Compounding frequency and time value: You plan to invest $2,000 in an individual

Question

Compounding frequency and time value: You plan to invest $2,000 in an individual retirement arrangement (IRA) today at a nominal annual rate of 8%, which is expected to apply to all future years.

a. How much will you have in the account at the end of 10 years if interest is compounded (1) annually, (2) semiannually, (3) daily (assume 365-day year), and (4) continuously?

b. What is the effective annual rate (EAR) for each compoundings period in part a?

c. How much greater will your IRA balance be at the end of 10 years if interest is compounded continuously rather than annually?

d. How does the compounding frequency affect the future value and effective annual rate for a given deposit? Explain in terms of your findings in part a throughc.

Explanation / Answer

a.

(1) Annually

Amount at the end of 10 years = $2,000 * (1 + 8%)10

= $4,318

(2) Semi-annually

Amount at the end of 10 years = $2,000 * (1 + 8%/2)10*2

= $4,382

(3) Daily

Amount at the end of 10 years = $2,000 * (1 + 8%/365)10*365

= $4,451

(4) Continuously

Amount at the end of 10 years = $2,000 * e8%*10

= $4,451

b.

(1) Annually

EAR = [(1 + 8%)- 1] * 100%

= 8.00%

(2) Semi-annually

EAR = [(1 + 8%/2)2 - 1] * 100%

= 8.16%

(3) Daily

EAR = [(1 + 8%/365)365 - 1] * 100%

= 8.33%

(4) Continuously

EAR = [e8%- 1] * 100%

= 8.33%

c. The interest in case of continuous compounding is greater than that of annual compounding by = $4,451 - $4,318

= $133

d. As can be seen from the above results, both future value and effective annual rate increases with the increase in compounding frequency.

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