Perfect matrix

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), and the certainty equivalent approach does it by adjusting the numerators of the NPV formula, while the Penalized Present Value calculates the average NPV (µ) at the risk-free rate, penalizing it afterwards by subtracting t standard deviations of the NPV (tσ):

$PPV=\mu -t\sigma$

The PPV has many versions, a particularly pragmatic one can be reached by assuming we know the maximum or most optimistic NPV (b), the minimum or most pesimistic one (a) -calculating the NPVs at the risk-free rate-, and being these NPVs approximately normally distributed. In this case, approximately, we have:

$\mu \ ={\frac {a+b}{2}}$

and

$\sigma \ ={\frac {b-a}{6}}$

Assuming a reasonable t of 1.5:

$PPV={\frac {a+b}{2}}-1.5{\frac {b-a}{6}}=0.25b+0.75a$

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

Further details on this topic can be found in 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.