Question

pf.7 annuities - homework
score: 16.97/58 answered: 9/16
question 10
a. terrence deposits $100.00 every month into an account earning 6% interest compounded monthly. how much will terrence have in the account in 15 years?
terrence will have in the account in 15 years.
b. alternatively, terrence could make a single deposit into an account earning 6% compounded monthly for 15 years. how much would the lump sum deposit (single deposit) have to be in order to have saved the same amount of money in the account?
terrence would have to make a lump sum deposit of . hint
question help: video 1 video 2 video 3 video 4

Explanation:

Step1: Identify the type of problem (Part A)

This is an annuity problem. The formula for the future value of an ordinary annuity is $FV = P \times \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}}$, where $P$ is the payment per period, $r$ is the annual interest rate (in decimal), $n$ is the number of compounding periods per year, and $t$ is the number of years.
Here, $P = 100$, $r = 0.06$, $n = 12$ (monthly compounding), $t = 15$.

Step2: Calculate the future value (Part A)

First, calculate $\frac{r}{n}=\frac{0.06}{12}=0.005$.
Then, $nt = 12\times15 = 180$.
Next, $(1 + 0.005)^{180}\approx2.45409$.
Then, $(1 + 0.005)^{180}-1\approx1.45409$.
Then, $\frac{(1 + 0.005)^{180}-1}{0.005}=\frac{1.45409}{0.005}=290.818$.
Finally, $FV = 100\times290.818 = 29081.80$.

Step3: Identify the type of problem (Part B)

This is a present value problem. The formula for the present value of a lump sum is $PV=\frac{FV}{(1 + \frac{r}{n})^{nt}}$, where $FV$ is the future value we found in part A, $r = 0.06$, $n = 12$, $t = 15$.

Step4: Calculate the present value (Part B)

We know $FV = 29081.80$, $\frac{r}{n}=0.005$, $nt = 180$.
So, $(1 + 0.005)^{180}\approx2.45409$.
Then, $PV=\frac{29081.80}{2.45409}\approx11849.96$.

Answer:

A. $\$29081.80$ (rounded to two decimal places)
B. $\$11849.96$ (rounded to two decimal places)