Question

Cain buys a 3-year $1000 10% bond with annual coupons, redeemable at par. The price assumes an annual effective yield rate of 8%. First, find the price of this bond.

Every time Belial receives a payment, he deposits the money into a savings account, with an annual effective interest rate of i. At the end of three years, he puts the last payment he receives from the bond into the savings account and observes that overall, his yield rate was 9% (it is as if he put the original price amount to a savings account with 0.09 annual effective interest rate).

What must i be so that this scenario could happen?

Answer 1

YTM = 8% = 0.08

Coupon rate = 10% = 0.10

time to maturity of bond, n = 3 years

coupon value, C = coupon rate* par value = 0.10*1000 = 100

price of bond = present value of coupons + present value of maturity amount

Present value of coupons = C*PVIFA( 8% , 3 years)

PVIFA( 8% , 3 years) = present value interest rate factor of annuity

= [((1+YTM)ⁿ - 1)/((1+YTM)ⁿ*YTM)] = [((1.08)³ - 1)/((1.08)³*0.08)] = 2.57709699

Present value(PV) of coupons = C*PVIFA( 8% , 30 years) = 100*2.57709699= 257.70969872

PV of maturity amount = par value/(1+YTM)ⁿ = 1000/(1.08)³ = 793.83224102

Price of bond when YTM is 8% = 257.70969872 + 793.83224102 = $1051.541940

assuming the effective annual rate , r = 9%

if he had put the price of bond into the savings account, the value of this amount after 3 years is:

value of the amount in savings account, A = price of bond*(1+r)³ = 1051.541940*(1.09)³ = 1361.777307

payments from bond :

interest at the end of year 1 = i1 = 100

interest at the end of year 2 = i2 = 100

interest at the end of year 3 plus maturity amount ( par value) = x = 100 + 1000 = 1100

now the annual effective rate of savings account = i

then:

[i1*(1+i)² ] + [i2*(1+i)] + x = A

[100*(1+i)² ] + [100*(1+i)] + 1100 = 1361.777307

we have to find i , which satisfies the above equation, which we can find by trial and error

by trial and error we will find that , i = 19.35%