CHAPTER 5
Bonds, Equities and Inflation
1 Bond calculations
1.1 Notation. A borrower who issues a bond agrees to pay interest at a specified rate until a specified date,
called the maturity date, and at that time pays a fixed sum called the redemption value.
The interest rate on a bond is called the coupon rate. This rate is often quoted as a nominal rate convertible
semi-annually and is applied to the face or par value of the bond. The face and redemption values are often the
same. Let
f = face or par value of the bond
r = the coupon rate per year
C = the redemption value of the bond
n = the number of years until the redemption date
P = current price of the bond
i = the yield to maturity (this is the same as the internal rate of return)
The values of f , r, C and the date of redemption are specified by the terms of the bond and are fixed throughout
the lifetime of the bond.
The values of P and i vary throughout the lifetime of the bond. As the price of a bond rises, the yield falls.
Also, the price of a bond and hence its yield depend on the prevailing interest rates in the market and the risk
of default.
1.2 Finite redemption date.
Example 1.2a. Suppose a bond with face value £500,000 is redeemable at par after 4 years. The coupon rate is
10% p.a. convertible semi-annually.
(a) Find the price to obtain an effective yield of 10% p.a. (b) Find the price to obtain an effective yield of 15% p.a.
Solution. For this example, f = C = 500,000; r = 0.1, and n = 4. Each coupon payment is £25,000. The cash flow
in thousands is as follows:
Time (years) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Cash flow (£1,000’s) P 25 25 25 25 25 25 25 525
(a) The price P (in thousands) for a yield of i = 10% is given by
P =
25
1.10.5
+
25
1.1
+
25
1.11.5
+
25
1.12
+
25
1.12.5
+
25
1.13
+
25
1.13.5
+
525
1.14
Let ↵ = 1/
p
1.1. Then
P = 25(↵ + ↵2 + · · · + ↵8) + 500↵8 = 25↵1 ↵
8
1 ↵ + 500↵
8 =
1
1.14

25
1.14 1
1.11/2 1 + 500

= 503.868
Alternatively, let i = 0.1 and ⌫ = 1/(1 + i), then
P = 50a(2)4 ,i +
500
(1 + i)4
= 50
i
i(2)
a4 ,i + 500⌫
4 = 503.868
Hence the bond must be bought at a premium.
Note that commonly used values of i/i(m), ⌫n and an are provided in the tables.
(b) For this case
P =
1
1.154

25
1.154 1
1.151/2 1 + 500

= 433.792
Alternatively, P = 50a(2)4 ,i + 500/(1 + i)
4 = 50 ii(2)a4 ,i + 500/(1 + i)
4 = 433.792.
Hence the bond is sold at a discount.
ST334 Actuarial Methods cR.J. Reed Sep 27, 2016(9:51) Section 1 Page 81












































































































债券⾯值
每年的票⾯利率
债券的赎回价值
到期收益率
有限的赎回⽇期
Page 82 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
Here is the general result: suppose the coupon rate is nominal r% p.a. payable m times per year. Then the
price P in order to achieve an effective yield of i per annum is given by
P = fra(m)n,i +
C
(1 + i)n
=
fr
m
amn,i0 +
C
(1 + i0)mn
(1.2a)
where (1 + i0)m = 1 + i and i0 = i(m)m is the effective interest rate per period. If the price rises, then the yield
falls, and conversely.
If P = f = C, then using the relation a(m)n = (1 ⌫n)/i(m) and equation (1.2a) shows that i(m) = r. This
means that if the face value, redemption value and price are all the same, then the nominal yield convertible
mthly equals the coupon rate.
If the income tax rate is t1, then equation (1.2a) becomes:
P = fr(1 t1)a(m)n,i +
C
(1 + i)n
=
g(1 t1)
i(m)
⇥
C C⌫n⇤ + C⌫n (1.2b)
where the right hand version of equation(1.2b) is called Makeham’s formula. The first term is the NPV of the
interest payments and the second term is the NPV of the redemption value, C.
Example 1.2b. Suppose a bond is issued for 15 years with a coupon rate of 6% per annum convertible 6-monthly.
The bond is redeemable at 105%. (This means that C = f ⇥ 1.05.)
(a) Find the price per unit nominal 1 so that the effective yield at maturity is 7% per annum.
(b) Assume the income tax rate is 35%. Find the price per unit nominal so that the effective yield at maturity is
7% p.a.
Solution. For this example, we have f = 100, C = 105, r = 0.06, i = 0.07 and n = 15.
(a) The size of each coupon is 3. Hence
P = 6a(2)15 ,i +
105
(1 + i)15
= 6
i
i(2)
a15 ,i +
105
(1 + i)15
⇡ 93.64
by using tables. Hence the bond should be issued at a discount. Part (b):
P = 6⇥ 0.65⇥ a(2)15 ,i +
105
(1 + i)15
= 3.9
i
i(2)
a15 ,i +
105
1.0715
⇡ 74.19
1.3 Gross yield, redemption yield and flat yield.
The terms gross interest yield, direct yield, flat yield, current yield or gross running yield all refer to 100r/P1,
which is the ratio of 100r, the annual coupon per £100, to the current price P1 per unit nominal1. This is the
same as the annual coupon (fr) divided by the current price, P .
The terms net interest yield and net running yield refer to the same quantity but after allowing for tax: hence it
is 100(1 t1)r/P1, the ratio of the (after tax) annual coupon per £100 to the current price P1 per unit nominal.
This is the same as (1 t1)fr/P , the annual coupon after tax divided by the current price, P .
Example 1.3a. The current price of a bond with an annual coupon of 10% is £103 per £100 nominal. Then the
gross interest yield (also known as the flat yield or running yield) is 10/103 = 0.0971 or 9.71%.
The interest yields defined in the last paragraph are useful when a bond is undated and there are no redemption
proceeds. If there are redemption proceeds, then we should also take these into account when calculating the
yield.
The term gross redemption yield is the same as the yield to maturity defined in paragraph 1.1 and ignores
taxation. The term net redemption yield or net yield to redemption or net yield refers to the yield after
allowing for tax.
If the gross redemption yield on a bond is an effective 3% every six months, then the gross redemption yield is
6.09% per annum (because 1.032 1 = 0.0609). Alternatively, it is described as a gross redemption yield of
6% convertible 6-monthly.2
1 In the UK, per unit nominal means per £100.
2 In Europe, the gross redemption yield is the same as the internal rate of return as defined in paragraph 1.1. However, in
the USA and the UK, the gross redemption yield of a bond with 6-monthly coupons is defined to be twice the effective
yield for 6 months—what we have defined to be the gross redemption yield convertible 6-monthly. In actuarial questions,
the distinction will be clarified by using phrases such as “a gross redemption yield of 6.09% per annum effective” or “a
gross redemption yield of 6% per annum convertible 6-monthly”. See Basic Bond Analysis by Joanna Place which can
be downloaded from http://www.bankofengland.co.uk/education/Pages/ccbs/handbooks












































































































s
5 Bonds, Equities and Inflation Sep 27, 2016(9:51) Section 1 Page 83
Example 1.3b. Consider a US treasury bond which has 14 years to redemption and a coupon of 12% p.a. payable
6-monthly. What is the price of the bond (per $100) if the gross redemption yield is 8% convertible 6-monthly.
Solution. The yield is 4% effective for 6 months.
Time 0 1/2 1 3/2 2 5/2 · · · 27/2 14
Cash flow P 6 6 6 6 6 · · · 6 106
So let ⌫ = 1/1.042 and ↵ = ⌫1/2 = 1/1.04. Hence
P = 6(↵ + ↵2 + ↵3 + · · · + ↵27 + ↵28) + 100↵28 = 6↵1 ↵
28
1 ↵ + 100↵
28 = 133.326
In practice, the most common problem is finding the gross redemption yield when the current market price is
given. Suppose £P denotes the price per £100 face value, there are n years to redemption and coupons are paid
every 6 months. Let i0 denote the effective yield per 6 months and let ↵ = 1/(1 + i0). Then
P =
fr
2
(↵ + ↵2 + · · · + ↵2n) + 100↵2n = fr
2
↵
1 ↵2n
1 ↵ + 100↵
2n =
fr
2
1 ↵2n
i0
+ 100↵2n (1.3a)
Finding i0 may require successive approximation on a calculator. Equation (1.3a) implies
fr
2i0
+
↵2n
1 ↵2n (100 P ) = P and so 2i
0 =
fr
P
+
100 P
P
2i0
(1 + i0)2n 1 ⇡
✓
fr +
100 P
n
◆
1
P
(1.3b)
by using the approximation (1 + i0)2n ⇡ 1 + 2ni0. This approximation for i0 is useful as a starting value when
finding an approximate solution. It can be remembered as the gross redemption yield payable 6-monthly (2i0)
approximately equals (coupon per year + capital gain per year)/price. This approximation will be greater than
the exact value.
1.4 Perpetuity. Suppose a security is undated—this means that it has no final redemption date. Assume the
coupon is r% payable half-yearly. The cash flow is as follows:
Time 0 1/2 1 3/2 2 5/2 3 7/2 · · ·
Cash flow P fr/2 fr/2 fr/2 fr/2 fr/2 fr/2 fr/2 · · ·
Assume an income tax rate of t1. Then the price P to achieve an effective yield of i per annum is:
P = fr(1 t1)a(2)1 = fr(1 t1)i(2)
1.5 Effect of capital gains tax on bond prices. Suppose we have a bond with face value f , redemption
value C, coupon rate r% payablem times per year for n years. Let
g =
fr
C
=
annual coupon
redemption value
Then P (n, i), the price of a bond with an effective annual yield of i, is given by equation (1.2b):
P (n, i) = (1 t1)fra(m)n,i + C⌫n (1.5a)
Using a(m)n,i = (1 ⌫n)/i(m) gives
P (n, i) = (1 t1)gCa(m)n,i + C
h
1 i(m)a(m)n,i
i
(1.5b)
= C +
⇥
(1 t1)g i(m)
⇤
Ca(m)n,i (1.5c)
Capital gains tax is levied on the difference between the sale and purchase price of a security. It is levied only
when the security is sold. Suppose that capital gains tax is levied at the rate t2; this alters the price of the bond
if the yield remains at i. Let the new price of the bond be P2(n, i). Equation (1.5c) shows that:
• If i(m)  (1 t1)g, then P C. There is no capital gain and so P2(n, i) = P (n, i).
• If i(m) > (1 t1)g, then P (n, i) < C. Hence there is a capital gain3. Hence
P2(n, i) = (1 t1)fra(m)n,i + C⌫n t2
⇥
C P2(n, i)
⇤
⌫n
3 Aide me´moire. If the coupon rate is very high (and hence g is large) then the bond will be in high demand; hence its price
will be high and there will be no capital gain.












































































































