7. 7524% 3. 297774
11. 68 10- Apr- 2006 114. 4 7. 9700% 3. 753991
11. 90 28- May- 2007 116. 6 8. 2733% 4. 463083
Por t f ol i o
Val u e 776. 34
Por t f ol i o Du r at i on 2. 781662
The port f olio durat ion is t he weight ed durat ion of t he bonds, using t he market values as
weight s. I t is comput ed as
Sumproduct ( market price, durat ion) / sum ( market price)
= 2. 7816
3. Usi n g t h e same dat a as i n Qu est i on 2, i f t h e ex pect at i on i s t h at yi el d w ou l d
i n cr ease by 50 basi s poi n t s, w h at w ou l d be t h e ex pect ed ch an ge i n t h e val u e of
t h e por t f ol i o?
An sw er : We can use t he mdu r at i on f unct ion in Excel, and comput e t he modif ied
durat ion of all t he bonds, and f ind t he port f olio modif ied durat ion, using a similar
met hod as in Answer 2. We would arrive at a number 2. 6763 as t he port f olio s
modif ied durat ion.
A 50bp increase in yield will reduce t he value of t he port f olio by
173
2. 6763* . 05 = 1. 3381%
I n rupee t erms t hat would be Rs. 776. 34 * 1. 3381%
= Rs. 10. 3888
174
Ch apt er 16
Fi x ed I n come Der i vat i ves
16. 1 Wh at ar e Fi x ed- I n come Der i vat i ves?
Fixed income derivat ives are securit ies t hat derive t heir value f rom some bond price,
int erest rat e or an underlying bond market variable. I n t erms of volumes globally, t hey
account f or a maj or proport ion of derivat ives market s. They are import ant because t hey
enable banks t o separat e f unding/ liquidit y decisions f rom int erest - rat e sensit ivit y
decisions.
16. 1. 1 For w ar d Rat e Agr eemen t s
Spot Rat es an d For w ar d Rat es
We already have discussed Spot or Zero- Coupon int erest rat es. A spot int erest rat e
is t he int erest rat e on an invest ment st art ing t oday and ending af t er some ( say n)
years. This is a pure int erest rat e i. e. it is assumed t hat t here are no coupon payment s
bet ween t oday and n years. This is also t he yield on a zero coupon bond of t he
corresponding mat urit y. I n t he absence of zero coupon bonds, t he spot rat es can be
est imat ed f rom t he yields on coupon bearing bonds by a process called boot st rapping
A f orward rat e is t he int erest rat e cont ract ed t oday on an invest ment t hat will be
init iat ed af t er some t ime ( n years) . I n ot her words, t hey are rat es implied by current
spot rat es f or periods in t he f ut ure.
Consider t he f ollowing example:
Ti me Spot Rat e ( an n u al i zed)
1 year 6%
2 year 7%
3 year 8%
This means t hat Rs. 100 invest ed t oday will give Rs. 106 at t he end of one year.
Rs 100 invest ed t oday will give Rs, ( 100* ( 1+ 7/ 100)
2
) t hat is Rs. 114. 49 at t he end of
t wo years and Rs ( 100* ( 1+ 8/ 100)
3
) t hat is Rs. 125. 97 at t he end of t hree years.
The quest ion we must ask is as f ollows:
What would be t he amount I would receive on Rs. 100 invest ed af t er one year at t he
end of t wo years?
Not ice t hat t he payof f f rom t he above invest ment would come at t he end of t wo years
f rom t oday.
We can re- creat e t he above invest ment s using present int erest rat es. For t he sake of
simplicit y, we assume t hat bid- ask spreads are negligible.
1. Borrow t oday in an amount t hat will give Rs. 100 af t er one year. This amount is
Rs. 100/ ( 1. 06) t hat is Rs. 94. 34.
2. I nvest t he same amount f or a period of t wo years. At t he end of t wo years, t he
payof f will be Rs. 94. 34* ( 1. 07)
2
t hat is Rs. 108.
175
At t he end of one year, Rs. 100 will have t o be paid out f or t he f irst borrowing. At t he
end of t wo
176
years Rs. 108 will f low in. I n t erms of cash f lows, t his is what it looks like:
Not ice t hat at Year 0, t here is an out f low and inf low of Rs. 94. 34 and hence t he net f low
is zero. I n ot her words, t he above series of f lows is t he same as:
I n ef f ect we have creat ed an invest ment where we will lend Rs. 100 af t er one year and
will get back Rs. 108 af t er t wo years. This means t hat t he int erest rat e f rom one year t o
t wo years f orward is 8%.
I n t he above example, t he 8% int erest rat e is called t he f orward rat e of int erest . I n
most circumst ances, t he f orward rat e of int erest is t he expect ed spot rat e f or t he
corresponding period.
Why is t he f orward rat e also t he expect ed spot rat e?
I f t he market had expect ed t he spot rat e ( f rom 1 t o 2 years) t o be less t han t he f orward
rat e indicat ed t oday, t hey would have heavily st art ed doing t he above t ransact ions t o
lock in t he great er f orward rat e. This would have meant addit ional demand f or borrowing
one- year money and lending t wo- year money. This would have pushed t he one- year
0
Year
1
Year
2
Years
Rs.
108
Rs.
100
Rs.
94.34
Rs.
94.34
0
Year
1
Year
2
Years
Rs.
108
Rs.
100
177
spot rat e up and t he t wo- year spot rat e down t ill t he implied f orward rat e was in line
wit h market expect at ions.
178
We can similarly const ruct t he 2- 3 year f orward rat e and t he 1- 3 year f orward rat e.
The rat es would be as f ollows:
Year Spot rat es
0- 1 6%
0- 2 7%
0- 3 8%
Year Forward Rat es
1- 2 8%
2- 3 10%
A point t o be not ed about f orward rat es is t hat t hey can never be negat ive. This applies
some rest rict ions on t he t erm st ruct ure of t he spot rat es. For inst ance, t he f ollowing
t erm st ruct ure cannot be possible
Year Spot rat es
0- 1 6%
0- 2 7%
0- 3 4%
This is because comput at ion shows t hat t he 2- 3 year f orward rat e is 1. 7%. I f t he
f orward rat e were negat ive, one can borrow t hree- year money and invest it f or t wo
years and sit on cash f rom year 2 t o year 3 t o make a risk f ree prof it .
Formula f or comput at ion of t he Forward Rat e:
I f we have t he n- year spot rat e as Rn and t he m- year spot rat e as Rm where m> n
And we want t o comput e t he f orward rat e Fmn f rom year n t o year m, t hen:
( 1+ Rn)
n
* ( 1+ Fmn )
m- n
= ( 1+ Rm)
m
Using t he above f ormula, Fmn can be comput ed.
16. 2 Mech an i cs of For w ar d Rat e Agr eemen t s
Forward Rat e Agreement s ( FRAs) are over t he count er derivat ive cont ract s t hat allow
count er- part ies t o lock int o a specif ied int erest rat e f or a f ut ure dat e. The buyer of an
FRA locks in a borrowing rat e while t he seller locks int o a lending rat e.
Typically t hese cont ract s are st ruct ured in such a way t hat t he dif f erence bet ween t he
market rat e and t he locked- in rat e is set t led.
Consider t he f ollowing example:
A and B ent er int o a forward rat e agreement of one year, st art ing one year f rom t oday,
f or a not ional amount of Rs. 100. Part y A is t he buyer i. e. it has locked int o a borrowing
rat e. The spot int erest rat es in t he market are t he same as t he ones ment ioned in t he
earlier example.
Pr i ci n g
The f irst quest ion t o ask is: What is t he most likely rat e at which t he Forward Rat e
Agreement will be cont ract ed?
