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title: "NPV theory"
canonical: "https://modelassist.epixanalytics.com/space/EA/26577726/NPV%20theory"
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Net Present Value
An NPV calculation attempts to determine the present value of a series of cashflows from a project that stretches out into the future. This present value is a measure of how much the company is gaining at today's money by undertaking the project: in other words, how much more the company itself will be worth by accepting the project.
An NPV calculation discounts future cashflows at a specified discount rate r that takes account of:
1. The time value of money (e.g. if inflation is running at 4%, £1.04 in a years time is only worth £1.00 today)
2. The interest that could have been earned over inflation by investing instead in a guaranteed investment
3. The extra return that is required over (1) and (2) to compensate for the degree of risk that is being accepted in this project.
Parts (1) and (2) are combined to produce the risk free interest rate, rf. This is typically determined as the interest paid by guaranteed fixed payment investments like government bonds with a term roughly equivalent to the duration of the project.
The extra interest r* over rf needed for part (3) is determined by looking at the uncertainty of the project. In risk analysis models, this uncertainty is represented by the spread of the distributions of cashflow for each period. The sum of r* and rf is called the risk-adjusted discount rate r.
The most commonly used calculation for the NPV of a cashflow series over n periods is as follows:
> Macro (mathblock)
where C*i* are the expected (i.e. average) values of the cashflows in each period and r is the risk-adjusted discount rate.
In our experience, NPV calculations performed in a risk analysis spreadsheet model are usually presented as a distribution of NPVs because the cashflow values selected in the NPV calculations are their distributions rather than their expected values. Theoretically, this is incorrect. Since an NPV is the net present value, it can have no uncertainty. It is the amount of money that the company values the project at today. The problem is that we have double counted our risk by first discounting at the risk-adjusted discounted rate r and then showing the NPV as a distribution (i.e. it is uncertain).
Two theoretically correct methods for calculating an NPV in risk analysis are discussed below, along with a more practical, but strictly speaking incorrect, alternative:
- *Theoretical approach 1:* Discount the cashflow distributions at the risk free rate
This produces a distribution of NPVs at rf and ensures that the risk is not double-counted. However, such a distribution is not at all easy to interpret since decision-makers will almost certainly never have dealt with risk free rate NPVs and therefore have nothing to compare the model output against.
- *Theoretical approach 2:* Discount the expected value of each cashflow at the risk-adjusted discount rate.
This is the application of the above formula. It results in a single figure for the NPV of the project. A risk analysis is run to determine the expected value and spread of the cashflows in each period. The discount rate is usually determined by comparing the riskiness associated with the project's cashflows against the riskiness of other projects in the company's portfolio. The company can then assign a discount rate above or below its usual discount rate depending on whether the project being analyzed exhibits more or less risk than the average. Some companies determine a range of discount rates (three or so) to be used against projects of different riskiness.
The major problems of this method are that it assumes the cashflow distributions are symmetric and that no correlation exists between cashflows. We have seen that distributions of costs and returns very often exhibit some form of asymmetry. In a typical investment project, there is also almost always some form of correlation between cashflow periods: for example, sales in one period will be affected by previous sales, a capital injection in one period often means that it doesn't occur in the next one (e.g. expansion of a factory) or the model may include a time series forecast of prices, production rates or sales volume that are autocorrelated. If there is a strong positive correlation between cashflows, this method will overestimate the NPV. Conversely, a strong negative correlation between cashflows will result in the NPV being underestimated. The correlation between cashflows may take any number of, sometimes complex, forms. We are not aware of any financial theory that provides a practical method for adjusting the NPV to take account of these correlations.
The practical approach:
The above two theoretical approaches are difficult to apply or interpret and beg an alternative. In practice, it is easier to apply the risk-adjusted discount rate r to the cashflow distributions to produce a distribution of NPVs. This method incorporates correlation between distributions automatically and enables the decision-maker to compare directly with past NPV analyzes.
As we have already explained, the problem associated with this technique is that it will double count the risk: firstly in the discount rate and then by representing the NPV as a distribution. However, if one is aware of this shortfall, the result is very useful in determining the probability of achieving the required discount rate (i.e. the probability of a positive NPV). The actual NPV to quote in a report would be the expected value of the NPV distribution.
Internal Rate of Return
The IRR of a project is the discount rate applied to its future cashflows such that it produces a zero NPV. In other words, it is the discount rate that exactly balances the value of all costs and revenues of the project. If the cashflows are uncertain, the IRR will also be uncertain and therefore have a distribution associated with it.
A distribution of the possible IRRs is useful to determine the probability of achieving any specific discount rate and this can be compared with the probability other projects offer of achieving the target discount rate. It is not recommended that the distribution and associated statistics of possible IRRs be used for comparing projects because of the properties of IRRs discussed below.
Problems in using IRR in risk analyzes
Unlike the NPV calculation, there is no exact formula for calculating the IRR of a cashflow series. Instead, a first guess is usually required, from which the computer will make progressively more accurate estimates until it finds a value that produces an NPV as near to zero as required.
If the cumulative cashflow position of the project passes through zero more than once, there is more than one valid solution to the IRR inequality. This is not normally a problem with deterministic models because the cumulative cashflow position can easily be monitored and the smaller of the two IRR solutions selected. However, a risk analysis model is dynamic, making it difficult to appreciate its exact behavior. Thus, the cumulative cashflow position may pass through zero and back in some of the risk analysis iterations and not be spotted. This can produce quite inaccurate distributions of possible IRRs. In order to avoid this problem, it may be worth including a couple of lines in your model that calculate the cumulative cashflow position and the number of times it passes through zero. If this is selected as a model output, you will be able to determine whether this is a statistically significant problem and alter the first guess to compensate for it.
IRRs cannot be calculated for only positive or only negative cashflows. IRRs are therefore not useful for comparing between two purely negative or positive cashflow options e.g. between hiring or buying a piece of equipment.
It is difficult to compare distributions of IRR between two options unless the difference is very large. Stochastic dominance tests will certainly be of little direct use. This is because a percentage increase in an IRR at low returns (e.g. from 3% to 4%) is of much greater real value than a percentage increase at high returns (e.g. from 30% to 31%). Consider the following illustration: I am offered payments of £20 a year for 10 years (i.e. £200 total) in return for a single payment now. I am asked to pay £200 - obviously a bad investment giving an IRR of 0%. I negotiate to drop the price and thereby produce a positive IRR. The figure above illustrates the relationship between the reduction in price I achieve and the resulting IRR. The reduction in price I achieve is directly equivalent to the increase in the present value of the investment, so the graph relates real value to IRR. As the savings I make approaches £200, the IRR approaches infinity. Clearly there is no straight line relationship between IRR and true value. It is therefore very difficult to compare the value of two projects in terms of the IRR distributions they offer. One project may offer a long right-hand tail that can easily increase the expected IRR but in real value terms this could easily be outweighed by a comparatively small diminishing of the left-hand tail of the other option.
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