达到实际收益率i1年时的P
赎回价值
Page 84 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
Hence
P2(n, i) =
(1 t1)fra(m)n,i + (1 t2)C⌫n
1 t2⌫n
and so
P2(n, i) =
8>><>>:
(1 t1)fra(m)n,i + C⌫n if i(m)  (1 t1)g
(1 t1)fra(m)n,i + (1 t2)C⌫n
1 t2⌫n if i
(m) > (1 t1)g
Example 1.5a. A bond with face value £1,000 is redeemable at par after 10 years and has a coupon rate of 6%
payable annually. The income tax rate is 40% and the capital gains tax rate is 30%. If the current price of the bond
is £800, what is the net yield?
Solution. After allowing for income tax of £24 on each coupon and capital gains tax of £60, the cash flow is as
follows:
Time 0 1 2 3 . . . 9 10
Cash flow 800 36 36 36 . . . 36 36 + 940
Hence the equation for i is given by
800 = 36a10 ,i + 940⌫
10 where ⌫ =
1
1 + i
The net income each year is £36 and this is 4.5% of the purchase price. There is a further gain on redemption. Hence
the net yield is greater than 4.5%. The gain on redemption is £140; if this amount was paid in equal instalments
over the 10 years, then the payment each year would be £50, which is 6.25% of the purchase price (this is the
approximation in equation (1.3b). Hence the net yield i satisfies 4.5 < i < 6.25. Now use tables:
if i = 0.05, 36a10 ,i + 940⌫10 = 847.339; if i = 0.055, 36a10 ,i + 940⌫10 = 821.660; and if i = 0.06,
36a10 ,i + 940⌫10 = 789.854.
Interpolating between i = 0.055 and i = 0.06 gives:
(x 0.055)/(0.06 0.055) = (f (x) f (0.055))/(f (0.06) f (0.055))
and hence x = 0.055 + 0.005(800 821.660)/(789.854 821.66) = 0.0584 or 5.8%.
1.6 Effect of term to redemption on the yield. Now consider how the yield from a bond, i, varies with n if
the price of the bond is fixed. Equation (1.5c) has the following consequence (see exercise 2):
Suppose bonds A and B have the same redemption value C, the same annual coupon fr, the same
purchase price P and the same coupon r% per annum payablem times per year.
Suppose bond A is redeemed after n1 years and bond B is redeemed after n2 years where n1 < n2. Let
i1 and i2 denote the yields of bonds 1 and 2 respectively.
• If P < C then i1 > i2 and so bond A with the shorter term has the higher yield.
• If P > C then i1 < i2 and so bond B with the longer term has the higher yield.
• If P = C then i1 = i2.
Informally, the result is obvious: if P < C, then the investor receives a capital gain of C P when the bond
is redeemed. It is clearly better to receive this amount sooner rather than later.
1.7 Optional redemption date. Sometimes a security does not have a fixed redemption date. The redemption
date may be at the borrower’s option4. This could be at any date, after a specified date or at any one of a series
of dates between two specified dates. The last possible redemption date is called the final redemption date; if
there is no such date then the security is said to be undated. For a bond, the borrower is the issuer of the bond.
An investor who purchases a bond with a variable redemption rate at the option of the borrower (or issuer)
cannot know the yield that will be obtained. It is possible to specify the following two amounts:
• The maximum price P to obtain a yield which is at least i0. The quantities f , r, t1, C andm are fixed known
numbers. Suppose the bond can be redeemed after n years for any n satisfying n1  n  n2 (where both n1
and n2 are multiples of 1/m and n2 is possibly infinite).
We think of P as a function such that if we are told n and i, then we can find P (n, i) by evaluating equa-
tion (1.5c).
4 It is also possible for a security to be redeemable at the lender’s option, but this is much less common.


































































































在票⾯价值可赎回
期限对赎回的影响
可选赎回⽇期 有时候证券没有固定的赎回⽇期还款⽇期可由借款⼈⾃⾏决定
获得⾄少io收益的最⼤价格P
•获得⾄少i0的收益的最⼤价格P。f, r, t1, C和m是固定的已知数。假设任意n满⾜n1≤n≤n2(其
中n1和n2都是1/m的倍数，n2可能是⽆限的)，n年后可以赎回键。我们认为P是⼀个函数，如
果我们已知n和i，那么我们可以通过计算⽅程(1.5c)来求出P (n, i)。
5 Bonds, Equities and Inflation Sep 27, 2016(9:51) Section 1 Page 85
First case: suppose i(m) < (1 t1)g, equivalently, suppose P (n, i) > C.
Then for a fixed price, a longer term implies a higher yield—see paragraph 1.6 above. So suppose n1 < n⇤
and P (n1, i) = P (n⇤, i⇤) for some i and i⇤; then i⇤ > i.
This means that whatever price we pay for the bond, the yield if the bond is redeemed at any date n⇤ with
n⇤ > n1 must be greater than i, the yield if it is redeemed at the earliest time n1.
If i(m) < (1 t1)g, the purchaser of the bond will receive no capital gain and should price the bond on
the assumption that redemption will take place at the earliest possible time n1. Then, no matter when
the bond is actually redeemed, the yield that will be obtained will be at least the value of the yield calculated
by assuming the bond is redeemed at the earliest possible time, n1.
Second case: suppose i(m) > (1 t1)g, equivalently suppose P (n, i) < C.
Then for a fixed price, a shorter term implies a higher yield—see paragraph 1.6 above. So suppose n⇤ < n2
and P (n2, i) = P (n⇤, i⇤) for some i and i⇤; then i⇤ > i.
If i(m) > (1 t1)g, the purchaser of the bond will receive a capital gain and should price the bond on
the assumption that redemption will take place at the latest possible time. Then, no matter when the
bond is actually redeemed, the yield that will be obtained will be at least the value of the yield calculated by
assuming the bond is redeemed at the latest possible time.
Third case: suppose i(m) = (1 t1)g, equivalently suppose P (n, i) = C for all n.
If i(m) = (1 t1)g, the yield is same whatever the redemption date.
• The minimum yield i if the price is P . Let P and C denote the price and redemption price per £100 face
value, respectively. By the above remarks, we have the following:
If P < C then it is better for the investor that the bond is redeemed sooner rather than later. Suppose i2
denotes the yield if the bond is redeemed at the latest possible time n2; then the yield will be at least i2
whatever the redemption date.
If P > C then it is better for the investor that the bond is redeemed later rather than sooner. Suppose i1
denotes the yield if the bond is redeemed at the earliest possible time n1; then the yield will be at least i1
whatever the redemption date.
If P = C then the yield will be i where i(m) = (1 t1)g whatever the redemption date.
1.8 Clean and dirty prices. Suppose investor A sells a bond to another investor, B, between two coupon
dates. Then B will be the registered owner of the bond at the next coupon date and so receive the coupon. But
A will feel entitled to the “accrued coupon” or “accrued interest”—this is the interest A earned by holding the
bond from the previous coupon date to the date of the sale. He therefore sells the bond for the dirty price—this
is the NPV of future cash flows from the bond5.
The clean price is defined to be equal to the dirty price minus the accrued interest. So we have
dirty price = NPV of future cash flows
clean price = dirty price accrued interest
The price quoted in the market is the clean price but the bond is sold for the dirty price.
The way that accrued interest is calculated depends on the bondmarket. For U.S. treasury bonds, the convention
is ACT/ACT. Hence:
accrued interest = value of next coupon⇥ days since last coupon
days between coupons
For eurobonds, the convention is 30/360. This implies that the accrued interest on the 31st of the month is
the same at the accrued interest on the 30th of the month. Also, the accrued interest for the 7 days from 22
February to 1 March would be 9/360 of the annual coupon. In the UK and Japan, the convention is ACT/365.
Hence:
accrued interest = annual coupon⇥ days since last coupon
365
Example 1.8a. A bond has a semi-annual coupon with payment dates of 27 April and 27 October. Suppose the
coupon rate is 10% per annum convertible semi-annually. Calculate the accrued interest per £100 if the bond is
purchased on 4 July. Assume ACT/365.
5 Of course, this NPV depends on the current yield which in turn depends on the general market sentiment about the bond.












































































































