Question

How much money should Sophie invest today into a fund that earns interest at 3.25% compounded quarterly, if she wants to receive $5,000 at the end of every 6 months for the next 5 years?

(the answer is not 47836.37)

Answer 1

Annual Interest Rate = 3.25% compounded quarterly

Quarterly Interest Rate = 3.25% / 4
Quarterly Interest Rate = 0.8125%

Effective Annual Rate = (1 + Quarterly Interest Rate)^4 - 1
Effective Annual Rate = (1 + 0.008125)^4 - 1
Effective Annual Rate = 1.03290 - 1
Effective Annual Rate = 0.03290 or 3.29%

Semiannual Interest Rate = (1 + Effective Annual Rate)^(1/2) - 1
Semiannual Interest Rate = (1 + 0.0329)^(1/2) - 1
Semiannual Interest Rate = 1.01632 - 1
Semiannual Interest Rate = 0.01632 or 1.632%

Semiannual Payment = $5,000
Semiannual Period = 10 (5 years)

Present Value = $5,000/1.01632 + $5,000/1.01632^2 + ... + $5,000/1.01632^9 + $5,000/1.01632^10
Present Value = $5,000 * (1 - (1/1.01632)^10) / 0.01632
Present Value = $5,000 * 9.158024
Present Value = $45,790.12

Therefore, Sophie needs to invest $45,790.12 today into a fund.