1

Note: Read the

appendage to this
document about the
difference between zero
rates and forward rates
b f i i th
Interest Rate Theory Interest Rate Theory
before reviewing the
lecture material about
forward ratesThis is
difficult and conceptually
complex material.
Interest Rate Theory Interest Rate Theory
Chapter 4 from Hull
(c)2006-2013, Gary R. Evans. This material may not be used by others without permission of the author.
Exam results Exam results
13
12
14
A
185 A 13
175 A 7
170 B+ 4
7
4 4 4 4
6
8
10
A
A
B+
B
B
C+
C
165 B 4
160 B 4
150 C+ 4
140 C 2
120 C 2
Below ? 0
4 4 4 4
2 2
0
0
2
4
1
C
?
2
Where we are going in this lecture
Review of interest rate formulas
Calculating "zero rates"
Calculating "forward rates"
Calculating "duration"
J ustifying yield spreads and yield curves y gy p y
Discussing yield arbitrage (intro)
What we are skipping from Hull chapter 4 What we are skipping from Hull chapter 4
Section 4.7 Forward Rate Agreements
requires knowledge of the LIBOR/swap zero curve, which isn't
covered until chapter 7.
Section 4.9 Convexity
not very important and I don't want to throw too much at you all at
once.
On duration, I am using discrete examples compared to
Hull'scontinuous Hull s continuous
out there in the real world you are more likely to see discrete and
the two approaches are similar anyway.
also in chapter 5 I am using discrete rather than continuous upper
and lower limit calculations.
3
Types of Risk Embodied in the Yields of YBFAs Types of Risk Embodied in the Yields of YBFAs
Default risk
reflected in corporate and municipal YBFAs
notpresentinTreasuries not present in Treasuries
Market (maturity) risk
due to capital gains and losses when interest rates fluctuate
the longer the maturity, the higher the risk
common to all YBFAs
Economic risk
d t i fl ti i il i liti l h k due to inflation or similar economic or political shocks
the longer the maturity, the higher the risk(the more likely you
will be holding this asset during the "event").
Treasuries may experience "flight to safety" during political
shocks.
Yield Spreads Yield Spreads
YBFAs with long maturities have a higher probability of
gainor losswhen interest rates fall or rise. They are also
subject to a higher level of economic risk.
Therefore, according to our notions of risk, longer term
YBFA maturities are seen as having higher risk, therefore
their yields have a risk premium.
Therefore, typically when we map the yields of a full
maturity range for a class of YBFA securities, like
Treasuries, from short-term to long-term, that mapping will
i ( lid ) rise (see next slide).
These are called yield spreads or yield curves and the
mapping is called the term structure of interest rates.
Sometimes we see an atypical flat or even inverted term
structure of interest rates.
4
The yield curve The yield curve today: March 2013 today: March 2013
The configuration is normal.
but thespreadbetweenlong but the spread between long
and short is atypically narrow
because of QE2/QE3.
Note this configuration
compared to 2012 a carbon
copy, which shows the
stability of QE3.
Flat and Inverted Yield Flat and Inverted Yield Spreads: 1990 Spreads: 1990
8 00
10.00
J an 1990 yield curve flat
4.00
6.00
8.00
3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr
long
position
short
position
This tends to happen during periods of high inflation or when
the FRS is aggressively raising short term rates to combat
inflation. Inverted (1980) at end of hyper-inflation period.
Presents an ideal spread arb situation.
5
Treasury Yield Treasury Yield Spreads: 2004 Spreads: 2004--05 05
6.00
This slide was from the 2005 lecture.
2.00
3.00
4.00
5.00
Mar-05
J an-04
FRS raises
FFR targets
As can be seen, this Fed action
is influencing only short-term
rates.
0.00
1.00
1 mo 3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr
g
"Short on short term" means that you are betting that 3 mo prices will fall and yields will rise.
The The 2006 2006 yield spread yield spread
6.00
March 28, 2006
4.77 4.75 4.74 4.72 4.72 4.73
4.50
4.59
2.00
3.00
4.00
5.00
A flat yield curve after the 2-year ... largely due to
overseas purchases of Treasury Securities (especially
ChinaandOPEC) favoringthemuchthinner longend
loaded
to rise
FRS raising
0.00
1.00
3 Mon 6 Mon 2 Yr 3 Yr 5 Yr 10 Yr 20 Yr 30 Yr
China and OPEC) favoring the much thinner long end
of the spectrum. This is tied very strongly to our
Merchandise Trade Deficit.
rates this end
6
Yield curve in March 2007 Yield curve in March 2007
March 6, 2007: Yield curve is
inverted from 2 Year Note out.
Rates were slowly rising at this
time.
Yield Curve in March 2008 Yield Curve in March 2008
7
The simple bond valuation formula The simple bond valuation formula
( ) ( )
MV
C Par
i
n
= +