脏价 未来现⾦流的净现值
净价 脏作应计利息
市场上的报价是清洁价格但⼈
券以肮脏价格售
Page 86 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
Solution. Number of days from 27 April to 4 July is 3 + 31 + 30 + 4 = 68. Hence accrued interest is 10⇥ 68/365 =
1.863.
Example 1.8b. A bond has a 9% coupon paid annually on August 11. Maturity is on 11 August 2005. The current
market yield for the bond is 8%. Find the dirty price and the clean price on (a) 8 June 2000; (b) 1 August 2000.
Assume 30/360.
Solution. (a) Using 30/360 gives 63 days between 8/06/00 and 11/08/00.
Time 8/06/00 11/08/00 11/08/01 11/08/02 11/08/03 11/08/04 11/08/05
Cash flow P 9 9 9 9 9 109
The price on 11/08/00 is given by 9(1 + ⌫ + ⌫2 + ⌫3 + ⌫4 + ⌫5) + 100⌫5 where ⌫ = 1/1.08. Hence the dirty price on
8/06/00 (which is the NPV of future cash flows on 8/06/00) is given by
P = ⌫63/360

9
✓
1 ⌫6
1 ⌫
◆
+ 100⌫5

= 111.48
The number of days from 11/08/99 to 8/06/00 is 297. Hence the accrued interest is 9 ⇥ 297/360 = 7.425. Hence
the clean price is 111.48 7.42 = 104.06. The clean price is the price quoted in the markets and financial press.
For part (b), we have 10 days instead of 63 and 350 days instead of 297. This gives a dirty price of 112.75, accrued
interest of 8.75 and a clean price of 104.00.
If a bond is quoted as cum dividend then the buyer will receive the next coupon payment; if a bond is quoted
as ex dividend then the buyer will not receive the next coupon payment (and the accrued interest is then
negative). UK gilts normally go ex dividend 7 working days before a coupon payment.
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.... ..... .....
time
.................
.................
.................
.................
.................
.................
....................
...................
...................
...................
...................
...................
...................
...................
...................
...................
...................
...................
....
............................... .....
.......
......
clean
price
cum dividend ex dividend
coupon coupon
date date
Figure 1.8a. Plot of dirty price against time
(PICTEX)
1.9
Summary. Bond calculations.
• Suppose coupon rate is nominal r% p.a. payable m times per year and i is the effective yield per
annum. Let g = annual coupon/redemption value = fr/C. Then the price P is:
P = fr(1 t1)a(m)n,i +
C
(1 + i)n
=
g(1 t1)
i(m)
⇥
C C⌫n⇤ + C⌫n
• Gross running yield = annual coupon
current price
=
fr
P
;
• Net running yield = annual coupon after tax
current price
=
(1 t1)fr
P
• Gross redemption yield (or yield to maturity) and net redemption yield.
• Effect of capital gains tax on yield. Suppose i is the yield.
If i(m) > (1 t1)g then capital gains tax is payable.
• Optional redemption date: at option of issuer.
If i(m) < (1 t1)g there is no capital gain; purchaser should assume early redemption.
If i(m) > (1 t1)g there is a capital gain; purchaser should assume late redemption.
• Clean and dirty price.












































































































5 Bonds, Equities and Inflation Sep 27, 2016(9:51) Exercises 2 Page 87
2 Exercises (exs5-1.tex)
1. Consider a bond with face value f , redemption value C, current price P which has a coupon rate of r% payable
m times per year for n years. Taxation is at the rate t1 and is paid at the end of each year—hence the annual tax is
t1fr. Find an equation for the yield to maturity.
2. Suppose bond A is redeemed after n1 years and bond B is redeemed after n2 years where n1 < n2. Suppose further
that both bonds have the redemption value C, the same annual coupon fr, the same purchase price P and both have
a coupon which is payablem times per year.
Let i1 and i2 denote the yields of bonds 1 and 2 respectively. Show the following:
(a) If P = C then i1 = i2. (b) If P < C then i1 > i2. (c) If P > C then i1 < i2.
3. Suppose the first coupon on an undated bond with current price P and face value f is delayed for t0 years. The
coupon rate is r% nominal p.a. payablem times per year and the income tax rate is t1.
Show that the effective annual yield, i, is given by
P = fr(1 t1)⌫t0 a¨(m)1 = fr(1 t1)⌫
t0
d(m)
where ⌫ =
1
1 + i
4. A bond with an annual coupon of 8% per annum has just been issued with a gross redemption yield of 6% per annum
effective. It is redeemable at par at the option of the borrower on any coupon payment date from the tenth anniversary
of issue to the twentieth anniversary of issue. The gross redemption yield on bonds of all terms to maturity is 6% per
annum effective.
Ten years after issue, the gross redemption yield on bonds of all terms to maturity is 10% per annum effective.
(a) Is the bond likely to be redeemed earlier or later than was assumed at issue?
(b) Is the gross redemption yield likely to be higher or lower than was assumed at issue?
(Institute/Faculty of Actuaries Examinations, September 1998 (adapted)) [3]
5. An investor purchases a bond, redeemable at par, which pays half-yearly coupons at a rate of 8% per annum. There
are 8 days until the next coupon payment and the bond is ex-dividend. The bond has 7 years to maturity after the
next coupon payment.
Calculate the purchase price to provide a yield to maturity of 6% per annum.
(Institute/Faculty of Actuaries Examinations, April, 2002) [4]
6. A fixed interest stock bears a coupon of 7% per annum payable half yearly on 1 April and 1 October. It is redeemable
at par on any 1 April between 1 April 2004 and 1 April 2010 inclusive at the option of the borrower.
On 1 July 1991 an investor purchased £10,000 nominal of the stock at a price to give a net yield of 6% per annum
effective after allowing for tax at 25% on the coupon payments.
On 1 April 1999 the investor sold the holding at a price which gave a net yield of 5% per annum effective to another
purchaser who is also taxed at a rate of 25% on the coupon payments.
(i) Calculate the price at which the stock was bought by the original investor.
(ii) Calculate the price at which the stock was sold by the original investor.
(Institute/Faculty of Actuaries Examinations, April 2000) [4]
7. An investor purchases a bond on the issue date at a price of £96 per £100 nominal. Coupons at an annual rate of 4%
are paid annually in arrears. The bond will be redeemed at par 20 years after the issue date.
Calculate the gross redemption yield from the bond.
(Institute/Faculty of Actuaries Examinations, September 2000) [4]
8. A new issue of a fixed interest security has a term to redemption of 20 years and is redeemable at 110%. The security
pays a coupon of 91/2% per annum payable half-yearly in arrears.
An investor who is liable to tax on all income at a rate of 23% and on all capital gains at a rate of 34% bought all
the stock at the date of issue at a price which gave the investor a yield to maturity of 8% per annum effective. What
price did the investor pay per £100 nominal of the stock?
(Institute/Faculty of Actuaries Examinations, April 1999) [5]
9. (i) Describe the risk characteristics of a government-issued, conventional, fixed-interest bond.
(ii) A particular government bond is structured as follows.
Annual coupons are paid in arrears of 8% of the nominal value of the bond. After 5 years, a capital
repayment is made, equal to half of the nominal value of the bond. The capital is repaid at par. The
repayment takes place immediately after the payment of the coupon due at the end of the 5th year. After
the end of the 5th year, coupons are only paid on that part of the capital that has not been repaid. At the end
of the 10th year, all the remaining capital is repaid.
Calculate the purchase price of the bond per £100 nominal, at issue, to provide a purchaser with an effective net
rate of return of 6% per annum. The purchaser pays tax at a rate of 30% on coupon payments only.
(Institute/Faculty of Actuaries Examinations, September 2003) [2+5=7]












































































































