How much should you deposit at the end of each month into an investment account that pays $$ 6.5 \% $$ compounded monthly to have $$ \$ 3 $$ million when you retire in 38 years? How much of the $$ \$ 3 $$ million comes from interest? (i) Click the icon to view some finance formulas.

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Let

$$
i=\frac{0.065}{12}\quad\text{(monthly interest rate)}
$$

and

$$
n=38\times12=456\quad\text{(total number of months)}.
$$

The formula for the future value of an annuity (with deposits made at the end of each period) is

$$
\text{FV}=D\left(\frac{(1+i)^n-1}{i}\right),
$$

where $$ D $$ is the monthly deposit. We want the account to have $$ \$3,\!000,\!000 $$ when you retire, so

$$
3,\!000,\!000 = D\left(\frac{(1+i)^n-1}{i}\right).
$$

Solving for $$ D $$ we have

$$
D=\frac{3,\!000,\!000\, i}{(1+i)^n-1}.
$$

Substitute $$ i=\frac{0.065}{12} $$ and $$ n=456 $$:

$$
D=\frac{3,\!000,\!000\left(\frac{0.065}{12}\right)}{\left(1+\frac{0.065}{12}\right)^{456}-1}.
$$

Now compute the numerical values step by step.

1. Compute the monthly rate:

$$
i=\frac{0.065}{12}\approx0.00541667.
$$

2. Compute $$(1+i)^n$$:

$$
(1+i)^{456}\approx\left(1.00541667\right)^{456}.
$$

It is easier to use logarithms:

$$
\ln\left((1.00541667)^{456}\right)=456\ln(1.00541667).
$$

Since $$\ln(1.00541667)\approx0.005402$$, we have

$$
456\times0.005402\approx2.465.
$$

Then

$$
(1.00541667)^{456}\approx e^{2.465}\approx11.77.
$$

3. Then

$$
(1+i)^n-1\approx11.77-1=10.77.
$$

4. Now, compute the numerator:

$$
3,\!000,\!000\times0.00541667\approx16,\!250.
$$

5. So the monthly deposit is approximately

$$
D\approx\frac{16,\!250}{10.77}\approx\$1,\!508.
$$

Thus, you must deposit about $$ \$1,\!508 $$ each month.

Next, the total amount you deposit over 456 months is

$$
\text{Total Contributions}=456\times1,\!508\approx\$687,\!648.
$$

Since the account will have $$ \$3,\!000,\!000 $$, the amount that comes from interest is

$$
\text{Interest} = 3,\!000,\!000 - 687,\!648\approx\$2,\!312,\!352.
$$

In summary:

$$
\boxed{D\approx\$1,\!508 \quad\text{per month, with about} \quad \$2,\!312,\!352 \quad \text{in interest.}}
$$
You need to deposit approximately \$1,508 at the end of each month. This will grow to \$3 million over 38 years, with about \$2,312,352 coming from interest.

How much should you deposit at the end of each month into an investment account that pays \( 6.5 \% \) compounded monthly to have \( \$ 3 \) million when you retire in 38 years? How much of the \( \$ 3 \) million comes from interest? (i) Click the icon to view some finance formulas.

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