Question

8. You plan to accumulate 150,000 at the end of 45 years by making the following deposits: (1) X at the end of years 1 - 10 (2) No deposits at the end of years 11 - 33 (3) Y at the end of years 34 - 45. The annual effective interest rate is 5%. X - Y = 120. Calculate X. A. 1,781 B. 1,681 C. 1,581 D. 1,481 E. 2,081

Answer 1

Sol :

To determine value of X at the end of year 10 as follows,

Rate of interest (r) = 5%

Number of years (n) = 10 years

Value of X at the end of year 10 = X((1+r)^n-1/r

Value of X at the end of year 10 = X (1+5%)^10 - 1/5%

Value of X at the end of year 10 = X (1.05)^10 -1/ 0.05

Value of X at the end of year 10 = 12.58X

Therefore value of the investment at the end of 45th year = 12.58X (1.05)^45-10 = 69.39X

Now Y invested at the end of 34th to 45th which is = 12 investments

Therefore value of the investment at the end of 45th year = Y(1.05)^12-1/0.05 = 15.92Y

Now 69.39X + 15.92Y = 150,000

X-Y = 120

So X = 120+Y

Substituting the above equation,

69.39 (120+Y) + 15.92Y = 150,000

= 8325.6 + 69.39Y + 15.92Y = 150,000

= 85.31Y = 150,000 - 8325.6

= 85.31Y = 141,674.4

Y = 141,674.4/85.31 = 1660.70

Now X-Y = 120

X = 120 + 1660.70

X = 1780.70

Therefore X = 1780.70 or 1781