179
The answer is obvious: I t should be t he f orward rat e implied by t odays int erest rat es
f rom year 1 t o year 2. We have earlier calculat ed t his at 8%. I f t he cont ract ed FRA rat e
is dif f erent , t hen one of t he part ies will carry out t he t wo t ransact ions ment ioned in t he
earlier example and benef it f rom it . This part y will have earned a risk- f ree prof it . I t is
unlikely t hat t he ot her part y will allow t hat t o happen.
I n pract ice, it is slight ly dif f erent because of bid- ask spreads bet ween lending and
borrowing rat es.
Suppose one year has passed by. Now t he one year spot int erest rat e is 7%. The
quest ion now is: Who has benef it ed f rom t he FRA and by how much?
The scenario now is as f ollows:
1. Part y A had locked int o a borrowing st art ing t oday and ending one year f rom
now at 8%.
2. Todays rat e is act ually 7%.
This means t hat part y A will lose out 1% at t he end of next year.
I n a t ypical FRA wit h net t ed out cash int erest payment s, t he amount t hat A would lose
will be discount ed at t he prevailing rat e ( 7%) and set t led. The FRA is t hen closed out . I t
is easy t o work out t hat t he part y t hat is long a FRA ( Borrower) receives a payment
when t he rat es go up and t he part y t hat has sold an FRA ( Lender) receives payment
when t he rat es go down.
The advant age of net t ing is t hat t he not ional amount s and int erest rat es need not be
act ually exchanged. This causes signif icant reduct ion is credit risk. However, one will
also f ind FRAs t hat are in t he nat ure of act ual lending. I n I ndia, t here is some amount of
f orward lending act ivit y bet ween banks and corporat es.
16. 3 I n t er est Rat e Fu t u r es
Fut ures are st andardized Forward cont ract s t hat are t raded on exchanges. The
count erpart in t his case will be t he exchange it self . These are cont ract s on eit her t he
level of int erest rat e of specif ied t enors, or on t he price of bonds of part icular mat urit y.
An example of t he f ormer are t he Euro- Dollar f ut ures cont ract s t raded on LI FFE. An
example of t he lat t er are t he T- Bond f ut ures t raded on CBOT. I n I ndia, int erest rat e
f ut ures have not been int roduced as yet .
There are several import ant dif f erences bet ween Fut ures and Forward cont ract s:
1. Fut ures are st andardized and available only f or cert ain t enors and dat es and
only on cert ain int erest rat e benchmarks. I n t hat sense, t heir usage is
rest rict ive.
2. Fut ures are t radable on t he exchange. Hence t hey are highly liquid inst rument s.
3. Fut ures are marked t o market daily and t he Prof it and Loss on t he cont ract is
paid out , bet ween t he part icipant and t he exchange.
Uses of FRAs an d Fu t u r es
As wit h any derivat ives cont ract s, FRAs and f ut ures have t hree main uses.
1. Hedging
2. Speculat ion
3. Arbit rage
180
Hedging: FRAs and Fut ures can be used t o remove uncert aint y about f ut ure int erest
rat es and hence reduce t he uncert aint y of f ut ure earnings.
For inst ance, suppose t he Financial Manager of a company knows t hat t here is going t o
be a large inf low of cash one year down t he line, which will have t o be invest ed. He is
also uncert ain about int erest rat es one year down t he line and want s t o remove t his
uncert aint y. A very good way t o do t his is t o sell a f orward rat e agreement st art ing one
year hence. This way, he can lock int o a f orward rat e t oday it self and remove t he
uncert aint y.
Speculat ion:
Suppose a speculat or f eels t hat int erest rat es are going t o f all drast ically in t he f ut ure,
t o a great er ext ent t han t hat implied by t he f orward rat es. He can ent er int o a f orward
rat e agreement and receive a locked in rat e. He st ands t o benef it if t he rat es indeed f all.
However, if t he rat es rise, he st ands t o lose. I n t his case, t he speculat or has t aken a
view t hat t he rat es will f all. I t is in t his sense t hat Forwards and Fut ures are j ust like
wagers on t he f ut ure levels of int erest rat es.
Sal i en t Poi n t s
1. A f orward rat e is t he int erest rat e on an invest ment t o be made at some point
in t he f ut ure.
2. A Forward Rat e Agreement is an over t he count er Forward cont ract bet ween
t wo part ies f or a specif ied int erest rat e at some point in t he f ut ure.
3. I nt erest Rat e Fut ures are st andardized f orward cont ract s on int erest rat es t hat
are t raded on an exchange.
4. Forward Rat e Agreement s and I nt erest Rat e Fut ures cont ract s can be used f or
hedging and speculat ion.
16. 4 I n t er est Rat e Sw aps
Wh at ar e i n t er est r at e sw aps ( I RS) ?
An I RS can be def ined as an exchange bet ween t wo part ies of int erest rat e obligat ions
( payment s of int erest ) or receipt s ( invest ment income) in t he same currency on an
agreed amount of not ional principal f or an agreed period of t ime.
The most common t ype of int erest rat e swaps are t he plain vanilla I RS. Current ly,
t hese are t he only kind of swaps t hat are allowed by t he RBI in I ndia. Dealing in Exot ics
or advanced int erest rat e swaps have not been permit t ed by t he RBI .
I n a plain vanilla swap, one part y agrees t o pay t o t he ot her part y cash f lows equal t o
t he int erest at a predet ermined f ixed rat e on a not ional principal f or a number of years.
I n exchange, t he part y receiving t he f ixed rat e agrees t o pay t he ot her part y cash f lows
equal t o int erest at a f loat ing rat e on t he same not ional principal f or t he same period of
t ime. Moreover, only t he dif f erence in t he int erest payment s is paid/ received; t he
principal is used only t o calculat e t he int erest amount s and is never exchanged.
An example will help underst and t his bet t er:
Consider a swap agreement bet ween t wo part ies, A and B. The swap was init iat ed on
July 1, 2001. Here, A agrees t o pay t he 3- mont h NSE- MI BOR rat e on a not ional principal
of Rs. 100 million, while B pays a f ixed 12. 15% rat e on t he same principal, f or a t enure
of 1 year.
181
We assume t hat payment s are t o be exchanged every t hree mont hs and t he 12. 15%
int erest rat e is t o be compounded quart erly. This swap can be depict ed diagrammat ically
as shown below:
MI BOR ( 3m)
12. 15%
An int erest rat e swap is ent ered t o t ransf orm t he nat ure of an exist ing liabilit y or an
asset . A swap can be used t o t ransf orm a f loat ing rat e loan int o a f ixed rat e loan, or vice
versa. To underst and t his, consider t hat in t he above example;
A had borrowed a 3 yr, 1 crore loan at 12%. This means t hat f ollowing t he swap, it will:
( a) Pay 12% t o t he lender,
( b) Receive 12. 15% f rom B
( c) Pay 3 mont h MI BOR
Thus, A s 12% f ixed loan is t ransf ormed int o a f loat ing rat e loan of MI BOR 0. 15%.
Similarly, if B had borrowed at MI BOR + 1. 50%, it can t ransf orm t his loan t o a f ixed rat e
loan @ 13. 65% ( 12. 15 + 1. 50) . Following f igure summarizes t his t ransact ion.
12% MI BOR ( 3m)
MI BOR ( 3m)
+ 1. 50% 12. 5%
An I RS can al so be u sed t o t r an sf or m asset s.
Ex ampl e
A f ixed- rat e earning bond can be t ransf ormed int o variable rat e earning asset and vice
versa. I n t he above example, it could be t hat A had a bond earning MI BOR+ 0. 5% and B
a bond earning 12. 5% int erest compounded quart erly. The swap would t hen result in A
receiving a f ixed income of 12. 65% and B receiving a variable income of MI BOR+ 0. 35%.
