many people prepare for retirement by making monthly contributions to a savings program. suppose that $2,500…

many people prepare for retirement by making monthly contributions to a savings program. suppose that $2,500 is set aside each year and invested in a savings account that pays 8% interest per year, compounded continuously. a. determine the accumulated savings in this account at the end of 25 years. b. in part (a), suppose that an annuity will be withdrawn from savings that have been accumulated at the eoy 25. the annuity will extend from the eoy 26 to the eoy 33. what is the value of this annuity if the interest rate and compounding frequency in part (a) do not change? click the icon to view the interest and annuity table for continuous compounding when i = 8% per year. a. the accumulated savings amount at the end of 25 years will be $191,778. (round to the nearest dollar.) b. the value of the annuity will be $ . (round to the nearest dollar.)

many people prepare for retirement by making monthly contributions to a savings program. suppose that $2,500 is set aside each year and invested in a savings account that pays 8% interest per year, compounded continuously. a. determine the accumulated savings in this account at the end of 25 years. b. in part (a), suppose that an annuity will be withdrawn from savings that have been accumulated at the eoy 25. the annuity will extend from the eoy 26 to the eoy 33. what is the value of this annuity if the interest rate and compounding frequency in part (a) do not change? click the icon to view the interest and annuity table for continuous compounding when i = 8% per year. a. the accumulated savings amount at the end of 25 years will be $191,778. (round to the nearest dollar.) b. the value of the annuity will be $ . (round to the nearest dollar.)

Answer

Explanation:

Step1: Recall annuity - future - value formula for continuous compounding

The formula for the future - value of an ordinary annuity with continuous compounding is $F = A\frac{e^{rt}-1}{e^{r}-1}$, where $A$ is the annual payment, $r$ is the annual interest rate, and $t$ is the number of years. Here, $A = 2500$, $r=0.08$, and $t = 25$.

Step2: Calculate the exponent terms

First, calculate $e^{rt}$ and $e^{r}$. $e^{rt}=e^{0.08\times25}=e^{2}\approx7.389056$, and $e^{r}=e^{0.08}\approx1.083287$.

Step3: Calculate the numerator and denominator

The numerator is $e^{rt}-1=e^{2}-1\approx7.389056 - 1=6.389056$. The denominator is $e^{r}-1\approx1.083287 - 1 = 0.083287$.

Step4: Calculate the future - value of the annuity

$F = 2500\times\frac{e^{2}-1}{e^{0.08}-1}=2500\times\frac{6.389056}{0.083287}\approx2500\times76.7159\approx191789.75\approx191790$.

For part (b), we first note that the accumulated amount at the end of 25 years is $F = 191790$. This amount will earn interest for an additional $n=33 - 25=8$ years with continuous compounding. The formula for continuous - compounding of a single amount is $F_2=P\times e^{rn}$, where $P$ is the principal amount, $r$ is the interest rate, and $n$ is the number of years. Here, $P = 191790$, $r = 0.08$, and $n = 8$.

Step1: Calculate the exponent term

$e^{rn}=e^{0.08\times8}=e^{0.64}\approx1.89648$.

Step2: Calculate the future - value of the annuity

$F_2=191790\times e^{0.64}=191790\times1.89648\approx363777$.

Answer:

a. $191790$ b. $363777$