Question

Question 4
Imagine that your goal is to retire 34 years from today with
$
1
,
000
,
000
$1,000,000 in savings. Assuming that you currently (i.e., today) have
$
5
,
000
$5,000 in savings, what rate of return must you earn on that savings to hit your goal?

(Hint: Solve your future value formula for the discount rate,
R
R)


0.4882
0.4882



4.8824
4.8824



0.1686
0.1686



0.2000
0.2000

Answer 1

To solve for the required rate of return to reach a future value of $1,000,000 with an initial investment of $5,000 over 34 years, we can use the future value formula:

\[
FV = PV \times (1 + r)^n
\]

Where:
- \( FV \) is the future value ($1,000,000),
- \( PV \) is the present value ($5,000),
- \( r \) is the annual interest rate (rate of return),
- \( n \) is the number of years (34).

We can rearrange the formula to solve for the rate of return \( r \):

\[
r = \left( \frac{FV}{PV} \right)^{\frac{1}{n}} - 1
\]

Now, substituting the known values into the formula:

\[
r = \left( \frac{1,000,000}{5,000} \right)^{\frac{1}{34}} - 1
\]

Calculating the fraction:

\[
\frac{1,000,000}{5,000} = 200
\]

Now, we substitute 200 into the formula:

\[
r = 200^{\frac{1}{34}} - 1
\]

Now we need to calculate \( 200^{\frac{1}{34}} \):

Using a calculator:

- \( 200^{\frac{1}{34}} \approx 1.0524 \)

Then,

\[
r \approx 1.0524 - 1 = 0.0524
\]

So, converting this to a percentage:

\[
r \approx 0.0524 \times 100 = 5.24\%
\]

After examining the options provided:

None of the options seem to match the calculated rate of return of approximately 5.24%. If the answer choices are strictly as you've provided, then it seems there's an inconsistency. However, the methodology used to calculate the rate of return is correct.

If you meant to list choices that might lead to this outcome or have a different context for the options, let me know for further clarification or correction!