Question

1. Deposits are to be made to a fund each January 1 and July 1 for the years 2020 through 2030. The deposit made on each July 1 will be 10.25% greater than the one made on the immediately preceding January 1. The deposit made each January 1 (except for January 1, 2020) will be the same amount as the delposit made on the immediately preceding July 1. The fund will be credited with interest at the nominal annual rate of 10 %.compounded semi-annually. On December 31, 2030 the fund will have a balance of 11,000. Determine the initial deposit to the fund. (D) 195 (E) 200 (A) 160 (B) 165 (C) 175

Answer 1

Let the initial deposit be 'X'

Interest rate with semi-annual compounding = 10%

Effective interest rate = (1+0.10/2)^2 = 10.25%

Present value of all January deposits = X + (1.1025X)/1.1025 + X(1.1025^2)/(1.1025^2) + ............X(1.1025^10)/(1.1025^10)

= 11X

Present value of all July deposits = X + (1.1025X)/1.1025 + X(1.1025^2)/(1.1025^2) + ............X*(1.1025^10)/(1.1025^10)

= 11X

Present value of July deposits on January 1 2020 = 11x * (1 + 10%/2)^1 = 11.55X

Total Present Value = 11x + 11.55X = 22.55X

So, 22.55X = 11,000/(1+10%/2)^22

=> X = $165