Given the uncertainty inherent in project forecasting and valuation,analysts will wish to assess the sensitivity of project NPV to the various inputs (i.e. assumptions) to the DCF model. In a typical sensitivity analysis the analyst will vary one key factor while holding all other inputs constant, ceteris paribus. The sensitivity of NPV to a change in that factor is then observed, and is calculated as a "slope": ΔNPV / Δfactor. For example, the analyst will determine NPV at various growth rates in annual revenue as specified (usually at set increments, e.g. -10%, -5%, 0%, 5%....), and then determine the sensitivity using this formula. Often, several variables may be of interest, and their various combinations produce a "value-surface",[11] (or even a "value-space"), where NPV is then a function of several variables. See also Stress testing.
Using a related technique, analysts also run scenario based forecasts of NPV. Here, a scenario comprises a particular outcome for economy-wide, "global" factors (demand for the product, exchange rates, commodity prices, etc...) as well as for company-specific factors (unit costs, etc...). As an example, the analyst may specify various revenue growth scenarios (e.g. 5% for "Worst Case", 10% for "Likely Case" and 25% for "Best Case"), where all key inputs are adjusted so as to be consistent with the growth assumptions, and calculate the NPV for each. Note that for scenario based analysis, the various combinations of inputs must be internally consistent (see discussion at Financial modeling), whereas for the sensitivity approach these need not be so. An application of this methodology is to determine an "unbiased" NPV, where management determines a (subjective) probability for each scenario – the NPV for the project is then the probability-weighted average of the various scenarios.
A further advancement is to construct stochastic or probabilistic financial models – as opposed to the traditional static and deterministic models as above.[10] For this purpose, the most common method is to use Monte Carlo simulation to analyze the project’s NPV. This method was introduced to finance by David B. Hertz in 1964, although it has only recently become common: today analysts are even able to run simulations in spreadsheet based DCF models, typically using an add-in, such as @Risk or Crystal Ball. Here, the cash flow components that are (heavily) impacted by uncertainty are simulated, mathematically reflecting their "random characteristics". In contrast to the scenario approach above, the simulation produces several thousand random but possible outcomes, or "trials"; see Monte Carlo Simulation versus “What If” Scenarios. The output is then a histogram of project NPV, and the average NPV of the potential investment – as well as its volatility and other sensitivities – is then observed. This histogram provides information not visible from the static DCF: for example, it allows for an estimate of the probability that a project has a net present value greater than zero (or any other value).
Continuing the above example: instead of assigning three discrete values to revenue growth, and to the other relevant variables, the analyst would assign an appropriate probability distribution to each variable (commonly triangular or beta), and, where possible, specify the observed or supposed correlation between the variables. These distributions would then be "sampled" repeatedly – incorporating this correlation – so as to generate several thousand random but possible scenarios, with corresponding valuations, which are then used to generate the NPV histogram. The resultant statistics (average NPV and standard deviation of NPV) will be a more accurate mirror of the project's "randomness" than the variance observed under the scenario based approach. These are often used as estimates of the underlying "spot price" and volatility for the real option valuation as above; see Real options valuation: Valuation inputs. A more robust Monte Carlo model would include the possible occurrence of risk events (e.g., a credit crunch) that drive variations in one or more of the DCF model inputs.
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