Question

Sylvia is considering depositing $600 at the end of each semi-annual period, for 5 years earning interest of 8%. She would like to know how large a one-time lump sum deposit she could make, at the same rate, to have the same amount of money after 5 years.

Sylvia later decides needs $ 11,000 in 8 years. She has the opportunity to make a one-time investment that will earn 10% compounded quarterly. How much must she invest to reach her goal?

Answer 1

The formula for determining the future value (F) of an annuity is F = P[(1+r)ⁿ-1]/r where P is the periodic payment, r is the rate of interest per period and n is the no. of periods.

Here, P = $ 600, r = 8/200 = 0.04 and n = 5*2 = 10. Then F = 600[(1.04)¹⁰-1]/(0.04) = 15000*0.480244284 = $ 7203.66 ( on rounding off to the nearest cent).

Suppose Sylvia deposits a one-time lump sum amount of $ x at the same rate, to have $ 7203.66 after 5 years. Then x(1.04)¹⁰ = 7203.66 so that x = 7203.66/1.480244284= $ 4866. 54. Thus, Sylvia needs to deposit an amount of $ 4866. 54, in one lump-sum.

If Sylvia needs $ 11000 in 8 years and . She has the opportunity to make a one-time investment that will earn 10% compounded quarterly, let her deposit $ x. Then x(1+10/400)^8*4 = 11000 or, x*2.203756938 = 11000 so that x = 11000/2.203756938 = 4991.48( on rounding off to the nearest cent).Thus, Sylvia needs to deposit an amount of $ 4991.48, in one lump-sum.