This can be shown diagrammat ically as f ollows:
MI BOR MI BOR ( 3m)
12. 50%
+ 0. 50%
12. 15%
Somet imes, a bank or f inancial int ermediary is involved in t he swap. I t charges a
commission f or t his. The t wo part ies of t en do not even know who t he ot her part y is. For
Par t y A Par t y B
Par t y A Par t y B
Par t y A Par t y B
182
t hem, t he int ermediary is t he count er- part y. For example, if a f inancial inst it ut ion
charging 20 basis point s were act ing as int ermediary, t he swap would look as f ollows:
12. 17% 12. 37%
MI BOR ( 3m) MI BOR ( 3m)
A Sw ap as a Combi n at i on of Bon ds
A swap can be int erpret ed as a combinat ion of bonds in such a way t hat t he receive
f ixed leg is short on a f loat ing rat e bond and long on a f ixed rat e bond and vice versa f or
t he receive f loat ing leg.
This has signif icant implicat ion on t he pricing and valuat ion of plain vanilla int erest rat e
swaps because a swap can be valued as a combinat ion of t he t wo:
An example will make t his very clear. Consider t he swap f or a not ional of Rs 100.
Part y A pays 3 mont h MI BOR and receives 12. 15% f or a period of t wo years. This is
equivalent t o A having a short posit ion in a 3 mont h MI BOR linked bond and a long
posit ion in a 2 year 12. 15% bond wit h quart erly payment s. 12. 15% is also t he going
swap rat e at t he t ime of incept ion of t he swap.
Assume t hat 1- mont h has passed since t he incept ion of t he bond. Hence t here are t wo
mont hs lef t f or t he int erest payment s t o be exchanged. Let us also assume t hat t he
swap has a look ahead conf igurat ion i. e. t he MI BOR t o be paid af t er t wo mont hs has
already been set .
1. The 12. 15% f ixed rat e bond can be valued according t o convent ional met hods
i. e. by discount ing each cash f low f rom t he bond by t he discount ing rat e f or t he
relevant period.
2. The MI BOR linked bond will reset t o par. This is because on t he next reset dat e,
t he coupon t hat will be f ixed ( MI BOR) will also be equal t o t he discount ing rat e
f or t he relevant period. Hence we have t he par value + MI BOR t o be discount ed
f or a period of t wo mont hs ( t ime t o reset ) .
The value of t he swap is simply t he dif f erence bet ween t he above t wo.
A sw ap as a st r i n g of FRAs or f u t u r es
A swap can also be int erpret ed as a st rip of FRAs or f ut ures cont ract s. Consider t hat
every t ime t he f loat ing index is reset an int erest rat e payment goes f rom one
count erpart y t o t he ot her in j ust t he same way t hat compensat ion is payable/ received
under an FRA. I n a similar way, as int erest rat e changes so t he value of a f ut ures
posit ion changes.
Consider a long f ut ures long posit ion and a short FRA posit ion remember t hese denot e
t he same obligat ion. Each posit ion gains if int erest rat es f all and loses if int erest rat es
rise. The risk/ ret urn prof ile is t hat of a swap- f loat ing rat e payer.
Similarly, f or a swap f ixed rat e payer t he posit ion is t he same as t hat f or a short f ut ures
posit ion and a long FRA posit ion. Each will lose if int erest rat es f all and gain if int erest
rat es rise.
Par t y A
Par t y
B
BANK
183
Pr i ci n g an I RS
I n order t o det ermine t he f ixed rat e or t he swap rat e t o be paid or received f or t he
desired int erest rat e swap, t he present value of t he f loat ing rat e payment s must equat e
t he present value of f ixed rat es. The t rut h of t his st at ement will become clear if we
ref lect on t he f act t hat t he net present value of any f ixed rat e or f loat ing rat e loan must
be zero when t hat loan is grant ed, provided, of course, t hat t he loan has been priced
according t o prevailing market t erms. However, we have already seen t hat a f ixed t o
f loat ing int erest rat e swap is not hing more t han t he combinat ion of a f ixed rat e loan and
a f loat ing rat e loan wit hout t he init ial borrowing and subsequent repayment of a
principal amount . Hence, in order t o arrive at an init ial f ixed rat e, we f ind t hat rat e f or
t he f loat ing leg t hat gives a zero present value f or t he ent ire swap. The market maker
t hen adds some spread so t hat t he present value t o t he market maker is slight ly
posit ive.
184
Wh y do f i r ms en t er i n t o i n t er est r at e sw aps?
Sw aps f or a com par at i ve advan t age
Comparat ive advant ages bet ween t wo f irms arise out of dif f erences in credit rat ing,
market pref erences and exposure.
Ex ampl e: Say, Firm A wit h high credit rat ing can borrow at a f ixed rat e of 12% and at
a f loat ing rat e of MI BOR + 20 bps. Anot her f irm B wit h a lower credit rat ing can borrow
at a f ixed rat e of 14 % and a f loat ing rat e of MI BOR + 150 bps.
Bef or e t h e Sw ap
Part y Fixed rat e loan Float ing rat e loan
A 12 % MI BOR + 0. 20%
B 14 % MI BOR + 1. 50%
Firm A has an absolut e advant age over f irm B in bot h f ixed and f loat ing rat es. Firm B
pays 200 bps more t han f irm A in t he f ixed rat e borrowing and only 120 bps more t han
A in t he f loat ing rat e borrowing. So, f irm B has a comparat ive advant age in borrowing
f loat ing rat e f unds.
Now, Firm A wishes t o borrow at f loat ing rat es and becomes t he f loat ing rat e payer in
t he swap arrangement . However, A act ually borrows f ixed rat e f unds in t he cash market .
I t is t he int erest rat e obligat ions on t his f ixed rat e f unds, which are swapped. At t he
same t ime, B wishes t o borrow at a f ixed rat e, and t hus will act ually borrow f rom t he
market at t he f loat ing rat e.
Then, bot h t he part ies will exchange t heir underlying int erest rat e exposures wit h each
ot her t o gain f rom t he swap. The calculat ion of t he gain f rom t he swap is shown below:
The gain t o f irm A, because it borrows in t he f ixed rat e segment is:
14% - 12% = 200 bps.
And, t he loss because f irm B borrows in t he f loat ing rat e segment is:
( MI BOR + 20 bps) ( MI BOR + 120 bps) = 130 bps.
Thus, t he net gain in t he swap = 200 120 = 70 bps. The f irms can divide t his gain
equally. Firm B can pay f ixed at 12. 15% t o f irm A and receive a f loat ing rat e of MI BOR
as illust rat ed below:
Af t er t h e Sw ap
MI BOR
12 . 15 %
12 % MI BOR + 150 bps
Par t y A Par t y B
185
Ef f ect ive cost f or f irm A = 12% + ( MI BOR 12)
= MI BOR - 15 bps
This result s int o a net gain of ( ( MI BOR + 20) - ( MI BOR - 15) ) i. e. , a gain of 35 bps.
Ef f ect ive cost f or f irm B = ( MI BOR + 150) + ( 12. 15% - MI BOR)
= 13. 65%
This result s int o a gain of ( 14% - 13. 65%) i. e. , a gain of 35 bps.
Thus, bot h t he part ies gain f rom ent ering int o a swap agreement .
As w e h ave seen , f i r ms can u se I RS t o t r an sf or m asset s an d l i abi l i t i es. Bu t
t h en , w h y don t f i r ms t ak e t h e desi r ed f or m of l oan or asset ( f i x ed or f l oat i n g)
i n t h e f i r st pl ace?
Ricardos comparat ive advant age t heory explains t his behavior t o some ext ent .
Cont inuing wit h t he same example, let us assume t hat As credit rat ing is bet t er t han
Bs, and A and B can raise loans f or f ixed and f loat ing rat es as given below:
Bef or e t h e Sw ap
Firm Fixed rat e loan rat e Float ing rat e loan rat e
A 12% MI BOR + 0. 20%
B 14% MI BOR + 1. 50%
Here, we see t hat t hough f irm A can borrow cheaply compared t o f irm B in bot h t he
market s, t he dif f erence in rat es available is not t he same. Firm B has a comparat ive
advant age in t he f loat ing rat e market because it pays only 1. 30% higher here,
compared t o t he 2% dif f erence in t he f ixed rat e market . So, f irm B will borrow at a
f loat ing rat e, and f irm A at f ixed rat e.
