## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 8 - Personal Finance - 8.4 Compound Interest - Exercise Set 8.4: 56

#### Answer

With quarterly compounding, the effective annual yield is 4.7% With monthly compounding, the effective annual yield is 4.7% With daily compounding, the effective annual yield is 4.7% Decreasing the length of the compounding period, that is, compounding the interest more often, increases the effective annual yield slightly. However, to see this difference, we should round off the effective annual yield to the nearest hundredth of a percent instead of the nearest tenth of a percent.

#### Work Step by Step

This is the formula we use when we find the effective annual yield $Y$: $Y = (1+\frac{r}{n})^{n}-1$ $Y$ is the effective annual yield $r$ is the stated interest rate $n$ is the number of times per year the interest is compounded We can find the effective annual yield when the interest is compounded quarterly. $Y = (1+\frac{r}{n})^{n}-1$ $Y = (1+\frac{0.046}{4})^{4}-1$ $Y = 0.0468$ With quarterly compounding, the effective annual yield is 4.7% We can find the effective annual yield when the interest is compounded monthly. $Y = (1+\frac{r}{n})^{n}-1$ $Y = (1+\frac{0.046}{12})^{12}-1$ $Y = 0.0470$ With monthly compounding, the effective annual yield is 4.7% We can find the effective annual yield when the interest is compounded daily. $Y = (1+\frac{r}{n})^{n}-1$ $Y = (1+\frac{0.046}{360})^{360}-1$ $Y = 0.0471$ With daily compounding, the effective annual yield is 4.7% Decreasing the length of the compounding period, that is, compounding the interest more often, increases the effective annual yield slightly. However, to see this difference, we should round off the effective annual yield to the nearest hundredth of a percent instead of the nearest tenth of a percent.

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