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Chapter 3: Problem 95

The effective yield of an investment plan is the percent increase in the balance after 1 year. Find the effective yield for each investment plan. Which investment plan has the greatest effective yield? Which investment plan will have the highest balance after 5 years? (a) \(7 \%\) annual interest rate, compounded annually (b) \(7 \%\) annual interest rate, compounded continuously (c) \(7 \%\) annual interest rate, compounded quarterly (d) \(7.25 \%\) annual interest rate, compounded quarterly

Short Answer

Expert verified
Plan (d) has the greatest effective yield of approximately \(7.4023\% \). This plan will also have the highest balance after 5 years.

Step by step solution

01

Calculate the effective yield for plan (a)

The formula for compound interest is \(A = P(1 + r/n)^(nt)\), where A is the amount of money accumulated after n years with interest compounded t times per year, P is the capital, r is the annual interest rate. For plan (a), the interest is compounded annually, so n=1. After one year, the effective yield, percentage increase in balance, is \(A-P = P[(1 + r/n)^(nt) - 1] = 0.07P\). Thus, the effective yield is \(7\%\).
02

Calculate the effective yield for plan (b)

The formula for continuous compound interest is \(A = Pe^{rt}\), where e is the base of the natural logarithm. After one year the effective yield is exactly the same as the nominal annual interest rate, \(7\%\).
03

Calculate the effective yield for plan (c)

For plan (c), the interest is compounded quarterly, so n=4. Applying the formula as in step 1 we get an effective yield of \(P[(1 + r/n)^(nt) - 1] = P[(1 + 0.07/4)^(4) - 1] = 0.071858P\). Thus, the effective yield is approximately \(7.1858\%\).
04

Calculate the effective yield for plan (d)

For plan (d), the interest is compounded quarterly and the interest rate is \(7.25\%\), so \(r=0.0725\) and \(n=4\). The effective yield becomes \(P[(1 + r/n)^(nt) - 1] = P[(1 + 0.0725/4)^(4) - 1] = 0.074023P\). Thus, the effective yield is approximately \(7.4023\%\).
05

Compare and determine the greatest effective yield

Comparing all the effective yields, we see that plan (d) has the highest effective yield of approximately \(7.4023\% \).
06

Determine the plan with highest balance after 5 years

We apply the compound interest formula for each plan with t set to 5. Because plan (d) has both a larger interest rate and the same compounding frequency as (c), plan (d) will also yield the highest balance after 5 years.

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