Question

suppose $5,000 is invested at an annual interest rate of 3%. the following table gives the future value of the investment after 10 years based on different compounding periods.
compounding future value in 10 years
annually $6,719.58
semiannually $6,734.28
quarterly $6,741.74
monthly $6,746.77
weekly $6,748.71
daily $6,749.21

1. what is the difference in total interest earned from compounding annually versus semiannually at the end of 10 years?

o $14.70
o $29.63
o $1,734.28

1. what is the difference in total interest earned from compounding weekly versus daily at the end of 10 years?

o $0
o $0.50
o $1,748.71

1. what appears to happen to the total interest earned as the number of compound periods per year increases?

o the total interest earned stays the same.
o the total interest earned continues to increase without limit.
o the total interest earned continues to increase, but not

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. Given $P = 5000$, $r=0.03$, and $t = 10$.

Step2: Calculate future value for annual compounding ($n = 1$)

$A_1=5000(1 +\frac{0.03}{1})^{1\times10}=5000(1.03)^{10}\approx6719.58$.

Step3: Calculate future value for semi - annual compounding ($n = 2$)

$A_2=5000(1+\frac{0.03}{2})^{2\times10}=5000(1.015)^{20}\approx6734.28$.

Step4: Find the difference for part 1

The difference between the total interest earned from semi - annual and annual compounding is $A_2 - A_1=6734.28 - 6719.58 = 14.70$.

Step5: Calculate future value for weekly compounding ($n = 52$)

$A_{weekly}=5000(1+\frac{0.03}{52})^{52\times10}\approx6748.71$.

Step6: Calculate future value for daily compounding ($n = 365$)

$A_{daily}=5000(1+\frac{0.03}{365})^{365\times10}\approx6749.21$.

Step7: Find the difference for part 2

The difference between the total interest earned from daily and weekly compounding is $A_{daily}-A_{weekly}=6749.21 - 6748.71=0.50$.

Step8: Analyze the effect of compounding periods for part 3

As the number of compounding periods per year increases, the total interest earned increases. This is because more frequent compounding means that interest is earned on interest more often. However, the increase in total interest earned slows down as the number of compounding periods becomes very large.

Answer:

1. $14.70$
2. $0.50$
3. The total interest earned increases as the number of compound - periods per year increases.