Present Value Ratio, GRR and Perpetuities
Posted by D Nathan Meehan September 24, 2011

### Present Value Ratio (PVR)

While NPV fails to deliver a measure of capital efficiency, the Present Value Ratio (PVR) index calculates a measure of investment efficiency that is very useful in ranking projects with significant capital investment. It is the ratio of the discounted (after tax) net cash generated by a project to the discounted pre-tax cash outlays (or investment).  Discounting for both measures is at the corporate discount rate. Note that the numerator is not revenue, but net cash generated. Operating expenses would be subtracted from the revenue along with taxes, royalties, etc. and not discounted back as part of the investment. Some companies use a version of PVR that is one plus this definition and are analogous to a discounted version of NTIR. A project with a PVR of one is equivalent to a project with an after tax (ATAX) PV equal to zero and a DCFROI equal to the discount rate.

PVR has many of the advantages of NPV in that there is no confusion about corporate reinvestment rates, no multiple solutions, etc. In the examples with Projects A through D, the PVR always ranks the projects in a way that generates the greatest NPV “bang for the buck.”

### Growth rate-of-return (GRR)

PVR has the weakness that it doesn’t have the same intrinsic feel of an interest rate as does DCFROI.  GRR is a measure that translates cash flows into an interest rate like measure that will always rank projects the same way as PVR.

To calculate GRR, all positive cash flows are compounded forward at the corporate discount rate to some time horizon; say t years in the future. Cash flows past that date are discounted back to the point t. This calculates the total equivalent amount of cash generated (say B) at time t assuming all cash flows are reinvested at the corporate discount rate. The negative net cash flows (excluding operating costs, taxes, etc.) are discounted back to time zero to get an equivalent time zero investment, I. If we were to put these I dollars in the bank and they grew to B at time t the interest rate required would be the GRR.  For ANEP compounding, the equation is: GRR requires some “getting used to” by management as it pushes high DCFROI projects downward and low (compared to the corporate discount rate) projects upward. There is no theoretically correct answer for what t should be and few people have an intrinsic feel for GRR.

### Perpetuity

A perpetuity is a series of cash payments that continues indefinitely. While there are no real perpetuities, the theoretical value of a perpetuity can be useful in approximating the value of certain cash flow streams including real estate and the terminal value of a going concern.  The valuation of a perpetuity assumes either constant periodic payments at regular time intervals infinitely into the future or payments that increase or decrease with a given growth rate, g. The value of the perpetuity is finite because payments received in the distant future are discounted to negligible present values.  The theoretical value of a perpetuity is: Where PV = Present Value of the Perpetuity, P = the periodic payment, and r = discount rate or interest rate. If the payments grow at rate g, the above equation becomes, for r>g: It is obvious that perpetuity approaches are not useful as the growth rate g approaches or exceeds the discount rate. If r=g, the PV becomes infinite as it is equivalent to an infinite series of effectively undiscounted cash flows.

Typical evaluations of a potential merger or acquisition forecast cash flows for five to ten years out and then to add a “perpetuity” value to account for the remaining life. If forecasts of net cash flow are made to ten years, then the perpetuity value would be calculated based on the tenth year’s cash flow. Perpetuity concepts yield potentially unrealistic values as the anticipated growth rate (g) approaches the cost of capital. Thus terminal values based on this approach should be compared with other approaches to valuations of going concerns.

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