Topic 10 - Term structure of Interest Rates

Question 1

Give a brief overview for Topic 10 - Term structure of Interest Rates

Answer 1

• Evaluate discrete and continuous spot and forward rates
• Theories of the term structure of interest rates
• Evaluate duration, convexity and the Redington’s
  conditions for immunisation

Question 2

What is a Discrete Time Spot Rate?

Answer 2

• Yield on a unit zero coupon bond with term n years, y,n
• Known as n-year spot rate of interest
• Effectively average interest rate over term
• If P,n is price of a zero coupon bond the equation is:
  P,n = 1/(1+y,n)^n = (1+y,n)^(-n) or
  (1+y,n) = P,n^(-1/n)

Question 3

Discrete Time Spot Rates

Example
Prices for a portfolio of zero coupon bonds are:
1yr = £93, 3yr = £80, 5yr = £68, all per £100 nominal
Calculate spot rates

Answer 3

1yr: 100(1+y,1)^-1 = 93 y1 = 7.5%
3yr: 100(1+y,3)^-3 = 80 y3 = 7.7%
5yr: 100(1+y,5)^-5 = 68 y5 = 8.0%

Variation of interest rates known as term structure of
interest rates

Question 4

What is a Discrete Time Forward Rate?

Answer 4

Discrete time forward rate ft,r is annual rate agreed at
t=0, for investment made at t>0 for period of r years

So if investor agrees (at t=0) to invest £100 at time t for
r years, the accumulated value at t + r is:
100(1+ft,r)r

ft,r is the average interest rate between times t and t + r

Question 5

Show how forward rates, spot rates and zero coupon bond prices are all connected using formulae and a timeline

Answer 5

(1+y,t)^t(1+ft,r)^r = (1+y,t+r)^(t+r) = 1/P,t+r

(1+ft,r)^r = (1+y,t+r)^(t+r)/(1+y,t)^t = P,t/P,t+r

Timeline available page 7 week 10 lecture notes

Question 6

Show how a one year forward rate is written

Answer 6

(1+f,t) = (1+y,t+1)^(t+1)/(1+y,t)^t = P,t/P,t+1

Question 7

Discrete Time Forward Rates
Example
The 3,5 and 7 year spot rates are 6%, 5.7% and 5% pa 
respectively and the 3 yr forward rate from t=4 is 
5.2%pa. Calculate:
y4
f5,2
f3
f3,4

Answer 7

So y,3 = 6%, y,5 = 5.7%, y,7 = 5% and f4,3 = 5.2%

y,4
(1+y,4)^4 = (1+y,7)^7/(1+f4,3)^3= (1.05)^7/(1.052)^3
Therefore y,4 = 4.85%

f5,2
(1+f5,2)^2 = (1+y,7)^7/(1+y,5)^5= (1.05)^7/(1.057)^5
Therefore f5,2 = 3.27%

f,3
1+f,3 = (1+y,4)^4/(1+y,3)^3 = (1.0485)^4/(1.06)^3
Therefore f,3 = 1.47%

f3,4
(1+f3,4)^4 = (1+y,7)^7/(1+y,3)^3 = (1.05)^7/(1.06)^3
Therefore f3,4 = 4.26%

Question 8

What is a Continuous Time Spot Rate?

Answer 8

This is the force of interest equivalent to an effective
rate of interest equal to the spot rate
If P,t is price of a zero coupon bond the equation is
P,t =e^(−txY,t)or
Y,t = −1/t ln(P,t)
Relationship of y,t & Y,t is same as i and δ
So y,t =e^(Y,t) -1 (analogous to i = e^(δ) -1)

Question 9

What is a Continuous Time Forward Rate?

Answer 9

This (Ft,r) is the force of interest equivalent to the 
annual forward rate of interest ft,r
Similar to the discrete forward rates
- e^(tY,t)e^(rFt,r) = e^[(t+r)Y,t+r]
Taking logs of both sides
tY,t + rFt,r = (t+r)Y,t+r 
Ft,r = (t+r)Y,t+r−tY,t  /r
Using Y,t = −1/t ln(P,)t
Ft,r = 1/r ln(P,t/P,t+r)

Question 10

Continuous Time Rates
Example
Prices for a portfolio of zero coupon bonds are
5yr = £70, 10yr = £47, 15yr = £30 all per £100 nominal
Calculate Y,10 and F5,10

Answer 10

100e^(−10Y,10) = 47
Y,10 = -0.1xln(0.47) =  7.55%

F5,10 = 1/10 ln(P,5/P,15) = 0.1 ln (0.7/0.3) = 8.47%

Question 11

Why do interest rates vary?

Answer 11

Due to expectations of lenders/borrowers

Question 12

Rates vary due to expectations of lenders/ borrowers.

What factors affect expectations?

Answer 12

• Supply and demand
• Base rates
• Interest rates in other countries
• Expected future inflation
• Tax rates
• Risks associated with changes in interest rates

Question 13

What is Expectations theory in regard to the term structure of interest rates?

