Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 8 - Personal Finance - 8.5 Annuities, Methods of Saving, and Investments - Exercise Set 8.5: 21

Answer

(a) We will have $\$30,000$ more from the lump-sum investment than from the annuity. (b) The lump-sum investment earns $\$30,000$ more interest than the annuity.

Work Step by Step

(a) This is the formula we use when we make calculations with compound interest: $A = P~(1+\frac{r}{n})^{nt}$ $A$ is the final amount in the account $P$ is the principal (the amount of money invested) $r$ is the interest rate $n$ is the number of times per year the interest is compounded $t$ is the number of years We can find the total amount in the account at the end of 20 years when we invest a lump sum at a rate of 5% compounded annually. $A = P~(1+\frac{r}{n})^{nt}$ $A = (\$30,000)~(1+\frac{0.05}{1})^{(1)(20)}$ $A = \$79,599$ After 20 years, there will be $\$79,599$ in the account. This is the formula we use to calculate the value of an annuity: $A = \frac{P~[(1+\frac{r}{n})^{nt}-1]}{\frac{r}{n}}$ $A$ is the future value of the annuity $P$ is the amount of the periodic deposit $r$ is the interest rate $n$ is the number of times per year the interest is compounded $t$ is the number of years $A = \frac{P~[(1+\frac{r}{n})^{nt}~-1]}{\frac{r}{n}}$ $A = \frac{(\$1500)~[(1+\frac{0.05}{1})^{(1)(20)}~-1]}{\frac{0.05}{1}}$ $A = \$49,599$ The value of the annuity is $\$49,599$ We can calculate the difference between the lump-sum investment and the value of the annuity. $\$79,599 - \$49,599 = \$30,000$ We will have $\$30,000$ more from the lump-sum investment than from the annuity.
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