Value of an Annuity

Question

Suppose that an employee contributes $100 to an annuity at the end of each month for 25 yrs. If the annuity earns 7%,

a. Determine the value of the annuity at the end of the 25-yr period.
b. How much interest will be earned?

Answer 1

The annuity as the name suggests has interest compounded annually. The employee contributes 12 * 100 = $1200 per year. The interest is 7%. At the end of the 1st year the value is 1200*(1+i), i = interest rate = 0.07. At the end of the 2nd year, he gets interest on the new contribution of $1200, in addition to interest on the 1st year's contribution, i.e. the total value is 1200*(1+i) + 1200*(1+i)². Continuing in  the same fashion, the value at the end of year 25 is, 1200*{ (1+i) + (1+i)²+ (1+i)³ + ... + (1+i)²⁵ }.

 

The sum of the series within { } is: (1+i)*((1+i)²⁵-1)/(1+i-1 = i) = ( (1+i)²⁶ - (1+i) ) / i = 67.6765

Therefore, the value is 67.6765*1200=81212

The interest earned is = value - principal = 81212 - 25*1200 = 51212.