Question

If a bond's yield to maturity does not change, the return on the bond each year will be equal to the yield to maturity. Confirm this for both a premium and a discount bond using a 4-year 4.2 percent coupon bond with annual coupon payments and a face value of $1,000.

a. Assume the yield to maturity is 3.2 percent.

What is the current value of the bond? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Bond price today            $

What will the bond value be in one year? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Bond price in one year            $

What is the rate of return for the first year? (Do not round intermediate calculations. Enter your answer as a percent rounded to 1 decimal place.)

Rate of return             %

b. Assume the yield to maturity is 5.2 percent.

What is the current value of the bond? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Bond price today            $

What will the bond value be in one year? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Bond price in one year            $

What is the rate of return for the first year? (Do not round intermediate calculations. Enter your answer as a percent rounded to 1 decimal place.)

Rate of return             %

Answer 1

a.

Value of Bond = Present value of all future cash inflows at the YTM rate.

$Rate of return for a period = \frac{C + BV_{1}-BV_{0}}{BV_{0}}$

where,

C= Coupon for the period

BV₁ = Bond value at the end one year

BV₀ = Bond value at beginning

Please refer below excel spreadsheet for calculations

E15 f(C9+J12-E13)/E13 2 Face Value$1,000.00 Coupon rate Maturit YTM 0.042 4 0.032 4 Bond Value on toda Bond value at end of one vear cash flow PVF(3.2%) $ 42.000.9690$ 40.70 $ 42.00 0.9389$ 39.44 $ 42.00 0.9098$38.21 $1,042.00 0.8816$ 918.65 ear | cash flow | PVF( 3.2%) | PV 1 $ 42.000.9690$40.70 2 $ 42.000.9389$39.44 3 $1,042.000.9098$948.04 ear PV 12 4 Bond value at end of one year$ 1,028.18 Bond value on toda $1,036.99 15 16 Rate of Return for first yean 0.032

Formula reference -

E15 Face Value Coupon rate Maturit YTM 1000 0.042 0.032 Bond Value on toda Bond value at end of one year ar cash flow PVF( 3.2% PV ar cash flow PVF( 32%) PV -1000 $C$3 -1000 $C$3 1000 $CS3 -1/1.032 B9 -1/1.032 B10 -1/1.032 B11 -1/1.032AB12 -C10 D10 C11 D11 -C12 D12 -SUM(E9:E1 1000 $C$3 1000 $C$3 1000 $C$3+1000 1/1.032AG9 1/1.032AG 10 1/1.032G11 H10*110 -1000 SC$3+C2 12 13 Bond value at end of one SUM(J9J11 Bond value on 15 Rate of Return for first year (C9+112-E13)/E13

In above calculation, we have

a. Bond Value today = $ 1,036.99

Bond value after one year = $ 1,028.18

Rate of return for first year = 3.2 %

We can see Bond is on premium and YTM remains same for one year then Rate of return for first year is same as YTM rate.

b.

If YTM is 5.2%, please find below spreadsheet for calculations

Formula reference -

In above calculation, we have

b. Bond Value today = $ 964.70

Bond value after one year = $ 972.87

Rate of return for first year = 5.2 %

We can see Bond is on discount and YTM remains same for one year then Rate of return for first year is same as YTM rate.

With above calculations we can confirm whether bond premium or discount, If yield to maturity does not change then rate of return for each year would be equal to YTM.