# Profitability Index Annuity

Joachim Kuczynski, 29 September 2022

Project ranking is an important issue in capital rationing, when the company has limited financial resources. A company can rank its projects with various measures: NPV, eNPV, PI, PIA, IRR, MIRR, Baldwin rate of return and many others. The profitability index annuity is a further development of the profitability index to consider the temporal cash flow distribution in the decision making process.

## Profitability Index

The profitability index (PI) of an investment project is the present value of all cash flows without investment (let us name them contribution cash flows), , divided by the present value of all investment cash flows, . Hereby it is important how we define the term investment. You can have a look at my proposal in this post. Hence we get for the profitibility index:

The shows how much value is created per investment. All projects with a are creating additional value to the investors, all projects with a are destroying value.

Sometimes the profitability index is defined net present value of the project devided by the present value of all investment cash flows. Let us name this definition of the profitability index PI*:

No matter which definition we take, the ranking of projects remains the same. The difference is only an “offset” of 1.

## Profitability Index Annuity

Most managers prefer projects in which the returns of the investment are in the beginning. They prefer early cash flow return. The profitability index cannot provide any information concerning the timing of the cash flows. The profitability index annuity (PIA) tries to solve that issue. It is defined as profitability index divided by the annuity factor :

An annuity is a sequence of equal cash flows paid each period for a specified number of periods . The sum of all the discount factors equals the annuity factor. With as the project lifetime and as annual discount rate, the annuity factor is defined as:

The previous equation is only valid, if all cash flows are discounted with the same discount rate in all periods. If you have different discount rates in the considered periods, the formula not valid any more but can be adapted easily. If you use the Component Cash Flow Procedure, there might be no solution or more than one possible solution for the PIA value. So take care when applying the PIA ranking!

## Conclusion

The annuity factor considers how much the investment project is influenced by the discounting effect. An investor that prefers early cash flow returns wants to have small discounting effects. The smaller the discounting effect the smaller the annuity factor and the higher the profitability index annuity. Looking at two projects with the same PI the investor prefers the project with the higher PIA.

Project rankings based on the profitability index annuity (PIA) have some desirable properties. Projects with shorter lives tend to have higher ranking PIAs, and short life implies rapid cash generation. Choosing projects ranked by their PIA values can help managers to invest in projects combining three desirable characteristics: High PV, low capital requirement and rapid cash generation.

In my point of view PIA is an additional useful figure to evaluate investments and projects. But we should not use it as the only measure. It relies on the term investment that we have to consider carefully. Always think about what is the true financial limitation in the company and then refer to that.

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