CHAPTER 6
Interest Rate Problems
1 Spot rates, forward rates and the yield curve
1.1 Basic idea. The interest rate offered on investments usually depends on the term or length of the invest-
ment. The term structure of interest rates is concerned with the analysis of the dependence of interest rates on
the length of the investment.
1.2 Spot rates and zero rates; zero coupon bonds. A £1 zero coupon bond of term n is an agreement to
pay £1 at the end of n years; zero coupon means that no interest payments are made. A zero coupon bond is
also called a pure discount bond.
The term structure of interest rates can be assessed from the prices of zero coupon bonds as follows.
Let Pt denote the price at issue of a £1 zero coupon bond which matures after a time period of length
t years. Let yt denote the yield to maturity of this bond—this is another name for the internal rate of
return. This yield, yt, is sometimes called the t-year spot rate of interest . Clearly
(1 + yt)tPt = 1 (1.2a)
and hence
yt = P
1/t
t 1
Usually, yt 6= yv when t 6= v. The function y : [0,1)! R with y(t) = yt is called the yield curve. An example
of a yield curve is given in figure 1.2a.
maturity year
yi
el
d
2000 2005 2010 2015 2020 2025 2030
0.05
0.06
0.07
0.08
Figure 1.2a. A Yield Curve.
Yield curves are plotted against the maturity date.
Yield curves usually rise gradually because long bonds
have higher yields than short bonds.
Note that yt, the t-year spot rate of interest or the t-year zero rate of interest is the effective rate of interest
per annum for an investment which lasts for t years.
Recall that , the nominal rate of interest compounded continuously or the force of interest, is defined by
i = e 1 or = log(1 + i) where i is the effective rate of interest per annum.
With continuous compounding, equation (1.2a) becomes
etYtPt = 1 and hence Yt = 1
t
logPt (1.2b)
where Yt is the t-year spot force of interest. Clearly, yt = eYt 1.
Thus Yt corresponds to , the nominal rate of interest compounded continuously, but now the dependence on
the length of time of the investment is considered.
Every fixed interest investment can be regarded as a combination of zero coupon bonds as the following exam-
ple illustrates.
ST334 Actuarial Methods cR.J. Reed Sep 27, 2016(9:51) Section 1 Page 101

即期利率 远期利率 收益率曲线
Page 102 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
Example 1.2a. Suppose a bond pays a coupon of size x at times 1, 2, . . . , n and has final redemption value C at
time n.
(a) Express P , the current price of the bond in terms of x, C and P1, P2, . . . , Pn where P1, P2, . . . , Pn are the prices
of £1 zero coupon bonds with different terms.
(b) Express the current price of the bond in terms of x, C and y1, y2, . . . , yn where y1, y2, . . . , yn are the correspond-
ing yields of the bonds.
(c) Show that the gross redemption yield, i, satisfies min{y1, y2, . . . , yn}  i  max{y1, y2, . . . , yn}.
Solution. The bond can be considered as a combination of n zero coupon bonds all with maturity value x and lengths
1, 2, . . . , n and one zero coupon bond with length n and maturity value C.
Hence the answer to (a) is: P = x(P1 + P2 + · · · + Pn) + CPn and the answer to (b) is
P = x

1
(1 + y1)
+
1
(1 + y2)2
+ · · · + 1
(1 + yn)n

+
C
(1 + yn)n
(c) This follows immediately from
x
(1 + y1)
+
x
(1 + y2)2
+ · · · + x
(1 + yn)n
+
C
(1 + yn)n
=
x
(1 + i)
+
x
(1 + i)2
+ · · · + x
(1 + i)n
+
C
(1 + i)n
Example 1.2b. Suppose we know the spot rate yk for k = 1, 2, . . . , 10. Find an expression for the price of a bond
with redemption value £100, annual coupon rate 8% and maturing in 10 years in terms of y1, y2, . . . , y10.
Solution. The cash flow is as follows:
Time 0 1 2 3 4 5 6 7 8 9 10
Cash flow 0 8 8 8 8 8 8 8 8 8 108
The price is
10X
k=1
8
(1 + yk)k
+
100
(1 + y10)10
1.3 Determining the spot rate. There are few zero coupon bonds and so the zero rate must also be calculated
from the prices of other financial instruments. Here are two possibilities:
• Determine y1 from the interest rate on one year Treasury bills. To find y2, suppose we observe the price P
of a two year bond with redemption value C, face value f and coupon rate r payable annually. Now the price
of the bond should equal the net present value of the cash flow stream, which is:
Time 1 2
Cash flow fr C + fr
Hence
P =
fr
1 + y1
+
fr + C
(1 + y2)2
As y1 is known, the value of y2 can be determined. Then the value of y3 can be determined from the price of a
three year bond, and so on.
• A zero coupon bond can also be constructed by subtracting bonds with identical maturity dates but different
coupons.
Suppose bond A is a ten year bond with price PA, coupon rA, face value and redemption value C.
Suppose bond B is a ten year bond with price PB , coupon rB , face value and redemption value C.
Without loss of generality, assume rA > rB .
Consider a portfolio of rBC of bond A and rAC of bond B. This portfolio will have a face value of
(rA rB)C, a price of P = rAPB rBPA and zero coupon. Hence
(1 + y10)10P = (rA rB)C
and y10 can be determined.
总赎回收益率
6 Interest Rate Problems Sep 27, 2016(9:51) Section 1 Page 103
1.4 Par yield of a bond. The n-year par yield of a bond is the coupon rate that causes the current bond price
to be equal to its face value, assuming the bond to be redeemed at par.
As usual, let f denote the face value, r the coupon rate, P the current price. Assume the coupon is payable
annually for n years. The cash flow is as follows:
Time 0 1 2 . . . n 1 n
Cash flow P fr fr . . . fr f + fr
We want P = f . This implies that
the n-year par yield, r, satisfies 1 = r
nX
k=1
1
(1 + yk)k
+
1
(1 + yn)n
(1.4a)
where yk is the k-year spot rate of interest.
The par yields give an alternative measure of the relationship between the yield and the term of an investment.
The difference between the n-year par yield and the n-year spot rate is called the n-year coupon bias. Thus the
n-year coupon bias is r yn where r is the n-year par yield.
1.5 Forward interest rates. Suppose fT,k denotes the interest rate per annum for money which will be
borrowed at some time T in the future for a period of time of length k. This is called a forward interest rate.
Clearly f0,k = yk.
......................................................................................................................................................................................................................................................................................................................................
....
...
....
.
....
...
....
.
....
...
....
.
..... .....
0 T
start of investment
T + k
end of investment
time
................................................................................................................................... .......
........
period of investment
Now £1 invested at time 0 grows to £(1 + yT+k)T+k at time T + k. Also £1 grows to £(1 + yT )T at time T . If
this amount of £(1 + yT )T is reinvested then it grows to £(1 + yT )T (1 + fT,k)r at time T + k. Hence
(1 + yT+k)T+k = (1 + yT )T (1 + fT,k)k
In particular
(1 + y2)2 = (1 + y1)(1 + f1,1)
(1 + y3)3 = (1 + y2)2(1 + f2,1) = (1 + y1)(1 + f1,1)(1 + f2,1)
and so on. Hence
(1 + yT )T = (1 + f0,1)(1 + f1,1) · · · (1 + fT1,1) (1.5a)
where f0,1 = y1 is the rate per annum for 1 year starting at time 0; f1,1 is the rate per annum for 1 year starting
at time 1, etc.1 Forward rates calculated from equations such as equation (1.5a) are called implied forward
rates as opposed to the market forward rates.
Example 1.5a. Suppose the one year zero rate is 0.07, or 7%, and the two year zero rate is 0.08 or 8%. Find the
implied one year forward rate for a period of length 1 year.
Solution. Recall that the “two year zero rate” is the rate per annum for an investment of length two years. So we are
given that y1 = 0.07 and y2 = 0.08. Using (1 + y2)2 = (1 + y0)(1 + f1,1) gives f1,1 = 1.082/1.07 1 = 0.090 or 9%.
1.6 Continuous compounding. For continuous compounding, we have
PT
PT+k
=
⇥
1 + fT,k
⇤k = ekFT,k
where the forward force of interest is FT,k = ln(1 + fT,k). Hence
FT,k =
lnPT lnPT+k
k
(1.6a)
=
(T + k)YT+k TYT
k
Note that F0,k = Yk where Yk is defined in equation (1.2b) on page 101.
1 If there is no possibility of confusion, the quantity fk,1 is often abbreviated to fk.
债券的票⾯收益
Page 104 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
Example 1.6a. Suppose the spot force of interest for an investment of 1 year is a nominal 7% per annum com-
pounded continuously and the spot force of interest for an investment of 6 months is a nominal 6% per annum
compounded continuously. Find the implied force of interest for an investment starting in 6 months’ time and lasting
for 6 months.
Solution. The general result is
e(t+k)F0,t+k = etF0,t ⇥ ekFt,k or (t + k)F0,t+k = tF0,t + kFt,k
In our case, t = k = 0.5, F0,1 = 0.07 and F0,0.5 = 0.06. Hence F0.5,0.5 = 0.08. So the answer is a nominal 8% per
annum compounded continuously.
1.7 A more sophisticated model with allowance for interest rates varying with time.
• Yields which depend on time and the length of the investment. The above equations are too simplistic:
we should really treat £P , the price of a £1 zero coupon bond, as a function of two variables instead of just
one. So, for 0  t  T , let P (t, T ) denote the value at time t of a zero coupon bond with face value 1 and
maturing at time T .
Clearly P : {(x, y) 2 R⇥ R : 0  x  y}! R with P (x, x) = 1 for all x 0.
This function P of two variables will be complicated: for example, the prices of 10 year bonds and 15 year
bonds will tend to move together in the short term. Hence the functions t! P (t, t + 10) and t! P (t, t + 15)
will be related. fs Corresponding to this function of two variables, we have the yield to maturity, y(t, T ) defined
by
[1 + y(t, T )]Tt P (t, T ) = 1 for all 0  t  T . (1.7a)
Note that y(t, T ) denotes the yield to maturity from a zero coupon bond that starts at time t and ends at time T .
We can now plot the yield curve at time t: this is a plot of the function T ! y(t, T ) for T t. Figure (1.2a)
on page 101 is such a plot with t = 2000.
This model reduces to the previous model if we assume the price just depends on the length of time between
issue and maturity and not on the date of issue. This is equivalent to assuming P (t, T ) = P (0, T t) for all t.
Then set Pt = P (0, t). As P (t, T ) determines Y (t, T ) and conversely, it follows that if P (t, T ) = P (0, T t)
for all t, then y(t, T ) = y(0, T t) for all t, and conversely. Set yt = y(0, t). And so we arrive at the simpler
model.
With continuous compounding, we have
e(Tt)Y (t,T )P (t, T ) = 1 where eY (t,T ) = 1 + y(t, T ) and so Y (t, T ) = lnP (t, T )
T t
• Forward rates. A forward rate is an interest rate agreed now for an investment which starts at some time in
the future.
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
....
...
....
.
....
...
....
.
....
...
....
.
....
...
....
.
..... .....
0 t
time of agreement
T
start of investment
T + k
end of investment
time
............................................................................................................................................................ .......
........
period of investment
Let ft,T,k denote the effective interest rate per annum agreed at time t for an investment starting at time T for
a period of length k. Hence the investment matures at time T + k. This means that £1 invested at time T will
accumulate to £(1 + ft,T,k)r at time T + k.
The discrete time forward rate can be related to the spot rate as follows: suppose we invest £1 at time t for
T t years and agree to invest the accumulated amount at time T for a further k years. Now the accumulated
amount at time T will be £ [1 + y(t, T )]Tt. This amount will accumulate to £ [1 + y(t, T )]Tt
⇥
1 + ft,T,k
⇤k at
time T + k. But £1 invested at time t for T + k t years will accumulate to £ [1 + y(t, T + k)]T+kt. Hence
[1 + y(t, T + k)]T+kt = [1 + y(t, T )]Tt
⇥
1 + ft,T,k
⇤k (1.7b)
and so, by equation (1.7a) ⇥
1 + ft,T,k
⇤k = [1 + y(t, T + k)]T+kt
[1 + y(t, T )]Tt
=
P (t, T )
P (t, T + k)
6 Interest Rate Problems Sep 27, 2016(9:51) Section 1 Page 105
More generally, repeated applications of equation (1.7b) gives:
[1 + y(t, T )]Tt = [1 + y(t, T 1)]T1t ⇥1 + ft,T1,1⇤
= (1 + ft,t,1)(1 + ft,t+1,1) · · · (1 + ft,T1,1) (1.7c)
Note that ft,t,1 = y(t, t+ 1) is the one-year spot interest rate—it is the one-year riskless rate prevailing at time t
for repayment at time t + 1.
Forward rates calculated from equations such as equation (1.7c) are called implied forward rates as opposed to
the market forward rates.
Relation to previous model. Suppose ft,T,k = f0,T,k for all t; i.e. the forward rate just depends on the start
and length of the investment and not when the agreement is made. Denote the common value ft,T,k for all t
by fT,k; hence fT,k is the forward rate for money which will be borrowed at some time T in the future for the
period of time of length k. The model then reduces to the simpler model at the start of this chapter.
• Continuous compounding. For continuous compounding, we have
P (t, T )
P (t, T + k)
=
⇥
1 + ft,T,k
⇤k = ekFt,T,k
where the forward force of interest is Ft,T,k = ln(1 + ft,T,k). Hence
Ft,T,k =
lnP (t, T ) lnP (t, T + k)
k
(1.7d)
=
(T + k t)Y (t, T + k) (T t)Y (t, T )
k
by equation (1.7a). Note that Ft,t,k = Y (t, t + k).
