59. Perpetual Annuity
Imagine that today you deposit $B ...

Question

59. Perpetual Annuity

Imagine 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) dt

where 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

Hello. In this video, if we want to receive $$6000 per year forever from a continuously compounded account with a 9% annual interest rate, how much should we deposit now? We want to write our answer as a whole number. So the first thing we're going to do in this problem is we're going to take our 9% annual interest rate and convert it to a decimal. 9% converted to a decimal, is going to give us an interest rate of 0.09%. Now, we want to receive 6000 $$per year forever, which is a sum of continuous years. So we're going to label a return function, call it P X, and P X is going to be defined as the sum of all these years, which is going to be an integral. Now, we're going to label our starting year to be zero, which is when we make our deposit. And since we want this return to be a lifetime return, we are going to label the endpoint as infinity. And we want to go ahead and use the function for a continuously compounded account with an annual interest rate. So we are going to integrate the function 6000 because we want our return to be 6000 multiplied by E raise the power of negative 0.09 TDT. So this is going to be the function that represents our return for the continuously compounded account. And so in order to figure out how much we should deposit now, we need to solve the following integral. So, the first thing we can do is we can move 6000 as a constant in front of the integral, and that is going to give us 6000 multiplied by the integral from 0 to infinity of Ease the power of negative 0.09 T DT. Now, there is an identity that we can use here in order to solve this following integral. If we have an integral from 0 to infinity of an exponential function raised to a negative constant kT DT, then this integral is going to evaluate to be 1 divided by k for all k that is greater than 0. So, what this means is that if we apply the identity to the integral, this integral is going to evaluate to be 1 divided by 0.09. And keep in mind that this is still being multiplied by 6000. And finally, if we multiply both of the values together, we will end up with a final value of 66,667. So what this means is that if we want to receive $$6000 per year with the interest rate, we need to deposit 66$$,667 now. And with that being said, the overall solution to this problem is going to be B. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Continuous Compounding

Perpetual Annuity and Present Value of Infinite Cash Flows

Improper Integral of an Exponential Decay Function

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