Answer 13

Expectations theory

• Attraction of short term and long term assets varies with expectation of future interest rates
• Expected fall in interest rates make short term assets less attractive and long term assets more attractive
• Short term yields will increase, long term will decrease
• Expected rise in interest will have converse effect

Question 14

What is Liquidity Preference in regard to the term structure of interest rates?

Answer 14

Liquidity Preference

• Long dated bonds more sensitive to interest rate changes
• Risk averse investors require greater compensation, via higher yield for greater risk of loss for longer term assets

Question 15

What is Market Segmentation in regard to the term structure of interest rates?

Answer 15

Market Segmentation

• Bonds of different terms attractive to different investors
• Investors choose assets that match their liabilities
• Supply of bonds may vary as issuer strategies may not tie in with investors.

Question 16

What is Interest rate risk?

Answer 16

The risk that assets and liabilities don’t

move by the same amount when rates change

Question 17

Why is Impact of changes in interest rates important to actuaries?

Answer 17

Actuaries value assets and liabilities over long time
periods
We need to consider the vulnerability of assets and
liabilities to changes in interest rates.

Question 18

State the formula we would use to measure interest rate sensitivity via the effective duration (volatility) test

Answer 18

If we define PV of a series of cash flows as
A =∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
Differentiate A with respect to i
ν(i) = -1/A.d/diA = -A’/A
ν(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘+1,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
Bottom line is familiar, this is the PV of a series of cash
flows payable annually in arrears.

Question 19

State the formula we would use to measure interest rate sensitivity for a series of regular cashflows via the duration (Discounted Mean Term) test

Answer 19

DMT is an estimate of the average duration of the cash
flows
τ = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
Top line looks familiar, this is an increasing annuity
DMT Similar to volatility actually equal to (1+i) ν(i)
This is DMT for regular cash flows what about a bond?

Question 20

Duration (Discounted Mean Term)
Consider the DMT for an eight year bond that pays
coupons annually in arrears at a rate of 6% pa and
redeemable at par. Assume i = 4% pa.

Answer 20

τ = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
τ = ∑8,t=1 6x𝑡xv^t + 100x8xv^8/ (∑8,t=1 6v^t + 100v^8)
τ = 6(Ia)8¬ + 100x8v^8 /  (6a8¬ + 100v^8 )
τ = 6x28.9133 + 800x1.04^-8 / (6x6.7327 + 100x1.04^-8)
τ = 758.03/113.465 = 6.68 years

Question 21

What do actuaries use DMT for?

Answer 21

We can use DMT to estimate the impact of a change in interest rates

Question 22

Consider the DMT for an eight year bond that pays
coupons annually in arrears at a rate of 6% pa and
redeemable at par. Assume i = 4% pa.
τ = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
τ = ∑8,t=1 6x𝑡xv^t + 100x8xv^8/ (∑n,𝑘=1 6v^t + 100v^8)
τ = 6(Ia)8¬ + 100x8v^8 / (6a8¬ + 100v^8 )
τ = 6x28.9133 + 800x1.04^-8 / (6x6.7327 + 100x1.04^-8)
τ = 758.03/113.465 = 6.68 years

We can use DMT to estimate the impact of a change in
interest rates
DMT of previous example was 6.68 years
Initial price of bond was 6a8 + 100v8 = £113.47 @ 4%pa
What is the price if interest rates increased to 5% pa?

Answer 22

6a8¬ + 100v^8 = 6x6.4632+100x1.05^-8 = £106.46

113.47(1.04/1.05)^6.68 = £106.44

Question 23

Duration (Discounted Mean Term)
Question
Calculate the DMT for a 12 year bond that pays
coupons annually in arrears at a rate of 8% pa and
redeemable at par. Assume i = 5% pa. Calculate the
change in the price of the bond if i increases to 6% pa.

Answer 23

Solution: Calculate the DMT for an 12 year bond that
pays coupons annually in arrears at a rate of 8% pa and
redeemable at par. Assume i = 5% pa.
τ = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
τ = ∑12,t=1 8x𝑡xv^t + 100x12xv^12/ (∑12,t=1 8v^t +100v^12)
τ = 8(Ia)12¬ + 100x12v^12 / (8a12¬ + 100v^12 )
τ = 8x52.4873 +1200x1.05^-12 / (8x8.8633 + 100x1.05^-12)
τ = 1088.10/126.59 = 8.595 years

Solution: Calculate the change in the price of the bond
if i increases to 6% pa.
Price at 5% = 8x8.8633 + 100x1.05^-12 = £126.59
DMT 8.595 years
Expected price at 6% pa = 126.59x(1.05/1.06)^8.595 =
£116.69
P@6% = 8x8.3838 + 100x1.06^-12 = £116.77v

Question 24

What is convexity?

Answer 24

Convexity gives a measure of the change in the duration of the bond when interest rates change

Question 25

What does positive convexity imply?

