Question: a. Use the appropriate formula to determine the periodic deposit.
b. How much of the financial goal comes from deposits and how much comes from interest?
Periodic Deposit Rate Time Financial Goal
$? at the end of each month 8.25% compounded monthly 45 years $1,000,000
1. **State the problem:**
We want to find the monthly periodic deposit needed to reach a financial goal of $1,000,000 in 45 years with an interest rate of 8.25% compounded monthly.
2. **Formula used:**
The future value of an ordinary annuity formula is:
$$FV = P \times \frac{(1 + r)^n - 1}{r}$$
where:
- $FV$ is the future value ($1,000,000$),
- $P$ is the periodic deposit (monthly deposit),
- $r$ is the monthly interest rate,
- $n$ is the total number of deposits.
3. **Calculate parameters:**
- Annual interest rate = $8.25\% = 0.0825$
- Monthly interest rate $r = \frac{0.0825}{12} = 0.006875$
- Number of years = $45$
- Number of months $n = 45 \times 12 = 540$
4. **Rearrange formula to solve for $P$:**
$$P = \frac{FV \times r}{(1 + r)^n - 1}$$
5. **Calculate $(1 + r)^n$:**
$$ (1 + 0.006875)^{540} = (1.006875)^{540} \approx 57.435 $$
6. **Calculate denominator:**
$$ (1.006875)^{540} - 1 = 57.435 - 1 = 56.435 $$
7. **Calculate periodic deposit $P$:**
$$ P = \frac{1,000,000 \times 0.006875}{56.435} = \frac{6875}{56.435} \approx 121.77 $$
So, the monthly deposit needed is approximately $\boxed{121.77}$.
8. **Calculate total amount deposited:**
$$ \text{Total deposits} = P \times n = 121.77 \times 540 = 65,743.80 $$
9. **Calculate interest earned:**
$$ \text{Interest} = FV - \text{Total deposits} = 1,000,000 - 65,743.80 = 934,256.20 $$
**Final answers:**
- Monthly deposit: $121.77$
- Total deposits: $65,743.80$
- Interest earned: $934,256.20$