NPV Sign and Time Scale

In this post I want to discuss the sign of NPV when time scale is shifted by an amount of \Delta t. The sign of NPV indicates whether an asset generates value to the fund providers (debt and equity) or not. If only the amount, but not the sign of NPV changes by a time shift, the decision to allocate the project or not does not change. That means that the decision itself does not depend on the time scale. In many books I have read that argument. But is this really true?

Project Cash Flow Procedure

The project cash flow procedure (PCFP) takes the same annual discount rate r for all cash flows C_i. By substituting \alpha = ln (1+r) we can write e^{- \alpha t} instead of (1+r)^{-t}. The NPV of the project at the “present” time t=0 without time shift, let´s name it NPV_0, is:

    \[NPV_0=\sum_{i}^{}E(C_i) e^{-\alpha t_i}\]

E(C_i) is the expected value of the i-th cash flow component. When we shift time by \Delta t, we get a new NPV, let`s call it NPV_ {\Delta t}:

    \[NPV_ {\Delta t} =\sum_{i}^{}E(C_i) e^{-\alpha (t_i+ {\Delta t} )}\]

    \[ NPV_ {\Delta t} = e^{ -\alpha  \Delta t} \sum_{i}^{}E(C_i) e^{-\alpha (t_i )}= e^{ -\alpha  \Delta t} NPV_0\]

Because e^{ -\alpha  \Delta t}>0, the sign of NPV_0 and NPV_ {\Delta t} is the same for all \Delta t. That means the decision, when based on the NPV sign, remains the same: Invest in case of positive NPV and do not invest in case of negative NPV. In the PCFP a time shift does not affect the decision.

Component Cash Flow Procedure

The component cash flow procedure (CCFP) takes the specific appropriate risk adjusted discount rate for each cash flow. That means that you have a specific discount rate r_i, or \alpha _ i = ln (1+r_i) respectively, for cash flow C_i. The NPV of the project without time shift, NPV_0, is:

    \[NPV_0=\sum_{i}^{}E(C_i) e^{-\alpha_i t_i}\]

When we shift time by \Delta t, we get a new NPV_ {\Delta t}:

    \[NPV_ {\Delta t} =\sum_{i}^{}E(C_i) e^{-\alpha_i (t_i+ {\Delta t} )}\]

We cannot make further simplifications because each term has an individual discount rate \alpha_i. That means that the sign of NPV with time shift does not have to be the same as the NPV without time shift, the sign can change. Hence the decision whether to allocate the project or investment can also change. Furthermore that means that we have to pay attention to take the right time scale in the calculation to come to the right decision.

Conclusion

In this post I described the advantages of the CCFP over the PCFP. The PCFP is only a simplification of the CCFP, PCFP can lead to wrong decisions. Because CCFP is the preferred and valid procedure of calculation, we can state that the sign of NPV can change by a time shift in general. The decision based on the NPV sign is only valid for the “present” time, which is t=0 in the DCF calculation. If you want the calculation to be the basis for an investment decision, you have to ensure that t=0 in the calculation is the point in time of the decision!

Several companies set t=0 at the beginning of the project`s revenues or the first investment cash flows. But this neglects the uncertainty and riskiness of cash flows from the decision point of time to their starting points. That is false in general, because of the arguments above. t=0 has to be the point in time of the decision. Otherwise you can generate false corporate decisions.

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