Question

If Ali wants to save for retirement and plans to withdraw $2,000 every month for the next 20 years, how much should he save today if the interest rate is 5 percent per year? Give the answer rounded to the nearest hundredths.

Answer 1

To determine how much Ali should save today to afford monthly withdrawals of $2,000 for 20 years, we can use the present value of an annuity formula.

The formula for the present value of an annuity is:

\[
PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r
\]

Where:
- \( PV \) is the present value (the amount to save today),
- \( PMT \) is the payment amount per period (monthly withdrawal),
- \( r \) is the interest rate per period,
- \( n \) is the total number of periods.

1. **Identify the variables:**
- Monthly withdrawal (\( PMT \)) = $2,000
- Annual interest rate = 5% → Monthly interest rate (\( r \)) = 5% / 12 = 0.41667% = 0.0041667 (as a decimal)
- Number of years = 20 → Total months (\( n \)) = 20 * 12 = 240

2. **Plug in the values:**

\[
PV = 2000 \times \left(1 - (1 + 0.0041667)^{-240}\right) / 0.0041667
\]

3. **Calculate \((1 + r)^{-n}\):**

\[
(1 + 0.0041667)^{-240} \approx (1.0041667)^{-240} \approx 0.3697
\]

4. **Calculating the PV:**

\[
PV = 2000 \times \left(1 - 0.3697\right) / 0.0041667
\]

\[
= 2000 \times (0.6303) / 0.0041667
\]

\[
= 2000 \times 151.2576 \approx 302515.20
\]

Therefore, Ali needs to save **$302,515.20** today to provide for his monthly withdrawals of $2,000 for 20 years at a 5% annual interest rate.