CM1 Ch 14: Term structure of interest rates

 Define the n-year discrete spot rate of interest.

 What is the discount factor from time n to time 0 using the discrete
n-year spot rate of interest?

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Discrete spot rate of interest

 The n-year discrete spot rate of interest, y n , is the yield on a unit

zero-coupon bond with term n years (ie the average annual interest rate
over the next n years, starting now).

 The discount factor from time n to time 0 (now) is:

1
(1 + y n )n

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
2

 What is the definition of a discrete forward rate of interest?

 What is the discount factor from time t + r to time t using the discrete
forward rate of interest?

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Discrete forward rate of interest

 The r-year discrete forward rate of interest, ft ,r , is the annual interest

rate agreed at time 0 for an investment made at time t > 0 for a period
of r years.

 The discount factor from time t + r to time t is:

1
(1 + ft ,r )r

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
2a
Two bonds paying annual coupons in arrears of 5% and redeemable at par
reach their redemption dates in exactly one and two years’ time, respectively.
The price of each of the bonds is £96 per £100 nominal.

(i) Calculate the 1-year spot rate.

(ii) Calculate the 1-year forward rate that applies from time 1.

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Obtaining spot and forward rates from bonds

-96 105 cashflow -96 5 105

0 1 time 0 1 2
y1 y1 f1, 1

From the first bond we have: 96 = 105(1 + y1)-1 fi y1 = 9.375%

From the second bond we have: 96 = 5(1 + y1)-1 + 105(1 + y1)-1(1 + f1,1)-1
fi f1,1 = 5%

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
3

How are continuous-time spot and forward rates related to discrete-time spot
and forward rates?

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Continuous-time spot and forward rates

The continuous-time spot/forward rate is the force of interest that is equivalent

to the annual effective (ie discrete-time) spot/forward rate.

So, if Yt is the t-year continuous-time spot rate and Ft , r is the r-year

continuous-time forward rate from time t, then:

Yt = ln(1 + y t )

Ft , r = ln(1 + ft , r )

eg a 3% pa discrete-time spot/forward rate is equivalent to a continuous-time

spot/forward rate of ln(1.03) = 2.9559% pa.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
4

How do we convert between spot and forward rates?

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Converting between spot and forward rates

We equate equivalent accumulation (or discount) factors. Considering the

time period from time 0 to time t and then forward to time t + r :

(1 + y t + r )t + r = (1 + y t )t (1 + ft , r )r

For example:

A(0,6)

0 2 6
A(0,2) A(2,6)

A(0,6) = A(0,2)A(2,6) fi (1 + y 6 )6 = (1 + y 2 )2 (1 + f2,4 )4

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
4a

At time 0 the 1-year spot rate is 8% pa, the 2-year spot rate is 9% pa, and the
3-year spot rate is 9½% pa effective.

(i) Calculate the 2-year (discrete-time) forward rate from time 1.

(ii) Calculate the continuous 2-year forward rate from time 1.

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Converting between spot and forward rates

9.5% pa
9% pa
8% pa

0 1 2 3

f1, 2 pa

A(0,3)
(i) A(0,1)A(1,3) = A(0,3) fi A(1,3) = A(0,1)

1.0953
fi (1 + f1,2 )2 = fi f1,2 = 10.26%
1.08

(ii) F1,2 = ln1.1026 = 9.77%

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
5

Define the instantaneous forward rate Ft , and give a formula for it in terms of
Pt , the price at time 0 of a zero-coupon bond with redemption payment 1 at
time t.

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Instantaneous forward rates

The instantaneous forward rate Ft is defined as:

Ft = lim Ft ,r
r Æ0

and can broadly be thought of as the forward force of interest applying over
the instant of time from t to t + h , where h is small.

In terms of Pt , the price at time 0 of a zero-coupon bond with redemption

payment 1 at time t:

1 d
Ft = - P
Pt dt t

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
6

What are the names of the three most popular theories that explain why
interest rates vary with term?

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Theories that explain why interest rates vary with term

 Expectations theory

 Liquidity preference theory

 Market segmentation theory

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
7

Describe the expectations theory.

