Question

Suppose you observe that the YTM on a 1-year zero-coupon bond is 5%, and the YTM on a 2-year zero-coupon bond is 6%. The YTM on a 2-year annual coupon bond with a 12% coupon rate is 5.8%. Assume all three bonds are riskless. If you were to repackage the 2-year coupon bond as two zero-coupon bonds, how much should you be able to sell them for? Please express your answer in dollars, rounded to the nearest cent

Answer 1

Let's assume for the purpose of this question that the par value (or face value) of each of the bond is 1,000. Such an assumption is customary in bond markets.

Price of one year zero coupon bond (ZCB) = P1 = FV/ (1 + y)ⁿ = 1,000 / (1 + 5%) = 952.38

Price of two year zero coupon bond (ZCB) = P2 = FV/ (1 + y)ⁿ = 1,000 / (1 + 6%)² =  890.00

Now please note the year wise cash flows from each of the bonds:

Bond	Year 1	Year 2
1 year ZCB	1,000
2 year ZCB		1,000
2 year coupon bond	120	1,120

Let's say we replicte the cash flows of 2 year coupon bond by creating a synthetic replicating portfolio of A number of 1 year ZCB and B number of 2 year ZCB.

Hence, cash flows in year 1 = A x 1,000

and that in year 2 = B x 1,000

If it's a  replicating portofiol for the 2 year coupon bond then yearwise cash flows should match.

Hence, A x 1,000 = 120

Hence, A = 0.12

and B x 1,000 = 1,120

Hence, B = 1.12

Hence, 1 number of 2 year coupon bond is = 0.12 number of 1 year ZCB + 1.12 number of 2 year ZCB

Hence, you should able to sell the repackaged portfolio at = 0.12 x P1 + 1.12 x P2 = 0.12 x 952.38 + 1.12 x 890.00 = 1,111.08