Question

In the bond market, we find the following Treasury bonds and their prices. Bond price $980 $98 $96 Maturity 2 years 1 year 2 years Face value $1,000 $100 $100 Coupon rate 10% 0% 0% a) Compute the YTMs for the above three bonds. b) Using the two zero coupon bonds, compute the forward rate that is applied for the period from the end of Year 1 to the end of Year 2. c) Suppose that we need the above coupon bond for your cash requirements. However, due to some reasons, we cannot buy the coupon bond. Therefore, instead of the coupon bond, we decide to buy 1 year and 2 year zero coupon bond. If this alternative investment has the same cash flows as the coupon bond, how many bonds we need to buy (i.e., XX 1 year bonds and OO 2 year bonds)? What is the cost for this alternative bond investment?

Answer 1

a) Calculating YTM of coupon bond

Coupon bond price = 980, Par value = 1000, Coupon rate = 10%. Years to maturity = 2

Coupon = Coupon rate x par value = 10% x 1000 = 100

We will find the ytm of coupon bond using rate function in excel

Formula to be used in excel: =rate(nper,-pmt,pv,-fv)

Using rate function in excel, we get YTM = 11.17%

Calculating YTM of 1 year zero coupon bond

Price of 1 year zero coupon bond = Par value / (1 + YTM)

98 = 100 / (1 + YTM)

YTM = (100/98) - 1 = 1.020408 -1 = 0.020408 = 2.0408% = 2.04%

Calculating YTM of 2 year zero coupon bond

Price of 2 year zero coupon bond = Par value / (1 + YTM)²

96 = 100 / (1 + YTM)²

YTM = (100/96)^1/2 - 1

YTM = (1.04167)^1/2 - 1 = 1.02062 - 1 = 0.02062 = 2.062% = 2.06%

b. Let F be the forward rate applied from end of year 1 to end of year 2

We know that

( 1 + YTM of two year zero coupon bond)² = (1 + YTM of 1 year zero coupon bond)(1 + F)

(1 + 2.06%)² = (1 + 2.04%)(1+F)

(1.0206)² = (1.0204)(1+F)

1.0416244 = (1.0204)(1+F)

F = (1.0416244/1.0204) - 1 = 1.02080 - 1 = 0.02080 = 2.08%

Forward rate applied from end of year 1 to end of year 2 is equal to 2.08%

c. Cash flow for coupon bond consist of coupon payments at the end of year 1 and year 2 and also face value payment at the end of year 2.

Cash flow of coupon bond at end of year 1 = Coupon payment of bond = 100

Cash flow of coupon bond at end of year 2 = Coupon payment of bond + Face value payment of bond = 100 + 1000 = 1100

To replicate these cash flow, we need to buy one 1 year zero coupon bond and eleven 2 year zero coupon bond

Cost of investment of this alternative bond investment = Cost of one 1 year zero coupon bond + cost of eleven 2 year zero coupon bond = 1 x 98 + 11 x 96 = 98 + 1056 = 1154

Cost of investment of this alternative bond investment is equal to $1154