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Expectations theory

The relative attraction of short and longer-term investments will vary according
to expectations of future movements in interest rates.

An expectation of a fall in interest rates will make short-term investments less

attractive and longer-term investments more attractive.

In these circumstances yields on short-term investments will rise and yields on

long-term investments will fall.

An expectation of a rise in interest rates will have the converse effect.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
8

Describe the liquidity preference theory.

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Liquidity preference theory

Long-dated bonds are more sensitive to interest-rate movements than short-

dated bonds.

Risk-averse investors will require compensation (in the form of higher yields)
for the greater risk of loss on longer bonds.

This theory would predict a higher return on long-term bonds than short-term
bonds.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
9

Describe the market segmentation theory.

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Market segmentation theory

Bonds of different terms attract different investors, (who will choose assets
that are similar in term to their liabilities).

For example, banks will invest in short-term bonds (since their liabilities are
very short term) and pension funds will invest in the longest-dated bonds
(since their liabilities are long-term).

Therefore, the demand for bonds will differ for different terms.

The supply of bonds will also vary by term, as governments’ and companies’
strategies may not correspond to investors’ requirements.

These different forces of supply and demand will lead to the term structure of
interest rates.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
10

Define the gross redemption yield.

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Gross redemption yield

The gross redemption yield is the effective interest rate that satisfies the
equation of value for a bond (where there is no allowance for tax).

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
10a

At time 0, the 1-year spot rate is 7% pa, the 2-year spot rate is 6¾% pa and
the 3-year spot rate is 6¼% pa. Calculate the price of a 3-year bond,
redeemable at par with annual coupons of 5% pa payable in arrears.

© IFE: 2019 

Price of bond

At time 0, the 1-year spot rate is 7% pa, the 2-year spot rate is 6¾% pa and
the 3-year spot rate is 6¼% pa. A 3-year bond is redeemable at par with
annual coupons of 5% pa payable in arrears.
5 5 105

0 1 2 3
7% pa
6.75% pa
6.25% pa

The price of the bond is:

5 5 105
P= + + = 96.60
1.07 1.06752 1.06253
© IFE: 2019
CM1 Ch 14: Term structure of interest rates
10b

Calculate the gross redemption yield of a 3-year bond, redeemable at par with
annual coupons of 5% pa payable in arrears. The price of the bond is 96.60.

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Gross redemption yield

The price of a 3-year bond, redeemable at par with annual coupons of 5% pa

payable in arrears is 96.60.

The gross redemption yield is the effective interest rate that solves the
equation of value:

96.60 = 5a3 + 100v 3

At 6.2%, the right-hand side is 96.804.

At 6.3%, the right-hand side is 96.544.

Interpolating, we get:

i - 6.2 96.60 - 96.804

= fi i = 6.28%
6.3 - 6.2 96.544 - 96.804
© IFE: 2019
CM1 Ch 14: Term structure of interest rates
11

 Define the n-year par yield.

 Define the coupon bias.

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Par yield and coupon bias

 The n-year par yield is the annual coupon rate, c, such that the price
paid for an n-year bond paying coupons annually in arrears and
redeemed at par is 100 per 100 nominal.
100
– 100 c c c c cashflows

0 1 2 3 n time

 The coupon bias is the difference between the n-year par yield and the
n-year spot rate.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
11a

The n-year forward rate for transactions beginning at time t and maturing at
time t + n is denoted as ft ,n . You are given:

f0,1 = 6.0% per annum

f0,2 = 6.5% per annum
f1,2 = 6.6% per annum

Calculate the 3-year par yield.

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Par yield

We are given that f0,1 = 6% pa , f0,2 = 6.5% pa and f1,2 = 6.6% pa .

100
c c c

0 1 2 3
6% pa
6.5% pa
6.6% pa

The par yield can be found from the equation:

c c 100  c
100     c  6.40
2
1.06 1.065 1.06  1.0662

So the par yield is 6.4%.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
12

 Define the discounted mean term.

 What is another name for the discounted mean term?

 State the formula for modified duration in terms of the discounted mean
term.