• Instantaneous forward rates. The instantaneous forward rate at time t for an investment commencing at
time T is defined by
Ft,T = lim
k!0
Ft,T,k
Using equation (1.7d) shows that
Ft,T = lim
k!0
logP (t, T ) logP (t, T + k)
k
= lim
k!0
logP (t, T + k) logP (t, T )
k
= @
@T
logP (t, T ) (1.7e)
= 1
P (t, T )
@P (t, T )
@T
Conversely, using P (T, T ) = 1 and integrating equation (1.7e) shows that
P (t, T ) = exp
✓

Z T
t
Ft,u du
◆
The instantaneous forward rate is a theoretical concept which is useful in modelling bond prices.
Note that any one of the functions P (t, T ), Y (t, T ) or Ft,T determines the other two. The short rate at time t or the
instantaneous rate at time t is defined to be
rt = Y (t, t) = lim
k!0
Y (t, t + k) = lim
k!0
Ft,t,k = Ft,t = @ logP (t, T )
@T

T=t
where the third equality follows from Ft,t,k = Y (t, t + k). Hence, rt, which is equal to Ft,t, is the instantaneous
forward rate at time t for an investment commencing at time t.
There is less information in the function rt than in the functions P (t, T ), Y (t, T ) or Ft,T .
Page 106 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
term in years
yi
el
d
0 5 10 15 20 25 30
0.05
0.06
0.07
0.08
Figure 1.8a. An Increasing Yield Curve.
term in years
yi
el
d
0 5 10 15 20 25 30
0.03
0.04
0.05
0.06
0.07
Figure 1.8b. A Decreasing Yield Curve.
term in years
yi
el
d
0 5 10 15 20 25 30
0.03
0.04
0.05
0.06
0.07
Figure 1.8c. A Humped Yield Curve.
1.8 Explaining the term structure of interest rates. The three common types of yield curves are as follows.
• Decreasing yield curve. Long term bonds have lower yields than short term bonds.
• Increasing yield curve. Long term bonds have higher yields than short term bonds. This is the most common
shape.
• Humped yield curve. Long term bonds have lower yields than short term bonds, but the very short term
bonds (less than 1 year) have lower yields than 1 year bonds.
There are three popular explanations of the term structure of interest rates:
(1) Expectations Theory. This theory postulates that the relative attraction of long and short term bonds
depends on the expectations of future movements in interest rates. If investors expect interest rates to fall, then
long term bonds will become more attractive; this will cause the yields on long term bonds to fall and lead to
a decreasing yield curve. Similarly, if investors expect interest rates to rise, then long term bonds will become
less attractive and so lead to an increasing yield curve.
(2) Liquidity Preference. This theory asserts that investors usually prefer short term investments over long
term ones. The reason for this is that investors anticipate they may need to sell their bonds soon and they know
that long term bonds are more sensitive to interest changes than short term bonds. Hence purchasing a short
term bond lessens the risk. This implies that longer-term bonds must offer higher yields as an inducement.
6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 2 Page 107
(3) Market Segmentation. Banks are interested in short term bonds because their liabilities are short term.
Pension funds are interested in long term bonds because their liabilities are long term. The market segmentation
theory postulates that the term structure arises from the different forces of supply and demand for bonds of
different lengths.
1.9
Summary. The simple model: interest rates constant in time.
• yt denotes the t-year spot rate of interest. This is the effective interest rate p.a. for an investment
which lasts for t years.
• Pt denotes the price in £ at issue of a £1 zero coupon bond which matures after t years. Hence
Pt(1 + yt)t = 1
• ft,k denotes the forward rate for an investment starting at time t for a period of k years. Then:
(1 + yt+k)t+k = (1 + yt)t(1 + ft,k)k and (1 + ft,k)k =
(1 + yt+k)t+k
(1 + yt)t
=
Pt
Pt+k
(1 + yt)t = (1 + f0,1)(1 + f1,1) · · · (1 + ft1,1)
where ft,1, sometimes written as ft, is the 1-year forward rate for an investment starting at time t for a
period of 1 year.
• Ft,k denotes the forward force of interest for an investment starting at time t for k years. Hence
Ft,k = ln(1 + ft,k) =
ln(Pt) ln(Pt+k)
k
Also
F0,k = ln(1 + yk) and e(t+k)F0,t+k = etF0,t ⇥ ekFt,k
• Ft denotes the instantaneous forward rate at time t. By definition
Ft = lim
k!0
Ft,k = lim
k!0
ln(1 + ft,k) = lim
k!0
ln(Pt) ln(Pt+k)
k
= d
dt
lnPt = 1
P
dP
dt
Conversely
Pt = exp
✓

Z t
0
Ft du
◆
The more sophisticated model: interest rates varying with time.
• y(t, T ) denotes the effective rate of interest per annum for an investment starting at time t and maturing
at time T .
• P (t, T ) denotes the price in £ at time t of a £1 zero coupon bond which matures at time T .
Hence P (t, T ) [1 + y(t, T )]Tt = 1.
• ft,T,k denotes the forward rate per annum at time t of an investment starting at time T and maturing
at time T + k.
Explaining the term structure of interest rates: expectations theory, liquidity preference, market
segmentation.
Par yield of a bond.
2 Exercises (exs6-1.tex)
1. At time 0 the 1-year spot rate is 8% p.a., the 2-year spot rate is 9% p.a. and the 3-year spot rate is 91/2% p.a. What is
the value of the 2-year forward rate from time 1? (Institute/Faculty of Actuaries Examinations, April 1997) [2]
2. The following n-year spot rates apply at time t = 0:
1 year spot rate of interest: 41/2% per annum effective
2 year spot rate of interest: 5% per annum effective
3 year spot rate of interest: 51/2% per annum effective
Calculate the 2 year forward rate of interest from time t = 1 expressed as an annual effective rate of interest.
(Institute/Faculty of Actuaries Examinations, April 2000) [2]
市场细分
Page 108 Exercises 2 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
3. The 1-year forward rates for transactions beginning at times t = 0, 1, 2 are fi, where
f0 = 0.06 f1 = 0.065 f2 = 0.07
Find the par yield for a 3-year bond. (Institute/Faculty of Actuaries Examinations, September 1997) [3]
4. 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.
Determine the 3-year par yield. (Institute/Faculty of Actuaries Examinations, September 1998) [3]
5. (i) The one-year forward rate applying at a particular point in time t is defined as ft,1. If ft,1 = 4%, calculate the
continuous time forward rate Ft,1 applying over the same period of time.
(ii) Define the instantaneous forward rate Ft.
(Institute/Faculty of Actuaries Examinations, September 2004) [2+2=4]
6. In a particular bond market, the two-year par yield at time t = 0 is 4.15% and the issue price at time t = 0 of a
two-year fixed interest stock, paying coupons of 8% annually in arrears and redeemed at 98, is £105.40 per £100
nominal. Calculate:
(a) the one-year spot rate;
(b) the two-year spot rate. (Institute/Faculty of Actuaries Examinations, April 2004) [6]
7. At time t = 0 the n-year spot rate of interest is equal to (2.25 + 0.25n)% per annum effective (1  n  5).
(a) Calculate the 2-year forward rate of interest from time t = 3 expressed as an annual effective rate of interest.
(b) Calculate the 4-year par yield.
(c) Without explicitly calculating the gross redemption yield of a 4-year bond paying annual coupons of 3.5%,
explain how you would expect this yield to compare with the par yield calculated in (b).
(Institute/Faculty of Actuaries Examinations, April 2006, adapted) [7]
8. The following n-year spot rates were observed at time t = 0:
1 year spot rate of interest: 4% per annum
2 year spot rate of interest: 5% per annum
3 year spot rate of interest: 6% per annum
4 year spot rate of interest: 7% per annum
5 year spot rate of interest: 71/2% per annum
6 year spot rate of interest: 8% per annum
(i) Define what is meant by an n-year spot rate of interest.
(ii) Calculate the two-year forward rate of interest at time t = 3.
(iii) Using the above n-year spot rates calculate the 6-year par yield at time t = 0.
(Institute/Faculty of Actuaries Examinations, April 1998) [2+2+4=8]
9. The force of interest (t) is a function of time and at any time, measured in years, is given by the formula
(t) = 0.04 + 0.001t
(i) (a) Calculate the accumulated value of a unit sum of money accumulated from time t = 0 to time t = 8.
(b) Calculate the accumulated value of a unit sum of money accumulated from time t = 0 to time t = 9.
(c) Calculate the accumulated value of a unit sum of money accumulated from time t = 8 to time t = 9.
(ii) Using your results from (i), or otherwise, calculate:
(a) The 8-year spot rate of interest from time t = 0 to time t = 8.
(b) The 9-year spot rate of interest from time t = 0 to time t = 9.
(c) f8,1, where f8,1 is the one-year forward rate of interest from time t = 8.
(Institute/Faculty of Actuaries Examinations, September 2003) [5+3=8]
10. Three bonds paying annual coupons in arrears of 7% and redeemable at 105 per £100 nominal reach their redemption
dates in exactly one, two and three years’ time, respectively. The price of each of the bonds is £98 per £100 nominal.
(i) Determine the gross redemption yield of the 3-year bond.
(ii) Calculate all possible spot rates implied by the information given.
(Institute/Faculty of Actuaries Examinations, September 2002) [3+5=8]
11. An individual is investing in a market in which a variety of spot rates and forward contracts are available.
If at time t = 0 he invests £1,000 for 2 years, he will receive £1,118 at time t = 2. Alternatively, if at time t = 0 he
agrees to invest £1,000 at time t = 1 for 2 years, he will receive £1,140 at time t = 3. However, if at time t = 0 he
agrees to invest £1,000 at time t = 1 for one year, he will receive £1,058 at time t = 2.
(i) Calculate the following rates per annum effective, implied by this data:
(a) The one year spot rate at time t = 0.
(b) The two year spot rate at time t = 0.
(c) The three year spot rate at time t = 0.
(ii) Calculate the three year par yield at time t = 0 in this market.
(Institute/Faculty of Actuaries Examinations, April 2003) [5+3=8]
6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 2 Page 109
12. The n-year spot rate of interest, yn is given by:
yn = 0.04 +
n
1000
for n = 1, 2 and 3.
(i) Calculate the implied one-year forward rates applicable at times t = 1 and t = 2.
(ii) Assuming that coupon and capital payments may be discounted using the same discount factors, and that no
arbitrage applies, calculate:
(a) The price at time t = 0 per £100 nominal of a bond which pays annual coupons of 3% in arrears and is
redeemed at 110% after 3 years.
(b) The 2-year par yield. (Institute/Faculty of Actuaries Examinations, April 2001) [9]
13. The forward rate from time t 1 to time t has the following values:
f0,1 = 4.0%, f1,2 = 4.5%, f2,3 = 4.8%
(i) Assuming no arbitrage, calculate:
(a) the price per £100 nominal of a 3-year bond paying an annual coupon in arrears of 5%, redeemed at par in
exactly 3 years, and
(b) the gross redemption yield from the bond.
(ii) Explain why the bond with a higher coupon would have a lower gross redemption yield, for the same term to
redemption. (Institute/Faculty of Actuaries Examinations, April, 2002) [7+2=9]
14. Bonds paying annual coupons of 6% annually in arrears and redeemable at par will be redeemed in exactly one year,
two years and three years respectively. The price of each of the bonds is £96 per £100 nominal.
(i) Determine the gross redemption yield of the 3 year bond.
(ii) Determine the discount factors ⌫(1), ⌫(2) and ⌫(3) that the market is using to discount payments due in 1, 2 and
3 years respectively.
(iii) Calculate f0,1, f1,2 and f2,3 where ft1,t is the forward interest rate from time t 1 to t.
(Institute/Faculty of Actuaries Examinations, September 1999) [3+3+4=10]
15. For a particular bond market, zero coupon bonds redeemable at par are priced as follows:
bonds redeemable in exactly one year are priced at 97;
bonds redeemable in exactly two years are priced at 93;
bonds redeemable in exactly three years are priced at 88;
and bonds redeemable in exactly four years are priced at 83.
(i) Assuming no arbitrage, calculate:
(a) the one-year, two-year, three-year and four-year spot interest rates
(b) the rate of return from a bond redeemable at par in 4 years’ time that pays a coupon of 4% annually in
arrears.
(ii) Explain why the four-year spot rate is greater than the rate of return from a bond redeemable at par in exactly
4 years’ time paying a coupon of 4% annually in arrears.
(iii) Explain the shape of the yield curve indicated by the spot rates calculated in (i)(a) using the liquidity preference
theory, if expectations of future short term interest rates are constant.
(Institute/Faculty of Actuaries Examinations, September 2004) [8+2+2=12]
16. Suppose ft,r is the forward rate applicable over the period t to t + r and it is the spot rate over the period 0 to t.
The gross redemption yield from a one year bond with a 6% annual coupon is 6% per annum effective; the gross
redemption yield from a 2 year bond with a 6% annual coupon is 6.3% per annum effective; and the gross redemption
yield from a 3 year bond with a 6% annual coupon is 6.6% per annum effective. All the bonds are redeemed at par
and are exactly one year from the next coupon payment.
(i) (a) Calculate i1, i2 and i3 assuming no arbitrage.
(b) Calculate f0,1, f1,1 and f2,1 assuming no arbitrage.
(ii) Explain why the forward rates increase more rapidly with term than the spot rates.
