Future Value of Income Streams

Business Calculus · Axiom Academy

EXAMPLE Future Value of Income Streams Calculating accumulated value with reinvested earnings f(t) = income rate, r = interest rate, T = time period Income received at time t earns interest for the remaining (T - t) years. The factor e^ r(T-t) captures this compound growth. You contribute 12,000/year to a retirement account earning 7% compounded continuously. What will you have after 30 years? Total contributions = 12,000 × 30 = 360,000. But future value = 1,223,040! The extra 863,040 is interest earned on your contributions and on previously earned interest. A business reinvests profits at rate f(t) = 40000 + 2000t dollars/year. If reinvested at 5%, what's accumulated after 8 years? As the business grows (increasing from 40,000 to 56,000/year over 8 years), the accumulated fund grows even faster due to compound interest on the increasing contributions. Parents save f(t) = 500e^ 0.03t dollars/month (increasing savings as income grows). At 6% interest, what's the fund worth after 18 years? Monthly rate → Annual: f(t) = 6000e^(0.03t) If you know the Present Value, you can find Future Value directly: Future Value = Present Value × e^(rT) First, calculate PV of 12,000/year for 30 years at 7%: This matches our direct calculation of 1,223,040 ✓ This relationship is useful for checking your work or when you've already calculated PV and need FV (or vice versa). It works because both PV and FV represent the same income stream, just valued at different points in time.

This is the written version of the interactive lesson above. See the full Business Calculus course.