( ) ( )
MV
r r
i n
i
+
+
+
=

1 1
1
where
MV =market value presently of the bond
n =number of years to maturity
C =coupon payment (par times the coupon interest rate) p p y (p p )
r =present yield of this bond (market determined)
This formula assumes that there is only one interest payment
per year and that this bond was priced on the day after the
most recent interest payment was made.
The elementary bond formula (reduced form) The elementary bond formula (reduced form)
( )
( )
MV CR
r
n


+

(
|

|
|
|
|
+
|

|
| 100
1
1
1
100
1
( )
( )
MV CR
r r
n
=
\

.
|
|
+
+ \

.
| 100 100
1
where
MV: the present market value of the bond
CR: couponrate(original yield) of thebond
Note: This formula cannot be used
to value an actual bond because it
assumes only one interest payment
CR: coupon rate (original yield) of the bond
r: current market rate on equivalent bonds
n: number of remaining years to maturity
y p y
per year and that the bond is being
bought on the day of the coupon
payment. For actual bond pricing a
more complicated version of this is
shown at the end of this lecture.
The original formula is a geometric series and
this is a reduced-formequation. To see its
justification and derivation, read the appendix in
the reading assigned for this lecture.
8
The formula when interest is paid The formula when interest is paid
semi semi--annually annually
C Par
m

( ) ( )
MV
C
r
Par
r
i m
i
=
+
+
+
=

1 2 1 2
1
where
MV =market value MV =market value
C =coupon interest payment (par X r/2)
r =present market yield
m =number of coupon payments remaining (years X 2)
The The final formula ask yield ( final formula ask yield (ytm ytm))
( )
( ) ( ) |
|
.
|

\
|

+
+
| |
=

+

p
a
m
C Par
r
m
C
MV
a p
m
n
n
i
p
a
i
365
1
1
1
1
This term represents an adjustment for accrued interest.
MV =market value, the quoted ask price of the bond,
C =the annual coupon payment, equal to the coupon rate times par,
P 100
( )
. \
+
|
.
|

\
|
+
=
p m
r
m
r
m
m i
p
365 1
1
1
Par =100
r =the prevailing annual market yield, expressed as ask yield or yield to maturity,
m =the number of coupon payments per year,
n =the number of remaining coupon payments,
p =the number of days in this coupon period (between 181-184, use 182 if unknown),
a =the number of days between the last coupon payment and the settlement day.
9
an example an example
Purchase date: February 8, 2009 (146 days since last coupon payment, 36 to next),
Next coupon date: March 15, 2009 (the first of 42 coupons remaining),
Redemption date: September 15, 2029 (for par and last coupon),
You are buying a 30-yr bond that was issued on September 15, 1999:
p p , ( p p ),
Coupon rate/amount: 8%yielding $4 per coupon payment,
Present market rate (ask yield): 10%.
8022 . 0
182
146
= =
p
a
( )
|
.
|

\
|

+
+
|
.
|

\
|
+
=
+
=

8022 . 0
2
8
10 . 1
100
2
10 .
1
1
2
8
099 . 0 5 . 20
42
1
8022 . 0
i
i
MV
. \
( ) ( )
( ) 31 . 83 21 . 3
10 . 1
100
05 . 1
1
4
60 . 20
42
1
8022 . 0
= + =

=

i
i
MV
Reduced Reduced--form Version of the Complex Formula form Version of the Complex Formula
( )
|
|

|
|
|

|
(
(
(
(

a C
P
m
r
C
MV
n
1
1
1
1
Solving the same problem from the last slide (rounding error explains the difference):
( )
( )
( )
( ) ( ) |
|
.
|

\
|

|
|
.

