Carlos set up a compound interest investment to pay his employee on retirement. He will deposit an initial $15 000 and add $240 to the investment every month. Interest will be paid at the rate of 4.85% per annum, compounding monthly.a) Carlos’ employee will retire in 18 years time. Use a financial solver to determine the amount of money that will be paid to the employee upon retirement. Write your answer correct to the nearest cent. (the dollar sign is not required)简答题
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Question

Carlos set up a compound interest investment to pay his employee on retirement. He will deposit an initial $15 000 and add $240 to the investment every month. Interest will be paid at the rate of 4.85% per annum, compounding monthly.a) Carlos’ employee will retire in 18 years time. Use a financial solver to determine the amount of money that will be paid to the employee upon retirement. Write your answer correct to the nearest cent. (the dollar sign is not required)简答题

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BlackTom AI · Accepted Answer

Future value = future value of the initial lump sum plus future value of the monthly ordinary annuity.

Relevant concepts:
- Monthly interest rate i = annual rate / 12.
- Number of periods N = years × 12.
- FV of a lump sum: P(1 + i)^N.
- FV of an ordinary annuity (end-of-period payments): PMT × [((1 + i)^N − 1) / i].

Step-by-step reasoning and calculations:
1. Convert the annual rate to a monthly rate and compute the number of months:
   - Annual rate r = 4.85% = 0.0485.
   - Monthly rate i = r/12 = 0.0485/12 = 0.004041666666666667 (which equals 97/24000).
   - Time t = 18 years → N = 18 × 12 = 216 months.

2. Compute the growth factor (1 + i)^N:
   - (1 + i) = 1.0040416666666667.
   - (1 + i)^216 ≈ 2.389862... (this is obtained with a financial calculator or accurate exponential calculation).

3. Future value of the initial $15,000:
   - FV_lump = 15000 × (1 + i)^216
   - FV_lump ≈ 15000 × 2.389862... ≈ 35,847.93 (rounded to cents).

4. Future value of the monthly $240 contributions (ordinary annuity):
   - FV_annuity = 240 × [((1 + i)^216 − 1) / i]
   - Using (1 + i)^216 ≈ 2.389862 and i = 97/24000,
     ((1 + i)^216 − 1) ≈ 1.389862,
     1/i = 24000/97 ≈ 247.4226804,
     so annuity factor ≈ 1.389862 × 247.4226804 ≈ 343.532...,
     FV_annuity ≈ 240 × 343.532... ≈ 82,532.89 (rounded to cents).

5. Add the two parts to get total future value:
   - FV_total = FV_lump + FV_annuity ≈ 35,847.93 + 82,532.89 ≈ 118,380.82.

Why this answer is correct:
- The calculation properly separates the single initial deposit (compounded for all periods) and the stream of equal monthly deposits (an ordinary annuity), uses the exact monthly interest rate and number of months, and applies standard compound-interest formulas. A financial solver (or a precise calculator) using N=216, monthly I%=4.85/12, PV = −15,000 (if entering sign convention) and PMT = −240 will return the same FV ≈ 118,380.82.

Other options: not applicable here (short-answer).

Conclusion: The amount available at retirement is 118,380.82, which matches the provided answer.

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