## 1.2 Present Value

The standard measure that delivers the expected value created by a business decision, incorporating the full probability distribution of possible results, or payoffs, is present value. The present value of a business decision is defined mathematically as follows:

The result of evaluating the right-hand side of this equation is the value (at time 0) created by pursuing a business decision (for example, going down a specific node on the decision tree in Figure 1.1). On the right-hand side of Equation 1, *E*(*CF*_{1}), *E*(*CF*_{2}), and *E*(*CF*_{3}) all denote expected cash flows in subsequent periods 1, 2, and 3 (the periods could be in months, quarters, years, and so on) due to undertaking the business decision, or project.^{3} Note that these are expected future cash flows. The value creation measure is evaluated at time 0, at the very start of the project. Therefore, we are evaluating what would be the value created if this project is undertaken. Because we are calculating the value of pursuing this business decision at time period 0, before the uncertainty of the project has resolved itself, we have to put in our expectations of future cash flows rather than actual, or realized, cash flows.

In the denominator of Equation 1 is the discount rate, *r*. The discount rate denotes the expected return that a business expects for taking on the risk associated with the project.

Equation 1 is modeling the riskiness of the project under consideration as a set of risky cash flows. These cash flows are modeled as a set of probability distributions (see Figure 1.2). The mean of each distribution is captured by the expected value of each cash flow. In Figure 1.2, the first bell-shaped curve, with a mean of *E*(*CF*_{1}), represents the probability distribution of the cash flows in period 1, the second curve (with a mean of *E*(*CF*_{2})) represents the distribution of cash flows in period 2, and so on.

Figure 1.2 Probability distributions modeling a set of risky cash flows

The variance of each distribution gives us the risk of the cash flows. In Figure 1.2, the variance is given by the “width” of the bell-shaped curve. The wider the curve, the greater the range of possible cash flows generated by the project for that period. The area under the curve shows the likelihood of a given range of cash flow outcomes. For example, the area under the first curve and between the points minus one standard deviation (–1 sd) and plus one standard deviation (1 sd) gives the likelihood that the cash flow from the project in the first period will fall within one standard deviation of the mean expected cash flow. In Equation 1, the variance, or risk, of cash flows is captured by the discount rate. The higher the risk, the higher the discount rate; that is, the more risk that a business decision entails, the higher the expected return the business expects as compensation for taking that risk (if it decides to do so).

The number of terms on the right-hand side of Equation 1 depends on the number of periods over which the business will be earning cash flows for undertaking this project. All future time periods in which the business realizes cash flows from undertaking this project, no matter how far out into the future these might be, need to be incorporated.

Although it might seem simplistic initially, this present value equation is the foundation for all of modern finance and, in particular, the topic of our book: financial valuation. Of course, the formula itself is easy to understand, but implementing it in practical business settings is the real challenge. In many ways, to do this well is an art. The goal of this book is to introduce the basics and a few advanced concepts about how to apply valuation to practical business problems that executives face.