Af t er t he swap deal, t he cost of t he f loat ing rat e loan t o f irm A will be MI BOR- 0. 15%, a
clean gain of 35 basis point s. Similarly, f irm B also gains 35 basis point s, because t he
cost of it s loan will be 13. 35% only, af t er t he swap. Thus, bot h part ies gain f rom t he
swap, as shown below:
Af t er t h e Sw ap
Fi r m Fi x ed r at e l oan r at e Fl oat i n g r at e l oan r at e Gai n
A - Mibor - 0. 15 % 35 bps
B 13. 65% - 35 bps
I n a perf ect market , however, t he spread bet ween f ixed and f loat ing rat es of f ered
should vanish due t o I RS. This is not seen in realit y, and spreads cont inue t o persist . So,
t he credit rat ings of t he f irms are not t he only crit eria by which lenders j udge f irms, and
t he comparat ive advant age t heory cont inues t o hold.
186
Sw aps f or Redu ci n g t h e Cost of Bor r ow i n g
Wit h t he int roduct ion of rupee derivat ives, t he I ndian corporat es can at t empt t o reduce
t heir cost of borrowing and t hereby add value. A t ypical I ndian case would be a
corporat e wit h a high f ixed rat e obligat ion.
MI PL, an AAA rat ed corporat e, 3 years back had raised 4-year funds at a fixed rat e of
18. 5%. Today a 364- day T- bill is yielding 10. 25%, as t he int erest rat es have come
down. The 3- mont h MI BOR is quot ing at 10%.
Fixed t o f loat ing 1 year swaps are t rading at 50 bps over t he 364- day T- bill vs. 6- mont h
MI BOR.
The t reasurer is of t he view t hat t he average MI BOR shall remain below 18. 5% f or t he
next one year. The f irm can t hus benef it by ent ering int o an int erest rat e f ixed f or
f loat ing swap, whereby it makes f loat ing payment s at MI BOR and receives f ixed
payment s at 50 bps over a 364- day t reasury yield i. e. 10. 25 + 0. 50 = 10. 75 %.
18 . 5 % 10. 75% MI BOR
MI BOR ( 3m)
The ef f ect ive cost f or MI PL = 18. 50 + MI BOR - 10. 75
= 7. 75 + MI BOR
At t he present 3m MI BOR is 10%, t he ef f ect ive cost is = 10 + 7. 75 = 17. 75%
The gain f or t he f irm is ( 18. 5 - 17. 75) = 0. 75 %
The risks involved f or t he f irm are:
Def ault / credit risk of part y B: Since t he count erpart y is a bank, t his
risk is much lower t han would arise in t he normal case of lending t o
corporat es. This risk involves losses t o t he ext ent of t he int erest rat e
dif f erent ial bet ween f ixed and f loat ing rat e payment s.
The f irm is f aced wit h t he risk t hat t he MI BOR goes beyond 10. 75%.
Any rise beyond 10. 75% will raise t he cost of f unds f or t he f irm.
Theref ore it is very essent ial t hat t he f irm hold a well- suggest ed view
t hat MI BOR shall remain below 10. 75%. This will require cont inuous
monit oring.
How does t he bank benef it out of t his t ransact ion?
The bank eit her goes f or anot her swap t o of f set t his obligat ion and in t he process earn a
spread. The bank may also use t his swap as an opport uni t y t o hedge it s own f loat ing
liabilit y. The bank may also leave t his posit ion uncovered if it is of t he view t hat MI BOR
shall rise beyond 10. 75%.
Tak i n g advan t age of f u t u r e vi ew s / specu l at i on
I f a bank holds a view t hat int erest rat e is likely t o increase and in such a case t he
ret urn on f ixed rat e asset s will not increase, it will pref er t o swap it wit h a f loat ing rat e
int erest . I t may also swap f loat ing rat e liabilit ies wit h a f ixed rat e.
MI PL Par t y B
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Ot her reasons f or using I RS are speculat ion on f ut ure int erest rat e movement s,
management of asset - liabilit y mismat ch, alt ering debt st ruct ure, of f - balance sheet gains,
and int erest risk management . I t has been observed t hat FRAs are more popular f or
hedging against int erest risks, while I RS are more popular f or speculat ion and
t ransf orming nat ure of asset s and liabilit ies.
Model Qu est i on s
1. An i n t er est r at e sw ap t r an sf or ms t h e n at u r e of ___________.
a) an exist ing liabilit y only
b) an exist ing asset only
c) an not ional liabilit y or an asset
d) an exist ing liabilit y or an asset
An sw er : d
2. A sw ap can be i n t er pr et ed as a st r i p of ___________.
a) f ixed rat e agreement s only
b) f ut ure cont ract s only
c) f ixed rat e agreement s or f ut ure cont ract s
d) None of t he above
An sw er : c
3. For w ar d r at es can n ot be ___________.
a) posit ive
b) negat ive
c) zero
d) higher t han spot rat e
An sw er : b
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Ch apt er 17
Gl ossar y of Debt Mar k et Ter ms
19
Accr u ed I n t er est
I f a coupon bearing securit y is t raded bet ween t wo coupon dat es, t he buyer has t o
compensat e t he seller by paying him t hat part of t he int erest which is due t o him f or t he
period f or which he has held t he securit y af t er t he immediat ely preceding coupon dat e.
The calculat ion of accrued int erest is done according t o t he day- count convent ion of t he
securit y or market ( See Day Count ) .
Ask Pr i ce
I n Financial Market s, market makers quot e bot h bid ( buy) price and ask ( of f er) price.
This indicat es t hat t he market maker, not knowing t he int ent ion of t he price t aker is
quot ing him rat es f or bot h buying as well as f or selling. The bid price is generally lower
t han t he sell price, as dict at ed by normal prof it mot ive ( Somet imes a dealer would quot e
t he same bid and ask, in which case t he price is called a "choice- price") . The dif f erence
bet ween t he bid and t he ask price is normally called t he "bid- ask spread". The spread
depends on many f act ors like liquidit y in t he inst rument quot ed, t he bias of t he dealer,
his eagerness or ot herwise t o t rade, market volat ilit y et c. A small dif f erence is
considered t o be a very f ine price as t he dealer is keeping very lit t le by way of his
prof it s. As example: A dealer quot ing 12. 50% 2004 might quot e 105. 15/ 20. This implies
t hat t he dealer is willing t o buy t he paper at 105. 15 while he is willing t o sell it at
105. 20. I n act ual pract ice t he price may be given as 15/ 20. I t is underst ood t hat market
part icipant s are aware of t he big f igure. I t should be kept in mind t hat while quot ing
int erest rat e rat es, t he bid is in f act higher t han t he of f er. For example a USD 3x6 FRA
can be quot ed as 5. 75/ 70, implying t hat t he dealer is willing t o buy t he FRA at a yield of
5. 70% while he will sell t he same FRA at an yield of 5. 75%.
Asset Back ed Secu r i t y
Any securit y t hat of f ers t o t he invest or an asset as t he collat eral is called Asset Backed
Securit y. The rat e of ret urn required by t he invest or f or such t ypes of bonds is generally
less compared t o bonds t hat of f er no collat eral.
Au ct i on
The process of issuing a securit y t hrough a price- discovery mechanism t hrough asking
f or bids. This is t he process f ollowed by t he RBI f or all t ypes of issues of debt market
paper by it .
Bal an ce Ten or
The un- expired lif e of t he securit y
19
This glossary has been downloaded and modif ied f rom www. debt onnet . com, t he f irst
int ernet based debt market port al in I ndia.
