Continuous Income Streams

Business Calculus · Axiom Academy

LESSON Continuous Income Streams Valuing money that flows continuously over time 1 What is a Continuous Income Stream? Many business scenarios involve money flowing in (or out) continuously over time rather than in discrete payments: Revenue from an online subscription service Profits from ongoing operations Interest earned on investments We model continuous income with a rate function f(t), which gives the rate of income (dollars per year) at time t. Total income over [0, T] = _0^T f(t)\,dt 2 Present Value vs. Future Value Money has a time value - 100 today is worth more than 100 in 5 years. We need two concepts: How much the income stream is worth today . What lump sum today equals this stream? How much will have accumulated by the end of the stream. Total including reinvested interest? 3 Present Value of Income Stream r = annual interest rate (decimal), T = time in years Income received at time t must be "discounted" back to today. The factor e^ -rt decreases as t increases - money further in the future is worth less today. Example: Constant Income Stream A franchise generates 50,000/year for 10 years. Interest rate is 5%. Find present value. Total income = 500,000 over 10 years, but present value is only 393,470. The difference ( 106,530) represents the time value of money. 4 Future Value of Income Stream Income earned at time t grows for (T-t) years

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