Questions: What is continuous compounding? How does the APY for continuous compounding compare to the APY for, say, daily compounding? Explain the formula for continuous compounding. Choose the correct answer below. A. Compounding at least once per year is called continuous compounding. The APY for continuous compounding is smaller than the APY for daily compounding. The formula for continuous compounding is also known as the compound interest formula. B. Compounding infinitely many times per year is called continuous compounding. The APY for continuous compounding is only slightly larger than the APY for daily compounding. The formula for continuous compounding is a special form of the compound interest formula. C. Compounding twice daily is called continuous compounding. The APY for continuous compounding is only slightly larger than the APY for daily compounding. The formula for continuous compounding is a special form of the simple interest formula. D. Compounding daily is also known as continuous compounding. The APY for continuous compounding is equivalent to the APY for daily compounding. The formula for continuous compounding is also known as the compound interest formula.

Solution Steps

Using the formula for continuous compounding, the amount $A$ after $t$ years is given by:

$$A = P \cdot e^{rt}$$

where:

• $P = 1000$ (principal),
• $r = 0.05$ (annual interest rate),
• $t = 1$ (time period in years),
• $e$ is the base of the natural logarithm.

Substituting the values:

$$A = 1000 \cdot e^{0.05 \cdot 1} \approx 1000 \cdot 1.051271096 \approx 1051.2711$$

Thus, the amount with continuous compounding is approximately $1051.27$.

The Annual Percentage Yield (APY) for continuous compounding is calculated as:

$$\text{APY} = \left( \frac{A}{P} - 1 \right) \cdot 100$$

Substituting the values:

$$\text{APY} = \left( \frac{1051.2711}{1000} - 1 \right) \cdot 100 \approx 5.1271\%$$

Thus, the APY for continuous compounding is approximately $5.13\%$.

For daily compounding, the amount $A$ is given by:

$$A = P \left(1 + \frac{r}{n}\right)^{nt}$$

where $n = 365$ (number of compounding periods per year). Substituting the values:

$$A = 1000 \left(1 + \frac{0.05}{365}\right)^{365 \cdot 1} \approx 1000 \cdot 1.051267496 \approx 1051.2675$$

Thus, the amount with daily compounding is approximately $1051.27$.

The APY for daily compounding is calculated similarly:

$$\text{APY} = \left( \frac{A}{P} - 1 \right) \cdot 100$$

Substituting the values:

$$\text{APY} = \left( \frac{1051.2675}{1000} - 1 \right) \cdot 100 \approx 5.1267\%$$

Thus, the APY for daily compounding is approximately $5.13\%$.

Final Answer