RISK ANALYSIS - University of Washington



RISK ANALYSIS TECHNIQUES

SENSITIVITY ANALYSIS

This involves observing what happens to a dependent variable (such as the objective function, NPV) as a change occurs in a particular variable. For example, NPV may depend on market size, market share, and sales price. Sensitivity analysis could tell us what happens to NPV as we change market size by one unit (any sized unit); or how NPV changes as we change market share by one unit; and so forth.

Use of Sensitivity Analysis: Sensitivity analysis provides an estimate of the impact of a change in a variable on NPV. This is helpful in determining whether a better estimate of that variable is important. So, we mind find that a change in variable X significantly affects NPV, but a change in Y does not. Then we might want to focus on X (see problems with sensitivity analysis below). This could mean getting a better fix on the probability distribution of X, or on influencing the probability distribution of X (e.g., by changing the nature of the investment, by risk management through hedging, etc.). In contrast, under these circumstances we will not likely be very interested in Y. Of course, when we decided that the NPV does not depend much on Y, we would want to make sure that we tested the effect of Y on NPV over a wide range of Y (because how NPV reacts to Y may depend on the level of Y).

Problems with Sensitivity Analysis: There are two key problems with sensitivity analysis.

• Interrelated Variables: The variables that describe a firm are generally interrelated. So examining the change in one variable at a time does not replicate reality and may be highly misleading. For example, suppose that we find that a small change in X (holding all other variables, including Y, constant) produces a big change in NPV. We might conclude that NPV varies a lot with X. But, what if, when X increases, Z also increases, and that an increase in Z reduces NPV (offsetting the effect of X)? To illustrate, suppose that X is sales and Y is costs. When sales increase (holding everything else, including costs, constant), profit and NPV go up. But, when sales go up, costs go up, and this may offset (or more than offset) the increase in sales; NPV may actually decline with the rise in sales due to the associated increase in costs. A sensitivity analysis of the effect of sales on NPV would completely ignore the related changes in costs.

• Probability: Another problem is that sensitivity analysis does not incorporate probability, only sensitivity. So, an increase in X might cause a big increase in NPV, but an increase in X may be extremely improbable. Focusing on X may therefore be a waste of time even though it appears to be a key driver of NPV.

SCENARIO ANALYSIS

Scenario analysis involves the examination of alternative scenarios that might unfold for an investment. If performed properly, this technique produces the same NPV as does the standard discounted expected cash flow NPV approach.

Use of Scenario Analysis

Signify the proper risk-adjusted discount rate (WACC) for analyzing the investment as k, the future time t expected cash flow as [pic], and the expected initial (time 0) outlay to undertake the investment as [pic]. The maximum possible life of the investment is signified as n. Equation (1) states the NPV as it is typically computed (signified here as NPV[standard]):

NPV[standard] = [pic] ( [pic] (1)

NPV[standard] in (1) is the project’s NPV as we normally compute it, which is project’s expected future cash flows discounted at the appropriate risk-adjusted discount rate k minus the expected initial outlay for the project. In most cases, the initial outlay is assumed to be known and equal to some amount [pic] (i.e., [pic] = [pic] in (1)).

An alternative way to compute an NPV, which we will refer to as NPV[scenarios] , is:

NPV[scenarios] = [pic] (2)

where m is the number of potential scenarios that might unfold for the investment. In (2), [pic]is the probability that scenario i will occur, and

[pic][scenario i] = [pic] (3)

In (3), [pic] is the initial outlay under scenario i, [pic] is the time t cash flow under scenario i, and n(i) is the life of the investment under scenario i. DCF[scenario i] in (3) is not an NPV; it is a computation that looks like an NPV. It is the discounted cash flows only for scenario i. The NPV of the investment takes into account all of the possible scenarios for the investment. The investment’s NPV is NPV[scenarios] in (2); NPV[scenarios] is the probability-weighted sum over all possible scenarios.

It can be shown that the two methods of computing an NPV ((2) and (3)) produce the same NPV, that is:

NPV[standard] = NPV[scenarios] (4)

To illustrate, assume a project with a known initial outlay of $100, that is, [pic] = $100 (all figures in $million). One of three scenarios might unfold: worst case, middle case, best case. Exhibit 1 below provides the data about these scenarios. Assume that the discount rate k = 10%.