189
Ban k Rat e
Bank Rat e is a direct inst rument of credit cont rol . I t is t hat int erest rat e or discount rat e
at which banks , f inancial inst it ut ions and ot her approved ent it ies in t he int erbank
market can get f inancial accommodat ion f rom t he cent ral bank of t he count ry . By hiking
t he bank rat e t he cent ral bank makes credit expensive and by lowering t he same t hey
make credit cheaper
Basi s Poi n t
One hundredt h of a percent age ( i. e. 0. 01) . As int erest rat es are generally sensit ive in
t he second place af t er t he decimal point , t he measure has large import ance f or t he debt
market .
Ben ch mar k Rat e
Benchmark rat es are rat es or t he prices of inst rument s t hat are t raded in t he market on
which are used f or pricing of ot her inst rument s. These rat es or prices are used as
benchmark f or f loat ing rat e inst rument s. Typically a benchmark rat e should sat isf y t he
f ollowing crit eria
1. The rat e should be available readily and should eit her be direct ly observable in
t he market or made available by a credible agency
2. The benchmark should be liquid so t hat count er- hedging st rat egies are readily
available
3. The rat e should be unique and leave no scope f or ambiguit y
The benchmark should be represent at ive of t he market . I nt ernat ionally t he most popular
benchmarks are t he LI BOR and t he US Treasury. I n I ndia, given t he paucit y of rat es t hat
sat isf y t he above crit erion, not many benchmarks exist , save t he MI BOR announced by
eit her t he NSE or Reut ers.
Bi d Pr i ce
See Ask Price
Bon d
A bond a promise in which t he I ssuer agrees t o pay a cert ain rat e of int erest , usually as
a percent age of t he bond' s f ace value t o t he I nvest or at specif ic periodicit y over t he lif e
of t he bond. Somet imes int erest is also paid in t he f orm of issuing t he inst rument at a
discount t o f ace value and subsequent ly redeeming it at par. Some bonds do not pay a
f ixed rat e of int erest but pay int erest t hat is a mark- up on some benchmark rat e.
Boot st r appi n g
Boot st rapping is an it erat ive process of generat ing a Zero Coupon Yield Curve f rom t he
observed prices/ yields of coupon bearing securit ies. The process st art s f rom observing
t he yield f or t he short est - t erm money market discount inst rument ( i. e. one t hat carries
no coupon) . This yield is used t o discount t he coupon payment f alling on t he same
mat urit y f or a coupon- bearing bond of t he next higher mat urit y. The result ing equat ion
is solved t o give t he zero yield ( also called spot yield) f or t he higher mat urit y period.
This process is cont inued f or all securit ies across t he t ime series. I f represent ed
algebraically, t he process would lead t o an n t h degree polynomial t hat is generally
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solved using numerical met hods. The most popular one being t he Newt on- Raphson
t echnique.
Cal l Mon ey
Borrowing or lending f or one day upt o 14 days, in t he int erbank market is known as call
money. Ent ry int o t his segment of t he market is rest rict ed t o not if ied part icipant s which
include scheduled commercial banks, primary dealers and sat ellit e dealers, development
f inancial inst it ut ions and mut ual f unds
Cal l Opt i on
See Opt ion
Cal l abl e Bon d
A Bond which has a Call covenant in it s t erms of issue, i. e. one in which t he I ssuer
reserves t he right t o buy- back t he issue is called a Callable Bond.
Cl ean Pr i ce
A Clean Price of a bond or securit y is t he discount ed value of all it s f ut ure cash f lows
( using a suit able discount rat e, which can be t he YTM or t he relevant spot rat e) .
However, if t he bond is t raded bet ween t wo coupon dat es, t he buyer of t he bond will
have t o compensat e t he seller f or t hat part of t he period bet ween coupons f or which t he
seller was owning t he bond ( See Accrued I nt erest ) . The price arrived at af t er adj ust ing
t he Clean Price f or t his f act or is called t he Dirt y Price.
Col l at er al i sed Bon d
Any f ixed income inst rument which has collat eral as a back up t o t he issue is called a
Collat eralised Bond. I n I ndia, relat ed t erminology is secured bonds or unsecured bonds.
Commer ci al Paper ( CP)
A Commercial Paper is a short t erm unsecured promissory not e issued by t he raiser of
debt t o t he invest or. I n I ndia Corporat es, Primary Dealers ( PD) , Sat ellit e Dealers ( SD)
and All I ndia Financial I nst it ut ions ( FI ) can issue t hese not es. For a corporat e t o be
eligible it must have a t angible net wort h of Rs 4 crore or more and have a sanct ioned
working capit al limit sanct ioned by a bank/ FI . I t is generally companies wit h very good
rat ing which are act ive in t he CP market , t hough RBI permit s a minimum credit rat ing of
Crisil- P2. The t enure of CPs can be anyt hing bet ween 15 days t o one year, t hough t he
most popular durat ion is 90 days. These inst rument s are of f ered at a discount t o t he
f ace value and t he rat e of int erest depends on t he quant um raised, t he t enure and t he
general level of rat es besides t he credit rat ing of t he proposed issue. While most of t he
issuing ent it ies have est ablished working capit al limit s wit h banks, t hey st ill pref er t o use
t he CP rout e f or f lexibilit y in int erest rat es. The credit rat ings f or CP are issued by
leading rat ing agencies. Recent ly t he quant um raised by t he issuer t hrough t he CP has
been excluded f rom t he ambit of bank f inance, but banks cont inue t o pref er earmarking
eit her t heir own limit s f or t he corporat e or t he consort ium limit s while subscribing t o t he
commercial paper.
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Con st i t u en t SGL A/ c
SGL account holders can have t wo SGL account s wit h RBI - SGL account no. 1 and SGL
account no. 2 . SGL account no. 1 is t he account f or t he own holdings of t he bank or t he
PD or SD who has t he direct account . SGL account no. 2 is f or t heir const it uent s . Those
who are not eligible f or direct SGL account wit h RBI , say , f or example , a Provident
Fund Trust , who is not eligible f or an SGL account can hold securit ies in demat f orm by
opening a const it uent SGL account wit h a Bank, PD or SD . Thr ough t he SGL account
no. 2 of t he part y who has direct account wit h RBI t he f acilit y will be made available t o
t he PF
Con ver t i bl e Bon d
A bond t hat is part ially or f ully convert ible int o equit y wit hin a specif ied period of t ime
f rom t he dat e of issue is known as a convert ible bond. I n such cases, t he bond does not
pay t he holder t hat part of t he mat urit y value t hat is earmarked f or conversion t o
equit y.
Con vex i t y
See in conj unct ion wit h Durat ion, PVBP and I mmunizat ion. Convexit y is anot her
measure of bond risk. The measure of Durat ion assumes a linear relat ionship bet ween
changes in price and durat ion. However, t he relat ionship bet ween change in price and
change in yield is not linear and hence t he est imat ed price change obt ained by durat ion
will give only an approximat e value. The error is insignif icant when t he change in yield is
small but does not hold t rue f or larger changes in yield, as t he act ual price- yield
relat ionship is convex. Convexit y is t he measure of t he curvat ure of t he price- yield
relat ionship. I t is also t he rat e of change of durat ion wit h a change in yield. A high
convexit y is of t en a desired charact erist ic as f or a given change in yield, posit ive or
negat ive, a bond' s percent age rise in price is great er t han t he percent age price loss.
While modif ied durat ion is used t o predict t he bond' s % change in price small change in
yields, modif ied durat ion and convexit y t oget her are used t o calculat e a bond' s %
change in price f or a large change in yield, as per t his relat ionship.
Cou pon
The rat e of int erest paid on a securit y, generally a f ixed percent age of t he f ace value, is
called t he coupon. The origin of t he t erm dat es back t o t he t ime when bonds had
coupons at t ached t o t hem, which t he invest or had t o det ach and present t o t he issuer t o
receive t he money.
