An investor borrows an amount at an annual effective interest rate of 5% and will repay all interest and principal in a lump sum at the end of 10 years. She uses the amount borrowed to purchase a 1000 par value 10-year bond with 8% semiannual coupons bought to yield 6% convertible semiannually. All coupon payments are reinvested at a nominal rate of 4% convertible semiannually. Calculate the net gain to the investor at the end of 10 years after the loan is repaid

$74.14

Explanation:

first we must calculate the market price of the bond:

0.03 = {40 + [(1,000 - MV)/20]} / [(1,000 + MV)/2]

0.03 x [(1,000 + MV)/2] = 40 + [(1,000 - MV)/20]

0.03 x (500 + 0.5MV) = 40 + 50 - 0.05MV

15 + 0.015MV = 90 - 0.05MV

0.065MV = 75

MV = 75 / 0.065 = $1,153.85

so the customer borrowed $1,153.85

in 10 years, the principal + interest will = $1,153.85 x (1 + 5%)¹⁰ = $1,879.50

the customer will receive:

20 semiannual payments of $40, the future value = $40 x 22.841 (FV annuity factor, 2%, 19 periods) + $40 = $953.64

bond's face value = $1,000

total money received = $1,000 + $953.64 = $1,953.64

net gains = $1,953.64 - $1,879.50 = $74.14