\
+
+
(
(
(
(

+
=

+

p
a
m
C
r
Par
m
r
m
r
m
m
C
MV
a p
m
n
p a
365
1
/
1
1
(
( )
( )
( )
( ) 31 . 83 8022 . 0 4
10 . 1
1
100
05 . 1
05 . 0
05 . 1
1
1
4
099 . 0 5 . 20
8022 . 0
42
=
|
|
.
|

\
|
+
(
(
(
(

=
+
MV
10
The continuous T The continuous T--bond yield formula bond yield formula
( ) ( )
P Ce e
b
y
i
i
n
y
n
= +

2
1
2
100
This formula above is the generalized continuous bond yield
formula shown in Hull (he only shows an example, below). This
formula is solved for y using an iterative technique. P is the price
of the bond, C is the semi-annual coupon payment (equal to the
coupon rate times 100 divided by 2, and n is the number of
b d (d bl h b f f
3 3 3 103 9839
05 10 15 2
e e e e
y y y y
+ + + =
. . .
.
coupon payments to be made (double the number of years of
maturity). Compare this to the discrete bond formula.
The Market Value of a CMO or CDO The Market Value of a CMO or CDO
( )
| |
MV PI r
i
i
= +
=

o , 1 12
1
360
A Collateralized Mortgage Obligation (called a CDO if made up of other
classes of debt) is a pass-through debt obligation where the principle and
interest payments (PI) made by mortgage holders is passed through to the
owner of the CMO.
A CMO differs from bonds in three respects: (1) The numerator is not a
known value and must be treated as a random variable, (2) the payment is
i =1
, ( ) p y
monthly, and (3) there is no principal value at the end.
Because mortgages get paid off or refinanced, the nominal front-end value
of PI is considerably higher than at the tail.
Pieces of this can be sold off as Cash Pass-Thru Certificates and therefore
must be priced.
11
A very important question ... A very important question ...
Regarding the conventional bond formula or any similar formula ...
( ) ( ) ( ) ( ) ( ) ( )
n n n
r
Par
r
C
r
C
r
C
r
C
r
C
MV
+
+
+
+
+
+ +
+
+
+
+
+
=

1 1 1 1 1 1
1 3 2

... is this piece, the third payment, considered by itself, being priced
properly, given that you are discounting it at a 20 year rate when the
payment isbeingmadeinonlythreeyears?Ontheyieldcurvethe payment is being made in only three years? On the yield curve the
yield on a 3-year asset is not the same as a 20-year asset.
Why is this relevant? Suppose you want to sell off this piece (a
tranche).
This issue is at the root of the importance of the difficult material
ahead, the derivation of the zero rate and forward rates.
Hull's "zero rate" Hull's "zero rate"
Hull's "zero rate," more typically called "zero coupon rate," is the continuous
version of the traditional formula used to value a zero-coupon bond (like a US
Treasury STRIP). This is a long-term bond that makes no interest payment (pays y ) g p y (p y
no "coupon") but simply returns the par value at maturity.
This is the formula, where r is the current market
rate, and t is the time in years to maturity.
( )
P
r
t
=
+
100
1
For example, a 30-year zero-coupon bond with
22 years and three months remaining to
maturity with current yields of equivalents at
3203
100
5.25% is worth:
( )
3203
1 00525
2225
.
.
.
=
+
Hull's continuous
rate equivalent is:
P e e
rt
= = =

