Question

Suppose that ​$1,700 is set aside each year and invested in a savings account that pays 8​% interest per​ year, compounded continuously.

a. Determine the accumulated savings in this account at the end of year 30.

b. Suppose that an annuity will be withdrawn from savings that have been accumulated at the end of year 30. The annuity will extend from the end of year 31 to the end of year 37. What is the value of this annuity if the interest rate and compounding frequency in Part​ (a) do not​ change?

Answer 1

(a)

Accomulated savings ($) = Future worth = 1,700 x [(e^rt) - 1] / [(e^r) - 1] = 1,700 x [(e^0.08x30) - 1] / [(e^0.08) - 1]

= 1,700 x [(e^2.4) - 1] ) / [(e^0.08) - 1] = 1,700 x (11.0232 - 1) / (1.0833 - 1) = 1,700 x (10.0232 / 0.0833)

= 204,586.86

(b)

Present value of annuity for 7 years = Accumulated savings at year 30

If required annuity be A per year, then

A x [1 - (e^0.08x7)^-1] / [(e^0.08) - 1] = 204,586.86

A x [1 - (e^0.56)^-1] / [(e^0.08) - 1] = 204,586.86

A x [1 - (e^-0.56)] / [(e^0.08) - 1] = 204,586.86

A x (1 - 0.5712) / (1.0833 - 1) = 204,586.86

A x (0.4288 / 0.0833) = 204,586.86

A x 5.1477 = 204,586.86

A = $39,743.67