(Institute/Faculty of Actuaries Examinations, September 2000) [12]
17. (i) (a) Explain what is meant by the “expectations theory” explanation for the shape of the yield curve.
(b) Explain how expectations theory can be modified by both “liquidity preference” and “market segmenta-
tion” theories.
(ii) Short-term, one-year annual effective interest rates are currently 10%; they are expected to be 9% in one year’s
time, 8% in two year’s time, 7% in three years’ time and to remain at that level thereafter indefinitely.
(a) If bond yields over all terms to maturity are assumed only to reflect expectations of future short-term
interest rates, calculate the gross redemption yields from 1-year, 3-year, 5-year and 10-year zero coupon
bonds.
(b) Draw a rough plot of the yield curve for zero coupon bonds using the data from part (ii)(a). (Graph paper
is not required.)
(c) Explain why the gross redemption yield curve for coupon paying bonds will slope down with a less steep
gradient than the zero coupon yield curve.
(Institute/Faculty of Actuaries Examinations, September 2003) [6+8=14]
Page 110 Exercises 2 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
18. In a particular bond market, n-year spot rates per annum can be approximated by the function 0.08 0.04e0.1n.
Calculate:
(i) The price per unit nominal of a zero coupon bond with term 9 years.
(ii) The 4-year forward rate at time 7 years.
(iii) The 3-year par yield. (Institute/Faculty of Actuaries Examinations, April 2007) [2+3+3=8]
19. The annual effective forward rate applicable over the period t to t + r is defined as ft,r where t and r are measured
in years. We have f0,1 = 4%, f1,1 = 4.25%, f2,1 = 4.5% and f2,2 = 5%. Calculate the following:
(i) f3,1
(ii) All possible zero coupon (spot) yields that the above information allows you to calculate.
(iii) The gross redemption yield of a 4-year bond redeemable at par with a 3% coupon payable annually in arrears.
(iv) Explain why the gross redemption yield from the 4-year bond is lower than the 1-year forward rate up to
time 4, f3,1. (Institute/Faculty of Actuaries Examinations, September 2007) [1+4+6+2=13]
20. The n-year spot rate of interest, in, is given by in = a bn for n = 1, 2 and 3, and where a and b are constants.
The one-year forward rates applicable at time 0 and at time 1 are 6.1% per annum effective and 6.5% per annum
effective respectively. The 4-year par yield is 7% per annum.
Stating any assumptions:
(i) calculate the values of a and b;
(ii) calculate the price per £1 nominal at time 0 of a bond which pays annual coupons of 5% in arrear and is
redeemed at 103% after 4 years. (Institute/Faculty of Actuaries Examinations, April 2008) [4+5=9]
21. Three bonds, paying annual coupons in arrears of 6% are redeemable at £105 per £100 nominal and reach their
redemption dates in exactly one, two and three years’ time respectively. The price of each bond is £103 per £100
nominal.
(i) Calculate the gross redemption yield of the three-year bond.
(ii) Calculate, to 3 decimal places, all possible spot rates implied by the information given, as annual effective rates
of interest.
(iii) Calculate, to 3 decimal places, all possible forward rates implied by the information given, as annual effective
rates of interest. (Institute/Faculty of Actuaries Examinations, September 2008) [3+4+4=11]
22. In a particular bond market, n-year spot rates can be approximated by the function 0.06 0.02e0.1n.
(i) Calculate the gross redemption yield for a 3-year bond which pays coupons of 3% annually in arrear, and is
redeemed at par. Show all workings.
(ii) Calculate the 4-year par yield. (Institute/Faculty of Actuaries Examinations, April 2012) [6+3=9]
23. (i) State the characteristics of a Eurobond.
(ii) (a) State the characteristics of a certificate of deposit.
(b) Two certificates of deposit issued by a given bank are being traded. A one-month certificate of deposit pro-
vides a rate of return of 12% per annum convertible monthly. A two-month certificate of deposit provides
a rate of return of 24% per annum convertible monthly.
Calculate the forward rate of interest per annum convertible monthly in the second month, assuming no
arbitrage. (Institute/Faculty of Actuaries Examinations, September 2012) [4+4=8]
24. Three bonds each paying annual coupons in arrear of 6% and redeemable at £103 per £100 nominal reach their
redemption dates in exactly one, two and three years’ time respectively.
The price of each bond is £97 per £100 nominal.
(i) Calculate the gross redemption yield of the 3-year bond.
(ii) Calculate the one-year and two-year spot rates implied by the information given.
(Institute/Faculty of Actuaries Examinations, April 2013) [3+3=6]
25. The force of interest, (t), is a function of time and at any time t, measured in years, is given by the formula
(t) = 0.05 + 0.002t.
(i) Calculate the accumulated value of a unit sum of money:
(a) accumulated from time t = 0 to time t = 7;
(b) accumulated from time t = 0 to time t = 6;
(c) accumulated from time t = 6 to time t = 7.
(ii) Calculate, using your results from part (i) or otherwise:
(a) the seven spot rate of interest per annum from time t = 0 to time t = 7;
(b) the six spot rate of interest per annum from time t = 0 to time t = 6;
(c) f6,1, where f6,1 is the one year forward rate of interest per annum from time t = 6.
(iii) Explain why your answer to part (ii)(c) is higher than your answer to part (ii)(a).
(iv) Calculate the present value of an annuity that is paid continuously at a rate of 30e0.01t+0.001t
2
units per annum
from t = 3 to t = 10. (Institute/Faculty of Actuaries Examinations, September 2013) [5+3+2+5=15]
6 Interest Rate Problems Sep 27, 2016(9:51) Section 3 Page 111
26. In a particular country, insurance companies are required by regulation to value their liabilities using spot rates of
interest derived form the government yield curve.
Over time t (measured in years), the spot rate of interest is equal to:
i = 0.02t for t  5.
An insurance company in this country has a group of annuity policies which involve making payments of £1m per an-
num for 4 years and £2m per annum in the fifth year. All payments are assumed to be paid halfway through the year.
(i) Calculate the value of the company’s liabilities.
(ii) Outline two reasons why the spot yield might rise with term to redemption.
(iii) Calculate the forward rate of interest from time t = 3.5 to time t = 4.5.
(Institute/Faculty of Actuaries Examinations, April 2015) [3+3+2=8]
27. Three bonds, each paying annual coupons in arrear of 3% and redeemable at £100 per £100 nominal, reach their
redemption dates in exactly one, two and three years time, respectively. The price of each bond is £101 per £100
nominal.
(i) Determine the gross redemption yield of the three-year bond.
(ii) Calculate the one-year, two-year and three-year spot rates of interest implied by the information given.
(iii) Calculate the one-year forward rate starting from the end of the second year, f2,1.
The pattern of spot rates is upward sloping throughout the yield curve.
(iv) Explain, with reference to the various theories of the yield curve, why the yield curve might be upward sloping.
(Institute/Faculty of Actuaries Examinations, September 2015) [3+5+2+4=14]
3 Vulnerability to interest rate movements
3.1 Background. Suppose an institution holds assets with net present value VA and has liabilities with net
present value VL. Assume that VA VL at time 0.
If the rate of interest falls, then both VA and VL will increase. If the rate of interest rises, then both VA and
VL will fall. Even for bonds with fixed coupons and fixed redemption values, a bondholder will make a gain
or loss when the yield changes because the current value of the bond changes. Clearly the institution wants to
ensure that VA VL however the interest rate behaves.
The concepts of duration and convexity give measures of the sensitivity of a cash flow to a movement in interest
rates. The concept of immunisation is a strategy for immunising a portfolio to changes in interest rates.
3.2 Duration or Macaulay duration or discounted mean term. Consider the following cash flow:
Time 0 t1 t2 . . . tn
Cash flow P ct1 ct2 . . . ctn
We assume that the cash flows ctk do not depend on i. Suppose the force of interest is and so the annual
effective rate of interest is i = e 1.
Clearly
P =
nX
k=1
ctke
tk =
nX
k=1
ctk
(1 + i)tk
The duration orMacaulay duration or discounted mean term is defined by
dM (i) = 1
P
dP
d
=
1
P
nX
k=1
tkctk
(1 + i)tk (3.2a)
Hence dM (i) is a weighted mean of the values tk where the weights are ctk/(1 + i)
tk . Hence the Macaulay
duration is the mean term of the cash flows weighted by their present values.
Now 1 + i = e. Hence
di
d
= 1 + i and
dP
di
=
dP
d
d
di
= ⌫
dP
d
and so dM (i) = (1 + i) 1
P
dP
di
(3.2b)
Clearly dM (i) is a measure of the sensitivity of P to the interest rate i.
Example 3.2a. Consider an n-year bond with face value f , redemption valueC and coupon rate r payable annually.
(a) Find the Macaulay duration, dM (i).
Page 112 Section 3 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
(b) Consider the special case of a bond which has a redemption value equal to its face value and is selling at par.
(c) Find the Macaulay duration of an n-year zero coupon bond with redemption value of £100.
Solution. (a) The cash flow is as follows:
Time 0 1 2 3 . . . n 1 n
Cash flow P fr fr fr . . . fr C + fr
Let ⌫ = 1/(1 + i) and ↵ = fr/C. Then
P =
nX
k=1
fr
(1 + i)k
+
C
(1 + i)n
= C↵an + C⌫n (3.2c)
dM (i) =
1
P
"
nX
k=1
kfr
(1 + i)k
+
Cn
(1 + i)n
#
=
↵(Ia)n + n⌫n
↵an + ⌫n
=
↵⌫(1 ⌫n n(1 ⌫)⌫n) + (1 ⌫)2n⌫n
↵⌫(1 ⌫)(1 ⌫n) + (1 ⌫)2⌫n
=
1
1 ⌫ ⌫
n n↵⌫ + 1 n(1 ⌫)
↵⌫(1 ⌫n) + (1 ⌫)⌫n (3.2d)
= 1 +
1
i
+
n(i ↵) (1 + i)
↵[(1 + i)n 1] + i (3.2e)
after some algebra. Alternatively, use P = fran,i + C⌫n. Hence
dP
d⌫
= fr(1 + 2⌫ + · · · + n⌫n1) + nC⌫n1 = fr(Ia)n + nC⌫
n
⌫
and hence
dP
di
= dP
d⌫
d⌫
di
= ⌫2
dP
d⌫
= ⌫
⇥
fr(Ia)n + nC⌫n
⇤
and so on.
(b) In this case, C = P = f ; hence ↵ = r. Substituting these equalities in equation (3.2c) gives ↵ = i. Hence
dM (i) =
1
P
"
nX
k=1
kfr⌫k + Cn⌫n
#
= i
nX
k=1
k⌫k + n⌫n =
1 ⌫n
1 ⌫
Alternatively, just substitute ↵ = i = (1 ⌫)/⌫ in equation (3.2d).
(c) The only payment is £100 at time n. Hence
dM (i) =
100n⌫n
100⌫n
= n
Example 3.2b. Consider an n-year bond with face value f , redemption value C and coupon rate r payable half-
yearly. Let i denote the effective annual yield and define i(2) in the usual way by (1 + i(2)/2)2 = 1 + i.
(a) Find the Macaulay duration, dM (i).
(b) Show that
dM (i) =
✓
1 +
i(2)
2
◆
1
P
dP
di(2)
Solution. Clearly,
P =
fr
2
2nX
k=1
1
(1 + i)k/2
+
C
(1 + i)n
Substituting into equation (3.2a) gives
dM (i) =
1
P
"
fr
2
2nX
k=1
k/2
(1 + i)k/2
+
Cn
(1 + i)n
#
For part (b), note that
di
di(2)
= 1 +
i(2)
2
Using this result in equation (3.2b) gives
dM (i) = (1 + i) 1
P
dP
di
=
✓
1 +
i(2)
2
◆2 1
P
dP
di
=
✓
1 +
i(2)
2
◆
1
P
dP
di(2)
6 Interest Rate Problems Sep 27, 2016(9:51) Section 3 Page 113
3.3 Effective duration or modified duration or volatility. Suppose i denotes the effective interest rate per
annum. Then:
P =
nX
k=1
ctk
(1 + i)tk
The effective duration, d(i) or volatility or modified duration of the cash flow is defined to be
d(i) = 1
P
dP
di
=
1
P
nX
k=1
tkctk
(1 + i)tk+1 (3.3a)
where the last equality assumes that the values of the cash flows ctj do not depend on i.
Thus the effective duration is also a measure of the rate of change of the net present value with i which does
not depend on the size of the net present value.
Note that
dM (i) = (1 + i)d(i)
Using the approximation f (x + h) ⇡ f (x) + hf 0(x) shows that if interest rates change from i to i + ✏ then
P (i + ✏) ⇡ 1 ✏d(i)P (i)
and hence
P (i + ✏) P (i)
P (i)
⇡ ✏d(i)
This leads to the following interpretation of the modified duration: if the yield i decreases by ✏ then the price
increases by the proportion 100✏d(i)%.
Example 3.3a. Suppose the modified duration of a bond is equal to 8. Suppose further that the bond’s yield
increases by 10 basis points. (A basis point is 0.01%.) Find the approximate change in the price of the bond.
Solution. The approximate change is ✏d(i) = 0.001⇥ 8 = 0.008. It decreases by 0.8% approximately.
3.4 Some results about duration.
• The Macaulay duration of a zero-coupon bond equals the time to maturity—by part (c) of example (3.2a).
• The duration increases as the yield decreases—see exercise 2 in section 4 below.
• The duration of a bond increases as the coupon rate decreases (other things being equal)—by equation (3.2e).
