Sensitivity Analysis & Decision Trees (Ch. 10)

ERRORS IN NPV ESTIMATES

·        Chapters 10, 11, 12 have a common theme: estimation error

·        Let’s say we’ve done everything we’re supposed to do: estimated net cashflows, estimated WACC, and computed NPV.  At that stage just because NPV is positive, that doesn’t necessarily mean you go ahead and accept the project.

·        First, you worry.  You worry about the fact that all you have is an estimate of NPV.  Even if you’ve done everything right, there is still going to be some estimation error.  You start with estimates of unit sales, sale price, unit cost, growth rates, WACC etc.  Each of those estimates is subject to estimation error.  Your sale price is not $6, it’s $6 ± x.  In the same way, your NPV is not $150, it’s $150  ± y.

·        Just because your estimate is positive, that doesn’t mean the true NPV is positive. 

·        Before you accept the project you worry about the possibility that you have come up with a positive NPV just because of estimation error, and the true NPV is negative.

·        Chapters 10, 11, 12 are about various things you can do to address this issue before you decide to accept the project.

IMPROVING ACCURACY OF NPV ESTIMATE

·        Ultimately, improving the accuracy of the NPV estimate is under your control.  Any estimate can be improved and made more accurate.  But this is always going to be costly.  It’s going to take time and effort by employees, or money spent on market research, etc.                                                                                             

·        You don’t want to blindly throw money into re-estimating different variables.

·        Sensitivity Analysis identifies what the critical variables are, the ones that have a major impact on NPV.  These are the ones you consider re-estimating.

·        Decision Trees gives us a framework for figuring out the benefit of re-estimation, i.e. of getting additional information).  So you compare the cost of getting information to the benefit, and you get it only when the NPV of getting info is positive.

SENSITIVITY ANALYSIS

·        The NPV estimate is based on using the expected value for different variables (like unit sales, sale price, unit cost, growth rates, etc).  For example, we may have assumed unit sales of 100,000 and a variable cost of $6

·        For each variable you now quantify the likely range for that variable.  Let’s say unit sales may lie between 85,000 and 115,000 and variable cost might fall between $5.75 and $6.30.

·        You’re saying this is the likely size of the errors you may have made in estimating these variables.   Unit sales are 100,000 ± 15,000

·        Taking each variable one at a time, you compute separately what the NPV will be if that variable takes the pessimistic value (e.g. variable cost turns out to be $6.30 or unit sales turn out to be 85,000) and what the NPV will be if that variable takes the optimistic value (variable cost turns out to be $5.75).

·        By doing this you are quantifying the impact of each variable’s likely estimation error on the NPV estimate.

·        You identify which variables NPV is sensitive too.  These are the variables you worry about, and consider re-estimating.

·        If errors in variable cost will make NPV fluctuate between $12,000 and $28,000, clearly we don’t care too much even if variable cost has been mis-estimated.

·        If errors in unit sales will make NPV fluctuate between -$5,000 and $44,000, we worry about whether we have over-estimated unit sales (and the true NPV is negative)

DECISION TREES

·        Sometimes a project involves a sequence of different decisions.  You make a decision today, then randomly there is an outcome better or worse than expected, then you face another decision, etc.

·        For example, today you have to make a decision whether to accept or reject a project.  At the expected market share of 10%, NPV is positive, so you accept.  Tomorrow, after you launch your product, actual market share may turn out to be 15% or 8% or some other number.  If it’s high, you may face a decision whether to expand; if it’s low, you may face a decision whether to scale down or abandon the project.

·        These future decisions to modify a project (scale it up, scale it down, abandon) are referred to as real options.

·        Real options allow you to boost your cashflows if the project does well, or cut your losses if it does badly.  Clearly if a project has such real options, it is more valuable.  The value of the real options should be accounted for when computing the NPV of the project.

·        Decision trees represent a simple way to account for the value of real options in computing NPV.

·        They also allow us to quantify the benefit of obtaining additional information about key variables.

·        Example 1: A farmer is considering whether to drill a well for water.  Currently buys water at a cost of $2,000 per year.  It will cost $10,000 to drill up to 200’.  The probability of striking water at 200’ is 0.6.  If he doesn’t strike water at 200’, it will cost an additional $4000 to drill to 250’.  If there’s no water at 200’ the probability you will find it at 250’ is only 0.1.  Should the farmer drill or just keep buying water, if the discount rate is 10%.  (Assume operating/maintenance costs for well are 0.)

DO NOT DRILL  PV of water costs = 2,000/.1 = 20,000

                                    FIND WATER  cost = 10,000

                                         prob 0.6                                          FIND WATER  14,000

            DRILL TO 200’                                   DRILL TO 250’             prob 0.1

                                                                                                      DON’T FIND     34,000

                                           DON’T FIND                                             prob 0.9

                                                prob 0.4

                       DO NOT DRILL  30,000

o       These problems are done backwards. 

o       Start with the last decision, B.  If you keep drilling, the expected cost of water = 0.1 * 14,000 + 0.9 * 34,000 = 32,000.  So the right decision is not to drill on.

o       This means that if you choose to drill today, and you don’t find water at 200’, your cost will be $30,000 (since you’ll stop drilling).

o       Then for the first decision, A, the expected cost of water = 0.6 * 10,000 + 0.4 * 30,000 = 18,000.

o       That’s better than buying water.  So the decision is to drill to 200’, but stop drilling if you don’t find water at 200’.

o       Note: till you figure out what your second decision will be, you don’t know what the relevant cashflows are for the first decision.  That’s why you have to take the second decision first.

·        Now let’s extend the example to see how we figure out the value of information.  Before deciding whether to drill to 200’, you can spend $3,000 to sink a test well that will tell you whether or not there’s water at 200’.  Is it worth paying $3,000 for this information?

o       The knee-jerk reaction would be:

§        Buying water costs $20,000

§        Drilling results in an expected cost of $18,000

§        If you do a test well for $3,000 first, the expected cost of water with drilling becomes $21,000

o       This is not the right way to think about it.  It takes the cost of the info into account but not the benefit.

                        SKIP TEST WELL    cost 18,000

                                                               FIND WATER  cost = 13,000

                                                                    prob ?

                              DRILL TEST WELL                           

                                       DON’T FIND   cost = 23,000

                                                                                 Prob ?

o       Expected cost if you drill a test well =     * 13,000 +     * 23,000 =

o       Cost of the info was $3,000.  What's the value? 

o       Where does the value come from?  Earlier, you were paying 10,000 to drill whether or not there was water at 200’.  With a probability of 0.4, the well turned out dry, and you wasted your $10,000.  This is what you save in exchange for spending $3,000 to drill the test well.  You drill the 200’ well only if there’s going to be water. The value of the info = your expected saving of

o       In general: