## Understanding Perpetuities

Problem:

You just received a grant (or won the lottery) and are presented with two options:

(A) Collect \$50,000 right away

(B) Receive \$10,500 at the end of each year for the next five years

Market interest rate is 5%. What is the correct answer?

Explanation:

In my backyard, I had planted a lime tree about five years back. Every year, during the summer months,  I would eagerly look for lime only to be disappointed that there was no lime. However, starting from the fifth year, the tree started producing so much lime. Me and my kiddo would have so much fun harvesting the lime and making freshly squeezed limeade. From then on the lime tree has been bearing fruit every year happily ever after. According to Merriam-Webster Dictionary, perpetuity means “the state of continuing forever or for a very long time”. In the world of finance, a perpetuity is an investment that receives periodic payments forever, just like my lime tree.

Simplest Perpetuity Formula:

Below is the formula for calculating the Present Value (PV) of a perpetuity: where “C” is the cash received at the end of each period, and “r” is the discount / interest rate.

For option B, the calculation can be done easily in Excel, PV = \$45,549.50.

In today’s dollars, Option A = \$50,000, and Option B = \$45,549.50.

The correct answer is Option A. Take the \$50,000 and run.

Growing Perpetuities:

Dividend paying stocks are good examples of perpetuity. You buy and hold dividend paying stocks and they pay dividends forever. For example, a company like General Mills has been paying dividends for the past 100+ years. In the case of stocks, healthy companies boost their dividends every year. In such a case, you can use the following formula to calculate the present value:

PV of Perpetuity = C / (r – g)

where “C” is the cash dividend at the end of the first period, and “r” is the discount / interest rate, “g” – is the rate at which the cash dividend increases every year.

To keep things simple, if a company pays \$1.00 cash dividend at the end of year 1, and the interest rate is 5%, and the “g” dividend growth rate is 2%, then:

PV of Perpetuity (aka Stock in this case) = \$1.00 / (0.05 – 0.02) =  \$33.33

In real world, nothing lasts forever. The formulas above help in approximate valuation of a stock or bond or future incoming cash flows. Without going into tedious math, a perpetuity helps by providing a quick and dirty method to estimate the Present Value (PV) of a investment that pays periodically forever.