Question

Christine​ O'Brien, who is​ self-employed, wants to invest ​$70,000 in a pension plan. One investment offers 8​% compounded quarterly. Another offers

7.75% compounded continuously.

a. Which investment will earn the most interest in 4 ​years?

b. How much more will the better plan​ earn?

c. What is the effective rate in each​ case?

d. If Ms.​ O'Brien chooses the plan with continuous​ compounding, how long will it take for her $70,000 to grow to 90,000​?

e. How long will it take for her $70,000 to grow to at least $90,000 if she chooses the plan with quarterly​ compounding? (Be​ careful; interest is added to the account only every​ quarter.)

(Note: Thank you for any help. I have been struggling with this question for nearly an hour. I can't figure it out.)

Answer 1

Amount Invested = P = $70000

Quarterly compounded Interest Rate r₁ = 8%

Continuously Compounded interest rate = r₂ = 7.75%

(a) Let the future value be F

Number of Years = n = 4

For Quarterly Compounded , F = P(1+r₁/4)⁴ⁿ = 70000(1+0.08/4)¹⁶ = $96095

For Continuously Compounded, F = Pe^r₂t = 70000*e^0.0775*4 = $95439.76

Hence, Quarterly Compounded Plan is better

(b) Difference = 96095 - 95439.76 = $655.24

(c) For Quarterly Compounded rate , EAR = (1+r₁/n)ⁿ - 1 = (1+0.08/4)⁴ - 1 = 0.0824 or 8.24%

For Continuously Compounded Rate, EAR = e^r₂ - 1 = e^0.0775 - 1 = 0.0806 or 8.06%

(d) Let the number of years taken be n

Interest rate = r₂ = 7.75%

Future Value = F = $90000

Present Value = $70000

=> F = Pe^r₂n

=> n = (1/r₂)* ln (F/P) = (1/0.0775)* ln (90000/70000) =  3.24 years

(e)

Let the number of years taken be n

Interest rate = r₂ = 7.75%

Future Value = F = $90000

Present Value = $70000

=> F = P(1+r₁/4)⁴ⁿ

=> n = (1/4)* ln (F/P)/ln(1+r₁/4) = (1/4)* ln (90000/70000)/ln(1+0.08/4) = 3.17 years