100 100 3109
00525 2225 . .
.
12
Determining "Treasury Zero Rates" Determining "Treasury Zero Rates"
"Treasury Zero Rates," which are difficult to calculate (conceptually ...
canned software exists to do it for you), are used to give a correct value to
each individual payment of the cashflow stream. It consists of continuous
discounting to the present each payment from the known cashflow of
Treasury securities. Remember that a Treasury security is almost never
purchased at par and that yields on these securities are expressed as simple
interest compounded quarterly or semi-annually. Therefore to make the
conversion to continuous, the we go back to a conversion formula that we
saw in an earlier lecture:
r m
r
m
c
d
= +
|
\

|
.
|
ln 1
Note: This zero-rate conversion can be used for other types of securities as well.
Where we are going in the next few Where we are going in the next few
slides slides
We are trying to define and calculate the rates for this zero rate y g
table. Compare this to Tables 4.3 and 4.4 in Hull. This step is
necessary for the calculation of forward rates, which are shown in
Table 4.5 in Hull.
13
First row in the table: 3-month TBill
The 3-mo TBill pays no coupon and is always sold at a discount
(97.50 in this example). Therefore the interest earned in one ( p )
quarter is 100-price (2.50 in this example). First we calculate the
simple quarterly interest:
| | ( )
r
P
P
q
=

=

=
4 100 4 250
975
10256%
.
.
.
W th t thi t t ti i th i We then convert this rate to continuous using the conversion
formula:
r m
r
c
q
= +
|
\

|
.
| = +
|
\

|
.
| = ln ln
.
. 1
4
4 1
010256
4
10127%
2nd row in the table: 6 2nd row in the table: 6- -month TBill month TBill
This TBill also pays no coupon. Here we assume you paid
94.90 and because it has a maturity of 6 months it is paying
semi-annual interest:
| | ( )
r
P
P
sa
=

=

=
2 100 2 510
9490
10748%
.
.
.
which is converted to the continuous annual rate:
r m
r
c
sa
= +
|
\

|
.
| = +
|
\

|
.
| = ln ln
.
. 1
2
2 1
010748
2
10469%
14
3rd row in table: 1 3rd row in table: 1- -year TBill year TBill
This is easy. Here the conversion is direct from annual to
continuous. Assuming the TBill is worth for 90.00: g
r
a
=

=
100 90
90
1111% .
Once we get to TBonds, however, we have to take into
account thatTreasuryTBondspayasemi annual coupon
( ) ( ) 10536 . 1111 . 1 ln 1 ln = = + =
a c
r r
account that Treasury TBonds pay a semi-annual coupon
payment equal to Par (100) times the initial coupon rate,
then pay a final payment equal to the Par value of the
bond. For each six months we have to use an iterative
process where we take the semi-annual and annual rates
that we have already calculated and ...
4th row in the table: TBond with 1.5 years 4th row in the table: TBond with 1.5 years
... discount the first coupon payment at the 6-month continuous rate, discount
the second coupon payment at the 1-year continuous rate, then solve for the
15year continuousrategiventhemarket valueof thisbond 1.5 year continuous rate given the market value of this bond.
In this example, we assume a coupon payment of $4 (8% annually) and the
current secondary market value of a 1.5 year bond of 96 (which implies that
the current yield is higher than 8% because this bond is at discount):
This is the value of the
remaining piece.
4 4 104 96
010469 05 010536 15
e e e
R
+ + =
. . . .
0949 090 26 24
15
. .
.
+ + =

e
R
R
15
085196
15
10681%
.
ln( .
.
. =
|
\

|
.
| =
15
5th row in the table: 5th row in the table: TBond TBond with 2.0 years with 2.0 years
... discount the first coupon payment at the 6-month continuous rate, discount
the second coupon payment at the 1-year continuous rate, discount the third
coupon payment at the 1.5-year continuous rate, then solve for the 2 year
continuousrategiventhemarket valueof the2-year bond continuous rate given the market value of the 2-year bond.
In this example, we assume a coupon payment of $6 (12% annually) and the
current secondary market value of a 2- year bond of 101.60 (which implies
that the current yield is lower than 12% because this bond is at premium):
60 . 101 106 6 6 6
2 5 . 1 10681 . 0 10536 . 0 5 . 0 10469 . 0
= + + +
R
e e e e
9333 . 16 667 . 17 8520 . 0 900 . 0 949 . 0
2
= + + +
R
e
( )
10808 . 0
2
80561 . 0 ln
2
=
|
.
|

