You want to be able to withdraw $$ \$ 30,000 $$ each year for 30 years. Your account earns $$ 10 \% $$ interest. a) How much do you need in your account at the beginning? b) How much total money will you pull out of the account? $$ \$ 900,000 $$ c) How much of that money is interest? $$ \$ $$ \$

Question

UpStudy AI · Accepted Answer

**Step 1. Present Value of an Annuity Formula**

We want the account to fund annual withdrawals of $$ A = \$30{,}000 $$ for $$ n = 30 $$ years at an annual interest rate of $$ i = 10\% = 0.10 $$. The present value (PV) needed is given by

$$
PV = A \times \frac{1 - (1 + i)^{-n}}{i}.
$$

---

**Step 2. Substitute the Given Values**

Substitute $$ A = 30000 $$, $$ i = 0.10 $$, and $$ n = 30 $$:

$$
PV = 30000 \times \frac{1 - (1.1)^{-30}}{0.10}.
$$

---

**Step 3. Compute $$ (1.1)^{-30} $$**

First, compute $$ (1.1)^{30} $$. It is approximately

$$
(1.1)^{30} \approx 17.45.
$$

Thus,

$$
(1.1)^{-30} \approx \frac{1}{17.45} \approx 0.0573.
$$

---

**Step 4. Plug the Value Back into the Formula**

Now substitute $$ (1.1)^{-30} \approx 0.0573 $$:

$$
PV = 30000 \times \frac{1 - 0.0573}{0.10}.
$$

Simplify the numerator:

$$
1 - 0.0573 = 0.9427.
$$

Thus,

$$
PV = 30000 \times \frac{0.9427}{0.10} = 30000 \times 9.427.
$$

---

**Step 5. Calculate the Present Value**

Multiplying,

$$
30000 \times 9.427 \approx 282810.
$$

So, the amount needed in the account at the beginning is approximately

$$
\$282{,}810.
$$

---

**Step 6. Total Money Withdrawn**

Over 30 years, the total withdrawals amount to

$$
30000 \times 30 = \$900{,}000.
$$

---

**Step 7. Total Interest Earned**

The interest earned is the difference between the total withdrawals and the initial deposit:

$$
\text{Interest} = 900000 - 282810 \approx \$617{,}190.
$$

---

**Final Answers:**

a) Approximately $$ \$282{,}810 $$  
b) $$ \$900{,}000 $$  
c) Approximately $$ \$617{,}190 $$ in interest
You need \$282,810 at the beginning. You'll withdraw a total of \$900,000 over 30 years, with \$617,190 of that being interest.

You want to be able to withdraw \( \$ 30,000 \) each year for 30 years. Your account earns \( 10 \% \) interest. a) How much do you need in your account at the beginning? b) How much total money will you pull out of the account? \( \$ 900,000 \) c) How much of that money is interest? \( \$ \) \$

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Economics Questions