Question

You are in charge of the bond trading and forward loan department of a large investment bank. You have the following YTM’s for five default-free pure discount bonds as displayed on your computer terminal: Years of Maturity 1 2 3 4 5 YTM 0.06 0.065 0.07 0.065 0.08 Where YTM denotes the yield to maturity of a default free pure discount bond (zero coupon bond) maturing at year j. a) A new summer intern from Harvard has just told you that he thinks that 3 year treasury notes with annual coupons of $100 and face value of $1,000 are trading for $1,000. Would you ask the intern to recheck the price of this coupon bond? If so, why? If there is one actually traded for $1,000, how would you take this opportunity? b) A client approaches you looking for an annualized quote on a forward loan of $5 million dollars to be received by the customer at the end of the third year and she will repay the loan at the end of the fifth year. How would you structure your holdings of pure discount bonds so you can exactly match the future cash flows of this loan? Please indicate the number of bonds to be purchased or sold and the involved cost/benefit in dollar terms. What is the corresponding annualized forward interest rate you quote for your client? c) Suppose that you purchased the bond in part 1(a) at the price you calculated. It is now one year later and you just received the first coupon payment on the bond. At this time, the yield to maturities up to 3 year pure discount bonds are Years to maturity 1 2 3 4 5 YTM 0.08 0.095 0.09 0.075 0.06 If you were to sell the bond now, what rate of return would you realize on your investment in the bond?

Answer 1

Part (a)

No arbitrage price of 3 year treasury notes with annual coupons of $100 and face value of $1,000 = PV of all future cash flows = 100 / (1 + y1) + 100 / (1 + y2)² + 1(100 + 1,000) / (1 + y3)³ where yi = YTM of the zero coupon bond maturing in year i

Hence, the no arbitrage price = 100 / (1 + 0.06) + 100 / (1 + 0.065)² + (100 + 1,000) / (1 + 0.07)³ = $  1,080.43

Since the price reported by intern is $ 1,000 which is significantly lower than the no arbitrage price calculated above, I will ask the intern to recheck the price.

If one such bond do exist. I will make use of the following arbitrage:

• Buy the actual bond at $ 1,000
• Short one tenth (or 0.10 number) of zero coupon bond maturing in 1 year
• Short one tenth (or 0.10 number) of zero coupon bond maturing in 2 years
• Short 1.1 number of zero coupon bond maturing in 3 year

This will result into an immediate payoff of $ 1,080.43 - 1,000 = $ 80.43 at t= 0 and all the future cash flows are matched. (Alternatively, we could have multiplied each of the position above by a factor of 10 to get rid of fraction such as 0.1, 1.1 etc).

Part (b)

You need $ 5 mn in hand at the end of year 3 to offer as a loan to your customer. You should therefore buy $ 5 mn / 1,000 = 5,000 nos. of zero coupon bonds maturing in three years from now.

Amount required by you today to buy 5,000 zero coupon bonds (ZCBs) maturing in 3 years = 5,000 x 1,000 / (1 + y3)³ = 5,000 x 1,000 / (1 + 0.07)³ = $  4,081,489.38

In order to get this money, we need to short ZCBs maturing in year 5. Price of a ZCB maturing in year 5 today = 1,000 / (1 + y5)⁵ = 1,000 / (1 + 0.08)⁵ = $  680.58

Hence, no. of 5 years ZCB to be short today = 4,081,489.38 / 680.58 =  5,997.05

Your customer will return the amount after two years i.e. at the end of year 5.

Effective rate = annualized two year forward rate at the end of year 3 = [(1 + y5)⁵ / (1 + y3)³]^1/2 - 1 = 9.52%

Part (c)

Purchase price, P₀ = $ 1,080.43

After 1 year, one more coupon is expected after 1 year and one coupon plus repayment is expected after two years.

Price, P₁ = PV of balance coupons + PV of repayment = 100 / (1 + 0.08) + (100 + 1,000) / (1 + 0.095) =  1,097.16

Hence, investment return = (P₁ + Coupon) / P₀ - 1= (1,097.16 + 100) / 1,080.43 - 1 = 10.80%