To find out the real amount you need to deposit annually in order to have [tex]$2.5 million in real dollars in 40 years given a nominal return rate of 10.3% and an inflation rate of 3.7%, follow these steps:
### Step 1: Calculate the Real Return Rate
The real return rate can be found using the formula for the real interest rate, which accounts for inflation:
\[ \text{Real Return} = \left( \frac{1 + \text{Nominal Return}}{1 + \text{Inflation Rate}} \right) - 1 \]
Given:
- Nominal Return = 10.3% (0.103)
- Inflation Rate = 3.7% (0.037)
So, the real return rate is:
\[ \text{Real Return} = \left( \frac{1 + 0.103}{1 + 0.037} \right) - 1 = 0.06364513018322082 \]
### Step 2: Use the Future Value of an Annuity Formula
You need to determine the annual deposit (P) using the future value of an annuity formula:
\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \]
Where:
- FV (Future Value) = $[/tex]2.5 million
- r (Real Return) = 0.06364513018322082
- n (Number of Years) = 40
Rearrange this formula to solve for P (annual deposit):
[tex]\[ P = \frac{FV \times r}{(1 + r)^n - 1} \][/tex]
### Step 3: Substitute the Given Values
Now plug in the values:
[tex]\[ P = \frac{2,500,000 \times 0.06364513018322082}{(1 + 0.06364513018322082)^{40} - 1} \][/tex]
### Step 4: Calculate the Annual Deposit
After substituting and calculating, the result for the annual deposit is:
[tex]\[ P = 14,733.105101594416 \][/tex]
### Conclusion
To achieve your goal of having [tex]$2.5 million in real dollars in an account in 40 years with a nominal return of 10.3% and an inflation rate of 3.7%, you must deposit approximately \$[/tex]14,733.11 each year.
