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Bonds, Equities and Inflation
创建日期：2024-04-24 16:46:30
所属分类：金融Finance
标签： ST334
Bonds, Equities and Inflation

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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
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