Question

Suppose you purchase a​ 30-year, zero-coupon bond with a yield to maturity of 8 %. You hold the bond for five years before selling it.

a. If the​ bond's yield to maturity is 8% when you sell​ it, what is the internal rate of return of your​ investment?

b. If the​ bond's yield to maturity is 9 % when you sell​ it, what is the internal rate of return of your​ investment?

c. If the​ bond's yield to maturity is 7 % when you sell​ it, what is the internal rate of return of your​ investment?

d. Even if a bond has no chance of​ default, is your investment risk free if you plan to sell it before it​ matures? Explain.

​Note: Assume annual compounding.

Answer 1

Zero coupon bonds are the bonds which do not pay coupon. They are generally sold at discount and mature at par.

Formula to calculate value of zero coupon bond

Price = FV*(1/(1+r)^n)

Where FV = Face value

r = Yield to maturity

n= years to maturity

In each of the cases we would calculate the purchase price and sale price at end of 5 years to calculate IRR. Assuming face value of bond is $100

a) Purchase price of zero coupon bond, years to maturity = 30 and yield to maturity = 8%

Purchase Price = $100*(1/(1+0.08)^30)

Purchase price =$100*0.09938

Purchase price = $9.94

Sale price of zero coupon bond, years to maturity = 25 and yield to maturity = 8%

Purchase Price = $100*(1/(1+0.08)^25)

Purchase price =$100*0.14602

Purchase price = $14.60

IRR = (Sale price/Purchase price)^(1/n) - 1

IRR = (14.60/9.94)^(1/5)-1

IRR = 8%

Since YTM to maturity is same when purchased and sold the IRR = YTM

The IRR of your investment if the bond's yield to maturity is 8% when you sell it is 8%

b) Purchase price of zero coupon bond is $9.94 when years to maturity = 30 and yield to maturity = 8%

Sale price of zero coupon bond, years to maturity = 25 and yield to maturity = 9%

Purchase Price = $100*(1/(1+0.09)^25)

Purchase price =$100*0.11597

Purchase price = $11.60

IRR = (11.60/9.94)^(1/5)-1

IRR = 3.14%

Since YTM to maturity has increased from 8% to 9% the IRR is less than YTM

The IRR of your investment if the bond's yield to maturity is 9% when you sell it is 3.14%

c) Purchase price of zero coupon bond is $9.94 when years to maturity = 30 and yield to maturity = 8%

Sale price of zero coupon bond, years to maturity = 25 and yield to maturity = 7%

Purchase Price = $100*(1/(1+0.07)^25)

Purchase price =$100*0.18425

Purchase price = $18.42

IRR = (18.42/9.94)^(1/5)-1

IRR = 13.13%

Since YTM to maturity has decreased from 8% to 7% the IRR is more than YTM

The IRR of your investment if the bond's yield to maturity is 7% when you sell it is 13.13%

d) Since the Yield to maturity changes therefore if bond is sold before maturity than also it is subject to risk

Therefore, even without default if you sell prior to maturity you are exposed to risk that the YTM may change.