Page 88 Exercises 2 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
10. A loan of nominal amount of £100,000 is to be issued bearing coupons payable quarterly in arrear at a rate of 5%
per annum. Capital is to be redeemed at 103 on a single coupon date between 15 and 20 years after the date of issue,
inclusive. The date of redemption is at the option of the borrower.
An investor who is liable to income tax at 20% and capital gains tax of 25% wishes to purchase the entire loan at
the date of issue. Calculate the price which the investor should pay to ensure a net effective yield of at least 4% per
annum. (Institute/Faculty of Actuaries Examinations, April 2005) [9]
11. A loan of nominal amount £100,000 is to be issued bearing interest payable half-yearly in arrears at a rate of 7% per
annum. The loan is to be redeemed with a capital payment of 110 per £100 nominal on a coupon date between 10
and 15 years after the date of issue, inclusive, the date of redemption being at the option of the borrower.
An investor who is liable to income tax at 25% and capital gains tax of 30% wishes to purchase the entire loan at the
date of issue at a price which ensures that the investor achieves a net effective yield of at least 5% per annum.
(i) Determine whether the investor would make a capital gain if the investment is held until redemption.
(ii) Explain how your answer to (i) influences the assumptions made in calculating the price the investor should pay.
(iii) Calculate the maximum price which the investor should pay.
(Institute/Faculty of Actuaries Examinations, September 2002) [3+2+5=10]
12. An investor purchased a bond with exactly 15 years to redemption. The bond, redeemable at par, has a gross
redemption yield of 5% per annum effective. It pays coupons of 4% per annum, half yearly in arrear. The investor
pays tax at 25% on the coupons only.
(i) Calculate the price paid for the bond.
(ii) After exactly 8 years, immediately after the payment of the coupon then due, this investor sells the bond to
another investor who pays income tax at a rate of 25% and capital gains tax at a rate of 40%. The bond is
purchased by the second investor to provide a net return of 6% per annum effective.
(a) Calculate the price paid by the second investor.
(b) Calculate, to one decimal place, the annual effective rate of return earned by the first investor during the
period for which the bond was held.
(Institute/Faculty of Actuaries Examinations, September 2005) [3+10=13]
13. Ten million pounds nominal of loan stock was issued on 1 March 1990. The stock is due to be redeemed in one
payment of 110% nominal on 1 March 2000. The stock pays interest at 10% per annum, payable half-yearly in
arrears on 1 September and 1 March.
(i) An investor, subject to income tax at 25% but not liable to capital gains tax, bought £10,000 nominal of the
stock at issue so as to obtain a net effective yield of 8% per annum, after tax. What price did the investor pay
per £100 nominal on the date of issue?
(ii) On 1 March 1998 the first investor, having received the interest due on that date, sold the stock to a second
investor who was subject to 40% tax on both income and capital gains. The price paid by the second investor
was calculated so as to earn him a net effective yield of 6% per annum, after tax. What price did the second
investor pay per £100 nominal?
(iii) Assuming that the first investor sold the whole of the stock of his loan to the second investor, what was the
effective net yield, after tax, earned on the first investment?
(Institute/Faculty of Actuaries Examinations, April 1998) [13]
14. An investor purchased a bond with exactly 20 years to redemption. The bond, redeemable at par, has a gross
redemption yield of 6%. It pays annual coupons, in arrears, of 5%. The investor does not pay tax.
(i) Calculate the purchase price of the bond.
(ii) After exactly 10 years, immediately after payment of the coupon then due, this investor sells the bond to another
investor. That investor pays income and capital gains tax at a rate of 30%. The bond is purchased by the second
investor to provide a net rate of return of 6.5% per annum.
(a) Calculate the price paid by the second investor.
(b) Calculate the annual effective rate of return earned by the first investor during the period for which the
bond was held. (Institute/Faculty of Actuaries Examinations, September, 2001) [3+10=13]
15. A fixed interest security pays coupons of 8% per annum half yearly on 1 January and 1 July. The security will
be redeemed at par on any 1 January between 1 January 2006 and 1 January 2011 inclusive, at the option of the
borrower.
An investor purchased a holding of the security on 1 January 2001, immediately after the payment of the coupon
then due, at a price which gave him a net yield of at least 5% per annum effective. The investor pays tax at 40% on
interest income and 30% on capital gains. On 1 January 2003 the investor sold the holding, immediately after the
payment of the coupon then due, to a fund which pays no tax at a price to give the fund a gross yield of at least 7%
per annum effective.
(i) Calculate the price per £100 nominal at which the investor bought the security.
(ii) Calculate the price per £100 nominal at which the investor sold the security.
(iii) Calculate the net yield per annum convertible half-yearly, which the investor actually received over the two
years the investor held the security. (Institute/Faculty of Actuaries Examinations, April 2003) [5+3+6=14]












































































































5 Bonds, Equities and Inflation Sep 27, 2016(9:51) Exercises 2 Page 89
16. A loan of nominal amount £10,000,000 is to be issued bearing a coupon of 8% per annum payable quarterly in
arrears. The loan is to be repaid at the end of 15 years at 110% of the nominal value. An institution not subject to
either income or capital gains tax bought the whole issue at a price to yield 9% per annum effective.
(i) Calculate the price per £100 nominal which the institution paid.
(ii) Exactly 5 years later, immediately after the coupon payment, the institution sold the entire issue of stock to an
investor who pays both income and capital gains tax at a rate of 20%. Calculate the price per £100 nominal
which the investor pays the institution to earn a net redemption yield of 7% per annum effective.
(iii) Calculate the net running yield per annum earned by this investor.
(iv) Calculate the annual effective rate of return earned by the institution in (i).
(Institute/Faculty of Actuaries Examinations, September 1999) [14]
17. A loan is issued bearing interest at a rate of 9% per annum and payable half-yearly in arrear. The loan is to be
redeemed at £110 per £100 nominal in 13 years’ time.
(i) The loan is issued at a price such that an investor, subject to income tax at 25% and capital gains tax at 30%,
would obtain a net redemption yield of 6% per annum effective. Calculate the issue price per £100 nominal of
the stock.
(ii) Two years after the date of issue, immediately after a coupon payment has been made, the investor decides to
sell the stock and finds a potential buyer who is subject to income tax at 10% and capital gains tax at 35%. The
potential buyer is prepared to buy the stock provided she will obtain a net redemption yield of at least 8% per
annum effective.
(a) Calculate the maximum price (per £100 nominal) which is the original investor can expect to obtain from
the potential buyer.
(b) Calculate the net effective annual redemption yield (to the nearest 1% per annum effective) that will be
obtained by the original investor if the loan is sold to the buyer at the price determined in (ii)(a).
(Institute/Faculty of Actuaries Examinations, April 2007) [5+10=15]
18. A loan of nominal amount of £100,000 is to be issued bearing interest payable quarterly in arrear at a rate of 8%. p.a.
Capital is to be redeemed at £105% on a coupon date between 15 and 20 years after the date of issue, inclusive, the
date of redemption being at the option of the borrower.
(i) An investor who is liable to income tax at 40% and tax on capital gains at 30% wishes to purchase the entire
loan at the date of issue. What price should she pay to ensure a net effective yield of at least 6% p.a?
Exactly 10 months after issue the loan is sold to an investor who pays income tax at 20% and capital gains at
30%. Calculate the price this investor should pay to achieve a yield of 6% p.a. on the loan:
(a) assuming redemption at the earliest possible date
(b) assuming redemption at the latest possible date.
(ii) Explain which price this second investor should pay to achieve a yield of at least 6% p.a.
(Institute/Faculty of Actuaries Examinations, April 1997) [17]
19. A fixed interest security pays coupons of 8% per annum half yearly on 1 January and 1 July. The security will be
redeemed at par on any 1 January from 1 January 2017 to 1 January 2022 inclusive, at the option of the borrower.
An investor purchases a holding of the security on 1 May 2011, at a price which gave him a net yield of at least
6% per annum effective. The investor pays tax at 30% on interest income and 25% on capital gains.
On 1 April 2013 the investor sold the holding to a fund which pays no tax at a price to give the fund a gross yield of
at least 7% per annum effective.
(i) Calculate the price per £100 nominal at which the investor bought the security.
(ii) Calculate the price per £100 nominal at which the investor sold the security.
(iii) Show that the effective net yield that the investor obtained on the investment was between 8% and 9% per
annum. (Institute/Faculty of Actuaries Examinations, April 2013) [5+3+6=14]
20. A loan of nominal amount £100,000 is to be issued bearing coupons payable quarterly in arrear at a rate of 7%
per annum. Capital is to be redeemed at 108% on a coupon date between 15 and 20 years after the date of issue,
inclusive. The date of redemption is at the option of the borrower.
An investor who is liable to income tax at 25% and capital gains tax at 35% wishes to purchase the entire loan at the
date of issue.
Calculate the price which the investor should pay to ensure a net effective yield of at least 5% per annum.
(Institute/Faculty of Actuaries Examinations, April 2008) [8]
21. A fixed-interest bond pays annual coupons of 5% per annum in arrear on 1 March each year and is redeemed at par
on 1 March 2025. On 1 March 2007, immediately after the payment of the coupon then due, the gross redemption
yield was 3.158% per annum effective.
(i) Calculate the price of the bond per £100 nominal on 1 March 2007.
On 1 March 2012, immediately after the payment of the coupon then due, the gross redemption yield on the bond
was 5% per annum.
(ii) State the new price of the bond per £100 nominal on 1 March 2012.












































































