Cr edi t Rat i n g
Credit Rat ing is an exercise conduct ed by a rat ing organisat ion t o explore t he credit
wort hiness of t he issuer wit h respect t o t he inst rument being issued or a general abilit y
t o pay back debt over specif ied periods of t ime. The rat ing is given as an alphanumeric
code t hat represent s a graded st ruct ure or credit wort hiness. Typically t he highest credit
rat ing is t hat of AAA and t he lowest being D ( f or def ault ) . Wit hin t he same alphabet
class, t he rat ing agency might
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CRR
This is t he acronym f or Cash Reserve Rat io. That part of t heir asset s which banks in
I ndia are required t o hold as Cash in balances wit h t he Reserve Bank of I ndia is called
t he Cash Reserve Rat io.
Cu r r en t Yi el d
Current Yield on a bond is def ined as t he coupon rat e divided by t he price of t he bond.
This is a very inadequat e measure of yield, as it does not t ake int o account t he ef f ect of
f ut ure cash f lows and t he applicat ion of discount ing f act ors on t hem.
Day Cou n t
The market uses quit e a f ew convent ions f or calculat ion of t he number of days t hat has
elapsed bet ween t wo dat es. I t is int erest ing t o not e t hat t hese convent ions were
designed prior t o t he emergence of sophist icat ed calculat ing devices and t he main
obj ect ive was t o reduce t he mat h in complicat ed f ormulae. The convent ions are st ill in
place even t hough calculat ing f unct ions are readily available even in hand- held devices.
The ult imat e aim of any convent ion is t o calculat e ( days in a mont h) / ( days in a year) .
The convent ions used are as below:
We t ak e t h e ex ampl e of a bon d w i t h Face Val u e 100, cou pon 12. 50% , l ast
cou pon pai d on 15t h Ju n e, 2000 an d t r aded f or val u e 5t h Oct ober , 2000.
A/ 360
I n t his met hod, t he act ual number of days elapsed bet ween t he t wo dat es is divided by
360, i. e. t he year is assumed t o have 360 days. Using t his met hod, accrued int erest is
3. 8888
A/ 365
I n t his met hod, t he act ual number of days elapsed bet ween t he t wo dat es is divided by
365, i. e. t he year is assumed t o have 365 days. Using t his met hod, accrued int erest is
3. 8356
A/ A
I n t his met hod, t he act ual number of days elapsed bet ween t he t wo dat es is divided by
t he act ual days in t he year. I f t he year is a leap year AND t he 29t h of February is
included bet ween t he t wo dat es, t hen 366 is used in t he denominat or, else 365 is used.
Using t his met hod, accrued int erest is 3. 8356
30/ 360
This is how t his convent ion is used in t he US. Break up t he earlier dat e as
D( 1) / M( 1) / Y( 1) and t he lat er dat e as D( 2) / M( 2) / Y( 2) . I f D( 1) is 31, change D( 1) t o 30.
I f D( 2) is 31 AND D( 1) is 30, change D( 2) t o 30. The days elapsed is calculat ed as Y( 2) -
Y( 1) * 360+ M( 2) - M( 1) * 30+ D( 2) - D( 1)
30/ 360 Eu r opean
This is t he variat ion of t he above convent ion out side of t he Unit ed St at es. Break up t he
earlier dat e as D( 1) / M( 1) / Y( 1) and t he lat er dat e as D( 2) / M( 2) / Y( 2) . I f D( 1) is 31,
change D( 1) t o 30. I f D( 2) is 31, change D( 2) t o 30. The days elapsed is calculat ed as
Y( 2) - Y( 1) * 360+ M( 2) - M( 1) * 30+ D( 2) - D( 1)
I n I ndian bond market s t he last convent ion is used. RBI while calculat ing yield in t he
SGL Transact ions f or T- Bills uses 364 as basis. This is probably because 364 is t he
longest t enure bill issued by it .
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Der i vat i ves
A Derivat ive is any inst rument t hat derives it s value f rom t he price movement of an
underlying asset . The most popular derivat ives include Opt ions, Fut ures and Swaps.
Given t he st eep progress made by comput ing devices and t he increased import ance of
quant it at ive t echniques t o t he f inancial market s, t he st ruct ure of derivat ives have
become severely complicat ed. I t is not uncommon t o f ind a combinat ion of several
opt ions on a swap which pay- out depending on t he occurrence of some event . The main
input f or pricing is volat ilit y in t he price of t he underlying asset , which has given rise t o
t he curious sit uat ion where t he asset volat ilit y is more heavily t raded t han t he derivat ive
it self . The applicat ion of derivat ive pricing has f ound it s way in valuat ion of any
cont ingent claim, f loat ing rat e not es, corporat e valuat ion and proj ect f inance.
Di r t y Pr i ce
Dirt y Price of a securit y is it s Clean Price plus Accrued I nt erest . Also see Clean Price,
Accrued I nt erest .
Di scou n t
The quant um by which a securit y is issued or is t raded below it s par value is called
Discount . Also see Discount Basis.
Di scou n t Basi s
Securit ies t hat do not carry a coupon are generally issued at a discount t o t heir f ace
value. Examples of such securit ies are T- Bills and Commercial Papers ( CP) .
Di scr i mi n at or y Pr i ce Au ct i on
See French Auct ion
Du r at i on
Durat ion is a measure of a bonds' price risk. I t is weight ed average of all t he cash- f lows
associat ed wit h a bond, weighed by t he proport ion of value due t o t he j t h payment in
t he cash- f low st ream, wit h sum of all j ' s equalling one. Durat ion measures t he
sensit ivit y of a bonds price t o a change in yield.
Du t ch Au ct i on
This is t he process of auct ion in which af t er receiving all t he bids a part icular yield is
det ermined as t he cut - of f rat e. All bids received at yields higher t han t he cut - of f rat e
( i. e. at higher prices) are rej ect ed. All bids received at yields below t he cut - of f rat e are
given allot ment at t he cut - of f rat e. The process is ident ical t o t hat of t he French Auct ion,
except f or t he f act t hat t here is no concept of allot ment at a premium. The Liquidit y
adj ust ment Facilit y ( LAF) of RBI is an example of such auct ion. Also see French Auct ion,
Winner' s Curse.
Fl oat i n g Rat e Not e
A Float ing Rat e Not e is an inst rument t hat does not pay a f ixed rat e of int erest on it s
f ace value. The int erest paid on such inst rument s is dependent upon t he value of a
benchmark rat e. The benchmark rat e is mut ually agreed upon by t he issuer and t he
invest or and has t o sat isf y some crit eria ( See Benchmark Rat es) . The int erest paid is
t ypically a mark- up on t he benchmark so agreed. An example would be a AAA rat ed
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corporat e issuer who issues a Not e t hat pays 30 bps above t he U. S. Treasury. I n I ndia a
very common inst rument of lat e has been an issue t hat pays a specif ied markup above
t he MI BOR.
Fr en ch Au ct i on
This is a process of auct ion in which af t er all t he bids are received, a part icular yield is
decided as t he cut - of f rat e. All bids t hat have been received at yields higher t han t he
cut - of f rat e ( i. e. at lower prices) are rej ect ed. All bids t hat have been received at below
t he cut - of f rat e ( i. e. at higher prices) are given f ull allot ment but at a premium f rom t he
price at t he cut - of f yield
Gi l t s
Anot her name f or government securit ies. The t erm ref lect s t he superior qualit y of t he
papers issued by t he government . The papers issued by t he Bank of England used t o
have gilt - edged borders and t he t erm gilt s originat ed f rom t here
Gr oss Pr i ce
See Dirt y Price
Ju n k Bon d
Any bond which has a credit rat ing below Baa/ BBB. These are bonds t hat are below
invest ment grade and carry very at t ract ive rat es of ret urn, commensurat e wit h t he high
credit risk.
