The risk free interest rate today (date 0) is 10% EAR. Mark offers to sell you a

Question

The risk free interest rate today (date 0) is 10% EAR. Mark offers to sell you a 2-year risk free zero coupon bond with face value $100 for $80. Before you can take action, Mark realizes he is making a mistake and the price of the bond is now its fair value. You buy the bond at the fair value, expecting to hold the bond for 2 years.

1 year later (on date 1), the interest rate is 20% EAR.

a) What is the fair value of the bond?

b) What should its price be?

Suppose there was no arbitrage (price = fair value) and you sold the bond at date 1.

c) What HPR did you make?

Suppose instead that on date 1, the interest was 5% EAR.

d) What is the fair value of the bond?

e) What should its price be?

Suppose there was no arbitrage (price = fair value) and you sold the bond at date 1.

f) What HPR did you make?

g) Suppose instead you hold the bond for 2 years. What is the AHPR (annualized HPR) did you make? Is the bond truly risk free?

Explanation / Answer

We had purchased $100 zero coupon bond expiring in 2 years at $80 today.

Using Financial calculator we can calculate the yield of this bond by using the following inputs :-

PV = -80, FV = 100, N=2, PMT = 0 (because it is zero coupon bond) Calculate I/Y = 11.8%.

Which means if we hold the bond for 2 years we will earn Annualised return of 11.8%.

a) If After 1 year the EAR rises to 20%, using the financial calculator we can calculate the price of bond in 1 year. FV = 100, N=1, PMT = 0, I/Y = 20% Compute PV = 83.33.

b) The price of bond should have increased by 11.8%. Which means after 1 year the price should have been 80*1.118 = 89.44

c) Holding period return will be increase in price divided by purchase price i.e. (83.33-80)/80 = 4.16%

d) If EAR reduced to 5% after Year 1, again using financial calculator we will calculate the PV at year 1. Inputting FV= 100, N=1, I/Y = 5%, PMT = 0, Calculate PV = 95.24

e) Its should have been the same as we calculated earlier i.e. $80 increased by 11.8% i.e. 89.44

f) Holding period return = (95.24-80)/80 = 19.05%

Practically only government bond is treated as risk free, so incase this bond is issued by government, and we hold it till the expiry of bond, this bond will be risk free because we will earn the same return which had been promised to us at the time of purchasing the bond, but in case we sold it earlier than that, it may not be risk free because the return may not be the same as expected as we had seen above.

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