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Discounted mean term

The discounted mean term is the average time of the cashflows, weighted
by present value:

 t ¥ PV  tcv t
t = DMT = =
 PV  cv t

Another name for the discounted mean term is the Macaulay duration.

The modified duration is defined as:

DMT
i ( p)
1+
p

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
12a

Calculate the discounted mean term, at an effective annual interest rate of

8%, for a 5-year fixed-interest security with coupons of 10% pa paid half-
yearly in arrears, redeemable at par.

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Discounted mean term

We will work in half-years, using an effective rate of interest of:

1
1.08 2 - 1 = 3.923%

PV = 5a10 + 100v 10 = 108.77

1 ¥ 5v + 2 ¥ 5v 2 +  + 10 ¥ 5v 10 + 10 ¥ 100v 10
DMT =
PV

5(Ia )10 + 1,000v 10 5 ¥ 42.203 + 1,000 ¥ 1.03923 -10

= = = 8.2
PV PV

So the discounted mean term is 4.1 years.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
13

 Define volatility.

 Give another name for volatility.

 State the relationship between the discounted mean term and the
volatility.

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Volatility

The volatility is defined to be:

d
PV (i )
PV ¢(i ) di
vol (i ) = - =-
PV (i ) PV (i )

where PV is the present value of the cashflows.

Another name for volatility is effective duration.

The discounted mean term and the volatility are connected by the equation:

DMT = (1 + i ) ¥ vol (i ) or, equivalently vol (i ) = v ¥ DMT

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
13a

A loan stock is issued that pays coupons annually in arrears at 8% pa and is

redeemable at par after 10 years.

Calculate the volatility of this stock on the date of issue at an effective rate of
interest of 8% pa.

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Volatility

A loan stock is issued that pays coupons annually in arrears at 8% pa and is

redeemable at par after 10 years. If i = 8% pa:

10
P (i ) = 8a10 + 100v 10 = 8 Â (1 + i )-t + 100(1 + i )-10
t =1
P (0.08) = 100

10
P ¢(i ) = -8 Â t (1 + i )-(t +1) - 1,000(1 + i )-11 = -8v (Ia )10 - 1,000v 11
t =1
P ¢(0.08) = -671.0

P ¢(0.08)
So the volatility is - = 6.71 .
P (0.08)

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
14

Consider a series of n cashflows, where Ck is the cashflow at time tk

( k = 1,2,..., n ). Prove that vol (i ) = v ¥ DMT .

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Proof that vol (i ) = v ¥ DMT

For this series of cashflows:

n n
PV (i ) = Â Ck v tk = Â Ck (1 + i )-tk
k =1 k =1

n
and: PV ¢(i ) = Â ( -tk )Ck (1 + i )-(tk +1)
k =1
So:
n n
-(t +1)
 ( -tk )Ck (1 + i ) k v ¥  tk Ck v tk
vol (i ) = - k =1 n
= k =1
n
= v ¥ DMT
-tk tk
 Ck (1 + i )  Ck v
k =1 k =1

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
15

Define convexity.

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Convexity

The convexity of a cashflow series is defined to be:

d2
PV (i )
PV ¢¢(i ) di 2
conv (i ) = =
PV (i ) PV (i )

Note that convexity is a measure representing the spread of payments around

the discounted mean term. Typically, the more spread out the cashflows, the
higher the convexity.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
15a

An investor has a liability of £100,000 to be paid in 7.247 years’ time.

Calculate the volatility and convexity of this liability at an effective rate of

interest of 8% pa.

© IFE: 2019 

Volatility and convexity

P (i ) = 100,000v 7.247 P (0.08) = 57,250.34

P ¢(i ) = -7.247 ¥ 100,000v 8.247 P ¢(0.08) = -384,160.36

P ¢¢(i ) = 8.247 ¥ 7.247 ¥ 100,000v 9.247 P ¢¢(0.08) = 2,933,491.19

384,160.36
So the volatility is: = 6.71
57,250.34

2,933,491.19
and the convexity is: = 51.24
57,250.34

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
16

State Redington’s conditions for immunisation.

