To find the total gain at the end of the second year for both accounts combined, we need to follow these steps:
- Calculate the amount placed in the savings account and the stock plan.
- Calculate the growth of the savings account after 2 years using compound interest.
- Calculate the value of the stock plan after the two years of changes.
- Combine the results and subtract the initial investment to find the total gain.
Step 1: Allocation of Funds
- Total savings: \( $6,300.00 \)
- Amount in savings account (30%): \[ 0.30 \times 6300 = 1890 \]
- Amount in stock plan (70%): \[ 0.70 \times 6300 = 4410 \]
Step 2: Growth of Savings Account
The savings account earns an APR of 3.6%, compounded annually. We can use the formula for compound interest: \[ A = P(1 + r)^t \] where \( A \) is the amount of money accumulated after n years, including interest, \( P \) is the principal amount (the initial amount of money), \( r \) is the annual interest rate (decimal), and \( t \) is the number of years the money is invested for.
For the savings account:
- \( P = 1890 \)
- \( r = 0.036 \)
- \( t = 2 \)
Calculating the amount: \[ A = 1890 \times (1 + 0.036)^2 \] \[ A = 1890 \times (1.036)^2 \] \[ A \approx 1890 \times 1.072656 \approx 2028.38 \]
Step 3: Growth of Stock Plan
The stock plan decreases by 2.5% in the first year and increases by 8.9% in the second year.
After the first year (decrease of 2.5%): \[ \text{Value after year 1} = 4410 \times (1 - 0.025) = 4410 \times 0.975 = 4307.75 \]
After the second year (increase of 8.9%): \[ \text{Value after year 2} = 4307.75 \times (1 + 0.089) = 4307.75 \times 1.089 = 4686.50 \]
Step 4: Combine and Calculate Total Gain
Now we combine the results from both accounts:
Total amount after 2 years: \[ \text{Total} = \text{Savings account} + \text{Stock plan} = 2028.38 + 4686.50 = 5714.88 \]
Total initial investment: \[ \text{Total investment} = 6300 \]
Total gain: \[ \text{Total gain} = \text{Total final amount} - \text{Total invested} = 5714.88 - 6300 = -585.12 \]
Thus, the total gain at the end of the second year for both accounts combined is a loss of approximately $585.12.