Duration is a measure of the riskiness of a bond; it is important for three reasons:
• it is a summary measure of the average payments which are due;
• it is a measure of the sensitivity of the price to interest rate changes;
• it is useful concept for immunising portfolios against interest rate changes.
Here are some standard examples:
Example 3.4a. Find the Macaulay duration and the effective duration of a constant perpetuity:
Time 0 1 2 3 . . .
Cash flow, c P (i) 1 1 1 . . .
Solution. Now P (i) = ⌫ + ⌫2 + ⌫3 + · · · = ⌫/(1 ⌫). Hence:
dM (i) =
1
P
(⌫ + 2⌫2 + 3⌫3 + · · ·) = 1
P
⌫
(1 ⌫)2 =
1
1 ⌫ =
1 + i
i
and d(i) =
1
i
Example 3.4b. Find the Macaulay duration of a constant annuity:
Time 0 1 2 · · · n 1 n
Cash Flow, c P (i) 1 1 · · · 1 1
Solution. P (i) = ⌫ + ⌫2 + · · · + ⌫n and
dM (i) =
⌫ + 2⌫2 + · · · + n⌫n
⌫ + ⌫2 + · · · + ⌫n =
1
1 ⌫
n⌫n
1 ⌫n =
1 + i
i
n
(1 + i)n 1
零息债券的期限等于到期⽇
Page 114 Section 3 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
3.5 Convexity. The convexity of the cash flow above is defined to be
c(i) =
1
P
d2P
di2
=
1
P
nX
k=1
tk(tk + 1)ctk
(1 + i)tk+2
The duration (first derivative) and convexity (second derivative) together give information about the local de-
pendence of P on i; they show how P changes when there is a small change in i. The second order Taylor
approximation is:
f (x + h) = f (x) + hf 0(x) +
h2
2
f 00(xu) where xu 2 (x, x + h)
Applying this to the net present value gives
P (i + ✏) ⇡ P (i) + ✏dP
di
+
✏2
2
d2P
di2
and hence
P (i + ✏) P (i)
P (i)
⇡ ✏d(i) + ✏
2
2
c(i)
Convexity measures how fast the slope of a bond’s price curve changes when the interest rate changes—
so higher positive convexity is good for the investor. This is illustrated in figure (3.5a): if the yield falls,
then the price of bond with higher convexity (CD) will rise more than the price of the bond with smaller
convexity (AB); conversely, if the yield rises, then the price of the bond with higher convexity (CD) will fall
less than the price of the bond with smaller convexity (AB). If two bonds have the the same price, the same
yield and the same same duration but different convexities, then the investor should prefer the bond with the
higher convexity—hence they should not have the same price!
i0 yield, i
price, P
P0
A
B
C
D
dP
di at i = i0
Figure 3.5a. Suppose the price of a bond is P0 when the yield is i0.
Then higher convexity (CD) is preferable to lower convexity (AB).
(wmf/convexity,0pt,0pt)
Finally, note that if the yield falls by the amount , then the rise in the price is larger than the fall in the price
if the yield increases by the amount.
3.6 Immunisation. Suppose a fund has assets with cash flow {atk} and net present value VA(i). Let dA(i)
denote the effective duration and cA(i) denote the convexity. Suppose further that the fund has liabilities with
cash flow {`tk} and net present value VL(i). Let dL(i) denote the effective duration and cL(i) denote the
convexity.
Suppose that at rate of interest i0 we have VA(i0) = VL(i0). The fund is immunised against small changes of
size ✏ in the interest rate if VA(i0 + ✏) VL(i0 + ✏).
Consider the difference S(i) = VA(i) VL(i). Taylor’s theorem gives
S(i0 + ✏) = S(i0) + ✏S0(i0) +
✏2
2
S00(i0) + · · ·
Using S(i0) = 0 shows that
凸性
凸性衡量了债券价格曲线在利率变化时的变化速度
收益率上⾼凸性债券的价格将
⽐低凸性债券的价格上涨更多
⼆收益率⼩向凸性债券价格下跌将
⼩于凸性更⼩的债券
⾼凸性⼼ 伏于低品性 AB
6 Interest Rate Problems Sep 27, 2016(9:51) Section 3 Page 115
S(i0 + ✏) = ✏S0(i0) +
✏2
2
S00(i0) + · · ·
Dividing by VA(i0) = VL(i0) gives
S(i0 + ✏)
VA(i0)
= ✏
S0(i0)
VA(i0)
+
✏2
2
S00(i0)
VA(i0)
+ · · ·
Now if we insist effective durations are equal at i0, we must have V 0A(i0) = V
0
L(i0) and so
S0(i0)
VA(i0)
= 0
and hence
S(i0 + ✏)
VA(i0)
=
✏2
2
S00(i0)
VA(i0)
+ · · · = ✏
2
2
[cA(i) cL(i)] > 0 provided that cA(i0) > cL(i0)
We have shown that if
(1) VA(i0) = VL(i0), the net present values are the same
(2) dA(i0) = dL(i0), the effective durations are the same
(3) cA(i0) > cL(i0), the convexity of the assets is greater than the convexity of the liabilities
then we have Redington immunisation at interest rate i0.
(Note that condition (3) implies that a decrease in interest rates will cause asset values to increase by more than
the increase in liability values; similarly an increase in interest rates will cause asset values to decrease by less
than a decrease in liability values.)
Redington immunisation is not usually a feasible method for organising assets. It will not usually be possible
to keep the effective durations equal; nor will it be possible to assess the future cash flows with any certainty.
3.7 Alternatively, differentiation with respect to the force of interest could be used. Recall equation (3.2b):
dP
d
= (1 + i)
dP
di
which implies dM (i) = (1 + i)d(i).
It also leads to
d2P
d2
=
di
d
dP
di
+ (1 + i)
d
d
dP
di
= (1 + i)
dP
di
+ (1 + i)
di
d
d2P
di2
and hence
1
P
d2P
d2
+ dM (i) = (1 + i)2c(i)
Hence the three conditions for Redington immunisation are equivalent to:
(1) VA(i0) = VL(i0), the net present values are the same
(2) dM (A)(i0) = dM (L)(i0), the Macaulay durations are the same
(3) cA(i0) > cL(i0), the convexity of the assets is greater than the convexity of the liabilities where now the
convexity is c = 1P
d2P
d2
.
Note that the three conditions above can be expressed in terms of the original cash flows as follows:
nX
k=1
atk
(1 + i)tk
=
nX
k=1
ltk
(1 + i)tk
nX
k=1
tkatk
(1 + i)tk
=
nX
k=1
tkltk
(1 + i)tk
nX
k=1
t2katk
(1 + i)tk
>
nX
k=1
t2kltk
(1 + i)tk
This is often the most useful version of the test conditions for numerical problems.
Page 116 Exercises 4 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
3.8
Summary.
• Macaulay duration or discounted mean term. This is the mean term of the cash flows weighted by
their present values.
dM (i) =
1
P
nX
k=1
tkctk
(1 + i)tk
= 1
P
dP
d
= (1 + i) 1
P
dP
di
• Effective duration or modified duration or volatility.
d(i) =
1
P
nX
k=1
tkctk
(1 + i)tk+1
= 1
P
dP
di
• Convexity.
c(i) =
1
P
d2P
di2
=
1
P
nX
k=1
tk(tk + 1)ctk
(1 + i)tk+2
• Immunisation. Redington immunisation occurs at interest rate i0 if
(1) VA(i0) = VL(i0), the net present values are the same
(2) dA(i0) = dL(i0), the effective durations are the same
(3) cA(i0) > cL(i0), the convexity of the assets is greater than the convexity of the liabilities.
Alternative conditions are:
nX
k=1
atk
(1 + i)tk
=
nX
k=1
ltk
(1 + i)tk
nX
k=1
tkatk
(1 + i)tk
=
nX
k=1
tkltk
(1 + i)tk
nX
k=1
t2katk
(1 + i)tk
>
nX
k=1
t2kltk
(1 + i)tk
4 Exercises (exs6-2.tex)
1. A particular bond has just been issued and pays coupons of 10% per annum paid half yearly in arrears and is redeemed
at par after 10 years. Find the duration of the bond in years, at a rate of interest of 5% per half year effective.
(Institute/Faculty of Actuaries Examinations, September 1999) [3]
2. By equation (3.2a), the Macaulay duration can be regarded as the expectation of a random variable, X , with the
following distribution:
X = tk with probability
ctk⌫
tkPn
k=1 ctk⌫
tk
for k = 1, 2, . . . , n.
Show that the derivative of dM (i) with respect to i is equal to ⌫2 where 2 = var[X]. Hence show that the
duration increases as the yield decreases2.
3. An insurance company has a continuous payment stream of liabilities to meet over the coming 20 years. The payment
stream will be at a rate of £10 million per annum throughout the period.
Calculate the duration of the continuous payment stream at a rate of interest of 4% per annum effective.
(Institute/Faculty of Actuaries Examinations, September 2004) [4]
4. (a) An investor holds a portfolio of fixed interest securities to meet future liabilities. State the conditions that need
to be met if the investor is to be immunised from small, uniform changes in the rate of interest.
(b) State the relationship between “volatility” and “duration” where “volatility” is defined as
1
A
dA
di
where A is the value of the portfolio of investments and i is the annual effective rate of interest.
(c) A perpetuity pays annual coupons in arrears. Show that the volatility of the perpetuity, as defined in (b) above,
is equal to 1/i where i is the rate of return. (Institute/Faculty of Actuaries Examinations, September 2004) [6]
5. A new management team has just taken over the running of a finance company. They discover that the company
has liabilities of £15 million due in 13 years’ time and £10 million in 25 years’ time. The assets consist of two zero
coupon bonds, one paying £12.425 million in 12 years’ time and the other paying £12.946 million in 24 years’ time.
The current interest rate is 8% per annum effective.
Determine whether the necessary conditions are satisfied for the finance company to be immunised against small
changes in the rate of interest. (Institute/Faculty of Actuaries Examinations, April 2003) [8]
2 See also page 81 of Bond Pricing and Portfolio Analysis by Olivier de La Grandville.
6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 4 Page 117
6. (i) State the features of a eurobond.
(ii) An investor purchases a eurobond on the date of issue at a price of £97 per £100 nominal. Coupons are paid
annually in arrear. The bond will be redeemed at par 20 years from the issue date. The rate of return from the
bond is 5% per annum effective.
(a) Calculate the annual rate of coupon paid by the bond.
(b) Calculate the duration of the bond. (Institute/Faculty of Actuaries Examinations, September 2005) [3+6=9]
7. An insurance company has liabilities of £10 million due in 10 years’ time and £20 million due in 15 years’ time,
and assets consisting of two zero-coupon bonds, one paying £7.404 million in 2 years’ time and the other paying
£31.834 million in 25 years’ time. The current interest rate is 7% per annum effective.
(i) Show that Redington’s first two conditions for immunisation against small changes in the rate of interest are
satisfied for this insurance company.
(ii) Determine the profit or loss, expressed as a present value, that the insurance company will make if the interest
rate increases immediately to 7.5% per annum effective.
(iii) Explain how you might have anticipated, before making the calculation in (ii), whether the result would be a
profit or loss. (Institute/Faculty of Actuaries Examinations, April 2004) [5+2+2=9]
8. An insurance company has liabilities of £87,500 due in 8 years’ time and £157,500 due in 19 years’ time. Its assets
consist of two zero coupon bonds, one paying £66,850 in 4 years’ time and the other paying £X in n years’ time.
The current interest rate is 7% per annum effective.
(i) Calculate the discounted mean term and convexity of the liabilities.
(ii) Determine whether values of £X and n can be found which ensure that the company is immunised against small
changes in the interest rate. (Institute/Faculty of Actuaries Examinations, April 2007) [5+5=10]
9. A small insurance fund has liabilities of £4 million due in 19 years’ time and £6 million in 21 years’ time. The
manager of the fund has sold the assets previously held and is creating a new portfolio by investing in the zero-
coupon bond market. The manager is able to buy zero-coupon bonds for whatever term he requires and has adequate
monies at his disposal.
(i) Explain whether it is possible for the manager to immunise the fund against small changes in the rate of interest
by purchasing a single zero-coupon bond.
(ii) In fact, the manager purchases two zero-coupon bonds, one paying £3.43 million in 15 years’ time and the other
paying £7.12 million in 25 years’ time. The current interest rate is 7% per annum effective.
Investigate whether the insurance fund satisfies the necessary conditions to be immunised against small changes
in the rate of interest. (Institute/Faculty of Actuaries Examinations, April 2005) [2+8=10]
10. A pension fund has liabilities of £3 million due in 3 years’ time, £5 million due in 5 years’ time, £9 million due
in 9 years’ time, and £11 million due in 11 years’ time. The fund holds two investments, X and Y . Investment X
provides income of £1 million payable at the end of each year for the next 5 years with no capital repayment.
Investment Y is a zero-coupon bond which pays a lump sum of £R at the end of n years (where n is not necessarily
an integer). The interest rate is 8% per annum effective.
(i) Investigate whether values of £R and n can be found which ensure that the fund is immunised against small
changes in the interest rate.
You are given that
P5
t=1 t
2⌫t = 40.275 at 8%.
(ii) (a) The interest rate immediately changes to 3% per annum effective. Calculate the revised present value of
the assets and liabilities of the fund.
(b) Explain your answer to (ii)(a). (Institute/Faculty of Actuaries Examinations, April 2006) [8+4=12]
11. An insurance company has a portfolio of annuity contracts under which it expects to pay £1 million at the end of
each of the next 20 years., followed by £0.5 million at the end of each of the following 20 years. The government
bond with the longest duration in which it can invest its funds pays a coupon of 10% per annum in arrears and is
redeemed at par in 15 years’ time. The yield to maturity of the government bond is 6% per annum effective and a
coupon payment has just been made.
