Question

pf.7 annuities - homework
score: 16.97/56 answered: 10/16
question 11
a. daniel deposits $400.00 every quarter into an account earning 4% interest compounded quarterly. how much will daniel have in the account in 20 years?
daniel will have in the account in 20 years.
b. alternatively, daniel could make a single deposit into an account earning 4% compounded quarterly for 20 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?
daniel 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 annuity (A)

This is a regular annuity (payments at the end of each period) with quarterly payments. 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:

• $P = 400$ (quarterly payment),
• $r = 0.04$ (annual interest rate),
• $n = 4$ (number of compounding periods per year),
• $t = 20$ (number of years).

Step2: Calculate the future value (A)

First, calculate $\frac{r}{n} = \frac{0.04}{4} = 0.01$, and $nt = 4 \times 20 = 80$.
Then, $(1 + 0.01)^{80} \approx 2.216715$.
Next, $\frac{(1.01)^{80} - 1}{0.01} \approx \frac{2.216715 - 1}{0.01} = 121.6715$.
Finally, $FV = 400 \times 121.6715 \approx 48668.60$.

Step3: Calculate the lump sum (B)

The formula for present value (lump sum) of a future amount is $PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}$. We know $FV \approx 48668.60$ from part A, $\frac{r}{n} = 0.01$, $nt = 80$.
So, $PV = \frac{48668.60}{(1.01)^{80}} \approx \frac{48668.60}{2.216715} \approx 21955.44$.

Answer:

A. $\$48668.60$ (rounded to two decimal places)
B. $\$21955.44$ (rounded to two decimal places)