8.3 The Zero-Coupon Equivalent of a Positive-Coupon Bond: Solution to Problem
[latex]1,000 = \frac{(40 \times 29.778) + 1,000}{(1 + RCY/p)^{np}}[/latex]
[latex](1 + RCY/p)^{np}=\frac{2,191.12}{1,000}[/latex]
= 2.1911 x
The ratio, “2.1911 x,” or “2.1911 times the denominator,” represents the relationship between the terminal- (or future-) value of the coupon inflows using an “exogenous” or external reinvestment rate. This differs from the bond’s price, which was calculated using the YTM. That is, for every dollar of investment, this bond will have a “terminal value” of $2.1911. Since, we now have just one outflow, and one inflow equivalent to the coupon annuity plus the maturity value (just like a Zero-coupon Bond!), the rate of return can be estimated using your simple future value interest rate tables. To do that, one would look across the 20-period row and find the closest factor to 2.1911. It can be found under the 4% column.
The mathematical solution is more precise and therefore preferred. Let’s solve for “R,” the bond’s “RCY,” using our familiar TVM formula.
FV = (PV) (1 + R) n
2.191 = ($1) (1 + R) 20
2.191 ÷ 1 = (1 + R) 20
R = (2.191) 1/20 – 1 = 0.04
This implies an RCY of 4% (i.e., R = .04) semi-annually or 8% p.a.! In this case, the RCY = YTM!
We have just converted a positive coupon to its zero-coupon bond equivalent under our reinvestment rate assumption. This bond is equivalent to purchasing a Zero-coupon bond for a price of $100, receiving no coupon cash flows, and – at the horizon or maturity – receiving one payment of $219.10, similar fashion to a Zero!