(i) (a) Calculate the duration of the insurance company’s liabilities at a rate of interest of 6% per annum.
(b) Calculate the duration of the insurance company’s assets at a rate of interest of 6% per annum effective, if
all the insurance company’s funds are invested in the government bond with the longest duration.
(ii) (a) Explain why the insurance company cannot immunise its liabilities by purchasing government bonds.
(b) Without any further calculations, state the circumstances under which the insurance company would make
a loss if there were a uniform change in interest rates. Explain why a loss would be made.
(Institute/Faculty of Actuaries Examinations, September 2003) [8+4=12]
Page 118 Exercises 4 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
12. (i) An investment provides income of £1 million payable at the end of each year for the next 10 years. There is
no capital repayment. If the interest rate is 7% per annum effective, show that the “discounted mean term” (or
“Macaulay duration”) of the investment is 4.9646 years.
(ii) An investment company has liabilities of £7 million due in 5 years’ time and £8 million due in 8 years’ time. The
company holds two investments,A andB. InvestmentA is the investment described in part (i) and InvestmentB
is a zero coupon bond which pays £X at the end of n years (where n is not necessarily an integer).
The interest rate is 7% per annum effective. Investigate whether values of £X and n can be found which ensure
that the investment company is immunised against small changes in the interest rate.
You are given that
P10
t=1 t
2⌫t = 228.451 at 7%. (Institute/Faculty of Actuaries Examinations, April 2001) [12]
13. (i) Prove that
(Ia)n =
a¨n n⌫n
i
A government bond pays a coupon half-yearly in arrears of £10 per annum. It is to be redeemed at par in exactly
10 years. The gross redemption yield from the bond is 6% per annum convertible half-yearly.
(ii) Calculate the duration of the bond in years.
(iii) Explain why the duration of the bond would be longer if the coupon rate were £8 per annum instead of £10 per
annum. (Institute/Faculty of Actuaries Examinations, April, 2002) [3+8+2=13]
14. A fixed interest security was issued on 1 January in a given year. The security pays half-yearly coupons of 4% per
annum. The security is redeemable at 110% 20 years after issue. An investor who pays both income tax and capital
gains tax at a rate of 25% buys the security on the date of issue. Income tax is paid on coupons at the end of the
calendar year in which the coupon is received. Capital gains tax is paid immediately on sale or redemption.
(i) Calculate the price paid by the investor to give a net rate of return of 6% per annum effective.
(ii) Calculate the duration of the net payments from the fixed interest security for an investor who pays income tax
as described above but who does not pay capital gains tax, at a rate of interest of 6% per annum effective.
(Institute/Faculty of Actuaries Examinations, September 2004) [6+8=14]
15. A pension fund has the following liabilities: annuity payments of £160,000 per annum to be paid annually in arrears
for the next 15 years and a lump sum of £200,000 to be paid in 10 years. It wishes to invest in two fixed interest
securities in order to immunise its liabilities. SecurityA has a coupon rate of 8% per annum and a term to redemption
of 8 years. Security B has a coupon rate of 3% per annum and a term to redemption of 25 years. Both securities are
redeemable at par and pay coupons annually in arrear.
(i) Calculate the present value of the liabilities at a rate of interest of 7% per annum effective.
(ii) Calculate the discounted mean term of the liabilities at a rate of interest of 7% per annum effective.
(iii) Calculate the nominal amount of each security that should be purchased so that both the present value and
discounted mean term of assets and liabilities are equal.
(iv) Without further calculation, comment on whether, if the conditions in (iii) are fulfilled, the pension fund is likely
to be immunised against small, uniform changes in the interest rate.
(Institute/Faculty of Actuaries Examinations, September 2006) [2+4+7+2=15]
16. A company incurs a liability to pay £1,000(1 + 0.4t) at the end of year t, for t equal to 5, 10, 15, 20 and 25. It
values these liabilities assuming that in the future there will be a constant effective interest rate of 7% per annum.
An amount equal to the total present value of the liabilities is immediately invested in 2 stocks:
Stock A pays coupons of 5% per annum annually in arrears and is redeemable in 26 years at par.
Stock B pays coupons of 4% per annum annually in arrears and is redeemable in 32 years at par.
The gross redemption yield on both stocks is the same as that used to value the liabilities.
(i) Calculate the present value of the liabilities.
(ii) Calculate the discounted mean term of the liabilities.
(iii) If the discounted mean term of the assets is the same as the discounted mean term of the liabilities, calculate the
nominal amount of each stock which should be purchased.
(Institute/Faculty of Actuaries Examinations, September 2002) [3+3+9=15]
17. An analyst is valuing two companies, Cyber plc and Boring plc. Cyber plc is assumed to pay its first annual dividend
in exactly 6 years. It is assumed that the dividend will be 6p per share. After the sixth year, annual dividends are
assumed to grow at 10% per annum compound for a further 6 years. Dividends are assumed to grow at 3% per annum
compound in perpetuity thereafter. Boring plc is expected to pay an annual dividend of 4p per share in exactly one
year. Annual dividends are then expected to grow thereafter by 0.5% per annum compound in perpetuity. The analyst
values dividends from both shares at a rate of interest of 6% per annum effective in a discounted dividend model.
(i) (a) Calculate the value of Cyber plc.
(b) Calculate the value of Boring plc.
6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 4 Page 119
There is a general rise in interest rates and the analyst decides it is appropriate to increase the valuation rate of interest
to 7% per annum effective.
(ii) Show that the percentage fall in the value of Cyber plc is greater than that in the value of Boring plc.
(iii) Explain your answer to part (ii) in terms of duration.
(Institute/Faculty of Actuaries Examinations, April, 2002) [8+6+2=16]
18. Let {Ctk} denote a series of cash flows at times tk for k = 1, 2, . . . , n.
(i) (a) Define the volatility of the cash flow series, and derive a formula expressing the volatility in terms of tk,
Ctk and ⌫.
(b) Define the convexity of the cash flow series and derive a formula expressing the convexity in terms of tk,
Ctk and ⌫.
(ii) A loan stock issued on 1March 1997 has coupons payable annually in arrear at 8% p.a. Capital is to be redeemed
at par 10 years from the date of issue.
(a) Show that the volatility of this stock at 1 March 1997 at an effective rate of interest of 8% p.a. is 6.71.
(b) At 1 March 1997 an investor has a liability of £100,000 to be paid in 7.247 years. Calculate the volatility
and convexity of this liability at 1 March 1997 at an effective rate of interest of 8% p.a.
(c) On 1 March 1997 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% p.a.
Given that the convexity of the loan stock at 1 March 1997 is 60.53, state with reasons whether the investor
will be immunised against small movements in interest rates on that date.
(iii) Calculate the present value of investor’s profit or loss at 1 March if, immediately after purchasing the loan stock
the rate of interest changes to 81/2% p.a. effective.
(Institute/Faculty of Actuaries Examinations, April 1997) [4+10+3=17]
19. An investor has to pay a lump sum of £20,000 at the end of 15 years from now and an annuity certain of £5,000 per
annum payable half-yearly in advance for 25 years, starting in 10 years’ time.
The investor currently holds an amount of cash equal to the present value of these two liabilities valued at an effective
rate of interest of 7% per annum.
The investor wishes to immunise her fund against small movements in the rate of interest by investing the cash in
two zero coupon bonds, bond X and bond Y . The market prices of both bonds are calculated at an effective rate of
interest of 7% per annum. The investor has decided to invest an amount in bondX sufficient to provide a capital sum
of £25,000 when bond X is redeemed in 10 years’ time. The remainder of the cash is invested in bond Y .
In order to immunise her holdings:
(i) Calculate the amount of money invested in bond Y .
(ii) Determine the term needed for bond Y and the redemption amount payable at the maturity date.
(iii) Without doing any further calculations, state which other condition needs to be satisfied for immunisation to be
achieved successfully. (Institute/Faculty of Actuaries Examinations, April 1998) [5+10+2=17]
20. (i) Let {Ctk} denote a series of cash flows at times tk for k = 1, 2, . . . , n and P (i) denote the present value of
these payments at an effective interest rate i so that P (i) =
Pn
k=1 Ctk/(1 + i)
tk .
(a) Define the discounted mean term (orMacaulay duration) of the cash flows in terms of of tk, Ctk and ⌫.
(b) Define the volatility (or effective duration) of the cash flows in terms of P (i) and show that for this series
of cash flows the discounted mean term = volatility⇥ (1 + i).
(ii) A fund has to provide an annuity of £60,000 per annum payable yearly in arrears for the next 9 years followed
by a final payment of £750,000 in 10 years’ time.
The fund has earmarked cash assets exactly equal to the present value of the payments and the fund manager
wants to invest these in two zero coupon bonds: Bond A which is repayable at the end of 5 years and Bond B
which is repayable at the end of 20 years.
The fund manager wants both the assets and the liabilities to have the same volatility. To achieve this, how
much should the manager invest in each of the bonds, given that an effective rate of interest of 7% per annum is
used to value both assets and liabilities?
(iii) State without doing any further calculations, what further condition would be required to ensure that the fund is
immunised against possible losses due to small changes in the effective rate of interest.
(Institute/Faculty of Actuaries Examinations, April 1999) [1+3+12+2=18]
21. A pension fund expects to make payments of £100,000 per annum at the end of each of the next 5 years. It wishes
to immunise these liabilities by investing in 2 zero coupon bonds which mature in 5 years and in 1 year respectively.
The rate of interest is 5% per annum effective.
(i) (a) Show that the present value of the liabilities is £432,948.
(b) Show that the duration of the liabilities is 2.9 years.
(ii) Calculate the nominal amounts of the 2 zero coupon bonds which must be purchased if the pension fund is to
equate the present value and duration of assets and liabilities.
Page 120 Exercises 4 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
(iii) (a) Calculate the convexity of the assets.
(b) Without calculating the convexity of the liabilities, comment on whether you think Redington’s immunisa-
tion has been achieved. (Institute/Faculty of Actuaries Examinations, September 2000) [18]
22. A variable annuity, payable annually in arrears for n years is such that the payment at the end of year t is t2. Show
that the present value of the annuity can be expressed as
2(Ia)n an n2⌫n+1
1 ⌫
A pension fund has a liability to pay £100,000 at the end of one year, £105,000 at the end of 2 years, and so on, the
amount increasing by £5,000 each year to £195,000 at the end of 20 years, this being the last payment. The fund
values these payments using an effective interest rate of 7% per annum. This is also the interest rate at which the
current prices of all bonds are calculated.
The fund invests an amount equal to the present value of these liabilities in the following two assets:
(A) a zero coupon bond redeemable in 25 years, and
(B) a fixed interest bond redeemable at par in 12 years’ time which pays a coupon of 8% per annum annually in
arrears.
(a) Calculate the present value and the duration of the liabilities.
(b) Calculate the amount of cash that should be invested in each asset if the duration of the assets is to equal that of
the liabilities. (Institute/Faculty of Actuaries Examinations, April 2000) [20]
23. An investor purchases a fixed interest security. The security pays coupons at a rate of 10% p.a. half-yearly in arrear,
and is to be redeemed at 110 in 20 years. The investor is subject to tax on the coupon payments at a rate of 25%.
(i) (a) Show that the price paid by the investor to obtain a rate of return of 10% p.a. effective is £81.76%.
(b) Calculate the effective duration (or volatility) of the security at 10% p.a. effective for this investor at the
purchase date.
(ii) The investor has two liabilities. The present value of the liabilities, at an effective rate of interest of 10% p.a. is
equal to the present value of the investor’s holding in the fixed interest security described above. The amount of
each of the two liabilities is the same. The second liability is due in 10 years.
(a) Show that the first liability must be due in 9.61 years in order that the effective duration (or volatility) of
the liabilities is equal to the effective duration of the assets.
(b) Calculate the convexity of the total of the two liabilities described above.
(c) Without any further calculations, state with reasons whether the fund comprising the asset and liabilities
described is immunised against small movements in the rate of interest.
(iii) One year after purchasing the fixed interest security, the price at which the security can be sold is still at a level
that will yield a rate of return of 10% p.a. effective. At this time, a second investor agrees a forward contract to
buy the security four years later, immediately after the coupon payment then due.
Calculate the forward price based on a “risk-free” rate of return of 10% p.a. effective and no arbitrage.
(Institute/Faculty of Actuaries Examinations, May 1999, Specimen Examination) [9+12+4=25]
24. A pension fund has liabilities to pay pensions each year for the next 60 years. The pensions paid will be £100m at the
end of the first year, £105m at the end of the second year, £110.25m at the end of the third year and so on, increasing
by 5% each year. The fund holds government bonds to meet its pension liabilities. The bonds mature in 20 years’
time and pay an annual coupon of 4% in arrears.
(i) Calculate the present value of the pension fund’s liabilities at a rate of interest of 3% per annum effective.
(ii) Calculate the nominal amount of the bond that the fund needs to hold so that the present value of the assets is
equal to the present value of the liabilities.
(iii) Calculate the duration of the liabilities.
(iv) Calculate the duration of the assets.
(v) Using your calculations in (iii) and (iv), estimate by how much more the value of the liabilities would increase
than the value of the assets if there were a reduction in the rate of interest to 1.5% per annum effective.
(Institute/Faculty of Actuaries Examinations, September 2007) [4+3+6+4+4=21]
25. (i) An investor is considering the purchase of an annuity, payable annually in arrear for 20 years. The first payment
is £500. Using a rate of interest of 8% per annum effective, calculate the duration of the annuity when:
(a) the payments remain level over the term;
(b) the payments increase at a rate of 8% per annum compound.
