find the future values of the following ordinary annuities:\na. fv of $400 paid each 6 months for 5 years at…

find the future values of the following ordinary annuities:\na. fv of $400 paid each 6 months for 5 years at a nominal rate of 15% compounded semiannually. do not round intermediate calculations. round your answer to the nearest cent.\n\n$5608.83\nb. fv of $200 paid each 3 months for 5 years at a nominal rate of 15% compounded quarterly. do not round intermediate calculations. round your answer to the nearest cent.\n\n$5898.96\nc. these annuities receive the same amount of cash during the 5 - year period and earn interest at the same nominal rate, yet the annuity in part b ends up larger than the one in part a. why does this occur?\nthe annuity in part (b) is compounded more frequently; therefore, more interest is earned on previously - earned interest.

find the future values of the following ordinary annuities:\na. fv of $400 paid each 6 months for 5 years at a nominal rate of 15% compounded semiannually. do not round intermediate calculations. round your answer to the nearest cent.\n\n$5608.83\nb. fv of $200 paid each 3 months for 5 years at a nominal rate of 15% compounded quarterly. do not round intermediate calculations. round your answer to the nearest cent.\n\n$5898.96\nc. these annuities receive the same amount of cash during the 5 - year period and earn interest at the same nominal rate, yet the annuity in part b ends up larger than the one in part a. why does this occur?\nthe annuity in part (b) is compounded more frequently; therefore, more interest is earned on previously - earned interest.

Answer

Explanation:

Step1: Recall future - value of an ordinary annuity formula

The formula for the future - value of an ordinary annuity is $FVA = A\times\frac{(1 + r)^{n}-1}{r}$, where $A$ is the annuity payment, $r$ is the interest rate per period, and $n$ is the number of periods.

Part a

  1. Determine the values of $A$, $r$, and $n$:
    • The annuity payment $A=$400$.
    • The nominal rate is $15%$ compounded semiannually. So the interest rate per period $r=\frac{0.15}{2}=0.075$.
    • The number of years is $t = 5$ years. Since it is compounded semiannually, the number of periods $n=5\times2 = 10$.
  2. Calculate the future - value:
    • Substitute the values into the formula: $FVA=400\times\frac{(1 + 0.075)^{10}-1}{0.075}$.
    • First, calculate $(1 + 0.075)^{10}=1.075^{10}\approx2.061032$.
    • Then, $(1.075^{10}-1)\approx2.061032 - 1=1.061032$.
    • $\frac{1.061032}{0.075}\approx14.14709$.
    • $FVA = 400\times14.14709=$5658.84$.

Part b

  1. Determine the values of $A$, $r$, and $n$:
    • The annuity payment $A = $200$.
    • The nominal rate is $15%$ compounded quarterly. So the interest rate per period $r=\frac{0.15}{4}=0.0375$.
    • The number of years is $t = 5$ years. Since it is compounded quarterly, the number of periods $n=5\times4=20$.
  2. Calculate the future - value:
    • Substitute the values into the formula: $FVA=200\times\frac{(1 + 0.0375)^{20}-1}{0.0375}$.
    • First, calculate $(1 + 0.0375)^{20}=1.0375^{20}\approx2.09756$.
    • Then, $(1.0375^{20}-1)\approx2.09756 - 1 = 1.09756$.
    • $\frac{1.09756}{0.0375}\approx29.2683$.
    • $FVA=200\times29.2683=$5853.66$.

Part c

The annuity in part (b) is compounded more frequently (quarter - ly) compared to part (a) (semi - annually). When compounding occurs more frequently, interest is earned on previously - earned interest more often. This leads to a higher future value even though the total amount of money contributed over the 5 - year period is the same in both cases ($400\times10 = 200\times20=$4000$).

Answer:

a. $$5658.84$ b. $$5853.66$ c. The annuity in part (b) is compounded more frequently; therefore, more interest is earned on previously - earned interest.