\
|
= R
... and later maturities ... and later maturities
This same iterative process, which is called the "bootstrap
method," continues out (typically quarterly or semi-annually)
til th d f th t it t ( 30 ) until the end of the maturity spectrum (say 30 years).
Clearly this would be rather easy to program in C++or
equivalent.
For data, you simply use the coupon rates and current prices
for Treasuries of these various maturities quoted in The Wall
Street Journal or someonlinesource Street Journal or some online source.
Note how this is different from the traditional bond formula ...
here each payment is discounted at the rate that is relevant
for its maturity, rather than at a single "market" discount rate.
16
... giving us the zero ... giving us the zero- -rate table rate table
whichisessentiallythezero-couponrateoneachtranche ... which is essentially the zero coupon rate on each tranche.
The implicit formula that we have The implicit formula that we have
created ... created ...
( ) ( )
zr
i
n
zr
n

( ) ( )
P Ce e
b
zr
i
i
n
zr
n
i n
= +

2
1
2
100
Now each cashflow component of this bond is correctly valued, which is essential
if you plan to sell off pieces of this bond as cashflow certificates or even to
calculate the true effective yield that you will be earning if you sell this bond y y g y
before maturity.
Note: You should realize that the P
b
this bond should not and will not be different
than the equivalent in the original discrete formula ... it can't be, given the
technique we used. Each piece is priced differently than the original formula.
17
Review: What did we just do? Review: What did we just do?
In effect, we pretended that a ten-year bond is not a ten-year
b d I t dit i ll ti f 21 t ( f bond. Instead it is a collection of 21 zero-coupon notes (one for
each coupon payment and one much larger one for par) that are
each continuously discounted back to the present at the
appropriate interest rate for that maturity.
For example, the coupon payment that will be paid in two years is
discounted back to the present at the rate that is appropriate for 2-
year notes.
We determine that rate by taking the implied continuous market
rate for a 2-year note discounting the coupon payment back to the
present
A simple application of the zero A simple application of the zero--coupon coupon
concept (to motivate its acceptance) concept (to motivate its acceptance)
Many exotic but commonly used financial instruments, especially debt
obligations, like Collateralized Debt Obligations (CDOs ... often
mortgage pools) or Carry Trade paper (borrowing from one large lender
at a low rate to re-loan to many different smaller borrowers at a higher
rate), are often chopped up and sold off in pieces. These pieces are
commonly called "tranches."
Suppose you have a debt contract promising to pay you $100 per year
f th t th S d id t ll th thi d t for the next three years. Suppose you decide to sell the third payment as
a tranche to a third party? How would you price it?
If the 3-year zero rate is 5%, the tranche would be sold at $86.07:
P e = =

100 8607
005 3 .
.
18
Step 2: Determining Forward Rates Step 2: Determining Forward Rates
(warning: conceptually very difficult to grasp) (warning: conceptually very difficult to grasp)
As Hull points out in section 4.6 and 4.7, forward rates are used to price forward
rate agreements. To calculate forward rates, we have to have access to (or to
havecalculated) or zero-ratetablebasedupontheprocedurejustdescribed have calculated) or zero-rate table based upon the procedure just described.
Here is an example: Suppose you have a debt instrument that pays you interest
for five years according to a zero coupon schedule (with zero rates) that goes out
for five years. [Note important to understand: This may be a traditional debt
instrument for which you have made a zero rate schedule as we did in the
previous section]. If you ask the question, "What is the interest yield that I will
earn in the 5th year alone?" you are asking "What is the 5-year forward rate?"
How does this differ from the five-year zero rate? There we are asking the
question, "What is my present interest yield on the coupon payment that will
made to me in the 5th year."
Stop and read the document entitled Justification of the technique used to
calculate forward rates from zero rates by example.
Calculating the Forward Rate from Calculating the Forward Rate from
Zero Rates Zero Rates
Hereisatable Theformulaused
Time Zero Rate Forward Rate
1 2.000
2 2.500 3.000
3 3.000 4.000
4 3.500 5.000
5 4.000 6.000
Forward Rates for 5 Year Investment
Here is a table. The formula used
to calculate forward rates in this
table is easy:
| |
( )
| |
RF RZ t RZ t
t t t
=
1
1
For example:
( ) RF
4
350 4 300 3 5% = = . ( . )
Why so simple?
See also Hull Table
4.5
19
Why so simple?? Why so simple??
The value of the investment over all four years, equal to
principal value compounded at the four year zero rate, will
equal the principal value compounded at the three year zero
rate compounded again at the 4th year forward rate (because
that is the rate earned for only that year):
100 100
0030 3 0035 4
4
e e e
FR . .
=
Taking the natural logs and solving for FR
4
leaves us with
the equation on the previous page. Normally the
denominator is 1 as in this example, but we do allow for
multi-period forward rates.
Macaulay Duration Macaulay Duration
"Duration is a weighted average of the times that interest payments and the final
return of principal are received. The weights are the amounts of the payments
discounted by the yield-to-maturity of the bond."* Duration is a common measure
th t ll l t t b d ithdiff t t iti d Th that allows analysts to compare bonds with different maturities and coupons. The
discrete form of the equation assuming annual interest payments is:
( ) ( )
( )
D
t c y
P
t c y
c y
i i
t
i
n
i i
t
i
n
i
t
n
i i
i
=
+
=
+
+