LAF
This is a f acilit y by which t he RBI adj ust s t he daily liquidit y in t he domest ic market s
( I ndia) eit her by inj ect ing f unds or by wit hdrawing t hem out . This met hod was made
ef f ect ive on t he 5t h June 2000 and is open f or Banks and Primary Dealers. This met hod
has replaced t he t radit ional met hod of ref inance based on f ixed rat es.
LI BOR
St ands f or London I nt erbank Of f ered rat e. This is a very popular bench mark and is
issued f or US Dollar, GB Pound, Euro, Swiss Franc, Canadian Dollar and t he Japanese
Yen. The mat urit y covers overnight t o 12 mont hs. The met hodology, very brief ly - t he
Brit ish Bankers Associat ion ( BBA) at 1100 hrs GMT asks 16 banks t o cont ribut e t he
LI BOR f or each mat urit y and f or each currency. The BBA weeds out t he best f our and
t he worst f our, calculat es t he average of t he remaining eight and t he value is published
as LI BOR. The f igures are put up in Reut ers on page LI BO and SWAP. The same is
available on TeleRat e page 3170.
Macau l ay Du r at i on
See Durat ion
Mar k To Mar k et
Mark t o Market or MTM is a very popular report ing and perf ormance measurement t ools
f or any invest ment . I n t his t echnique t he price at which t he invest ment was made is
compared wit h t he price which t he asset can realised if liquidat ed in t he market at t hat
moment . The dif f erence is eit her t he MTM gain or MTM loss depending upon t he current
195
wort h vis- - vis t he original price. Liabilit ies can also be made subj ect t o t he same
analysis as asset s. Periodicit y of MTM depends on t he liquidit y of t he market in which t he
asset is a class. For example currency and bond invest ment s are MTM- ed online while
ot her invest ment s like real est at e may be MTM- ed at higher int ervals.
MI BOR
St ands f or Mumbai I nt erBank Of f ered Rat e, it is closely modeled on t he LI BOR. Current ly
t here are t wo calculat ing agent s f or t he benchmark - Reut ers and t he Nat ional St ock
Exchange ( NSE) . The NSE MI BOR benchmark is t he more popular of t he t wo, ref lect ed
by t he larger number of deals t hat are t ransact ed using t his benchmark.
Modi f i ed Du r at i on
This is a slight variat ion t o t he concept of Durat ion. Modif ied Durat ion can be def ined as
t he approximat e percent age change in price f or a 1% change in yield. Mat hemat ically it
is represent ed as Mod. Durat ion = Durat ion / ( 1+ y/ n) , where n= number of coupon
payment s in t he year
Mu l t i pl e Pr i ce Au ct i on
See French Auct ion
Net Pr i ce
See Clean Price
Non Con ver t i bl e Deben t u r e ( NCD)
A Non Convert ible debent ure, as against a convert ible debent ure, is not convert ible,
eit her in part or t he whole, int o equit y on it s mat urit y.
Not i ce Mon ey
Money borrowed or lent in t he int erbank market f or a period beyond one day and upt o
14 days.
Open Mar k et Oper at i on s
One of t he maj or inst rument s of monet ary policy by which t he cent ral bank of a count ry
manipulat es short - t erm liquidit y and t hereby t he int erest rat es t o desired levels.
Generally open market operat ions involve purchase and sale of t reasury bills in t he open
market or conduct ing repos.
PLR
This is t he acronym f or Prime Lending Rat e. This is t he rat e at which a bank in I ndia
lends t o it s prime cust omer. The bank usually f ollows an int ernal credit rat ing syst em
and charges a spread over t he PLR f or non- prime cust omers.
Pr i ce Val u e of a Basi s Poi n t
See PVBP
Pr i mar y Deal er ( PD)
A Primary Dealer in t he securit ies market is an ent it y licensed by t he RBI t o carry on t he
business of securit ies and act as market maker in securit ies. I n t urn t he Primary Dealer
196
will enj oy cert ain privileges f rom t he RBI like ref inance f rom RBI at concessional rat es,
access t o t he int erbank call money market et c. The PD has t o give an annual
undert aking t o t he RBI on his level of part icipat ion in t he primary issues of government
securit ies. To qualif y f or Primary Dealership t he applicant company should have a
net wort h of Rs. 50. 00 crore and a f ew years of experience in t he securit ies market .
PVBP
Also called t he Price Value of a Basis Point or Dollar Value of 01. This is one way of
quant if ying t he sensit ivit y of a bond t o changes in t he int erest rat es. I f t he current price
of t he bond is P( 0) and t he price af t er a one basis point rise in rat es is P( 1) t hen PVBP is
- [ P( 1) - P( 0) ] . This can be est imat ed wit h t he help of t he modif ied durat ion of a bond, as
( Price of t he bond * modif ied durat ion* . 0001)
Repo
Repo or Repurchase Agreement s are short - t erm money market inst rument s. Repo is
not hing but collat eralized borrowing and lending. I n a repurchase agreement securit ies
are sold in a t emporary sale wit h a promise t o buy back t he securit ies at a f ut ure dat e at
specif ied price. I n reverse repos securit ies are purchased in a t emporary purchase wit h a
promise t o sell it back af t er a specif ied number of days at a pre- specif ied price. When
one is doing a repo , it is reverse repo f or t he ot her part y
Rever se Repo
See Repo
Ri sk Fr ee Rat e
An int erest rat e given out by an invest ment t hat has a zero probabilit y of def ault .
Theoret ically t his rat e can never exist in pract ice but sovereign debt is used as t he
nearest proxy.
Sat el l i t e Deal er
Second level t o t he Primary Dealers , Sat ellit e Dealers are licensed by RBI t o carry on
t he business of ret ailing in government securit ies . They are basically ent it ies who have
very good dist ribut ion net work f or ret ailing. The minimum net wort h t o qualif y f or
Sat ellit e Dealership is Rs. 5. 00 crore and a f ew years of experience in t he securit ies
market .
SGL
Subsidiary General Ledger Account is t he demat f acilit y f or government securit ies
of f ered by t he Reserve Bank of I ndia . I n t he case of SGL f acilit y t he securit ies remain in
t he comput ers of RBI by credit t o t he SGL account of t he owner. RBI of f ers SGL f acilit y
only t o banks, primary dealers and sat ellit e dealers
SLR
This is t he acronym f or St at ut ory Liquidit y Rat io. That part of t heir Net Demand and
Time liabilit ies ( NDTL) t hat a bank is required by law t o be kept invest ed in approved
securit ies is known as SLR. The approved securit ies are t ypically sovereign issues. The
maint enance of SLR ensures a minimum liquidit y in t he bank' s asset s.
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Spr ead
Spread is t he dif f erence bet ween t wo rat es of int erest s. I t is of t en generalised t o imply
t he dif f erence bet ween eit her price or yield. Spreads can be bet ween t wo risk classes or
can be bet ween t enors in t he same risk class. For example 130 bps bet ween AAA and
GOI means a 1. 30% spread bet ween a AAA issue and t hat made by t he Government of
I ndia. 5 paisa spread bet ween bid and ask means t hat in t he t wo way price quot ed t he
dif f erence bet ween t he buy and sell price is 5 paisa 60 bps spread bet ween 3 mont h T
Bill over 10 Year means t hat t he dif f erence bet ween t he yield in t he 3 mont h Treasury
Bill and t hat on a 10 Year paper of t he same risk class in 60 basis point s.
