### Expected Monetary Value and Value at Risk

Before we dive into probability and expected monetary value (EMV), we will introduce a motivational problem from the petroleum industry. You are exploring the possibility of drilling a potential new oil field. You can either do the drilling yourself, or you can “farm out” the drilling operation to a partner. The field in which you are proposing to drill may or may not have oil – you don’t know until you drill and find out. If you drill yourself, you take on the risk if the field is not a producer, but if the field is a producer, you don’t need to share your profits with anyone. If you farm out the drilling operation, you are not exposed to any losses if the field is not a producer, but if the field is a producer, the drilling company will take the lion’s share of the profits. Table 6.1 shows the net present value of the prospective oil field.

Field is Dry | Field is a Producer | |
---|---|---|

Drill Yourself | -$250,000 | $500,000 |

Farm Out | $0 | $50,000 |

Clearly, if you knew the field was a producer you would want to drill yourself (and if you knew it wasn’t, then you would not want to drill at all). But you don’t know this before you make your drilling decision. What should you do?

Suppose that you had enough information on the productivity of wells drilled in a similar geology to estimate that the probability of a dry hole was 65% and the probability of a producing well was 35%. Do these probabilities make your life any easier?

These decision problems can be solved by calculating a quantity known as the “expected monetary value” (EMV) – basically a probability-weighted average of net present values of different outcomes. Formally, the EMV is defined by determining probabilities of each distinct or “mutually exclusive” outcome, determining the NPV under each of the possible outcomes, and then weighting each possible value of the NPV by its probability. In mathematical terms, if Z is some alternative; Y_{1}, Y_{2}, …, Y_{n} represent a set of possible outcomes of some uncertain variable; X_{1}, X_{2}, …, X_{n} represent the NPVs associated with each of the possible outcomes; and P(Y_{1}), P(Y_{2}), …, P(Y_{n}) represent the probabilities of each of the outcomes, then the EMV is defined by:

The alternative with the highest EMV would be the option chosen. A decision-maker who chooses among alternatives in this way would be called an “expected-value decision-maker.”

Some things to remember about probabilities:

- A probability is a number between zero and one. So P(Y
_{j}) = 0.05 means that there is a 5% chance of outcome*j*occurring. If P(Y_{j}) = 0, it means that outcome*j*never occurs and if P(Y_{j}) = 1, it means that outcome*j*always occurs. - If a set of outcomes is mutually exclusive (meaning that multiple outcomes cannot occur) and exhaustive (meaning that the set captures all possible outcomes), then the probabilities of all outcomes in that set would be equal to one. Rolling a six-sided die with the outcomes {1,2,3,4,5,6} is an example. This set of outcomes is mutually exclusive because the face of the die cannot show two numbers at once (unless the die is crooked). The set of outcomes is exhaustive since it contains all possible outcomes of the die being rolled. In this case, P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1. The set of outcomes {1,2,3} is an example of non-exhaustive outcomes, since the die could show a 4, 5 or 6 upon being rolled.

Now, back to our oil field problem. We’ll describe the problem again, using the language of expected monetary value. There are two alternatives – to drill yourself or to farm-out. The uncertainty is in the outcome of the drilling process. The set of mutually exclusive and exhaustive outcomes is {dry hole, producer}. These are the only two possible outcomes, regardless of whether you choose to drill yourself or farm-out. The NPVs of each alternative, under each possible outcome, are shown in Table 10.1. To decide whether to drill or farm out, you would calculate the EMV of each option as follows:

$\begin{array}{l}EMV(\text{Drill})=P(\text{dryhole})\times -\$250,000+P(\text{producer})\times \$500,000\\ =0.65\times -\$250,000+0.35\times \$500,000=\$12,500.\\ EMV(\text{Farmout})=P(\text{dryhole})\times -\$0+P(\text{producer})\times \$50,000\\ =0.65\times -\$0+0.35\times \$50,000=\$17,500.\end{array}$

In this case, you should choose to farm out the drilling operation.

The basic idea behind EMV is fairly straightforward, assuming that you can actually determine the relevant probabilities with some precision. But the meaning of the EMV is a little bit subtle and requires some degree of care in interpretation. Let’s take a very basic situation – a coin flip. Suppose that we were to flip a coin. If it shows heads, you must pay me $1. If it shows tails, then I must pay you $1. The EMV of this game, assuming that heads and tails have equal probabilities, is $0. (See if you can figure out why, based on the EMV equation and the fact that P(heads) = 0.5 and P(tails) = 0.5.) But if you think about this for a minute, how useful is the EMV? If you play the coin-flipping game once, you will never ever have an outcome where the payoff to you is $0. The payoff will either be that you gain or lose one dollar.

If you look at the EMVs from the oil-field problem, you will see the same thing. The EMV of drilling is $12,500 but there is no turn of events under which you would wind up earning $12,500 – if you drill, it would either be that you lose $250,000 or gain $500,000. Similarly, if you farm out you will never earn exactly $17,500. You will either lose nothing (payoff of $0) or you will gain $50,000. So what does the EMV mean when it tells you that farming-out is the better option?

It’s important to remember that the EMV is a type of average. If you were to play the coin-flip game or the oil-field game a large number of times under identical circumstances, and make the same decision each time (i.e., to drill or to farm out), then over the long run you would expect to wind up with $12,500 if you choose to drill and $17,500 if you choose to farm out. While the EMV may be useful for gamblers or serial investors, using EMV needs to be done with some care for stand-alone projects in the face of uncertainty.

One potential alternative to calculating EMV when probabilities are known (or can be estimated) is to use those probabilities to describe the likelihoods of gaining or losing certain amounts of money. This is the idea behind the “value at risk” for an investment project or a portfolio of projects. The value at risk (VaR) describes the amount of money that will be gained or lost with some probability, typically worst-case situations (like describing the amount of money that would be gained or lost with a 5% probability). From a decision perspective, you might want to avoid investment opportunities with a large VaR (given some probability). Many times, VaR will have a duration associated with it, most often when calculated in reference to portfolios of financial assets (like stocks or bonds).

Wikipedia has a very descriptive entry for VaR. The introductory section of the Damadoran VaR paper is written in clearer English, but gets into the details quickly after the first few pages.

To force the example a little bit, here is how VaR might be applied in our simple oil field problem. The field will either yield a producer (probability 35%) or a dry hole (probability 65%). Those are the only two outcomes and probabilities for this problem. If we think about the negative outcome – the dry hole – and ask how much of the value of the project might be at risk if the hole turns out to be dry, then we can calculate VaR just by using the NPVs from Table 10.1. (In this example, there is no time dimension to VaR since the project is a one-shot deal.)

If we choose the option to drill ourselves, the 65% VaR would be -$250,000. If we choose to farm-out the drilling operation, the 65% VaR would be $0. We might choose to farm out simply because the VaR technique tells us that our extreme losses would be lower if we farm-out than if we drill the well ourselves.