Page 90 Section 3 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
A tax-free investor purchased the bond on 1 March 2007, immediately after payment of the coupon then due, and
sold the bond on 1 March 2012 immediately after payment of the coupon then due.
(iii) Calculate the gross annual rate of return achieved by the investor over this period.
(iv) Explain, without doing any further calculations, how your answer to part (iii) would change if the bond were
due to be redeemed on 1 March 2035 (rather than 1 March 2025). You may assume that the gross redemption
yield at both the date of purchase and the date of sale remains the same as in parts (i) and (ii) above.
(Institute/Faculty of Actuaries Examinations, April 2012) [3+1+2+3=9]
22. A fixed interest security pays coupons of 4% per annum, half-yearly in arrear and will be redeemed at par in exactly
ten years.
(i) Calculate the price per £100 nominal to provide a gross redemption yield of 3% per annum convertible half
yearly.
(ii) Calculate the price, 91 days later, to provide a net redemption yield of 3% per annum convertible half-yearly, if
income tax is payable at 25%. (Institute/Faculty of Actuaries Examinations, September 2013) [2+2=4]
23. An investor is considering the purchase of two government bonds, issued by two countries A and B respectively,
both denominated in euro.
Both bonds provide a capital repayment of e100 together with a final coupon payment ofe6 in exactly one year. The
investor believes that he will receive both payments from the bond issued by country A with certainty. He believes
that there are four possible outcomes for the bond from country B, shown in the table below.
Outcome Probability
No coupon or capital payment 0.1
Capital payment received, but no coupon payment received 0.2
50% of capital payment received, but no coupon payment received 0.3
Both coupon and capital payments received in full 0.4
The price of the bond issued by country A is e101.
(i) Calculate the price of the bond issued by country B to give the same expected return as that for the bond issued
by country A.
(ii) Calculate the gross redemption yield from the bond issued by country B assuming that the price is as calculated
in part (i).
(iii) Explain why the investor might require a higher expected return from the bond issued by country B than from
the bond issued by country A. (Institute/Faculty of Actuaries Examinations, September 2013) [3+1+2=6]
3 Equity calculations
3.1 Calculating the yield. Owners of equities receive a stream of dividend payments. The sizes of these
dividend payments are uncertain and depend on the performance of the company. In the UK they are often
paid six-monthly with an interim and a final dividend.
Consider the following special case of annual dividends (the results for six-monthly dividend payments are
obtained in the same manner). Suppose the present price of a share just after the annual dividend has been paid
is P . Suppose dividends are paid annually and dk is the estimated gross dividend in k years time. Then the
cash flow is displayed in the following table:
Time 0 1 2 3 . . .
Cash flow P d1 d2 d3 . . .
The yield i of the share given by:
P =
1X
k=1
dk
(1 + i)k
If dividends are assumed to grow by a constant proportion each year, then the cash flow is:
Time 0 1 2 3 . . .
Cash flow P d1 d1(1 + g) d1(1 + g)2 . . .
and so the yield i of the share given by:
P =
1X
k=1
d1(1 + g)k1
(1 + i)k
=
d1
i g












































































































5 Bonds, Equities and Inflation Sep 27, 2016(9:51) Section 4 Page 91
Example 3.1a. The next dividend on an equity is due in 6 months and is expected to be 6 pence. Subsequent
dividends will be paid annually. Assume the second, third and fourth dividends are each 10% greater than their
predecessor. Assume also that dividends subsequent to the fourth grow at the rate of 5% per annum. Calculate the
NPV of the dividend stream assuming the investor’s yield is 12% per annum.
Solution. The cash flow is as follows:
Time 0 1/2 3/2 5/2 7/2 9/2 . . .
Cash flow (in pence) 0 6 6⇥ 1.1 6⇥ 1.12 6⇥ 1.13 6⇥ 1.13 ⇥ 1.05 . . .
Let ⌫ = 1/(1 + i) = 1/1.12. Hence
NPV = 6⌫1/2 + 6(1.1)⌫3/2 + 6(1.1)2⌫5/2 + 6(1.1)3⌫7/2 +
1X
k=5
6(1.1)3(1.05)k4⌫(2k1)/2
= 6⌫1/2 + 6(1.1)⌫3/2 + 6(1.1)2⌫5/2 + 6(1.1)3⌫7/2 + 6(1.1)3⌫9/2
1.05
1 1.05⌫
3.2 Other measures. Estimating future dividends involves a large amount of subjectivity. Consequently,
other numerical measures for comparing equities are usually used—mainly the dividend yield and the P/E
ratio.
The dividend yield (often described simply as the yield) is given by
Dividend yield =
gross annual dividend per share
current share price
The dividend yield is an indication of the current level of income from a share.
The P/E ratio or price/earnings ratio is given by
P/E ratio =
current share price
net annual profit per share
The net annual profit per share is the last annual profits of the company divided by the number of shares. For
example, suppose the share price is 400p and the earnings per share is 20p. Then the P/E ratio is 20; i.e. the
shares are on a multiple of 20, or the shares sell at 20 times earnings.
Yields and P/E ratios move in opposite directions. If the share price rises, the yield falls and the P/E ratio rises.
Typically a high P/E ratio suggests a growth company and a low P/E ratio suggests a company with more static
profits. (Belief that a company is a growth company implies belief in higher dividends in the future and hence
a higher share price.) Further information about these measures is given in other courses.
4 Real and money rates of interest
4.1 Idea. The real rate of interest is the rate of interest after allowing for inflation; the money rate of interest
is the rate of interest without allowing for inflation.
Example 4.1a. An amount of £100 grows to £120 in 1 year. Inflation is 5% p.a. Find the money and real rates of
interest.
Solution. The money rate of interest is 20% p.a. In real terms, £100 grows to £120/1.05 = £114.29 in 1 year. Hence
the real rate of interest is 14.29% p.a.
Suppose iM denote the money rate of interest, iR denotes the real rate of interest and q denotes the inflation
rate, then
1 + iM = (1 + q)(1 + iR) or, equivalently, iR =
iM q
1 + q
(4.1a)
Note that iM = iR(1 + q) + q ⇡ iR + q if iR and q are small.
Suppose £1 grows to £A(t) over the interval (0, t). The amount £A(t) may be worth only £A⇤(t) at time 0 after
allowing for inflation, where A⇤(t) < A(t). The rate of interest needed for £1 to grow to £A⇤(t) is the real rate
of interest and is, in this case, less than the money rate of interest.
The term deflation means that the rate of inflation is negative and A⇤(t) > A(t). The real rate of interest is
then higher than the money rate of interest.












































































































Page 92 Section 4 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
Example 4.1b. Find the real rate of interest for the following cash flow using the inflation index Q(t):
Time 0 1 2 3
Cash flow a0 = 100 a1 = 8 a2 = 8 a3 = 108
Inflation Index, Q(t) 150 156 166 175
Solution. The cash flow should be adjusted as follows:
Cash flow a0 = 100 a1 = 8⇥ Q(0)Q(1) a2 = 8⇥ Q(0)Q(2) a3 = 108⇥ Q(0)Q(3)
This leads to a real rate of interest iR given by
100 =
8⇥ 150
156
⌫R +
8⇥ 150
166
⌫2R +
108⇥ 150
175
⌫3R where ⌫R =
1
1 + iR
and hence iR = 2.63
In general, consider the cash flow
Time 0 t1 t2 . . . tn
Cash flow 0 Ct1 Ct2 . . . Ctn
Inflation Index Q(0) Q(t1) Q(t2) . . . Q(tn)
The equation for the yield is given by
nX
k=1
Ctk
Q(0)
Q(tk)
⌫tk = 0 which leads to
nX
k=1
Ctk
Q(tk)
⌫tk = 0 (4.1b)
where, as usual, ⌫ = 1/(1 + i) and i is the yield. Equation (4.1b) shows that the yield does not depend on the
base value of the inflation index.
4.2 Special case: constant inflation assumption. Suppose we assume that the rate of inflation will be q%
per annum. Consider the following cash flow:
Time 0 t1 t2 . . . tn
Cash flow 0 Ct1 Ct2 . . . Ctn
Inflation Index 1 (1 + q)t1 (1 + q)t2 . . . (1 + q)tn
The real yield iR is given by
nX
k=1
Ctk
(1 + q)tk
⌫tkR = 0 where ⌫R =
1
1 + iR
The money yield iM is given by
nX
k=1
Ctk⌫
tk
M = 0 where ⌫M =
1
1 + iM
Note that ⌫R = (1 + q)⌫M which is just another version of equation (4.1a): (1 + iR)(1 + q) = 1 + iM .
4.3 Inflation linked payments. Suppose we have the series of payments c = {ct1 , . . . , ctn} and the payment
ctk due at time tk is adjusted in line with some inflation index Q(t). This means that the payment at time tk
will become
cq,tk = ctk
Q(tk)
Q(0)
Hence the NPV of the series of inflation adjusted payments cq = {cq,1, . . . , cq,n} using the rate of interest i is
NPVi(cq) =
nX
k=1
cq,tk⌫
tk where ⌫ =
1
1 + i
Consider adjusting the interest rate for inflation. If iR denotes the real rate of interest corresponding to the
money rate of interest iM , then 1 + iM = (1 + iR)Q(1)/Q(0) and in general (1 + iM )tk = (1 + iR)tkQ(tk)/Q(0).
This implies that even if iM is constant over time, iR is a function of t.
This result generalises as follows: suppose we have both a discrete payment stream, c, and a continuous
payment stream, ⇢. If NPVi(c, ⇢) denotes the net present value at time 0 using interest rate i, then we know
that
NPVi(c, ⇢) =
X
k
ctk⌫
tk +
Z 1
0
⇢(u)⌫u du












































































































