# Penalized present value

The Penalized Present Value (PPV) is a method of capital budgeting under risk developed by Fernando Gómez-Bezares in the 1980s.

## Description

The risk-adjusted rate of return method penalizes risk by increasing the discount rate when calculating the Net Present Value (NPV); the certainty equivalent approach does it by adjusting the numerators of the NPV formula (see Valuation using discounted cash flows #Basic formula for firm valuation using DCF model). Contrasting to both, PPV calculates the average NPV (µ) at the risk-free rate, penalizing it afterwards by subtracting t standard deviations of the NPV (tσ): ${\displaystyle PPV=\mu -t\sigma }$

The PPV has many versions, a particularly pragmatic one can be reached by assuming (a) we know the maximum or most optimistic NPV, (b) the minimum or most pessimistic one, (c) these NPVs are approximately normally distributed, and may be calculated via the risk-free rate. Under these approximations, we then have: ${\displaystyle \mu \ ={\frac {a+b}{2}}}$ and ${\displaystyle \sigma \ ={\frac {b-a}{6}}}$. Then, assuming a reasonable t of 1.5: ${\displaystyle PPV={\frac {a+b}{2}}-1.5{\frac {b-a}{6}}=0.25b+0.75a}$

Therefore, given that we are risk-averse, we weight more the worst case than the most favorable one. Obviously other weights could be applied. According to this criterion, the decision maker will look for investments with positive PPVs, and if a choice is needed, he or she will choose the investment with the highest PPV.

## References

• Gómez-Bezares, F. (1993): "Penalized present value: net present value penalization with normal and beta distributions", in Aggarwal, ed., Capital budgeting under uncertainty, Prentice-Hall, Englewood Cliffs, New Jersey, pages 91–102.
• Gómez-Bezares, F. and F.R. Gómez-Bezares (2013): “An analysis of risk treatment in the field of finance”, in C.-F. Lee & A.C. Lee, eds., Encyclopedia of finance, Springer, New York, 2ª ed., pages 705-711.