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 b. in part (a), suppose that an annuity will be withdrawn from savings that have been accumulated at the eoy 25.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 $32456. (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 b. in part (a), suppose that an annuity will be withdrawn from savings that have been accumulated at the eoy 25.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 $32456. (round to the nearest dollar.)

Answer

Explanation:

Step1: Identify the annuity - future - value formula for continuous compounding

The formula for the future - value of an ordinary annuity with continuous compounding is $F = A\frac{e^{rn}-1}{e^{r}-1}$, where $A$ is the annual payment, $r$ is the annual interest rate, and $n$ is the number of years. Given $A = 2500$, $r=0.08$, and $n = 25$. $F = 2500\times\frac{e^{0.08\times25}-1}{e^{0.08}-1}$

Step2: Calculate the exponent values

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

Step3: Calculate the numerator and denominator

The numerator is $e^{2}-1\approx7.389056 - 1=6.389056$. The denominator is $e^{0.08}-1\approx1.083287 - 1 = 0.083287$.

Step4: Calculate the future - value of the annuity

$F = 2500\times\frac{6.389056}{0.083287}\approx2500\times76.7127 = 191781.75\approx191782$.

For part (b), we first note that the accumulated amount at the end of year 25 is $F = 191782$ (from part (a)). Now, we have an ordinary annuity from year 26 to year 33 ($n = 8$ years) with the same interest rate $r = 0.08$. The present - value of this annuity at the start of year 26 (end of year 25) is $P = A\frac{1-(1 + r)^{-n}}{r}$, and we want to find the value of the annuity. But we can also use the future - value formula for the annuity from year 26 to year 33. The annual payment $A$ is still $2500$, $r = 0.08$, and $n = 8$. $F_{new}=A\frac{(1 + r)^{n}-1}{r}=2500\times\frac{(1 + 0.08)^{8}-1}{0.08}$

Step1: Calculate $(1 + 0.08)^{8}$

$(1 + 0.08)^{8}=1.08^{8}\approx1.85093$.

Step2: Calculate the numerator

$(1 + 0.08)^{8}-1\approx1.85093-1 = 0.85093$.

Step3: Calculate the future - value of the new annuity

$F_{new}=2500\times\frac{0.85093}{0.08}=2500\times10.636625 = 26591.5625\approx26592$. The total value of the annuity is the sum of the accumulated value at the end of year 25 and the future - value of the annuity from year 26 to year 33. But if we assume we are just looking for the value of the annuity from year 26 to year 33, the value of the annuity is $26592$. If we want the combined value (including the amount accumulated at year 25), we add the two amounts: $191782+26592=218374$. But since the problem seems to be asking for the value of the annuity from year 26 - 33, we focus on the $26592$ value for the annuity part. However, if we consider the whole problem context, we need to re - evaluate.

Let's re - do part (b) using the fact that we have an annuity starting at the end of year 26 and ending at the end of year 33. The annual payment $A = 2500$, $r=0.08$, $n = 8$. The future - value of an ordinary annuity formula is $F=A\frac{(1 + r)^{n}-1}{r}$. $F = 2500\times\frac{(1.08)^{8}-1}{0.08}$ $(1.08)^{8}=1.85093$. $\frac{(1.08)^{8}-1}{0.08}=\frac{1.85093 - 1}{0.08}=\frac{0.85093}{0.08}=10.636625$. $F = 2500\times10.636625=26591.56\approx26592$.

Answer:

a. $191782$ b. $26592$