5 Bonds, Equities and Inflation Sep 27, 2016(9:51) Exercises 5 Page 93
Now suppose every cash payment is adjusted to allow for inflation at the rate of q per unit time, then
NPVi(cq, ⇢q) =
X
k
ctk (1 + q)
tk⌫tk +
Z 1
0
⇢(u)(1 + q)u⌫u du
Using the relation 1 + iM = (1 + q)(1 + iR), or ⌫R = (1 + q)⌫M , shows that
NPViM (cq, ⇢q) = NPViR(c, ⇢) where (1 + iM ) = (1 + iR)(1 + q). (4.3a)
The NPV using the real rate of interest and payments not adjusted for inflation is the same as the NPV of
inflation adjusted payments using the money rate of interest.
If i0 is the yield of a project when inflation is ignored, then NPVi0 (c, ⇢) = 0. Using equation (4.3a) implies that
NPVi1 (cq, ⇢q) = 0 where (1 + i1) = (1 + i0)(1 + q). Hence the yield of the project when payments are adjusted
for inflation is i1 = (1 + q)i0 + q.
Projects which appear unprofitable when interest rates are high, may become profitable when only a small
allowance is made for inflation.
4.4 Index linked bonds. Clearly, if the payments on a bond are linked precisely to the inflation index, then
the real yield of the bond will be the same whatever happens to the the rate of inflation. However, sometimes
the calculation is more complicated than this. Index linked UK government securities have coupons linked to
the value of the inflation index 8 months before the payment is made. The real yield must be calculated using
the inflation index at the times the actual payments are made.
Example 4.4a. Suppose an index-linked bond with face value f is redeemable after n years with redemption
value C. The coupon rate is r% per annum convertible semi-annually. The coupon payments are adjusted by using
the inflation index Q. If the current price of the bond is P , find an equation for the money yield iM .
Solution. The cash flow is as follows:
Time 0 1/2 1 3/2 . . . (2n 1)/2 n
Cash flow (unadjusted) P fr/2 fr/2 fr/2 . . . fr/2 fr/2 + C
Inflation Index Q(0) Q(1/2) Q(1) Q(3/2) . . . Q(n 1/2) Q(n)
The equation for the money yield, iM is
P =
2nX
k=1
fr
2
Q(k/2)
Q(0)
⌫k/2M + C
Q(n)
Q(0)
⌫nM
4.5
Summary.
• Equities: calculating the yield from forecasted dividends; the dividend yield; the P/E ratio.
• Real rate of interest and money rate of interest: (1 + iM ) = (1 + q)(1 + iR).
• Adjusting coupon payments using an inflation index.
5 Exercises (exs5-2.tex)
1. On the assumption of constant future price inflation of 2.5% per annum, an investor has purchased a UK government
index-linked bond at a price to provide a real rate of return of 2% per annum effective if held to redemption. Explain
why the real yield to redemption will be lower if the actual constant rate of inflation is higher than the assumed 2.5%
per annum. (Institute/Faculty of Actuaries Examinations, April, 2002) [2]
2. An investor has earned a money rate of return from a portfolio of bonds in a particular country of 1% per annum
effective over a period of 10 years. The country has experienced deflation (negative inflation) of 2% per annum
effective during the period. Calculate the real rate of return per annum over the 10 years.
(Institute/Faculty of Actuaries Examinations, September 2005) [2]
3. Dividends payable on a certain share are assumed to increase at a compound rate of 3% per half-year. A dividend
of £5 per share has just been paid. Dividends are paid half-yearly. Find the value of the share to the nearest £1,
assuming an effective rate of interest of 8% p.a. (Institute/Faculty of Actuaries Examinations, May 1999) [3]
4. Dividends payable on a share are assumed to increase at a compound rate of 21/2% per half-year. A dividend of £2
per share has just been paid. Dividends are payable half-yearly. Find the value of the share to the nearest £1 assuming
an effective rate of interest of 7% per annum. (Institute/Faculty of Actuaries Examinations, April 1999) [3]












































































































Page 94 Exercises 5 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
5. (i) Define the characteristics of a government index-linked bond.
(ii) Explain why most index-linked securities issued carry some inflation risk in practice.
(Institute/Faculty of Actuaries Examinations, September 2003) [2+2=4]
6. An investor has invested a sum of £10m. Exactly one year later, the investment is worth £11.1m. An index of prices
has a value of 112 at the beginning of the investment and 120 at the end of the investment. The investor pays tax at
40% on all money returns from investment. Calculate:
(a) The money rate of return per annum before tax.
(b) The rate of inflation.
(c) The real rate of return per annum after tax. (Institute/Faculty of Actuaries Examinations, September 2006) [4]
7. An investor purchased a holding of ordinary shares two months before payment of the next dividend was due.
Dividends are paid annually and it is expected that the next dividend will be a net amount of 12p per share. The
investor anticipates that dividends will grow at a constant rate of 4% per annum in perpetuity.
Calculate the price per share that the investor should pay to obtain a net return of 7% per annum effective.
(Institute/Faculty of Actuaries Examinations, April 2003) [4]
8. An investor is considering the purchase of 100 ordinary shares in a company. Dividends from the share will be paid
annually. The next dividend is due in one year and is expected to be 8p per share. The second dividend is expected
to be 8% greater than the first dividend and the third dividend is expected to be 7% greater than the second dividend.
Thereafter dividends are expected to grow at 5% per annum compound in perpetuity.
Calculate the present value of this dividend stream at a rate of interest of 7% per annum effective.
(Institute/Faculty of Actuaries Examinations, April 2000) [5]
9. A loan of £20,000 was issued, and was repaid at par after 3 years. Interest was paid on the loan at the rate of 10%
per annum, payable annually in arrears. The value of the retail price index at various times was as follows:
At the date the loan was made 245.0
One year later 268.2
Two years later 282.2
Three years later 305.5
Calculate the real rate of return earned on the loan. (Institute/Faculty of Actuaries Examinations, April 1997) [5]
10. An equity pays annual dividends and the next dividend, payable in 10 months time, is expected to be 5p. Thereafter,
dividends are expected to grow by 3% per annum compound. Inflation is expected to be 2% per annum throughout.
Calculate the value of the equity assuming that the real yield obtained is 2.5% per annum, convertible half-yearly.
(Institute/Faculty of Actuaries Examinations, September 2002) [5]
11. An investment which pays annual dividends is bought immediately after a dividend payment has been made. Divi-
dends are expected to grow at a compound rate of g per annum and price inflation is expected to be at a rate of e per
annum. If the dividend payment expected at the end of the first year is d per unit invested and if it is assumed the
investment is held indefinitely, show that the expected real rate of return per annum per unit invested is given by
i0 = (d + g e)/(1 + e). (Institute/Faculty of Actuaries Examinations, September 1999) [5]
12. An ordinary share pays annual dividends. The next dividend is expected to be 10p per share and is due in exactly
9 months’ time. It is expected that subsequent dividends will grow at a rate of 5% per annum compound and that
inflation will be 3% per annum. The price of the share is 250p and dividends are expected to continue in perpetuity.
Calculate the expected effective real rate of return per annum for an investor who purchases the shares.
(Institute/Faculty of Actuaries Examinations, September 2000) [5]
13. A share has a price of 750p and has just paid a dividend of 30p. The share pays annual dividends. Two analysts agree
that the next expected dividend will be 35p but differ in their estimates of long-term dividend growth. One analyst
expects no dividend growth and the other analyst expects constant dividend growth of 10% per annum, after the next
dividend payment.
(i) Derive a formula for the value of a share in terms of the next expected dividend (d1), a constant rate of dividend
growth (g), and the rate of return from the share (i).
(ii) Calculate the rates of return the two analysts expected from the share.
(Institute/Faculty of Actuaries Examinations, September 1999) [3+3=6]
14. An ordinary share pays annual dividends. The next dividend is expected to be 5p per share and is due in exactly
3 months’ time. It is expected that subsequent dividends will grow at a rate of 4% per annum compound and that
inflation will be 1.5% per annum. The price of the share is 125p and dividends are expected to continue in perpetuity.
Calculate the effective real rate of return per annum for an investor who purchases the share.
(Institute/Faculty of Actuaries Examinations, September 2003) [7]












































































































