Question

Smith can repay $300, 000 one of two ways.
? (i) By 25 level annual payments at the end of each year, starting one year after the

loan is made, at some unknown loan interest rate.

? (ii) By 25 annual interest payments to the lender at an effective annual interest rate jL = 11% along with 25 level annual deposits into a sinking fund earning an effective annual interest rate of jI = 13% (at the end of each year, starting one year after the loan is made).

Find the loan interest rate (as an effective annual interest rate) in (i) to make the schemes equivalent (that is, so that the total amount paid at the end of each year for both options are equal. )

Answer 1

(ii)

i = effective interest on the loan per payment period

i = 11%

j = effective interest earned in the sinking fund per pay period

j = 13%

n = 25 years

P: principal = $300000

X: Payment towards sinking fund

Where a(n,j) is the sum of all discounted values at j interest rate

a(25,13%) = (1/13%)*(1-1/(1+13%)^25) = 7.329

X = $300000*(11%-13%+1/7.329) = $34933.28

(i) Let r be the effective annual interest rate

300000 = (34933.28/r)*(1-1/(1+r)^25)

Solving the above equation, we get

r = 10.73% (annual effective interest rate)