Future Value of an Income Stream The chef-owner of the Vegan...

Future Value of an Income Stream The chef-owner of the Vegan Restaurant expects that in the first 5 years of its operation, revenue will be generated at the rate of $R(t)=100+20 t$ thousand dollars/year, where $t$ is the number of years after the grand opening of the restaurant. If the prevailing rate of interest for the period is $5 \% /$ year compounded continuously, what will be the value of the restaurant's revenue stream at the end of 5 years?

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Key Concepts

Continuous Compounding

Continuous compounding refers to the process of calculating interest assuming that it is applied and added to the principal continuously, rather than at discrete intervals. In finance, this leads to an exponential growth factor, where the accumulation factor is typically expressed as e^(rt), with r representing the interest rate and t the time period.

Integration for Continuous Cash Flows

When cash flows occur continuously over time, integration is used to sum these flows over a given period. This mathematical process allows us to compute the total value of a continuously changing income stream by accounting for the contribution of infinitesimally small intervals, each adjusted for the time value of money.

Future Value Calculation for Income Streams

The future value calculation involves determining how much a series of cash flows will be worth at a specific future date, taking into account the effect of interest accumulation over time. In the context of continuous income streams, each small cash flow is compounded from its time of receipt to the target future date using an appropriate accumulation factor derived from continuous compounding.

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