Question

59. Perpetual AnnuityImagine that today you deposit $$B in a savings account that earns interest at a rate of *p*% per year compounded continuously (see Section 7.2). The goal is to draw an income of I $$per year from the account forever. The amount of money that must be deposited is:B = I × ∫(from 0 to ∞) e^(-rt) dtwhere r = p/100.Suppose you find an account that earns 12% interest annually, and you wish to have an income from the account of $5000 per year. How much must you deposit today?

Accepted Answer

Identify the given variables: the interest rate is $$p = 12\%$$, so the continuous compounding rate is $$r = \frac{p}{100} = \frac{12}{100} = 0.12$$, and the desired income per year is $$I = 5000$$ dollars. Recall the formula for the amount to deposit today to receive a perpetual income $$I$$ with continuous compounding interest rate $$r$$:

$$B = I 	imes \$$int_0$$^{\infty} e$$^{-rt}$$ \, dt$$ Evaluate the integral $$\$$int_0$$^{\infty} e$$^{-rt}$$ \, dt. $$This is an improper integral of an exponential decay function, which converges because $$r \u003e 0. $$The integral evaluates to:

$$\$$int_0$$^{\infty} e$$^{-rt}$$ \, dt = \left[ -\frac{1}{r} e$$^{-rt}$$ ight]_0$$^{\infty}$$ = \frac{1}{r}$$ Substitute the value of the integral back into the formula for $$B$$:

$$B = I 	imes \frac{1}{r} = \frac{I}{r}$$ Finally, plug in the known values $$I = 5000$$ and $$r = 0.12$$ into the formula to find the amount $$B$$ that must be deposited today:

$$B = \frac{5000}{0.12}$$

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Concetti chiave

Continuous Compounding

Perpetual Annuity and Present Value of Infinite Cash Flows

Improper Integral of an Exponential Decay Function