If you receive $110 each quarter for 19 years and the discount rate is 0.07, what is the future value?

Question

If you receive $110 each quarter for 19 years and the discount rate is 0.07, what is the future value?

BlackTom AI · Accepted Answer

We start by identifying the cash flow details and the appropriate formula. The problem gives a quarterly payment of 110 dollars for 19 years, with a discount rate of 7% per year. Since payments are quarterly, we must convert the annual rate to a quarterly rate and count the total number of quarters.

Step 1: Convert the annual rate to a quarterly rate.
- Quarterly rate i = 0.07 / 4 = 0.0175.

Step 2: Determine the number of payment periods.
- There are 4 quarters per year, for 19 years: n = 19 × 4 = 76 periods.

Step 3: Use the future value of an ordinary annuity formula for level payments.
- FV = P × [ ( (1 + i)^n − 1 ) / i ], where P = 110, i = 0.0175, n = 76.

Step 4: Compute (1 + i)^n.
- (1 + 0.0175)^76 ≈ (1.0175)^76. This exponentiation captures the compound growth of each quarterly payment until the end of the 76th period. A precise calculation yields a value around 3.736 to 3.74 for (1.0175)^76, which reflects substantial growth over 76 quarters.

Step 5: Substitute back into the formula.
- ( (1.0175)^76 − 1 ) / 0.0175 ≈ (3.736 − 1) / 0.0175 ≈ 2.736 / 0.0175 ≈ 156.4.
- Then FV ≈ 110 × 156.4 ≈ 17,204 approximately.

Step 6: Refine and acknowledge typical rounding.
- Using a more exact (1.0175)^76 value from a calculator can yield a factor around 156.0 to 156.5, which would produce an FV in the vicinity of 17,200. The provided numerical result of 17,209.01 aligns with a precise evaluation of the same expression.

Conclusion (not stated as final here): The key method is to treat the payments as a quarterly annuity, convert the annual rate to a quarterly rate, compute the total number of periods, apply the future value formula for an ordinary annuity, and then evaluate with careful rounding to reach the given figure.

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If you receive $110 each quarter for 19 years and the discount rate is 0.07, what is the future value?

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