STRI PS
STRI PS is t he acronym f or Separat e Trading of Regist ered I nt erest and Principal of
Securit ies. The STRI PS program let s invest ors hold and t rade t he individual int erest and
principal component s of eligible Treasury not es and bonds as separat e securit ies. When
a Treasury f ixed- principal or inf lat ion- indexed not e or bond is st ripped, each int erest
payment and t he principal payment becomes a separat e zero- coupon securit y. Each
component has it s own ident if ying number and can be held or t raded separat ely. For
example, a Treasury not e wit h 10 years remaining t o mat urit y consist s of a single
principal payment at mat urit y and 20 int erest payment s, one every six mont hs f or 10
years. When t his not e is convert ed t o STRI PS f orm, each of t he 20 int erest payment s
and t he principal payment becomes a separat e securit y. STRI PS are also called zero-
coupon securit ies because t he only t ime an invest or receives a payment during t he lif e
of a STRI P is when it mat ures.
A f inancial inst it ut ion, government securit ies broker, or government securit ies dealer can
convert an eligible Treasury securit y int o int erest and principal component s t hrough t he
commercial book- ent ry syst em. Generally, an eligible securit y can be st ripped at any
t ime f rom it s issue dat e unt il it s call or mat urit y dat e. Securit ies are assigned a st andard
ident if icat ion code known as a CUSI P number. CUSI P is t he acronym f or Commit t ee on
Unif orm Securit y I dent if icat ion Procedures. Just as a f ully const it ut ed securit y has it a
unique CUSI P number, each STRI PS component has a unique CUSI P number. All int erest
STRI PS t hat are payable on t he same day, even when st ripped f rom dif f erent securit ies,
have t he same generic CUSI P numbers. However, t he principal STRI PS f rom each not e
or bond have a unique CUSI P number. STRI PS component s can be reassembled or
"reconst it ut ed" int o a f ully const it ut ed securit y in t he commercial book- ent ry syst em. To
reconst it ut e a securit y, a f inancial inst it ut ion or government securit ies broker or dealer
must obt ain t he appropriat e principal component and all unmat ured int erest component s
f or t he securit y being reconst it ut ed. The principal and int erest component s must be in
t he appropriat e minimum or mult iple amount s f or a securit y t o be reconst it ut ed. The
f lexibilit y t o st rip and reconst it ut e securit ies allows invest ors t o t ake advant age of
various holding and t rading st rat egies under changing f inancial market condit ions t hat
may t end t o f avour t rading and holding STRI PS or f ully const it ut ed Treasury securit ies.
Ter m Mon ey
Money borrowed and lent f or a period beyond 14 days is known as t erm money
198
Tr easu r y Bi l l s
Treasury Bills are short - t erm obligat ions of t he Treasury/ Government . They are
inst rument s issued at a discount t o t he f ace value and f orm an int egral part of t he
money market . I n I ndia t reasury bills are issued t wo mat urit ies 91 days and 364 days.
Un i f or m Pr i ce Au ct i on
See Dut ch Auct ion
WDM Segmen t
The Nat ional St ock Exchange of I ndia has t wo t rading segment s , one is t he Capit al
Market s Segment and t he ot her is t he Wholesale Debt Market Segment . The Capit al
Market s Segment is meant f or equit ies t rading whereas all t he t rades in debt
inst rument s are put t hrough t he WDM Segment . t he WDM represent s t he only f ormal
screen- based t rading and report ing mechanism f or secondary market t rades in debt
inst rument s.
Wi n n er s Cu r se
I n a French auct ion, every successf ul bidder is one whose bid is equal or higher t han t he
cut - of f price. Theref ore, successf ul bidders have t o pay a premium on t he cut - of f price,
on being successf ul in t he auct ion. This is called t he winners curse in t reasury auct ions.
Yi el d Cu r ve
The relat ionship bet ween t ime and yield on a homogenous risk class of securit ies is
called t he Yield Curve. The relat ionship represent s t he t ime value of money - showing
t hat people would demand a posit ive rat e of ret urn on t he money t hey are willing t o part
t oday f or a payback int o t he f ut ure. I t also shows t hat a Rupee payable in t he f ut ure is
wort h less t oday because of t he relat ionship bet ween t ime and money. A yield curve can
be posit ive, neut ral or f lat . A posit ive yield curve, which is most nat ural, is when t he
slope of t he curve is posit ive, i. e. t he yield at t he longer end is higher t han t hat at t he
short er end of t he t ime axis. This result s as people demand higher compensat ion f or
part ing t heir money f or a longer t ime int o t he f ut ure. A neut ral yield curve is t hat which
has a zero slope, i. e. is f lat across t ime. This occurs when people are willing t o accept
more or less t he same ret urns across mat urit ies. The negat ive yield curve ( also called an
invert ed yield curve) is one of which t he slope is negat ive, i. e. t he long t erm yield is
lower t han t he short t erm yield. I t is not of t en t hat t his happens and has import ant
economic ramif icat ions when it does. I t generally represent s an impending downt urn in
t he economy, where people are ant icipat ing lower int erest rat es in t he f ut ure.
Yi el d Pi ck - Up
Yield pick up or yield give up ref ers t o t he yield gained or lost at t he t ime of init iat ion of
a t rade primarily in bonds and debent ures. Suppose one sold 12. 50 % GOI 2004 at a
yield of 10. 00 per cent and moved int o 11. 83 % GOI 2014 at a yield of 11. 25 per cent
t he yield pick up is t o t he t une of 125 basis point s. I f one did exact ly t he reverse of t his
t he yield give up is t o t he ext ent of 125 bps. These concept s are ordinarily used in bond
swap evaluat ion.
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Yi el d To Mat u r i t y
Yield t o Mat urit y ( YTM) is t hat rat e of discount t hat equat es t he discount ed value of all
f ut ure cash f lows of a securit y wit h it s current price. I n a way, it is anot her way of
st at ing t he price of a securit y as ot her t hings remaining const ant , t he price is a direct
f unct ion of t he YTM. The def iciency of YTM is t hat it assumes t hat all int ermediat e and
f inal cash f low of t he securit y is re- invest ed at t he YTM, which ignores t he shape of t he
yield curve. This makes YTM applicable as a measure f or an individual securit y and t o
dif f erent bonds in t he same risk class. The YTM, given it s inst rument - specif ic nat ure
does not provide unique mapping f rom mat urit y t o int erest rat e space. I t is used
primarily f or it s simplicit y of nat ure and ease of calculat ion. More sophist icat ed t raders
would use t he Zero Coupon Yield Curve ( ZCYC) f or valuat ion. See Zero Coupon Yield
Curve.
Zer o Cou pon Bon d
A Zero Coupon Bond ( ZCB) is one t hat pays no periodic int erest ( does not carry a
coupon) . These bonds are t ypically issued at a discount and redeemed at f ace value. The
discount rat e, appropriat ed over t he lif e of t he bond is t he ef f ect ive int erest paid by t he
issuer t o t he invest or. I n I ndia, t he spect rum of ZCB is virt ually non- exist ent beyond one
year. Upt o one year, t he Treasury Bills issued are proxied f or ZCB. Also see Zero Coupon
Yield Curve.
Zer o Cou pon Yi el d Cu r ve
The Zero Coupon Yield Curve ( also called t he Spot Curve) is a relat ionship bet ween
mat urit y and int erest rat es. I t dif f ers f rom a normal yield curve by t he f act t hat it is not
t he YTM of coupon bearing securit ies, which get s plot t ed. Represent ed against t ime are
t he yields on zero coupon inst rument s across mat urit ies. The benef it of having zero
coupon yields ( or spot yields) is t hat t he def iciencies of t he YTM approach ( See Yield t o
Mat urit y) is removed. However, zero coupon bonds are generally not available across
t he ent ire spect rum of t ime and hence st at ist ical est imat ion processes are used. The
NSE comput es t he ZCYC f or t reasury bonds using t he Nelson- Seigel procedure, and
disseminat es t his inf ormat ion on an everyday basis. The zero coupon yield curve is
usef ul in valuat ion of even coupon bearing securit ies and can be ext ended t o ot her risk
classes as well af t er adj ust ing f or t he spreads. I t is also an import ant input f or robust
measures of Value at Risk ( VaR) .