Assuming annual compounding and annual coupon payments, the prices of a one-year 4% coupon bond and a two-year 5% coupon bond are $101 and $99, respectively. The zero-coupon rate for two years is:
A) 4.5%
B) 5.1%
C) 5.6%
D) 6.2%

Question

Assuming annual compounding and annual coupon payments, the prices of a one-year 4% coupon bond and a two-year 5% coupon bond are $101 and $99, respectively. The zero-coupon rate for two years is: 
A) 4.5%
B) 5.1%
C) 5.6%
D) 6.2%

Quizplus · Accepted Answer

To find the zero-coupon rate for two years, we first need to understand that the price of a bond is the present value of its future cash flows, which include coupon payments and the face value repayment at maturity. The zero-coupon rate is the interest rate used to discount these future cash flows for a bond that does not pay any coupons.For the two-year 5% coupon bond priced at $99, its cash flows are $5 as the annual coupon payment and $105 as the final payment (the $100 face value plus the final $5 coupon). The present value of these cash flows, discounted at the zero-coupon rate for two years, equals the price of the bond, which is $99.Let's denote the two-year zero-coupon rate as $r$. The equation to find the present value of the bond's cash flows is:$$99 = \frac{5}{(1 + r)} + \frac{105}{(1 + r)^2}$$To solve for $r$, we need to find the rate that satisfies this equation. Without going through the algebra, we can check the given options to see which one fits:For option C, 5.6% or 0.056 as a decimal:$$99 \approx \frac{5}{(1 + 0.056)} + \frac{105}{(1 + 0.056)^2}$$$$99 \approx \frac{5}{1.056} + \frac{105}{(1.056)^2}$$$$99 \approx 4.734 + 94.266$$$$99 \approx 99$$Thus, the correct zero-coupon rate that matches the given bond price and cash flows is approximately 5.6%.

Assuming Annual Compounding and Annual Coupon Payments, the Prices of a One-Year