Question

Suppose you purchase a ten-year bond with 6 percent annual coupons. You
hold the bond for four years, and sell it immediately after receiving the fourth
coupon. If the bond's yield to maturity was 4.5% when you purchased and 7%
when you sold the bond. What is your annual rate of return on the bond in
each of the following situations:
a) All coupons were immediately spent when received.
b) All coupons were reinvested in a bank account, which pays 2 percent
interest until the bond is sold.

Answer 1

a. First we will calculate the purchase of bond

Assume Face value = $1000, Initial Yield to maturity at time of purchase = 4.5%, Years to maturity of bond = 10 years

Annual Coupon payment = face value x coupon rate = 1000 x 6% = $60

Price of a bond can be found out finding present value of future cash flows of bond. Future cash flows of bond consist of coupons paid annually and face value payment at maturity of bond. PV function in excel can be used to find the purchase price of bond or present value of cash flows

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

96 Calculating Purchase price of bond 97 Face value (fv) 98 Annual Coupon payment (pmt) 99 Yield to maturity (rate) 100 No of years to maturity (nper) 101 Purchase Price of bond (pv) 1000] 60.00| 4.50%! 101 =PV(H99,H100,-H98,-H97) PV(rate, nper, pmt, [fr], [type]) 102 103

Using PV function in excel, we get Purchase Price of bond = $1118.6907

Now we will find the selling price of bond after 4 years

Yield to maturity after 4 years = 7%, No of years to maturity = 6

We will now use PV function in excel to find selling price of bond

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

Calculating Selling price of bond 97 Face value (fv) 98 Annual Coupon payment (pmt) 99 Yield to maturity (rate) 100 No of years to maturity (nper) 101 Selling Price of bond (pv) 102 1000] 60.00 7.00% |=PV(H99,H100,-H98,-H97) PV(rate, nper, pmt, [fv], [type]) 103

Using PV function in excel, we get selling price of bond = $952.3346

If coupons are spent, then there will be no reinvestment income from coupons

So , Annual return of bond = (Selling price of bond / Purchase price bond)^{1/holding period of bond in years} - 1 = (952.3346 / 1118.6907)^1/4 - 1 = (0.851293)^1/4 - 1 = 0.960549 - 1 = -0.039451 = -3.9451% = -3.95% (rounded to two decimal places)

So Annual return of bond = -3.95%

b) If coupons are reinvested then there will be reinvestment income from coupons. Reinvestment income can be calculated by finding future value at end of year 4 from reinvestment of coupons at rate of 2%

Future value at end of year 4 from reinvestment of coupons = FC = 60(1+2%)³ + 60(1+2%)² + 60(1+2%) + 60 = 60 x 1.061208 + 60 x 1.0404 + 60 x 1.02 + 60 = 63.67248 + 62.424 + 61.20 + 60 = 247.29648

Annual return of bond = [(Selling price + FC) / Purchase price]^{1/holding period in years} - 1 =[(952.3346 + 247.29648) / 1118.6907]^1/4 - 1 = [1199.63108 / 1118.6907]^1/4 - 1 = 1.017617 - 1 = 0.017617 = 1.7617%

So Annual return of bond = 1.7617%