Question

Donald takes out a loan to be repaid with annual payments of $500 at the end of each year for 2n years. The annual effective interest rate is 4.94%. The sum of the interest paid in year 1 plus the interest paid in year n + 1 is equal to $720. Calculate the amount of interest paid in year 10.

Answer 1

Present value of annuity = P*[1 - (1/(1+r))^n / r ]

P = annual payments

r = rate of interest

n = number of periods

let 1 / (1+r) = v

so value of loan at t = 0 is = 500 / 4.94% [1 - v^2n]

interest in 1st year = (loan amount at t = 0 ) * interest rate

interest in 1st year =(500 / 4.94% [1 - v^2n]) * 4.94% = 500*[1 - v^2n]..............(1)

from the above we can say that interest in xth year = 500*[1 - v^(2n-(x-1))]

interest in (n+1)th year = 500*[1 - v^2n-((n+1)-1)]

= 500*[1 - v^n]..........(2)

in the given question (1) + (2) = $720

so, 500*[1 - v^2n] + 500*[1 - v^n] = 720

1000 - 500v^2n - 500v^n = 720

v^2n + v^n - 0.56 = 0

let v^n = x

x^2 + x - 0.56 = 0

by solving x we get x = v^n = 0.4

interest in 10th period using above derived formula of 'interest for xth period' we get

= 500*[1 - v^2n-(10-1)]

= 500*[1 - v^2n-9]

= 500*[1 - v^2n*v^-9]

we know v = 1 / (1+4.94%)

v^n = 0.4

interest = 500*[1 - (0.4)^2*(1+4.94%)^9]

interest paid in 10th year = $376.53