Answer 25

Positive convexity implies that DMT is an increasing
function of i
So value will increase more when i falls than it will
decrease when i increases.

Question 26

State the formula we would use to measure the convexity for an asset with a series of regular cashflows

Answer 26

c(i) = -1/A.d^2/di^(2)A = -A’’/A
c(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x (𝑡𝑘+1)𝑣^(𝑡,𝑘+2,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘

Question 27

Calculate the convexity for an asset with the following
cash flows at i = 7% pa:
£9,000 at t = 7
£12,000 at t = 13

Answer 27

PV = 9000x1.07^-7 +12000x1.07^-13 = £10,584.32

c(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x (𝑡𝑘+1)𝑣^(𝑡,𝑘+2,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘

c(i) = 9000x7x8x1.07^-9 +12000x13x14x1.07^-15/10584.32
c(i) = 1065724.71/10584.32 = 100.689

Question 28

Question
An asset provides the following cash flows:
£8,000 at t =4
£19,000 at t = 15
Calculate the PV, volatility, DMT and convexity at i =7% pa.

Answer 28

PV = 8000x1.07^-4 +19000x1.07^-15 = £12,989.64

ν(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘+1,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
ν(i) = 8000x4x1.07^-5 +19000x15x1.07^-16/ 12989.64
ν(i) = 119354.91/12989.64 = 9.189

τ = (1+i)ν(i) = 1.07x9.189 = 9.832 years

c(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x (𝑡𝑘+1)𝑣^(𝑡,𝑘+2,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
c(i) = 8000x4x5x1.07^-6 +19000x15x16x1.07^-17/ 12989.64
c(i) = 1550193.98/12989.64 = 119.34

Question 29

What is the ideal scenario for an actuary when producing a portfolio of assets?

Answer 29

Ideal scenario is to select portfolio of assets that protect against any changes in interest rates

Question 30

The ideal scenario for an actuary is to select portfolio of assets that protect against any changes in interest rates. However, what is the reality regarding this task?

Answer 30

• Almost impossible to select portfolio that generate
  identical cash flows to that of the liabilities
• Possible to select portfolio that offers some protection

Question 31

How do we define a portfolio surplus?

Answer 31

S(i) = V,A(i) - V,L(i)

Question 32

What is immunisation?

Answer 32

Three conditions that a portfolio must satisfy in order to protect S(i)

Question 33

What are the three conditions of immunisation?

Answer 33

Condition 1
V,A(i,0) = V,L(i,0)
- PV of assets at i,0 = PV of liabilities at i,0

Condition 2
v,A(i,0) = v,L(i,0) or V’,A(i0) = V’,L(i0)
- Volatility of assets at i,0 = Volatility of liabilities at i,0

Condition 3
c,A(i,0) > c,L(i,0) or V’’,A(i,0) = V’’,L(i,0)
- Convexity of assets at i,0 > Convexity of liabilities at i,0

Question 34

A fund must make the following payments at i = 6% pa
 £40,000 at t = 5
 £40,000 at t = 7
Show that immunisation can be achieved by purchasing
the following portfolio
 A nominal amount Y of a 4 year zero coupon bond
 A nominal amount Z of a 9 year zero coupon bond

Answer 34

Condition 1
V,A(i,0) = V,L(i,0)
PV liabilities @ 6% pa
40000v^5 + 40000v^7 = £56,492.61
PV assets @ 6% pa
Yv^4 + Zv^9 = £56,492.61 .........(1)

Condition 2
v,A(i,0) = v,L(i,0) or V’,A(i,0) = V’,L(i,0)
Volatility liabilities @ 6% pa
(40000x5v^6 + 40000x7v^8)/ V,L(i,0) = 316667.57/ V,L(i,0)
Volatility assets @ 6% pa
(Yx4v^5 + Zx9v^10)/ V,A(i,0) = 316667.57/ V,A(i,0)
Yx4v^5 + Zx9v^10 = 316667.57 …………………(2)

Eqn 1 = Yv^4 + Zv^9 = £56,492.61
Eqn 2 = Yx4v^5 + Zx9v^10 = 316667.57
Solve 1 & 2 simultaneously   
Multiply (1) by 4v and rearrange 
(1) becomes Yx4v^5 = -Zx4v^10 + £56,492.61x4v
56492.61x4v + Zx5v^10 = 316667.57
Z = £37,066.21
Y = £43,622.59

Condition 3
c,A(i,0) > c,L(i,0) or V’’,A(i,0) = V’’,L(i,0)
Convexity liabilities @ 6% pa
(40000x5x6v^7 + 40000x7x8v^9)/V,L(i,0) 
= 2123921/56,492.61 = 37.5964
Convexity assets @ 6% pa
(Yx4x5v^6 + Zx9x10v^11)/ V,A(i,0)
= 2372386/56,492.61 = 41.9946

All three conditions hold for this portfolio at i = 6% pa