Question

On 1/1/1995 a firm issued a 20-year bond with a face value of $1,000, coupon rate 6%, paid semi-annually, trading at the price of $975. You bought the bond on 3/12/1999 at a yield of 8%. You sell the bond on 4/15/2005 at a yield of 5 3/8%. You were careful to invest all the coupons at a yield of 7 7/8% for all the (whole or partial) semi-annual periods of the holding period.

(a) Calculate: the YTM at the issue date.

(b) Calculate: the price you paid upon purchase.

(c) Calculate: the price you received upon sale.

(d) Calculate: the annualized rate of return (bond-equivalent yield) you earned during the holding period.

(e) Consider the alternative reinvestment strategy: You were careful to invest all the coupons as follows: first 4 semi-annual periods at 7 7/8% p.a.; the next four semi-annual periods at 8 3/8% p.a.; the next four semi-annual periods at 5 1/8% p.a.; the remaining semi-annual periods at 5 3/8% p.a.

Calculate: the annualized rate of return (bond-equivalent yield) you earned during the holding period.

Answer 1

The Solution to the part a)

YTM at the issue date will be: [Coupon + (Redemption Value - Bond Price)/remaining life] / (Redemption Value + Bond Price )/2

= [ $ 1000 * 6% + ( $ 1000 - $ 975) / 20 ] / ( $ 1000 + $ 975)/2

= $ 61.25 / $ 987.50

= 6.20%p.a.

The Solution to the part b)

The price of a bond is the sum total of present values of all coupon payments and the present value of maturity value.

Further, the cash flows value and the discount rate should always be consistent i.e. if  cash flows are received semiannually, time interval and required rate of return is also compounded semiannually,

Thus,

The coupon rate will be 6%/2 = 3%

YTM will be 8%/2 = 4%

And; years to maturity will be 20 years - 4.17 years = 15.83 years*2 = 31.67 years

The purchase price of the Bond as on March 12, 1999, will be = PV of all coupon payments + PV of maturity value

PV of all coupon payments = Coupon payment * cumulative discount factor for 31.67 years at the current YTM

= ($ 1000 * 3%) * (1-1/ (1+YTM)^N) / YTM

= $ 30 * (1 - 1/(1+4%)^31.67)/ 4%

= $ 30 * 17.7798

= $ 533.39

PV of maturity value = Face Value * Discount factor for 31.67th year at the current YTM

= $ 1000 * 1/ (1+YTM)^N

= $ 1000 * 0.2888

= $ 288.81

The purchase price of the Bond as on March 12, 1999 will be = PV of all coupon payments + PV of maturity value

= $ 533.39 + $ 288.81

= $ 822.20

The Solution to the part c)

As on April 15, 2005,

The coupon rate will be 6%/2 = 3%

YTM will be 5.375%/2 = 2.6875%

And; years to maturity will be 20 years - 10.25 years = 9.75 years*2 = 19.50 years

The price of the Bond received on April 15, 2005, will be = PV of all coupon payments + PV of maturity value

PV of all coupon payments = Coupon payment * cumulative discount factor for 19.50 years at the current YTM

= ($ 1000 * 3%) * (1-1/ (1+YTM)^N) / YTM

= $ 30 * (1 - 1/(1+2.6875%)^19.50)/ 2.6875%

= $ 30 * 15.0243

= $ 450.73

PV of maturity value = Face Value * Discount factor for 19.50th year at the current YTM

= $ 1000 * 1/ (1+YTM)^N

= $ 1000 * 0.5962

= $ 596.22

The price received on the sale of the Bond as on April 15, 2005 will be = PV of all coupon payments + PV of maturity value

= $ 450.73 + $ 596.22

= $ 1046.95

The Solution to the part d)

The coupon payments during the holding period were reinvested at 7.875% semi-annually

Hence,

Bond Equivalent yield = n * holding period rate

where n = number of compounding in a year

= 2* 7.875%

= 15.75% p.a.

Hence the bond equivalent yield during the holding period is 15.75%