To find out how much Henry Bowie needs to deposit today to have enough money for the engine overhaul in 6 years, we can follow these detailed steps:
### Step 1: Calculate the Expected Cash Outflow
First, we need to calculate the expected value of the cash outflows based on their probabilities. This is done by multiplying each cash outflow by its corresponding probability and summing them up.
Given cash outflows and their probabilities:
- \[tex]$330 with probability \(10\%\) (or \(0.1\))
- \$[/tex]490 with probability [tex]\(30\%\)[/tex] (or [tex]\(0.3\)[/tex])
- \[tex]$710 with probability \(50\%\) (or \(0.5\))
- \$[/tex]780 with probability [tex]\(10\%\)[/tex] (or [tex]\(0.1\)[/tex])
The expected cash outflow (E) is calculated as follows:
[tex]\[ E = (330 \times 0.1) + (490 \times 0.3) + (710 \times 0.5) + (780 \times 0.1) \][/tex]
Calculating each term:
[tex]\[ 330 \times 0.1 = 33 \][/tex]
[tex]\[ 490 \times 0.3 = 147 \][/tex]
[tex]\[ 710 \times 0.5 = 355 \][/tex]
[tex]\[ 780 \times 0.1 = 78 \][/tex]
Adding them together:
[tex]\[ E = 33 + 147 + 355 + 78 = 613 \][/tex]
So, the expected cash outflow is \[tex]$613.
### Step 2: Calculate the Present Value
Next, we need to find out how much Henry should deposit today to cover this expected cash outflow in 6 years with an annual interest rate of \(9\%\) compounded annually.
We use the Present Value (PV) formula:
\[ PV = \frac{FV}{(1 + r)^n} \]
where:
- \( FV \) is the future value (\$[/tex]613)
- [tex]\( r \)[/tex] is the annual interest rate ([tex]\(0.09\)[/tex])
- [tex]\( n \)[/tex] is the number of years (6)
Plugging in the values we get:
[tex]\[ PV = \frac{613}{(1 + 0.09)^6} \][/tex]
Calculating the denominator:
[tex]\[ (1 + 0.09)^6 = 1.689478 \][/tex]
Now, calculate the present value:
[tex]\[ PV = \frac{613}{1.689478} \approx 365.5118713769593 \][/tex]
### Step 3: Round the Deposit Amount
Finally, we round the deposit amount to 0 decimal places as specified in the problem:
[tex]\[ \text{Deposit amount} \approx 366 \][/tex]
### Conclusion
To ensure Henry has enough money in 6 years, he should deposit \$366 today in an account earning [tex]\(9\%\)[/tex] interest compounded annually.
