Math 2000 Solutions 6

Math 2000 - Solutions 6
Spring 2002

Unit 4A 10. If we invest $40,000 at an APR of 8.5% for 30 years, we accumulate a balance of $\$40,000 \times (1+0.085)^{30}= \$462,330.07$ . 12. After 10 years José has $\$1500 \times (1+0.056)^{10}= \$2586.61$ ; after 30 years he has $\$1500 \times (1+0.035)^{30}= \$7691.46$ . After 10 years Marta has $\$1500 \times (1+0.057)^{10}= \$2611.21$ ; after 30 years she has $\$1500 \times (1+0.057)^{30}= \$7912.99$ . To compare Marta's and José's balances, we must compute absolute relative differences. After 10 years, Marta has $24.60 more than José, and since $\frac{\$24.60}{\$2586.61} = 0.0095$ , Marta's balance is 0.95% higher than José's. After 30 years the change is more dramatic: Marta has $221.53 more than José, and since $\frac{\$221.53}{\$7691.46} = 0.029$ , Marta's balance is 2.9% higher than José's. Consequently, a small interest rate difference gets magnified over time. 20. Investing $15,000 at an APR of 7.8% compounded monthly for 15 years yields

$\begin{displaymath} A=\$15000(1 + \frac{0.078}{12})^{120}=\$48,147.25. \end{displaymath}$

26. Comparing Annual Yields. If we invest $1000 for 1 year at an APR of 8% compounded quarterly, we end up with

$\begin{displaymath} A=\$1000 (1+\frac{0.08}{4})^{4}=\$1082.43, \end{displaymath}$

and an APY of $\frac{\$82.43}{\$1000}=0.08243$ , or 8.24%. If the interest is compounded monthly, we get

$\begin{displaymath} A=\$1000 (1+\frac{0.08}{12})^{12}=\$1083.00, \end{displaymath}$

and an APY of 8.30%. Daily compounding leads to

$\begin{displaymath} A=\$1000 (1+\frac{0.08}{365})^{365}=\$1083.28, \end{displaymath}$

and an APY of 8.33%. Compounding monthly as opposed to quarterly increases the APY noticably, but increasing the frequency of compounding from monthly to daily has a smaller effect. Unit 4B 8. Depositing $200 monthly for 18 years at 7.5% yields

$\begin{displaymath} A=\$200 \times \frac{(1+\frac{0.075}{12})^{12\times 18}-1}{\frac{0.075}{12}} =\$90,920.11. \end{displaymath}$

This is more than twice the total amount of deposits made over the 18 years, which is $\$200 \times 12 \times 18 = \$43,200$ . 12. George deposits $40 monthly for 10 years at 7%, yielding

$\begin{displaymath} A=\$40 \times \frac{(1+\frac{0.07}{12})^{12\times 10}-1}{\frac{0.07}{12}} = \$6923.39. \end{displaymath}$

Overall, he deposits $\$40 \times 12 \times 10 = \$4800$ . Harvey deposits $150 quarterly for 10 years at 7.5%, yielding

$\begin{displaymath} A=\$150 \times \frac{(1+\frac{0.075}{4})^{4\times 10}-1}{\frac{0.075}{4}} = \$8818.79. \end{displaymath}$

Overall, he deposits $\$150 \times 4 \times 10 = \$6000$ . Thus we see that Harvey comes out ahead, despite his less frequent compounding. This is because Harvey's APR is higher than George's, and overall he deposits more than George does.