Growing Perpetuity

We have seen that a perpetuity represents an infinite stream of future cash flows. However, we have also seen that as time passes the value of these cash flows constantly diminishes. $100 may be able to buy us quite a few goods today, but in 50 years time $100 will not be nearly as valuable as it is today. It is for this reason that receiving infinite payments is not enough. The payments must also keep growing at a certain rate. This will ensure that they are not considerable behind inflation. This is the idea behind a growing perpetuity. The same has been explained in detail in this article:

Growing Infinite Payments

As already stated, a growing perpetuity involves payments that do not remain fixed. Instead these payments keep on growing at the same constant rate of growth. So, if the rate of growth of the payments is 7%, each payment will be 7% more than the payment received before it.

Present Value of a Growing Perpetuity

The present value of a growing perpetuity can be derived from a complex mathematical calculation. This is because a growing perpetuity is also an infinite series which has a finite sum. For our purposes, we can just remember the formula required for our calculation.

Present Value (Growing Perpetuity) = D / (R - G)

Where:

    D = Expected cash flow in period 1

    R = Expected rate of return

    G = Rate of growth of perpetuity payments

However, we need to understand that for this formula to hold true, G must always be greater than R. If G is less than R or equal to R, the formula does not hold true. This is because, the stream of payments will cease to be an infinitely decreasing series of numbers that have a finite sum.

Examples:

Growing perpetuities are found in a variety of places in our daily lives. Some of them have been mentioned below:

  • College endowment funds need to be growing perpetuities. This is because with the passage of time, tuition and other expenses will become more and more expensive. Hence the college endowment funds must keep growing to meet the scholarship demands represented by growing expenses.

  • Stock valuations always assume a growing perpetuity for their terminal value calculation. Without the concept of a growing perpetuity it would be impossible to value a stock.

  • Loss of Real Value of Money: Since the formula assumes that the growth rate of the perpetuity will always be less than the required rate of return, it is implying a loss scenario. This is because, no matter what the case, the growth rate can, by definition, never be more than the required rate of return.

    The growing perpetuity, thus assumes that we will lose a small amount of the real value of money every year. Just like the perpetuity, a growing perpetuity can only be summed up because the series is infinitely decreasing. The growing perpetuity assumes that we will lose the real value of money at a slower rate as compared to an ordinary perpetuity.


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