Exhibit 1. Three Scenarios (all dollar amounts in $million)

| | | | |[pic] for all |

|Scenario |Probability |[pic] |[pic] |t > 2 |

|Worst case |.2 |$5 |$7 |$9 |

|Middle case |.5 |$6 |$8 |$12 |

|Best case |.3 |$8 |$10 |$14 |

So, in the worst case (which has a 20% probability of occurring), in the first year the cash flow will be $5, in the second year the cash flow will be $7, and for all years after that (in perpetuity) the cash flow will be $9. Similarly for the middle case and the best case.

Solving the problem the usual way using equation (1), the expected cash flows and NPV are:

[pic] = .2($5) + .5($6) + .3($8) = $6.40 (5a)

[pic] = .2($7) + .5($8) + .3($10) = $8.40 (5b)

[pic] = .2($9) + .5($12) + .3($14) = $12.00 (5c)

NPV[standard] = [pic] ( [pic]

= [pic]+ [pic]+ [pic][pic]( $100

= $11.93 (6)

Now let’s use equation (2). The DCF[scenario i] computations for the three scenarios are:

DCF[worst case] = [pic]+ [pic]+ [pic][pic]( $100 = ( $15.29 (7a)

DCF[middle case] = [pic]+ [pic]+ [pic][pic]( $100 = $11.24 (7b)

DCF[best case] = [pic]+ [pic]+ [pic][pic]( $100 = $31.24 (7c)

NPV[scenarios] = [pic]

= (( $15.29)(.2) + ($11.24)(.3) + ($31.24)(.3)

= $11.93 (8)

Problem with Scenario Analysis: The major problem is that there may be many plausible scenarios and computing the variable values for each is a tedious matter unless a complex computer model is used. But, if a complex computer model is used, why not go the full mile and use simulation?

SIMULATION

Simulation is the artificial replication of something real. Monte Carlo simulation involves assigning probability distributions to some or all of the variables that are input in the simulation, and then generating a probability distribution output for one or more variables (such as cash flow). Simulation can be very useful in evaluating a large investment project.

What Should Be Simulated? Have the computer simulate (produce as output) each future period’s cash flow probability distribution, and the parameters of that distribution (mean, variance, correlation with the market, autocorrelation, etc.). These simulated data are inputs in the determination of the investment’s risk-adjusted discount rate (k) and NPV.

If k is Uncertain: There is generally some degree of uncertainty about the appropriate k to use in the NPV computation. A good way to deal with this is to assign (or simulate) a probability distribution for k. Then we can generate a probability distribution of NPVs using the various possible k’s to discount the expected cash flows. If all the NPVs for all plausible k’s are positive, we can probably conclude that we have a winner. If all the NPVs for all plausible k’s are negative, we can probably infer that we have a loser. If the NPV is positive for some k’s and negative for others, we have a problem. In this case, we may want to gather more information in order to improve the estimate of the k and the cash flows.

If k is Known: If the expected cash flows have been estimated and the appropriate k to discount those expected cash flows is known, the single unique NPV of the project can be computed using (1). For any given discount rate and series of expected cash flows, there is a single NPV. The discount rate adjusts for the risk of the project; and the expected (mean) cash flow estimates are affected by the entire probability distribution of the cash flows (i.e., that expected cash flow takes into account low cash flow outcomes as well as high cash flow outcomes). The NPV is our current estimate of the net gain from adopting the investment.

Sometimes it is argued that, even if the expected cash flows and the k are known, it is useful, in assessing a project’s risk, to generate a probability distribution of the project’s “NPV” computed using the risk-free rate as the discount rate. This exercise is not productive. An NPV probability distribution in this case is meaningless because discounting a risky cash flow using the risk-free rate produces a number that has no economic meaning. Furthermore, the NPV of a project does not have a positive variance probability distribution if the k and the expected cash flows are known. Under these conditions, there is only one NPV (the probability distribution assigns a probability of 1.0 to the computed NPV).

10/26/2003

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