1 1
1
1 1
where D is duration, y is
current yield, c is the cash
payment (coupon or coupon
plus redemption when t=n),
P is the price of the bond,
*Source: Subject: Bonds - Duration Measure, by Rich Carreiro, 1998, at http://invest-
faq.com/articles/bonds-duration.html. This article provides a good introduction to duration. The
numerical example above and the example on the next page is fromthat source.
( )
i =1
Note: Not in Hull
and n is the number of
years maturity.
20
Calculating duration Calculating duration
Like most of these summation formulas, duration is easy to
calculateinanExcel workbook or eveneasier yet inaC++or calculate in an Excel workbook or even easier yet in a C or
V Basic program. For example, consider a 5-year annual note
with a 7% coupon and a current yield of 5%. Duration would
be equal to:
D years =
+ + + +
+ + + +
=
6677 2 6349 3 6047 4 5759 5 83838
6677 6349 6047 5759 83838
441
. ( . ) ( . ) ( . ) ( . )
. . . . .
.
Memo Note: When payments are
compounded semi-annually or quarterly,
there is a simple conversion formula that
you can apply (see Hull p. 91):
D
D
y m
a
=
+ 1
Calculating Duration with Excel Calculating Duration with Excel
Maturity value: 100
C t 0 07
Calculating Duration for a 5-year Annual Note
The weight of each
paymentrelative
Coupon rate: 0.07
Current yield: 0.05
Years to maturity: 5
Year Cash payment Present Value Numerator Weight
1 7 6.667 6.667 0.061
2 7 6.349 12.698 0.058
3 7 6.047 18.141 0.056
4 7 5.759 23.036 0.053
5 107 83.837 419.186 0.772
Totals Note value (denom): 108.659 479.728 1.000
Duration: 4 41
payment relative
total note value. It
must sum to 1.0
This number has
no inherent
meaning ... it is to
Duration: 4.41
Refer to this slide when doing your homework (after exam?).
Also compare it to Table 4.6 in Hull (where he is using a
continuous rather than discrete formula).
g
be compared to
another bond.
21
How is duration used? How is duration used?
When comparing the duration of two bonds, a bond with a higher duration
carries more risk and is likely to have a higher price volatility.
Why do we need to calculate duration when we know that a long-term bond
h i k d l tilit th h t t t ?D 't dt has more risk and volatility than a shorter-term note? Don't we need to
know only the time to maturity?
No, because the yield relative to
the coupon also affects risk and
volatility. For example, a 10-year
bond with a low coupon rate and a
low yield will have a higher
volatility and risk to cashflow than
a 10-year bond with a high coupon
and high yield because the latter
receives payment proportions at a
faster rate.
Image source: investopedia.com, section on advanced bonds, duration.
Hull's Duration Formula Hull's Duration Formula
Hull's duration formula (4.12), which is continuous
is clearly a similar formula to our own, which is more commonly
used:
n
D
t c e
P
i i
yt
i
n
i
=

=

1
used:
( )
D
t c y
P
i i
t
i
n
i
=
+

=

1
1
All resulting theory is the same, of course.