5 Bonds, Equities and Inflation Sep 27, 2016(9:51) Exercises 5 Page 95
15. An investor is considering investing in the shares of a particular company.
The shares pay dividends every 6 months, with the next dividend being due in exactly 4 months’ time. The next
dividend is expected to be d1, the purchase price of a single share is P , and the annual effective rate of return
expected from the investment is 100i%. Dividends are expected to grow at a rate of 100g% per annum from the level
of d1 where g < i. Dividends are expected to be paid in perpetuity.
(i) Show that
P =
d1(1 + i)1/6
(1 + i)1/2 (1 + g)1/2
(ii) The investor delays purchasing the shares for exactly 2 months at which time the price of a single share is £18,
100g% = 4% and d1 = £0.50. Calculate the annual effective rate of return expected by the investor to the nearest
1%. (Institute/Faculty of Actuaries Examinations, April 2001) [7]
16. (i) Describe the characteristics of ordinary shares (equities).
(ii) The prospective dividend yield from an ordinary share is defined as the next expected dividend divided by the
current price. A particular share is expected to provide a real rate of return of 5% per annum effective. Inflation
is expected to be 2% per annum. The growth rate of dividends from the share is expected to be 3% per annum
compound. Dividends will be paid annually and the next dividend is expected to be paid in 6 months’ time.
Calculate the prospective dividend yield from the share.
(Institute/Faculty of Actuaries Examinations, September 2004) [4+4=8]
17. The following investments were made on 15 January 1993.
Investment A: £10,000 was placed in a special savings account with a 5-year term. Invested money was accumulated
at 31/2% per annum effective for the first year, 41/2% per annum effective for the second year, 51/2% per annum
effective for the third year, 61/2% per annum effective for the fourth year, and 71/2% per annum effective for the fifth
year.
Investment B: £10,000 was placed in a zero coupon bond with a 5-year term in which the redemption proceeds were
the amount invested multiplied by the ratio of the Retail Price Index for the month two months prior to that in which
redemption fell to the Retail Price Index for the month two months prior to the date of investment; this amount was
further increased by 23/4% compound for each complete year money was invested.
Investment C: An annuity payable annually in arrears for 5 years was purchased for £10,000 to yield 61/4% per
annum effective.
The Retail Price Index at various times was as follows:
November 1992 237.6
January 1993 240.0
January 1994 250.0
January 1995 264.4
January 1996 266.6
January 1997 270.4
November 1997 274.0
January 1998 275.6
(i) What is the amount of the annual income yielded by the annuity?
(ii) Calculate the real rate of return per annum earned on investment A and on investment B over the period 15 Jan-
uary 1993 to 15 January 1998. The investor is not liable for tax.
(iii) Determine which of the three investments yielded the highest real rate of return per annum over the period
15 January 1993 to 15 January 1998. (Institute/Faculty of Actuaries Examinations, April 1998) [1+4+3=8]
18. An investor purchases a bond 3 months after issue. The bond will be redeemed at par ten years after issue and pays
coupons of 6% per annum annually in arrears. The investor pays tax of 25% on both income and capital gains (with
no relief for indexation).
(i) Calculate the purchase price of the bond per £100 nominal to provide the investor with a rate of return of 8% per
annum effective.
(ii) The real rate of return expected by the investor from the bond is 3% per annum effective. Calculate the annual
rate of inflation expected by the investor. (Institute/Faculty of Actuaries Examinations, April, 2002) [6+2=8]
19. In a particular country, income tax and capital gains tax are both collected on 1 April each year in relation to gross
payments made during the previous 12 months.
A fixed interest bond is issued on 1 January 2003 with term of 25 years and is redeemable at 110%. The security
pays a coupon of 8% per annum, payable half-yearly in arrears.
An investor, who is liable to tax on income at a rate of 25% and on capital gains at a rate of 30%, bought £10,000
nominal of the stock at issue for £9,900.
(i) Assuming an inflation rate of 3% per annum over the term of the bond, calculate the net real yield obtained by
the investor if they hold the stock to redemption.
(ii) Without doing any further calculation, explain how and why your answer to (i) would alter if tax were collected
on 1 June instead of 1 April each year. (Institute/Faculty of Actuaries Examinations, April 2004) [9+2=11]












































































































Page 96 Exercises 5 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
20. An ordinary share pays annual dividends. The next dividend is due in exactly 8 months’ time. This dividend is
expected to be £1.10 per share. Dividends are expected to grow at a rate of 5% per annum compound from this level
and are expected to continue in perpetuity. Inflation is expected to be 3% per annum. The price of the share is £21.50.
Calculate the expected effective annual real rate of return for an investor who purchases the share.
(Institute/Faculty of Actuaries Examinations, April 2007) [7]
21. A government issued a number of index-linked bonds on 1 June 2000 which were redeemed on 1 June 2002. Each
bond had a nominal coupon rate of 3% per annum, payable half-yearly in arrears, and a nominal redemption price
of 100. The actual coupon and redemption payments were indexed according to the increase in the retail price index
between 6 months before the bond issue date and 6 months before the coupon or redemption dates.
The values of the retail price index in the relevant months were:
Date Retail Price Index
December 1999 100
June 2000 102
December 2000 107
June 2001 111
December 2001 113
June 2002 118
(i) An investor purchased £100,000 nominal at the issue date, and held it until it was redeemed. The issue price
was £94 per £100 nominal.
Calculate all the investor’s cash flows from this investment, before tax.
(ii) The investor is subject to income tax at a rate of 25% and capital gains tax at a rate of 35%. When calculating
the amount of capital gain which is subject to tax, the price paid for the investment is indexed in line with the
increase in the retail price index between the month in which the investment was purchased and the month in
which it was redeemed.
(a) Calculate the investor’s capital gains tax liability in respect of this investment.
(b) Calculate the net effective yield per annum obtained by the investor on these bonds.
(Institute/Faculty of Actuaries Examinations, April 2004) [3+6=9]
22. At time t = 0 an investor purchased an annuity-certain which paid her £10,000 per annum annually in arrear for three
years. The purchase price paid by the investor was £25,000.
The value of the retail price index at various times was as shown in the table below:
Time t (years) t = 0 t = 1 t = 2 t = 3
Retail price index 170.7 183.3 191.0 200.9
(i) Calculate, to the nearest 0.1%, the following effective rates of return per annum achieved by the investor from
her investment in the annuity:
(a) the real rate of return; (b) the money rate of return.
(ii) By considering the average rate of inflation over the three-year period, explain the relationship between your
answers in (a) and (b) in (i). (Institute/Faculty of Actuaries Examinations, April 2005) [7+2=9]
23. An ordinary share pays annual dividends. A dividend of 25p per share has just been paid. Dividends are expected to
grow by 2% next year and by 4% the following year. Thereafter, dividends are expected to grow at 6% per annum
compound in perpetuity.
(i) State the main characteristics of ordinary shares.
(ii) Calculate the present value of the dividend stream described above at a rate of interest of 9% per annum effective
from a holding of 100 ordinary shares.
(iii) An investor buys 100 shares in (ii) for £8.20 each. He holds them for two years and receives the dividends
payable. He then sells them for £9 immediately after the second dividend is paid.
Calculate the investor’s real rate of return if the inflation index increases by 3% during the first year and by 3.5%
during the second year assuming dividends grow as expected.
(Institute/Faculty of Actuaries Examinations, April 2006) [4+4+4=12]
24. On 9 October 1997 an investor, not liable to tax, had the choice of purchasing either:
(A) 71/2% Treasury Stock at a price of £107 per £100 nominal repayable at par in 2006.
(B) 2% Index Linked Treasury Stock 2006 at a price of £203 per £100 nominal. The RPI base figure for indexing
was 69.5 and the index applicable to the next coupon (payable on 8 April 1998) was 158.5. (This may also be
taken as the latest known value of the RPI on 9 October.)
Assume both stocks are redeemable on 8 October 2006; and that both coupons are paid half-yearly in arrear on
8 April and 8 October. Assuming that the RPI will grow continuously at a rate of 2.5% per annum from its latest
known value,
(i) Show that the real yield from stock (A) as at 9 October 1997 is 4% per annum effective.
(ii) By considering the series of future receipts from stock (B), derive a formula for the present value of £100
nominal of that stock.
(iii) Calculate the real yield as at 9 October 1997 from stock (B).
(Institute/Faculty of Actuaries Examinations, September 1998) [5+6+2=13]












































































































5 Bonds, Equities and Inflation Sep 27, 2016(9:51) Exercises 5 Page 97
25. (i) Describe the characteristics of an index-linked government bond.
(ii) On 1 July 2002, the government of a country issued an index-linked bond of term 7 years. Coupons are paid
half-yearly in arrears on 1 January and 1 July each year. The annual nominal coupon is 2%. Interest and capital
payments are indexed by reference to the value of an inflation index with a time lag of 8 months.
You are given the following values of the inflation index:
Date Inflation index
November 2001 110.0
May 2002 112.3
November 2002 113.2
May 2003 113.8
The inflation index is assumed to increase continuously at the rate of 21/2% per annum effective from its value
in May 2003. An investor, paying tax at the rate of 20% on coupons only, purchased the stock on 1 July 2003,
just after a coupon payment has been made.
Calculate the price to this investor such that a real net yield of 3% per annum convertible half-yearly is obtained
and assuming that the investor holds the bond to maturity.
(Institute/Faculty of Actuaries Examinations, September 2006) [3+10=13]
26. On 15 March 1996, the government of a country issued an index-linked bond of term 6 years. Coupons are payable
half-yearly in arrears, and the annual nominal coupon rate is 3%.
Interest and capital payments are indexed by reference to the value of an inflation index with a time lag of 8 months.
A tax-exempt investor purchased the stock at £111 per £100 nominal on 16 September 1999, just after the coupon
payment had been made.
You are given the following values of the inflation index:
Date Inflation Index
July 1995 110.5
March 1996 112.1
July 1999 126.7
September 1999 127.4
(i) Calculate the amount of the coupon payment per £100 nominal stock on 15 March 2000.
(ii) Calculate the effective real annual yield to the investor on 16 September 1999. You should assume that the
inflation index will increase from its value in September 1999 at the rate of 4% per annum effective.
(iii) Without doing any further calculations, explain how your answer to (ii) would alter, if at all, if the inflation
index for July 1995 had been more than 110.5. (Institute/Faculty of Actuaries Examinations, April 2001) [17]
27. On 15 May 1997, the government of a country issued an index-linked bond of term 15 years. Coupons are payable
half-yearly in arrears, and the annual nominal coupon rate is 4%.
Interest and capital payments are indexed by reference to the value of a retail price index with a time lag of 8 months.
The retail price index value in September 1996 was 200 and in March 1997 was 206.
The issue price of the bond was such that, if the retail price index were to increase at a rate of 7% p.a. from
March 1997, a tax exempt purchaser of the bond at the issue date would obtain a real yield of 3% p.a. convert-
ible half-yearly.
(i) (a) Derive the formula for the price of the bond at issue to a tax-exempt investor.
(b) Show that the issue price of the bond is £111.53%.
(ii) An investor purchases a bond at a price calculated in (i) and holds it to redemption. The retail price index
increases continuously at 5% per annum from March 1997. A new tax is introduced such that the investor pays
tax at 40% on any real capital gain, where the real capital gain is the difference between the redemption money
and the purchase price revalued according to the retail price index to the redemption date. Tax is only due if the
real capital gain is positive.
Calculate the real annual yield convertible half-yearly actually obtained by the investor.
(Institute/Faculty of Actuaries Examinations, September 1997) [12+7=19]
28. The shares of a company currently trade at £2.60 each, and the company has just paid a dividend of 12p per share.
An investor assumes that dividends will be paid annually in perpetuity and will grow in line with a constant rate of
inflation. The investor estimates the assumed inflation rate from equating the price of the share with the present value
of all estimated gross dividend payments using an effective interest rate of 6% per annum.
(i) Calculate the investor’s estimation of the effective inflation rate per annum based on the above assumptions.
(ii) Suppose that the actual inflation rate turns out to be 3% per annum effective over the following 12 years, but
that all the investor’s other assumptions are correct.
Calculate the investor’s real rate of return per annum from purchase to sale, if she sold the shares after 12 years
for £5 each immediately after a dividend has been paid. You may assume that the investor pays no tax.
(Institute/Faculty of Actuaries Examinations, April 2008) [4+6=10]












































































