(ii) Explain why the answer in (i)(b) is higher than the answer in (i)(a).
(Institute/Faculty of Actuaries Examinations, April 2008) [6+2=8]
26. An insurance company is considering two possible investment options.
The first investment option involves setting up a branch in a foreign country. This will involve an immediate outlay
of £0.25 million, followed by investments of £0.1 million at the end of one year, £0.2 million at the end of two years,
£0.3 million at the end of three years and so on until a final investment is made of £1 million in 10 years’ time. The
6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 4 Page 121
investment will provide annual payments of £0.5 million for 20 years with the first payment at the end of the 8th year.
There will be an additional incoming cash flow of £5 million at the end of the 27th year.
The second investment option involves the purchase of 1 million shares in a bank at a price of £4.20 per share. The
shares are expected to provide a dividend of 21p per share in exactly one year, 22.05p per share in two years and so
on, increasing by 5% per annum compound. The shares are expected to be sold at the end of 10 years, just after a
dividend has been paid, for £5.64 per share.
(i) Determine which of the options has the higher net present value at a rate of interest of 7% per annum effective.
(ii) Without doing any further calculations, determine which option has the higher discounted mean term at a rate
of interest of 7% per annum effective. (Institute/Faculty of Actuaries Examinations, September 2008) [9+2=11]
27. A company has a liability of £400,000 due in 10 years’ time.
The company has exactly enough funds to cover the liability on the basis of an effective interest rate of 8% per annum.
This is also the interest rate on which current market prices are calculated and the interest rate earned on cash.
The company wishes to hold 10% of its funds in cash, and to invest the balance in the following securities:
• a zero-coupon bond redeemable at par in 12 years’ time
• a fixed-interest stock which is redeemable at 110% in 16 years’ time bearing interest at 8% per annum payable
annually in arrear.
(i) Calculate the nominal amounts of the zero-coupon bond and the fixed-interest stock which should be purchased
to satisfy Redington’s first two conditions for immunisation.
(ii) Calculate the amount which should be invested in each of the assets mentioned in (i).
(iii) Explain whether the company would be immunised against small changes in the rate of interest if the quantities
of stock in part (i) are purchased. (Institute/Faculty of Actuaries Examinations, September 2008) [10+2+2=14]
28. A company has the following liabilities:
• annuity payments of £200,000 per annum to be paid annually in arrear for the next 20 years;
• a lump sum of £300,000 to be paid in 15 years.
The company wishes to invest in two fixed-interest securities in order to immunise its liabilities.
Security A has a coupon rate of 9% per annum and a term to redemption of 12 years.
Security B has a coupon rate of 4% per annum and a term to redemption of 30 years.
Both securities are redeemable at par and pay coupons annually in arrear. The rate of interest is 8% per annum
effective.
(i) Calculate the present value of the liabilities.
(ii) Calculate the discounted mean term of the liabilities.
(iii) Calculate the nominal amount of each security that should be purchased so that Redington’s first two conditions
for immunisation against small changes in the rate of interest are satisfied for this company.
(iv) Describe the further calculations that will be necessary to determine whether the company is immunised against
small changes in the rate of interest. (Institute/Faculty of Actuaries Examinations, April 2012) [3+4+8+2=17]
29. (i) Describe three theories that have been put forward to explain the shape of the yield curve.
The government of a particular country has just issued five bonds with terms to redemption of one, two, three, four
and five years respectively. The bonds are redeemed at par and have coupon rates of 4% per annum payable annually
in arrear.
(ii) Calculate the duration of the one year, three year and five year bonds at a gross redemption yield of 5% per an-
num effective.
(iii) Explain why a five year bond with a coupon rate of 8% per annum would have a lower duration than a five year
bond with a coupon rate of 4% per annum.
Four years after issue, immediately after the coupon payment then due the government is anticipating problems
servicing its remaining debt. The government offers two options to the holders of the bond with an original term of
five years:
Option 1: the bond is repaid at 79% of its nominal value at the scheduled time with no final coupon payment being
paid.
Option 2: the redemption of the bond is deferred for 7 years from the original redemption date and the coupon rate
is reduced to 1% per annum for the remainder of the existing term and the whole of the extended term.
Assume the bonds were issued at a price of £95 per £100 nominal.
(iv) Calculate the effective rate of return per annum from Option 1 and 2 over the total life of the bond and determine
which would provide the higher rate of return.
(v) Suggest two other considerations that bond holders may wish to take into account when deciding which options
to accept. (Institute/Faculty of Actuaries Examinations, September 2012) [7+6+2+6+2=23]
30. Two investment projects are being considered.
(i) Explain why comparing the two discounted payback periods or comparing the two payback periods are not
generally appropriate ways to choose between two investment projects.
Page 122 Exercises 4 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
The two projects each involve an initial investment of £3 million. The incoming cash flows from the two projects are
as follows:
Project A. In the first year, Project A generates cash flows of £0.5 million. In the second year, it will generate cash
flows of £0.55 million. The cash flows generated by the project will continue to increase by 10% per annum until the
end of the sixth year and will then cease. Assume that all cash flows are received in the middle of the year.
Project B. Project B generates cash flows of £0.64 million per annum for 6 years. Assume that all cash flows are
received continuously throughout the year.
(ii) (a) Calculate the payback period from project B.
(b) Calculate the discounted payback period from project B at a rate of interest of 4% per annum effective.
(iii) Show that there is at least one “cross-over point” for projects A and B between 0% per annum effective and
4% per annum effective where the cross-over point is defined as the rate of interest at which the net present
value of the two projects is equal.
(iv) Calculate the duration of the incoming cash flows from projects A and B at a rate of interest of 4% per annum
effective.
(v) Explain why the net present value of projectA appears to fall more rapidly than the net present value of projectB
as the rate of interest increases. (Institute/Faculty of Actuaries Examinations, September 2012) [3+5+6+6+2=22]
31. An insurance company has liabilities of £6 million due in 8 years’ time and £11 million due in 15 years’ time. The
assets consist of two zero-coupon bonds, one paying £X in 5 years’ time and the other paying £Y in 20 years’ time.
The current interest rate is 8% per annum effective. The insurance company wishes to ensure that it is immunised
against small changes in the rate of interest.
(i) Determine the values of £X and £Y such that the first two conditions for Redington’s immunisation are satisfied.
(ii) Demonstrate that the third condition for Redington’s immunisation is also satisfied.
(Institute/Faculty of Actuaries Examinations, April 2013) [8+2=10]
32. A pension fund has liabilities to meet annuities payable in arrear for 40 years at a rate of £10 million per annum.
The fund is invested in two fixed interest securities. The first security pays annual coupons of 5% and is redeemed
at par in exactly 10 years’ time. The second security pays annual coupons of 10% and is redeemed at par in exactly
5 years’ time. The present value of the assets in the pension fund is equal to the present value of the liabilities of the
fund and exactly half the assets are invested in each security. All assets and liabilities are valued at a rate of interest
of 4% per annum effective.
(i) Calculate the present value of the liabilities of the fund.
(ii) Calculate the nominal amount of each security purchased by the pension fund.
(iii) Calculate the duration of the liabilities of the pension fund.
(iv) Calculate the duration of the assets of the pension fund.
(v) Without further calculations, explain whether the pension fund will make a profit or loss if interest rates fall
uniformly by 1.5% per annum effective.
(Institute/Faculty of Actuaries Examinations, September 2013) [1+6+3+4+2=16]
33. A fixed interest security, redeemable at par in 10 years, pays annual coupons of 9% in arrear and has just been issued
at a price to give an investor who does not pay tax a rate of return of 7% per annum effective.
(i) Calculate the price of the security at issue.
(ii) Calculate the discounted mean term (duration) of the security at issue.
(iii) Explain how your answer to part (ii) would differ if the annual coupons on the security were 3% instead of 9%.
(iv) (a) Calculate the effective duration (volatility) of the security at the time of issue.
(b) Explain the usefulness of effective duration for an investor who expects to sell the security over the next
few months. (Institute/Faculty of Actuaries Examinations, April 2015) [2+3+2+3=10]
34. An insurance company has sold a pension product to an individual. Under the arrangement, the individual is to
receive an immediate annuity of £500 per year annually in arrear for 12 years. The insurance company has invested
the premium it has received in a fixed-interest bond that pays coupons annually in arrear at the rate of 5% per annum
and which is redeemable at par in exactly 8 years.
(i) Calculate the duration of the annuity at an interest rate of 4% per annum effective.
(ii) Calculate the duration of the bond at an interest rate of 4% per annum effective.
(iii) State with reasons whether the insurance company will make a profit or a loss if there is a small increase in
interest rates at all terms. (Institute/Faculty of Actuaries Examinations, September 2015) [2+3+2=7]
6 Interest Rate Problems Sep 27, 2016(9:51) Section 5 Page 123
5 Stochastic interest rate problems
5.1 One investment. Suppose £1 is invested at time 0 for n time units. Suppose further that the interest rates
are i1 for time period 1, . . . , in for time period n. Let £Sn denote the accumulated amount at time n. Then
Sn = (1 + i1)(1 + i2) · · · (1 + in)
Now suppose ij is a random variable with mean µj and variance 2j for j = 1, . . . , n. Suppose further that i1,
. . . , in are independent—this is clearly an unreasonable assumption!!! Then
E[Sn] =
nY
j=1
E[1 + ij] =
nY
j=1
(1 + µj)
E[S2n] =
nY
j=1
E[1 + 2ij + i2j] =
nY
j=1
(1 + 2µj + 2j + µ
2
j)
and an expression for var(Sn) can be obtained by using
var(Sn) = E[S2n] E[Sn]2
If µj = µ and 2j =
2 for j = 1, . . . , n, then E[Sn] = (1 + µ)n, E[S2n] = (1 + 2µ + µ2 + 2)n and so
var(Sn) = (1 + 2µ + µ2 + 2)n (1 + µ)2n
5.2 Repeated investments. Now consider the following sequence of investments:
Time 0 1 2 3 . . . n 1 n
Cash flow, c 1 1 1 1 . . . 1 0
Let An denote the value at time n. Then
An =
nX
j=1
nY
k=j
(1 + ik) = (1 + i1) · · · (1 + in) + · · · + (1 + in1)(1 + in) + (1 + in)
and hence
An = (1 + in)(1 +An1) (5.2a)
where in and An1 are independent because An1 depends only on i1, i2, . . . , in1. Let µn = E[An]. Clearly
µ1 = 1 + µ and E[A21] = 1 + 2µ + µ2 + 2. Taking expectations of equation (5.2a) gives:
µn = (1 + µ)(1 + µn1) for n 2.
and hence by induction
µn = (1 + µ)n + (1 + µ)n1 + · · · + (1 + µ) = (1 + µ) [(1 + µ)
n 1]
µ
= (1 + µ)sn,µ and sn,µ is tabulated for common n and µ.
Thus E[An] is just the accumulated value s¨n,µ = (1 + µ)sn,µ calculated at the mean rate of interest.
Calculating the variance of An is straightforward—squaring equation (5.2a) gives
A2n = (1 + 2in + i
2
n)(1 + 2An1 +A
2
n1)
Using the fact that in is independent of An1 and taking expectations gives
E[A2n] = (1 + 2µ + µ2 + 2)(1 + 2µn1 + E[A2n1]) for n 2.
and this gives a recurrence relation for E[A2n]. Then use var[An] = E[A2n] µ2n.
Example 5.2a. The yield on a fund is thought to be uniform between 2% and 6%.
(a) For a single premium of £1, find the mean and standard deviation of the accumulation after a term of 5 years.
(a) Suppose there is an annual premium of £1. Find the mean and standard deviation of the accumulation after a term
of 5 years.
Solution. In this case S5 = (1 + i1)(1 + i2)(1 + i3)(1 + i4)(1 + i5) where i1, i2, . . . , i5 are i.i.d. U [0.02, 0.06]. Clearly
µ = E[i] = 0.04 and the density of i is fi(x) = 25 for i 2 [0.02, 0.06]. The variance of i is given by
2 = var[i] =
Z 0.06
0.02
25(x 0.04)2 dx = 25(x 0.04)
3
3
0.06
0.02
=
4
30,000
Page 124 Section 5 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
It follows that E[S5] = 1.045 = 1.21665 and E[S25 ] = (1 + 2µ + 2 + µ2)5 = 1.481157. Hence var[S5]1/2 = 0.03033.
(b) Now µn = E[An] = (1 + µ)sn,0.04 = 1.04sn,0.04. In particular, µ5 = E[A5] = 1.04s5 ,0.04 = 5.63298.
Next, using A1 = 1 + i1 gives E[A21] = E[(1 + i1)2] = 1 + 2µ + µ2 + 2 = 1.08173. Denote this quantity by ↵.
Then E[A22] = ↵(1 + 2µ1 + E[A21]) = ↵(1 + 2.08s1 ,0.04 + E[A21]) = 4.5019.
Then E[A23] = ↵(1 + 2µ2 + E[A22]) = ↵(1 + 2.08s2 ,0.04 + E[A22]) = 10.5416.
Then E[A24] = ↵(1 + 2µ3 + E[A23]) = ↵(1 + 2.08s3 ,0.04 + E[A23]) = 19.5085.
Then E[A25] = ↵(1 + 2µ4 + E[A23]) = ↵(1 + 2.08s4 ,0.04 + E[A24]) = 31.7393.
Using var[A5] = E[A25] E[A5]2 gives var[A5]1/2 = 0.09441.
