# Monthly Archives: January 2012

## What Does the Discount Rate Really Mean?

From Finance 101, to compute the present value of future cash flows, you must discount the future cash flows by a discount rate. For example, if you expect to receive a payment of $\mathrm{CF}_1$ in one year, and you use a yearly rate of  r to discount, the present value would be given by

$\mathrm{PV} = \frac{\mathrm{CF}_1}{1+r}.$

Normally, the discount rate accounts for the time value of money. When computing the present value of uncertain future cash flows, the discount rate must also incorporate a risk premium associated with the cash flow: the higher the uncertainty, the higher the risk premium. But how does the rate reflect the cash flow uncertainty? This is where many people are confused, and rules of thumb abound. Determining the “appropriate” discount rate is something of an art form.

However, there is a very simple interpretation of the discount rate that is often easier to establish: that of incorporating a probability of success. Suppose that, with probability p, you will indeed receive the cash flow, and with probability 1-p, you will receive nothing. Letting r represent only the time value of money, the present value is given by

$\mathrm{PV} = p\frac{\mathrm{CF}_1}{1+r} + (1-p) 0.$

With a bit of algebra, we can transform this as follows:

$\mathrm{PV} = \frac{\mathrm{CF}_1}{(1+r)/p} = \frac{\mathrm{CF}_1}{1+(1+r-p)/p} = \frac{\mathrm{CF}_1}{1+\tilde{r}}.$

where $\tilde{r}=(1+r-p)/p$. In other words, we have incorporated into a new discount rate $\tilde{r}$ the separate effects of the time value of money r and the probability of success p.

An entrepreneur (or investor) may more easily estimate (or believe to accurately estimate) the probability of success than the proper risk premium to add to the time value of money. Conversely, from a given discount rate, we can work the gears in reverse and compute the implied probability of success:

$p = \frac{1+r}{1+\tilde{r}}.$

The discount rate associated with a given time value of money and probability of success is given in this handy table. The shaded area corresponds to cases often encountered in practice.