# Present Value

A present value computation is a means of converting a stream of future cash flows to a lump sum. In current appraisal practice, a future stream of cash flows from a firm whose equity is being appraised (called a “subject company” or SC) is converted to a present value using a special type of rate called a discount rate. This discount rate is built up from three components: inflation, opportunity cost, and risk.

Are future cash payments equivalent to present values of cash payments? Suppose an SC is expected to generate \$1,000 net cash flows in each of the next five years. How much would that investment be worth today? One simple answer might be to add the five year’s expected future cash flows and say the investment is worth \$5,000. What, if anything, would be wrong with this simple addition? We can think of a few things.

First, the purchasing power of money generally declines over time. A dollar today will buy more goods and services than it will in the future. This is inflation. So we need a method of converting the future expected cash flows to reflect the expected decline in purchasing power.

It is important to stress that the inflation effect involves compounding. This means that if I start out with \$1 today and inflation is expected to be say 3% each year I will have .97 cent of current purchasing power after one year. In the second year I will lose another 3% of purchasing power but the loss of purchasing power will not be measured against the present value of \$1 but against the value at the end of the first year of .97 cents. This compounding effect is captured by what is called a discount factor:

Let

Let r = the inflation rate (stated as a decimal percentage)

n = the number of future periods of cash flow

fv = the annual face value amount of expected cash flows

pv= present value discount factor = 1/(1+ r) n

Example. Assume that an SC will generate cash flows of \$1,000 in each of the next five years. Here is the present value computation assuming an inflation rate of 3% : How do we interpret this computation? If the annual inflation rate is 3% then collecting \$1,000 at the end of one year will provide the same amount of purchasing power as \$971 does today. Collecting \$1,000 at the end of year two will provide the same amount of purchasing power as \$943 does today and so on through year five. Adding the present values for each of the five year’s expected revenue of \$1,000 per year gives us the net present value of \$4,580. Had we simply added the five payments of \$1,000 we would have ignored the loss of purchasing power due to inflation.

There is a second reason why simply adding up the future cash flow does not work. By investing a lump sum of cash today, as would be expected in the acquisition of an SC, the investor is foregoing the present use of that cash. Presumably this cash would have been available to purchase some other investment opportunity.

A very plausible assumption is that the alternative investment would have a return at least equal to the safest possible investment available. Usually the rate of return considered safest is the rate of return on US treasury bonds. What this implies is that in the conversion of future cash flows to present values we have to take into account not merely the loss of purchasing power due to inflation but also the interest income that would have been earned had the investment not been made. This investment income foregone is often termed an opportunity cost.

We are quickly confronted with a practical measurement problem. It is relatively easy to predict the rate of inflation to reflect loss of purchasing power because good historical data exists and reasonably accurate short-term estimates of future rates of inflation are also available. It is also a simple matter to find the interest rate associated with the safest investment opportunities (short or mid-term US treasury bonds). But what is less clear is whether to simply add the treasury rate to the expected inflation rate.

Historically, the treasury rates were set so as to include the effects of inflation. However, in recent years, treasury rates have clearly been less than the rate of inflation so now the rates should be added. Clearly some care and thought must be given to determining how to combine the inflation rate and risk-free rate. That said, these problems are relatively minor compared to the biggest problem associated with present value discounting: providing a deeper level of present value discounting to reflect the risk associated with the SC investment.

It seems reasonable that potential investors in a closely held SC consider the relative risk of the investment. It makes perfect economic sense to demand a higher rate of return on investments that are deemed more risky. The relationship of increasing rates of return for increased risk is quite pronounced in the case of publicly traded debt instruments such as government and corporate bonds.

Well-established and sometimes credible rating agencies measure the level of risk associated with various debt instruments on ordinal scales. Safer rated debt generally returns lower interest rates than those rated less safe. For this reason, all variants of the income method add what is termed a risk premium to the risk-free treasury rate and inflation rate in developing an overall discount rate to apply to future expected cash flow. The risk premium is the additional rate of return required to accept the risk of the investment.

Example. Assume a five year expected cash flow of \$1,000. Further assume a 3% annual inflation rate and a risk free treasury rate of 3%. Finally assume that it is determined that a rational investor should demand a 10% return to reflect the risk associated with the investment. When an income method is utilized all three of these rates are added together (or built up) and the combined rate is used to develop a discount factor which then yields a present value for the equity investment as follows: The additional risk factor is called a risk premium. Click here for a discussion of how risk premiums are determined. The risk factor  is also termed a “hurdle rate”. It is so named because only by offering a rate equal to or above the risk premium does it make economic sense to make the investment. Unfortunately, the principle of demanding a greater rate of return for riskier investment rates in the context of investing in closely held firms is not necessarily rational. In fact, allowing hurdle rates to dictate investments in closely held firms where specialized expertise and experience is required violates a very fundamental economic principle which states that sunk costs should not be factored into investment decisions. For a discussion of the irrationality of hurdle rates click here.

For more on the subtle challenges in applying present value computations to future cash flows click here.

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