5.3 The lognormal distribution. Calculating the distribution function of a product of independent random
variables is usually not possible—however, there is one case where it is possible and that is the case when the
factors have independent lognormal distributions.
Definition. The random variable Y : (⌦,F ,P) ! ( (0,1),B(0,1) ) has the lognormal distribution with
parameters (µ,2) iff ln(Y ) has the normal distribution N (µ,2).
Here are two memorable properties of the lognormal distribution:
• The product of independent lognormals is lognormal.
Suppose Yi ⇠ lognormal(µi,2i ) for i = 1, . . . , n and Y1, . . . , Yn are independent. Using the standard result
about the distribution of the sum of independent normal random variables gives
nY
i=1
Yi = Y1 ⇥ · · ·⇥ Yn ⇠ lognormal(
nX
i=1
µi,
nX
i=1
2i )
In particular, if Y1, . . . , Yn are i.i.d. lognormal(µ,2) then
Q
Yi ⇠ lognormal(nµ, n2).
• Mean and variance of a lognormal.
If Z ⇠ N (µ,2) then the moment generating function of Z is
E[etZ] = etµ+
1
2 t
22 for all t 2 R
If Y ⇠ lognormal(µ,2) then Y = eZ where Z ⇠ N (µ,2). Hence
E[Y ] = E[eZ] = eµ+
1
2
2
and E[Y 2] = E[e2Z] = e2µ+22
and so
var[Y ] = e2µ+2 (e2 1)
Example 5.3a. Suppose £1 is invested at time 0 for n time units. Suppose further that the interest rate is ij for the
time period (j 1, j) where j = 1, . . . , n and the i1, i2, . . . , in are independent and 1 + ij ⇠ lognormal(µj ,2j ).
(a) Let £Sn denote the accumulated amount at time n. Find the distribution of Sn.
(b) Let £Vn denote the present value of £1 due at the end of n years. Find the distribution of Vn.
Solution. (a) Now Sn = (1 + i1)(1 + i2) · · · (1 + in) and lnSn =
Pn
j=1 ln(1 + ij).
But ln(1 + ij) ⇠ N (µj ,2j ) and the 1 + ij are independent. Hence lnSn ⇠ N (
Pn
j=1 µj ,
Pn
j=1
2
j ).
Hence
Sn ⇠ lognormal(
nX
j=1
µj ,
nX
j=1
2j )
(b) Now Vn(1 + i1)(1 + i2) · · · (1 + in) = 1 and hence lnVn =
Pn
j=1(1 + ij). Hence
Vn ⇠ lognormal(
nX
j=1
µj ,
nX
j=1
2j )
Example 5.3b. Suppose Y : (⌦,F ,P)! ( (0,1),B(0,1) ) has the lognormal distribution with parameters (µ,2)
and E[Y ] = ↵ and var[Y ] = . Express µ and 2 in terms of ↵ and .
Solution. We know that ↵ = eµ+ 12
2
and = e2µ+
2
(e
2 1). Hence
2 = ln
✓
1 +

↵2
◆
and eµ =
↵p
1 + /↵2
=
↵2p
+ ↵2
or µ = ln
↵2p
+ ↵2
6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 6 Page 125
6 Exercises (exs6-3.tex)
1. ln(1 + it) is normally distributed with mean 0.06 and standard deviation 0.01. Find Pr[it  0.05].
(Institute/Faculty of Actuaries Examinations, September 1999) [3]
2. The rate of interest in any year has a mean of 6% and standard deviation of 1%. The yield in any year is independent
of the yields in all previous years. Find the mean and standard deviation of the accumulated value at time 12 of an
investment of £10 at time 0.
(Institute/Faculty of Actuaries Examinations, September 1997, adapted) [3]
3. An individual purchases £100,000 nominal of a bond on 1 January 2003 which is redeemable at 105 in 4 years’ time
and pays coupons of 4% per annum at the end of each year.
The investment manager wishes to invest the coupon payments on deposit until the bond is redeemed. It is assumed
that the rate of interest at which the coupon payments can be invested is a random variable and the rate of interest in
any one year is independent of that in any other year.
Deriving the necessary formulæ, calculate the mean value of the total accumulated investment on 31 December 2006
if the annual effective rate of interest has an expected value of 51/2% in 2004, 6% in 2005 and 41/2% in 2006.
(Institute/Faculty of Actuaries Examinations, April 2004) [5]
4. The yields on an insurance company’s funds in different years are independent and identically distributed. Each year
the distribution of (1 + i) is lognormal with parameters µ = 0.075 and 2 = 0.00064, where i is the annual yield on
the company’s funds.
Find the probability that a single investment of £2,000 will accumulate over 10 years to more than £4,500.
(Institute/Faculty of Actuaries Examinations, April 1999) [5]
5. An investment bank models the expected performance of its assets over a 5 day period. Over that period, the return
on the bank’s portfolio, i has a mean value of 0.1% and standard deviation 0.2%. Also 1+i is lognormally distributed.
Calculate the value of j such that the probability that i is less then or equal to j is 0.05.
(Institute/Faculty of Actuaries Examinations, September 2000) [5]
6. Let it denote the rate of interest earned in the year t 1 to t. Each year the value of it is 8% with probability 0.625,
4% with probability 0.25 and 2% with probability 0.125. In any year, it is independent of the rates of interest earned
in previous years. Let S3 denote the accumulated value of 1 unit for 3 years.
Calculate the mean and standard deviation of S3. (Institute/Faculty of Actuaries Examinations, April 1998) [5]
7. An investment bank models the expected performance of its assets over a 5 day period. Over that period, the return
on the bank’s portfolio, i, has a mean value of 0.15% and standard deviation 0.3%. The quantity 1 + i is lognormally
distributed.
Calculate the value of j such that the probability that i is greater than or equal to j is 0.9.
(Institute/Faculty of Actuaries Examinations, September 2003) [6]
8. In any year t, the yield on a fund of investments has mean jt and standard deviation st. In any year, the yield is
independent of the value in any other year. The accumulated value, after n years, of a unit sum of money invested at
time 0 is Sn.
(i) Derive formulæ for the mean and variance of Sn if jt = j and st = s for all years t.
(ii) (a) Calculate the expected value of S8 if j = 0.06.
(b) Calculate the standard deviation of S8 if j = 0.06 and s = 0.08.
(Institute/Faculty of Actuaries Examinations, September 2003) [5+3=8]
9. The expected annual effective rate of return from an insurance company’s investments is 6% and the standard devi-
ation of annual returns is 8%. The annual effective returns are independent and (1 + it) is lognormally distributed,
where it is the return in the tth year.
(a) Calculate the expected value of an investment of £1 million after 10 years.
(b) Calculate the probability that the accumulation of the investment will be less than 90% of the expected value.
(Institute/Faculty of Actuaries Examinations, September 2004) [8]
10. The rate of interest earned in the year from time t1 to t is denoted by it. Assume (1+ it) is lognormally distributed.
The expected value of the rate of interest is 5%, and the standard deviation is 11%.
(i) Calculate the parameters of the lognormal distribution of (1 + it).
(ii) Calculate the probability that the rate of interest in the year from time t 1 to t lies between 4% and 7%.
(Institute/Faculty of Actuaries Examinations, September 1997) [4+4=8]
Page 126 Exercises 6 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
11. In any year, the rate of interest on funds invested with a given insurance company is independent of the rates of
interest in all previous years.
Each year the value of 1 + it where it is the rate of interest earned in the tth year, is lognormally distributed. The
mean and standard deviation of it are 0.07 and 0.20 respectively.
(i) Determine the parameters µ and 2 of the lognormal distribution of 1 + it.
(ii) (a) Determine the distribution of S15, where S15 denotes the accumulation of one unit of money over 15 years.
(b) Determine the probability that S15 > 2.5.
(Institute/Faculty of Actuaries Examinations, April 2000) [5+4=9]
12. The annual rates of interest from a particular investment, in which part of an insurance company’s funds is invested,
are independent and identically distributed. Each year, the distribution of (1 + it), where it is the rate of interest
earned in year t, is log-normal with parameters µ and 2.
it has mean value 0.07 and standard deviation 0.02, the parameter µ = 0.06748 and 2 = 0.0003493.
(i) The insurance company has liabilities of £1m to meet in one year from now. It currently has assets of £950,000.
Assets can be invested in the risky investment described above or in an investment which has a guaranteed
return of 5% per annum effective. Find, to two decimal places, the probability that the insurance company will
be unable to meet its liabilities if:
(a) All assets are invested in the investment with the guaranteed return.
(b) 85% of assets are invested in the investment which does not have the guaranteed return and 15% of assets
are invested in the asset with the guaranteed return.
(ii) Determine the variance of return from the portfolios in (i)(a) and (i)(b) above.
(Institute/Faculty of Actuaries Examinations, September 1998) [7+3=10]
13. An insurance company has just written contracts that require it to make payments to policyholders of £1,000,000 in
5 years’ time. The total premiums paid by policyholders amounted to £850,000. The insurance company is to invest
half the premium income in fixed interest securities that provide a return of 3% per annum effective. The other half
of the premium income is to be invested in assets that have an uncertain return. The return from these assets in year t,
it, has a mean value of 3.5% per annum effective and a standard deviation of 3% per annum effective. The quantities
(1 + it) are independently and lognormally distributed.
(i) Deriving all necessary formulæ, calculate the mean and standard deviation of the accumulation of the premiums
over the 5-year period.
(ii) A director of the company suggests that investing all the premiums in the assets with an uncertain return would
be preferable because the expected accumulation of the premiums would be greater than the payments due to
the policyholders. Explain why this may be a more risky investment policy.
(Institute/Faculty of Actuaries Examinations, September 2005) [9+2=11]
14. £10,000 is invested in a bank account which pays interest at the end of each year.
The rate of interest is fixed randomly at the beginning of each year and remains unchanged until the beginning of the
next year. The rate of interest applicable in any one year is independent of the rate applicable in any other year.
During the first year the rate of interest per annum effective will be one of 3%, 4% or 6% with equal probability.
During the second year, the rate of interest per annum effective will be either 5% with probability 0.7 or 4% with
probability 0.3.
(i) Assuming that interest is always reinvested in the account, calculate the expected accumulated amount in the
bank account at the end of 2 years.
(ii) Calculate the variance of the accumulated amount in the bank account at the end of the 2 years.
(Institute/Faculty of Actuaries Examinations, September 2002) [4+7=11]
15. £1,000 is invested for 10 years. In any year, the yield on the investment will be 4% with probability 0.4, 6% with
probability 0.2 and 8% with probability 0.4 and is independent of the yield in any other year.
(i) Calculate the mean accumulation at the end of 10 years.
(ii) Calculate the standard deviation of the accumulation at the end of 10 years.
(iii) Without carrying out any further calculations, explain how your answers to parts (i) and (ii) would change (if at
all) if:
(a) the yields had been 5%, 6% and 7% instead of 4%, 6% and 8% per annum, respectively; or
(b) the investment had been made for 12 years instead of 10 years.
(Institute/Faculty of Actuaries Examinations, April 2003) [2+5+4=11]
16. The annual yields from a particular fund are independent and identically distributed. Each year, the distribution of
1 + i is log-normal with parameters µ = 0.07 and 2 = 0.006, where i denotes the annual yield on the fund.
(i) Find the mean accumulation in 10 years’ time of an investment in the fund of £20,000 at the end of each of the
next 10 years, together with £150,000 invested immediately.
(ii) Find the single amount which should be invested in the fund immediately to give an accumulation of at least
£600,000 in 10 years’ time with probability 0.99.
(Institute/Faculty of Actuaries Examinations, April 2001) [12]
6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 6 Page 127
17. In any year the yield on funds invested with a given insurance company has mean value j and standard deviation s,
and it is independent of the yields in all previous years.
(i) Derive formula for the mean and the variance of the accumulated value after n years of a single investment of 1
at time 0.
(ii) Let it be the rate of interest earned in the tth year. Each year the value of (1 + it) has a lognormal distribution
with parameters µ = 0.08 and = 0.04.
Calculate the probability that a single investment of £1,000 will accumulate over 16 years to more than £4,250.
(Institute/Faculty of Actuaries Examinations, April 1997) [6+6=12]
18. In any year the yield on a fund of speculative investments has a mean value j and standard deviation s. In any year,
the yield is independent of the value in any other year.
(i) Derive formulæ for the mean and variance of the accumulated value after n years, Sn, of a unit investment at
time 0.
(ii) Suppose that the random variable (1 + i) where i is the annual yield of the fund in any year, has a lognormal
distribution. Then if the mean value and standard deviation of i are given by j = 6% and s = 1%:
(a) Find the parameters µ and 2 of the lognormal distribution for (1 + i).
(b) If S12 is the random variable denoting the accumulation at the end of 12 years of a unit investment show
that S12 has a lognormal distribution with parameters 0.69869 and 0.0010679.
(c) Calculate the expected value of S12 and calculate the probability that the accumulation at the end of 12 years
will be at least double the original investment.
(Institute/Faculty of Actuaries Examinations, September 1999) [5+8=13]
19. A company is adopting a particular investment strategy such that the expected annual effective rate of return from
investments is 7% and the standard deviation of annual returns is 9%. Annual returns are independent and (1 + it) is
lognormally distributed where it is the return in the tth year. The company has received a premium of £1,000 and
will pay the policyholder £1,400 after 10 years.
(i) Calculate the expected value and standard deviation of an investment of £1,000 over 10 years, deriving all the
formulæ that you use.
(ii) Calculate the probability that the accumulation of the investment will be less than 50% of its expected value in
10 years’ time.
