108156.pdf

10. Variable Interest Rates 219

points on day 180,and remains at this level until the end of the year. If

a bond is bought at the beginning of the year, on which day should it

be sold to produce a logarithmic return of 4.88% or more?

An investment in coupon bonds is more complicated. Even if the bond is
kept to maturity, the coupons are paid in the meantime and can be reinvested.
The return on such an investment depends on the interest rates prevailing at
the times when the coupons are due. First consider the relatively simple case
of an investment terminated as soon as the first coupon is paid.

Example 10.2

Let us invest the sum of $1,000 in 4-year bonds with face value $100 and $10
annual coupons. A coupon bond of this kind can be regarded as a collection
of four zero-coupon bonds maturing after 1, 2, 3 and 4 years with face value
$10, $10, $10 and $110, respectively. Suppose that such coupon bonds trade at
$91.78, which can be expressed as the sum of the prices of the four zero-coupon
bonds,
91 .78 = 10e−y(0)+ 10e−^2 y(0)+ 10e−^3 y(0)+ 110e−^4 y(0).

(The length of a time step isτ= 1.) This equation can be solved to find the
yield,y(0)∼=12%. We can afford to buy 10.896 coupon bonds. After one year
we cash the coupons, collecting $108.96, and sell the bonds, which are now
3-year coupon bonds. Consider three scenarios:

1. After one year the interest rate remains unchanged,y(1) = 12%, the coupon
   bonds being valued at

10e−^0.^12 + 10e−^2 ×^0.^12 + 110e−^3 ×^0.^12 ∼= 93. 48

dollars, and we shall receive 108.96 + 1, 018. 52 ∼= 1 , 127 .48 dollars in total.

2. The rate drops to 10%. As a result, the coupon bonds will be worth

10e−^0.^1 + 10e−^2 ×^0.^1 + 110e−^3 ×^0.^1 ∼= 98. 73

dollars each. We shall end up with $1, 184 .63.

3. The rate goes up to 14%, the coupon bonds trading at $88.53. The final
   value of our investment will be $1, 073 .51.

Exercise 10.5

Find the ratey(1) such that the logarithmic return on the investment in

Example 10.2 will be a) 12%, b) 10%, c) 14%.