4.3 Maturity Pull: Market Yield Change for a Zero-Coupon Bond

Using the question posed at the end of the prior page, how much of a change in YTM would be required for the bond’s price to offset its maturity pull, in one year? In other words, at what point in YTM will the investor actually risk losing money, assuming his investment horizon is one-year and he chooses to invest in a five-year zero? More specifically, by how much would YTM have to rise before the investor would lose money, assuming a one-year holding period?

The easy answer to this problem involves illustrating the zero-coupon bond component of the prior illustration. Here, the investor bought a five-year zero, compounded semiannually at a discount of 10%, for a present value/price of $613.90 (i.e., $1,000 ÷ [1 + .05] ¹⁰ = $613.90). If he held the bond for a year, the discount rate would have to rise to at least 12.6% before he would lose one year’s accretion, thereby offsetting the, otherwise “normal,” maturity pull. At that discount rate, using round (discount rate) numbers, the bond would be worth $613.38 and the investor, if he were to sell the bond, would lose the accretion he would otherwise have earned.

This solution can be stated mathematically:

[latex]$613.90 = $1,000 ÷ (1 + R)^{8}[/latex]

[latex]$613.90 = \frac{1,000.00}{(1+R)^{8}}[/latex]

[latex](1 + R)^{8}=\frac{$1,000.00}{$613.90}[/latex]

[latex](1 + R) = (1.6289)^{1/8}[/latex]

[latex]R = (1.6289)^{1/8} - 1[/latex]

R ≈ 6.3% (semiannually) and 12.6% (annually)

Rates have to rise to 12.6% before the market wipes out the accretion. You may think of a YTM of 12.6% as a sort of break-even point. The investor will lose money if he sells after one-year at a YTM greater than 12.6%.

Note that this is simply a problem with the time value of money. Here the exponent was changed to eight in order to represent the passage of one-year’s time (two half-year periods).