(iii) The company has invested £1,200 to meet its liability in 10 years’ time. Calculate the probability that it will
have insufficient funds to meet its liability.
(Institute/Faculty of Actuaries Examinations, April, 2002) [9+8+3=20]
20. An investment fund provides annual rates of return, which are independent and identically distributed, with annual
accumulation factors following the lognormal distribution with mean 1.04 and variance 0.02.
(i) An investor knows that she will have to make a payment of £5,000 in 5 years’ time.
Calculate the amount of cash she should invest now in order that she has a 99% chance of having sufficient cash
available in 5 years’ time from the investment to meet the payment.
(ii) Comment on your answer to (i). (Institute/Faculty of Actuaries Examinations, April 2004) [9+3=12]
21. (i) In any year, the interest rate per annum effective on monies invested with a given bank has mean value j and
standard deviation s and is independent of the interest rates in all previous years.
Let Sn be the accumulated amount after n years of a single investment of 1 at time t = 0.
(a) Show that E[Sn] = (1 + j)n.
(b) Show that var[Sn] = (1 + 2j + j2 + s2)n (1 + j)2n.
(ii) The interest rate per annum effective in (i), in any year, is equally likely to be i1 or i2 where i1 > i2. No other
values are possible.
(a) Derive expressions for j and s2 in terms of i1 and i2.
(b) The accumulated value at time t = 25 years of £1 million invested with the bank at time t = 0 has expected
value £5.5 million and standard deviation £0.5 million. Calculate the values of i1 and i2.
(Institute/Faculty of Actuaries Examinations, April 2005) [5+8=13]
22. An actuarial student has created an interest rate model under which the annual effective rate of interest is assumed
to be fixed over the whole of the next 10 years. The annual effective rate is assumed to be 2%, 4% and 7% with
probabilities 0.25, 0.55 and 0.2 respectively.
(a) Calculate the expected accumulated value of an annuity of £800 per annum payable annually in advance over
the next 10 years.
(b) Calculate the probability that the accumulated value will be greater than £10,000.
(Institute/Faculty of Actuaries Examinations, April 2006) [4]
Page 128 Exercises 6 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
23. The rate of interest is a random variable that is distributed with mean 0.07 and variance 0.016 in each of the next
10 years. The value taken by the rate of interest in any one year is independent of its value in any other year. Deriving
all necessary formulæ, calculate:
(i) The expected accumulation at the end of 10 years, if one unit is invested at the beginning of 10 years.
(ii) The variance of the accumulation at the end of 10 years, if one unit is invested at the beginning of 10 years.
(iii) Explain how your answers in (i) and (ii) would differ if 1,000 units had been invested.
(Institute/Faculty of Actuaries Examinations, September 2006) [3+5+1=9]
24. £80,000 is invested in a bank account which pays interest at the end of each year. Interest is always reinvested in the
account. The rate of interest is determined at the beginning of each year and remains unchanged until the beginning
of the next year. The rate of interest applicable in any one year is independent of the rate applicable in any other year.
During the first year, the annual effective rate of interest will be one of 4%, 6% or 8% with equal probability.
During the second year, the annual effective rate of interest will be either 7% with probability 0.75 or 5% with prob-
ability 0.25.
During the third year, the annual effective rate of interest will be either 6% with probability 0.7 or 4% with probabil-
ity 0.3.
(i) Derive the expected accumulated amount in the bank account at the end of 3 years.
(ii) Derive the variance of the accumulated amount in the bank account at the end of 3 years.
(iii) Calculate the probability that the accumulated amount in the bank account is more than £97,000 at the end of
3 years. (Institute/Faculty of Actuaries Examinations, April 2007) [5+8+3=16]
25. The expected effective annual rate of return from a bank’s investment portfolio is 6% and the standard deviation of
annual effective returns is 8%. The annual effective returns are independent and (1 + it) is lognormally distributed,
where it is the return in year t. Deriving any necessary formulæ:
(i) calculate the expected value of an investment of £2 million after 10 years;
(ii) calculate the probability that the accumulation of the investment will be less than 80% of the expected value.
(Institute/Faculty of Actuaries Examinations, September 2007) [6+3=9]
26. An insurance company holds a large amount of capital and wishes to distribute some of it to its policyholders by way
of two possible options:
Option A. £100 for each policyholder will be put into a fund from which the expected annual effective rate of return
from the investments will be 5.5% and the standard deviation of annual returns 7%. The annual effective rates of
return will be independent and (1 + it) is lognormally distributed, where it is the rate of return in year t. The policy
holder will receive the accumulated investment at the end of ten years.
Option B. £100 will be invested for each policyholder for five years at a rate of return of 6% per annum effective.
After five years, the accumulated sum will be invested for a further five years at the prevailing five-year spot rate.
This spot rate will be 1% per annum effective with probability 0.2, 3% per annum effective with probability 0.3,
6% per annum effective with probability 0.2, and 8% per annum effective with probability 0.3. The policyholder will
receive the accumulated investment at the end of ten years.
Deriving any necessary formulæ:
(i) Calculate the expected value and the standard deviation of the sum the policyholders will receive at the end of
the ten years for each of options A and B.
(ii) Determine the probability that the sum the policyholders will receive at the end of ten years will be less than
£115 for each of options A and B.
(iii) Comment on the relative risk of the two options from the policyholders’ perspective.
(Institute/Faculty of Actuaries Examinations, April 2008) [17+5+2=24]
27. A pension fund holds an asset with current value £1 million. The investment return on the asset in a given year is
independent of returns in all other years. The annual investment return in the next year will be 7%with probability 0.5
and 3% with probability 0.5. In the second and subsequent years, annual investment returns will be 2%, 4% and 6%
with probability 0.3, 0.4 and 0.3, respectively.
(i) Calculate the expected accumulated value of the asset after 10 years, showing all steps in your calculation.
(ii) Calculate the standard deviation of the accumulated value of the asset after 10 years, showing all steps in your
calculation.
(iii) Without doing any further calculations, explain how the mean and variance of the accumulation would be
affected if the returns in years 2 to 10 were 1%, 4% or 7% with probability 0.3, 0.4 and 0.3 respectively.
(Institute/Faculty of Actuaries Examinations, September 2008) [3+4+2=9]
28. The annual yields from a fund are independent and identically distributed. Each year, the distribution of 1 + i is
lognormal with parameters µ = 0.05 and 2 = 0.004, where i denotes the annual yield on the fund.
(i) Calculate the expected accumulation in 20 years’ time of an annual investment in the fund of £5,000 at the
beginning of each of the next 20 years.
(ii) Calculate the probability that the accumulation of a single investment of £1 made now will be greater than its
expected value in 20 years’ time. (Institute/Faculty of Actuaries Examinations, April 2012) [5+5=10]
6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 6 Page 129
29. An individual wishes to make an investment that will pay out £200,000 in 20 years’ time. The interest rate he will
earn on the invested funds in the first 10 years will be either 4% per annum with probability of 0.3 or 6% per annum
with probability 0.7. The interest rate he will earn on the invested funds in the second 10 years will also be either
4% per annum with probability of 0.3 or 6% per annum with probability 0.7. However, the interest rate in the second
10 year period will be independent of that in the first 10 year period.
(i) Calculate the amount the individual should invest if he calculates the investment using the expected annual
interest rate in each 10 year period.
(ii) Calculate the expected value of the investment in excess of £200,000 if the amount calculated in part (i) is
invested.
(iii) Calculate the range of the accumulated amount of the investment assuming the amount calculated in part (i) is
invested. (Institute/Faculty of Actuaries Examinations, September 2012) [2+3+2=7]
30. A cash sum of £10,000 is invested in a fund and held for 15 years. The yield on the investment in any year will be
5% with probability 0.2, 7% with probability 0.6 and 9% with probability 0.2, and is independent of the yield in any
other year.
(i) Calculate the mean accumulation at the end of 15 years.
(ii) Calculate the standard deviation of the accumulation at the end of 15 years.
(iii) Without carrying out any further calculations, explain how your answers to parts (i) and (ii) would change (if at
all) if:
(a) the yields had been 6%, 7% and 8% instead of 5%, 7% and 9% per annum, respectively.
(b) the investment had been for 13 years instead of 15 years.
(Institute/Faculty of Actuaries Examinations, April 2013) [2+5+4=11]
31. An insurance company has just written contracts that require it to make payments to policyholders of £10 million in
five years’ time. The total premiums paid by policy holders at the outset of the contracts amounted to £7.85 million.
The insurance company is to invest the premiums in assets that have an uncertain return. The return from these assets
in year t, it, has a mean value of 5.5% per annum effective and a standard deviation of 4% per annum effective.
(1 + it) is independently and lognormally distributed3.
(i) Calculate the mean and standard deviation of the accumulation of the premiums over the five-year period. You
should derive all necessary formulæ. (Note. You are not required to derive the formulæ for the mean and
variance of a lognormal distribution.)
A director of the insurance company is concerned about the possibility of a considerable loss from the investment
strategy suggested in part (i). He therefore suggests investing in fixed-interest securities with a guaranteed return of
4% per annum effective.
(ii) Explain the arguments for and against the director’s suggestion.
(Institute/Faculty of Actuaries Examinations, September 2013) [9+3=12]
32. In any year, the yield on investments with an insurance company has mean j and standard deviation s and is inde-
pendent of the yield in all previous years.
(i) Derive formulæ for the mean and variance of the accumulated value after n years of a single investment of 1 at
time 0 with the insurance company.
Each year the value of (1 + it), where it is the rate of interest earned in the tth year, is lognormally distributed. The
rate of interest has a mean value of 0.04 and standard deviation of 0.12 in all years.
(ii) (a) Calculate the parameters µ and 2 for the lognormal distribution of (1 + it).
(b) Calculate the probability that an investor receives a rate of return between 6% and 8% in any year.
(iii) Explain whether your answer to part (ii)(b) looks reasonable.
(Institute/Faculty of Actuaries Examinations, April 2015) [5+8+2=15]
3 This question is badly phrased. See Mathematical Writing by Donald E. Knuth, Tracy Larrabee and Paul M. Roberts.
ST334 Actuarial Models
Chapter 6: Interest Rate Problems
Learning objectives:
To demonstrate an understanding of the concepts and be able to perform calcula-
tions relating to spot and forward interest rates, duration(s), convexity and immu-
nisation, and stochastic interest rate problems (single and repeated investment).
1. Spot interest rates
• The term structure of interest rates is concerned with the analysis of
dependence of interest rates on the length and starting time of the in-
vestment.
• yt denotes the t-year spot rate of interest. It is the effective interest rate
per annum for an investment which lasts t years.
• We have (1+yt)tPt = 1, where Pt is the price at issue of a zero-coupon
bond paying £1 at maturity in t years.
• In continuous time, etYtPt = 1 whereYt is the t-year spot rate of interest
under continuous compounding.
2. Par yield of a bond
The n-year par yield of a bond is the coupon rate r that sets the bond
price P to its face value f , assuming it is ”redeemed at par” so we have
C = f .
3. Forward interest rates
• ft,k denotes the forward interest rate for an investment starting at time
t for a period of k years. Then: (1+ yT )T (1+ fT,k)k = (1+ yT+k)T+k.
• In continuous time,
e(T+k)YT+k = eTYT ekFT,k
where FT,k denotes the forward interest rate for an investment starting
at time T for a period of k years, under continuous compounding.
Or in terms of zero-coupon bonds,
PT
PT+k
= ekFT,k
1
4. Duration, convexity and immunisation
• For a cash flow {ct1 ,ct2 , · · · ,ctn}, its NPV with force of interest d (or
effective yield i) is
P= Ânk=1 ctked tk = Â
n
k=1
ctk
(1+i)tk .
The Duration or Macaulay duration or discounted mean term:
is the average or mean term of cashflows weighted by present values.
dM(i) = 1P dPdd = 1PÂnk=1
tkctk
(1+i)tk =(1+ i) 1P dPdi .
The effective duration or modified duration:
d(i) = 1P dPdi = 1PÂnk=1
tkctk
(1+i)tk+1
.
Hence dM(i) = d(i)(1+ i).
• The convexity c(i) of the cash flows is
c(i) =
1
P
d2P
di2
=
1
P
n
Â
k=1
ctktk(tk+1)
(1+ i)tk+2
• Redington immunisation is a strategy that ensures that a change in
interest rates will not affect the value of a portfolio. Redington immu-
nisation occurs at interest rate i0 if:
NPVA(i0) = NPVL(i0),
dA(i0) = dL(i0),
cA(i0)> cL(i0). There is a set of testable conditions in the notes.
5. Stochastic interest rate problems
• One investment:
If Sn = (1+ i1)(1+ i2)...(1+ in) denotes the accumulation of £1 over
n time periods, and i j, j = 1, ...,n are independent random variables
with mean µ j and variance s2j , then
E[Sn] =Pnj=1(1+µ j)
E[S2n] =Pnj=1(1+2µ j+s2j +µ2j )Repeatedinvestments :
An is value at time n, of repeated investment of £1 at each time from
time 0 to time n1. Then
An = Ânj=1Pnk= j(1+ ik) and hence An = (1+ in)(1+An1).
µn =E[An] = (1+µ)snµ and E[A2n] = (1+2µ+µ2+s2)(1+2µn1+
E[A2n1]),n 2.
• Lognormal distribution
If Y is lognormal(µ,s2), then Y = eZ where Z is Normal(µ,s2), then
2
EY = EeZ = eµ+
1
2s
2
EY 2 = Ee2Z = e2µ+2s
2
VarY = e2µ+s
2
(es
21).