Question

Suppose that there are three zero-coupon bonds, each with a face value of $100 and no default risk. The 1-yr bond has a price of $94, the 2-yr bond at $90 and the 3-yr bond at $83.

a. What are their spot yields, their yields to maturity?

b. What is the price of a 3-yr default-free bond with a 5% annual coupon?

c. What is the forward rate on a 1-yr zero coupon bond 2 years from now?

d. What is the term premium embedded in the forward rate if the expected rate is 8%?

Answer 1

Available Information

There are 3 zero-coupon bonds

Face Value of Bond	$100	$100	$100
Years to Maturity (n)	1	2	3
Current Price	$94	$90	$83

a. Calculation of Spot Yields/Yield to Maturity = (Face Value/Current Price)1/n - 1

In case of Zero Coupon Bond, Spot yields are identical to Yields to Maturity and are calculated the same ways as the YTM. Both the factors determine the return on a bond.  

Yields to maturity is the total rate of return that have been earned by a bond including interest payment and original principal at the maturity.

Spot rate is the rate of return earned when it is bought and sold on the secondary market without interest payment.

Face Value of Bond	$100	$100	$100
Years to Maturity (n)	1	2	3
Current Price	$94	$90	$83
Calculation of Spot Yields	($100/$94)1/1 - 1	($100/$90)1/2 - 1	($100/$83)1/3 - 1
Spot Yields % / Yield to Maturity	6.383	5.409	6.408

b. Available Information

Years to Maturity (n) = 3

Coupon Rate = 5%

Current Price = Face Value / (1 + Coupon Yield)n

Current Price = $100/(1+0.05)3

Current Price = $115

c. Computation Forward rate on a 1-yr zero coupon bond 2 years from now =

Spot Yield on 1 Year Bond = 6.383 % and Years to Maturity (n) = 1

Spot Yield on 2 Year Bond = 5.409% and Years to Maturity (n) = 2

Forward Rate = ((1 + Spot Yield on 2 Year Bond)n/ (1 + Spot Yield on 1 Year Bond)n) - 1

Forward Rate = ((1 + 0.05409)2/ (1 + 0.06383)1) - 1

Forward Rate = 4.563