Page 98 Exercises 5 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
29. An investor is interested in purchasing shares in a particular company. The company pays annual dividends, and a
dividend of 30 pence per share has just been made. Future dividends are expected to grow at the rate of 5% per annum
compound.
(i) Calculate the maximum price per share that the investor should pay to give an effective return of 9% per annum.
(ii) Without doing any further calculations, explain whether the maximum price paid will be higher, lower or the
same if:
(a) after consulting the managers of the company, the investor increases his estimate of the rate of growth of
future dividends to 6% per annum.
(b) as a result of a government announcement, the general level of future price inflation in the economy is now
expected to be 2% per annum higher than previously assumed.
(c) general economic uncertainty means that, whilst the investor still estimates future dividends will grow at
5% per annum, he is now much less sure about the accuracy of this assumption.
(Institute/Faculty of Actuaries Examinations, April 2013) [4+6=10]
30. An ordinary share pays dividends on each 31 December. A dividend of 35p per share was paid on 31 December 2011.
The dividend growth is expected to be 3% in 2012, and a further 5% in 2013. Thereafter, dividends are expected to
grow at 6% per annum compound in perpetuity.
(i) Calculate the present value of the dividend stream described above at a rate of interest of 8% per annum effective
for an investor holding 100 shares on 1 January 2012.
An investor buys 100 shares for £17.20 each on 1 January 2012. He expects to sell the shares for £18 on 1 Jan-
uary 2015.
(ii) Calculate the investor’s expected real rate of return. You should assume that dividends grow as expected and
use the following values of the inflation index:
Year 2012 2013 2014 2015
Inflation index at start of year: 110.0 112.3 113.2 113.8
(Institute/Faculty of Actuaries Examinations, April 2012) [4+5=9]
31. Mrs Jones invests a sum of money for her retirement which is expected to be in 20 years’ time. The money is
invested in a zero coupon bond which provides a return of 5% per annum effective. At retirement, the individual
requires sufficient money to purchase an annuity certain of £10,000 per annum for 25 years. The annuity will be
paid monthly in arrear and the purchase price will be calculated at a rate of interest of 4% per annum convertible
half-yearly.
(i) Calculate the sum of money the individual needs to invest at the beginning of the 20 year period.
The index of retail prices has a value of 143 at the beginning of the 20 year period and 340 at the end of the 20 year
period.
(ii) Calculate the annual effective real return the individual would obtain from the zero coupon bond.
The government introduces a capital gains tax on zero coupon bonds of 25% of the nominal capital gain.
(iii) Calculate the net annual effective real return to the investor over the 20 year period before the annuity com-
mences.
(iv) Explain why the investor has achieved a negative real rate of return despite capital gains tax only being a tax on
the profits from an investment. (Institute/Faculty of Actuaries Examinations, September 2013) [5+2+3+2=12]
32. A 91-day government bill is purchased for £95 at the time of issue and is redeemed at the maturity date for £100.
Over the 91 days, an index of consumer prices rises from 220 to 222.
Calculate the effective real rate of return. (Institute/Faculty of Actuaries Examinations, September 2008) [3]
33. A tax advisor is assisting a client in choosing between 3 types of investment. The client pays tax at 40% on income
and 40% on capital gains.
Investment A requires the investment of £1 million and provides an income of £0.1 million per year in arrears for
10 years. Income tax is deducted at source. At the end of the 10 years, the investment of £1 million is returned.
In investment B, the initial sum of £1 million accumulates at the rate of 10% per annum compound for 10 years.
At the end of 10 years, the accumulated value of the investment is returned to the investor after deduction of capital
gains tax.
Investment C is identical to investment B except that the initial sum is deemed, for tax purposes, to have increased
in line with the index of consumer prices between the date of the investment and the end of the 10 year period. The
index of consumer prices is expected to increase by 4% per annum compound over the period.
(i) Calculate the net rate of return from each of the investments.
(ii) Explain why the expected rate of return is higher for Investment C than for Investment B and is higher for
Investment B than for Investment A. (Institute/Faculty of Actuaries Examinations, September 2008) [7+3=10]












































































































5 Bonds, Equities and Inflation Sep 27, 2016(9:51) Exercises 5 Page 99
34. An ordinary share pays annual dividends. The next dividend is expected to be 6p per share and is due in exactly
6 months’ time. It is expected that subsequent dividends will grow at a rate of 6% per annum compound and that
inflation will be 4% per annum. The price of the share is 175p and dividends are expected to continue in perpetuity.
Calculate the expected effective real rate of return per annum for an investor who purchases the share.
(Institute/Faculty of Actuaries Examinations, April 2015) [6]
35. (i) State the characteristics of an equity.
An investor was considering investing in the shares of a particular company on 1 August 2014. The investor assumed
that the next dividend would be payable in exactly one year and would be equal to 6 pence per share. Thereafter,
dividends will grow at a constant rate of 1% per annum and are assumed to be paid in perpetuity. All dividends will
be taxed at a rate of 20%. The investor requires a net rate of return from the shares of 6% per annum effective.
(ii) Derive and simplify as far as possible a general formula which will allow you to determine the value of a share
for different values of:
• the next expected dividend;
• the dividend growth rate;
• the required rate of return;
• the tax rate.
(iii) Calculate the value of one share to the investor.
The company announces some news that makes the shares more risky.
(iv) Explain what would happen to the value of the share, using the formula derived in part (ii).
The investor bought 1,000 shares on 1 August 2014 for the price calculated in part (iii). He received the dividend
of 6 pence on 1 August 2015 and paid the tax due on the dividend. The investor then sold the shares immediately
for 120 pence. Capital gains tax was charged on all gains of at a rate of 25%. On 1 August 2014, the index of retail
prices was 123. On 1 August 2015, the index of retail prices was 126.
(v) Determine the net real return earned by the investor.
(Institute/Faculty of Actuaries Examinations, September 2015, corrected) [4+5+2+3=14]


































































































































Excercise5



1.The bond would have an (eight month) indexation lag. This means that if future

inflation is higher than the assumption, the investor would not receive inflation

compensation for the whole of the time between the purchase date and the date on

which a payment would be received.









5. If f = the rate of inflation; j = the real rate of return and i = the money rate of return,

then j = ( i-f )/(1 + f ). In this case, f = 2%, i= 1% and therefore j = 3.061%
ST334 Actuarial Models
Chapter 5: Bonds, Equities and Inflation
Learning objectives:
To demonstrate an understanding of the concepts and be able to perform calcula-
tions relating to bond pricing using Makeham’s formula, equity calculations and
relationship between money and real interest rates.
1. Bond calculations
• Makeham’s formula:
P= f r ·a(m)n ,i +
C
(1+ i)n
=
f r
m
·a
nm , i
(m)
m
+
C
(1+ i(m)m )
nm
• Gross interest yield/direct yield/flat yield/current yield: f rP = 100rP1
• If P= f =C, then r = i(m).
• Makeham’s formula (with income tax):
P= f r(1 t1) ·a(m)n ,i +
C
(1+ i)n
=C(1 vn)[(1 t1)g
i(m)
1]+C, g= f r/C
• Net interest yield: f r(1t1)P = 100r(1t1)P1
• If the redemption date is optional,
– If C > P (or equivalently, i(m) > (1 t1)g), there is a capital gain
at the redemption date. The worst case is the redemption money
repaid at the LATEST possible date.
– If C < P (or equivalently, i(m) < (1 t1)g), there is a capital loss
at the redemption date. The worst case is the redemption money
repaid at the EARLIEST possible date.
– IfC= P (or equivalently, i(m) = (1 t1)g), no capital gain or loss.
It is irrelevant when the bond is redeemed.
1

– Makeham’s formula (with income tax and capital gain tax):
P= f r(1 t1) ·a(m)n ,i +Cvn
if there is no capital gain.
P=
f r(1 t1) ·a(m)n ,i +(1 t2)Cvn
1 t2vn
if there is a capital gain.
2. Equity calculations
• The yield of a share is i such that P= Âk=1 dk(1+i)k
• Dividend yield, P/E ratio.
3. Real and money interest rates
Money interest rates iM and real interest rate iR:
(1+ iM) = (1+q)(1+ iR)
2