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You Save $15,000. You place 1/3 in a savings account earning 4.6% APR compounded annually. You then invest 1/4 of the remaining balance in a 3 year

Treasury bond earning 5.2% APR compounded annually and the rest ina stock plan. Your stock plan increases value 3% the first year, decreases 8% the second year
and increases 6% in value the third year. What are the balances for each account by the end of the third year and the total gain on your original saved amount?

1 answer

Let's break down your financial plan step by step:

  1. Initial Savings: You have saved $15,000.

  2. Savings Account:

    • You place \( \frac{1}{3} \) of $15,000 into a savings account: \[ \text{Savings Account Investment} = \frac{1}{3} \times 15000 = 5000 \]

    • This savings account earns an APR of 4.6% compounded annually. The formula for compound interest is: \[ A = P(1 + r)^t \] where:

      • \( A \) is the amount of money accumulated after \( t \) years, including interest.
      • \( P \) is the principal amount (the initial amount of money).
      • \( r \) is the annual interest rate (decimal).
      • \( t \) is the time the money is invested for in years.

      For our savings account: \[ A = 5000 \times (1 + 0.046)^3 \] Calculating \( (1 + 0.046)^3 \): \[ (1.046)^3 \approx 1.143724 \] Now calculate \( A \): \[ A \approx 5000 \times 1.143724 \approx 5718.62 \]

  3. Remaining Balance after putting money in the savings account: \[ \text{Remaining Balance} = 15000 - 5000 = 10000 \]

  4. Treasury Bond:

    • You invest \( \frac{1}{4} \) of this remaining balance: \[ \text{Treasury Bond Investment} = \frac{1}{4} \times 10000 = 2500 \]
    • The bond earns an APR of 5.2% compounded annually. Using the compound interest formula: \[ A = 2500 \times (1 + 0.052)^3 \] Calculating \( (1 + 0.052)^3 \): \[ (1.052)^3 \approx 1.165172 \] Now calculate \( A \): \[ A \approx 2500 \times 1.165172 \approx 2912.93 \]

  5. Investment in Stock Plan:

    • The remainder after the savings account and the bond is: \[ \text{Stock Investment} = 10000 - 2500 = 7500 \]
    • The stock plan has changes in value over three years:
      • Year 1: Increases by 3%: \[ 7500 \times 1.03 = 7725 \]
      • Year 2: Decreases by 8%: \[ 7725 \times 0.92 = 7105.50 \]
      • Year 3: Increases by 6%: \[ 7105.50 \times 1.06 = 7521.83 \]

  6. Calculating Final Balances:

    • Savings Account: \( \approx 5718.62 \)
    • Treasury Bond: \( \approx 2912.93 \)
    • Stock Plan: \( \approx 7521.83 \)

    The total balance at the end of three years: \[ \text{Total Balance} = 5718.62 + 2912.93 + 7521.83 \approx 16153.38 \]

  7. Calculating Total Gain: \[ \text{Total Gain} = \text{Total Balance} - \text{Original Amount} \] \[ \text{Total Gain} \approx 16153.38 - 15000 = 1153.38 \]

Summary of Results:

  • Savings Account Balance: $5,718.62
  • Treasury Bond Balance: $2,912.93
  • Stock Plan Balance: $7,521.83
  • Total Balance after 3 Years: $16,153.38
  • Total Gain: $1,153.38