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Redington’s conditions for immunisation

 PVassets = PVliabs

 volassets = volliabs or DMTassets = DMTliabs

¢
or PVassets = PVliabs
¢

 conv assets > conv liabs ¢¢

or PVassets > PVliabs
¢¢

These conditions must hold at the current interest rate.

If a fund is immunised then small changes in the interest rate will result in a
profit being made on the fund.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
16a

A loan stock issued on 1 March 2018 has coupons payable annually in arrears
at 8% pa. Capital is to be redeemed at par 10 years from the date of issue.
The volatility of this stock at 1 March 2018 at an effective rate of interest of
8% pa is 6.71. At 1 March 2018 an investor has a liability of £100,000 to be
paid in 7.247 years’ time, which also has a volatility of 6.71 and convexity of
51.24.

On 1 March 2018 the investor decides to invest a sum equal to the present
value of the liability in the loan stock, where the present value of the liability
and the price of the loan stock are both calculated at an effective rate of
interest of 8% pa. Given that the convexity of the loan stock at 1 March 2018
is 60.53, state with reasons whether the investor will be immunised against
small movements in interest rates on that date.

© IFE: 2019 

Immunisation

The present value of the assets is equal to the present value of the liabilities
and the volatility of the assets is equal to the volatility of the liabilities, so
conditions 1 and 2 are satisfied.

Since the convexity of the assets is 60.53 and the convexity of the liabilities is
51.24, the third condition is also satisfied and immunisation has been
achieved.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates
16b

An investor has to pay a lump sum of £20,000 in 15 years’ time. The investor
wishes to immunise these liabilities by investing in two zero-coupon bonds,
Bond X and Bond Y. The effective rate of interest is 7% per annum. The
investor has decided to invest an amount in Bond X sufficient to provide a
capital sum of £10,000 when Bond X is redeemed in ten years’ time.

Determine the term needed for Bond Y.

© IFE: 2019 

Immunisation – term for Bond Y

If Y is the amount invested in Bond Y, then equating present values:

PVL = PVA fi 20,000v 15 = 10,000v 10 + Y

fi Y = 2,165.43

The DMT of the assets must equal the DMT of the liabilities and so if n is the
term of Bond Y then:

10 X + nY 15 ¥ 20,000v 15 - 10 ¥ 10,000v 10
= 15 fi n =
X +Y 2,165.43
fi n = 26.7 years

The spread of the assets around the DMT, and hence the convexity, is greater
than that of the liabilities and so immunisation is achieved.
© IFE: 2019
CM1 Ch 14: Term structure of interest rates
17

List the practical problems faced when trying to immunise a fund.

© IFE: 2019 

Practical problems with immunising a fund
 Requires continuous rebalancing of the asset portfolio to keep the
asset and liability PV/volatilities equal. This will take time and incur
dealing costs.
 There may be options or uncertainties in the asset or liability cashflows,
meaning that the cashflows are estimated rather than being known with
certainty.
 Assets may not exist to provide the necessary overall asset volatility to
match the liability volatility.
 Redington’s theory only provides immunisation against small changes in
the interest rate.
 Redington’s theory assumes a flat yield curve (ie the interest rate is the
same at all durations) and that the interest rate changes by the same
amount at all durations, which may not be the case in reality.

© IFE: 2019
CM1 Ch 14: Term structure of interest rates

Summary Card

Discrete-time spot rate of interest Card 1

Discrete-time forward rate of interest Card 2
Continuous-time spot/forward rates Card 3
Converting between spot and forward rates Card 4
Instantaneous forward rate Card 5
Theories of the term structure of interest rates: Card 6
Expectations theory Card 7
Liquidity preference theory Card 8
Market segmentation theory Card 9
Gross redemption yield (GRY) Card 10

© IFE: 2019 

Par yield and coupon bias Card 11
Discounted mean term (Macaulay duration) Card 12
Volatility (effective duration) Card 13
Proof that vol (i ) = v ¥ DMT Card 14

Convexity Card 15
Redington’s conditions for immunisation Card 16
Practical problems with immunisation Card 17

© IFE: 2019