Question

The yield-to-maturity (YTM) on one-year bond with zero coupon and face value $ 1000 is 5 %. The YTM on two-year bond with 5 % coupon paid annually and face value $ 1000 is 6 %. (i) What are the current prices of these bonds? (ii) Find Macaulay durations of these bonds. Consider a third bond which is a zero coupon two-year bond with face value $ 1000. (iii) What must be the price of the third bond so that the Law of One Price holds. Explain where you use the LOOP. (iv) Find the yield-to-maturity on the thrid bond with the price from part (iii).

Answer 1

YTM on one year zero coupon bond is 5% and FV = $1000

YTM = 6% on two year bond ; Coupon =5%; FV = $1000

i) Price of zero coupon bond = 1000 / (1 + 0.05) = $952.38

Price of two year bond = 50/1.06 +1050 / (1.06)² = 47.16 + 934.49 = $981.65

ii) Macaulay duration of zero coupon bonds = 1 (since the duration of zero coupon bond is equal to its time to maturity)

Macaulay duration of coupon bond = [1*50/1.06 + 2*1050 / (1.06)²] / 981.65 = 1.95

iii) A third bond which is a two year zero coupon bond with FV = $1000 and law of one price holds when arbitrage opportunities exist. We consider YTM as 5% similar to zero coupon bond in part (i) because of LOOP.

so, the price of this bond P = $952.38

iv) YTM = 1000 / ( 1 + YTM)² = 952.38

YTM = 2.47%