Nonstandard Investments as a Challenge for Multiple Interest Rate Analysis?

1.0 Introduction

This post contains some musing on corporate finance and its relation to the theory of production.

2.0 Investments, the NPV, and the IRR

In finance, an investment project or, more shortly, an investment, is a sequence of dated cash flows. Consider an investment in which these cash flows take place at the end of n successive years. Let C_t; t = 0, 1, ..., n - 1; be the cash flow at the end of the tth year here, counting back from the last year in the investment. That is, C_{n - 1} is the cash flow at the end of the first year in the investment, and C₀ is the last cash flow.

The Net Present Value (NPV) of an investment is the sum of discounted cash flows in the investment. Let r be the interest rate used in time time discounting, and suppose all cash flows are discounted to the end of the first year in the investment. Then the NPV of the illustrative investment is:

NPV₀(r) = C_{n - 1} + C_{n - 2}/(1 + r) + ... + C₀/(1 + r)^{n - 1}

If the above expression is multiplied by (1 + r)^{n - 1}, one obtains the NPV of the investment with every cash flow discounted to the last year in the investment:

NPV₁(r) = C_{n - 1}(1 + r)^{n - 1} + C_{n - 2}(1 + r)^{n - 2} + ... + C₀

For the next step, I need some sign conventions. Let a positive cash flow designate revenues, and a negative cash flow be a cost. Suppose, for now, that the (temporally) first cash flow is a cost, that is negative. Then (-1/C_{n - 1}) NPV₁(r) is a polynomial in (1 + r), with unity as the coefficient for the highest-order term. All other terms are real.

Such a polynomial has n - 1 roots. These roots can be real numbers, either negative, zero, or positive. They can be complex. Since all coefficients of the polynomial are real, complex roots enter as conjugate pairs. Roots can be repeating. At any rate, the polynomial can be factored, as follows:

NPV₁(r) = (-C_{n - 1})(r - r₀) (r - r₁)... (r - r_{n - 1})

where r₀, r₁, ..., r_{n - 1} are the roots of the polynomial. Note that the interest rate appears only in terms in which the difference between the interest rate and one root is taken. And all roots appear on the Right Hand Side. I am going to call an specification of NPV with these properties an Osborne expression for NPV.

Suppose, for now, that at least one root is real and non-negative. The Internal Rate of Return (IRR) is the smallest real, non-negative root. For notational convenience, let r₀ be the IRR.

3.0 Standard Investments in Selected Models of Production

A standard investment is one in which all negative cash flows precede all positive cash flows. Is there a theorem that an IRR exists for each standard investment? Perhaps this can be proven by discounting all cash flows to the end of the year in which the last outgoing cash flow occurs. Maybe one needs a clause that the undiscounted sum of the positive cash flows does not fall below the undiscounted sum of the negative cash flows.

At any rate, an Osborne expression for NPV has been calculated for standard investments characterizing two models of production. As I recall it, Osborne (2010) illustrates a more abstract discussion with a point-input, flow-output example. Consider a model in which a machine is first constructed, in a single year, from unassisted labor and land. That machine is then used to produce output over multiple years. Given certain assumptions on the pattern of the efficiency of the machine, this example is of a standard investment, with one initial negative cash flow followed by a finite sequence of positive cash flows.

On the other hand, I have presented an example for a flow-input, point-output model. Techniques of production are represented as finite series of dated labor inputs, with output for sale on the market at a single point in the time. Each technique is characterized by a finite sequence of negative cash flows, followed by a single positive cash flow.

In each of these two examples, the NPV can be represented by an Osborne expression that combines information about all roots of a polynomial. Thus, basing an investment decision on the NPV uses more information than basing it on the IRR, which is a single root of the relevant polynomial.

4.0 Non-standard Investments and Pitfalls of the IRR

In a non-standard investment at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments can highlight three pitfalls in basing an investment decision on the IRR:

• Multiple IRRs: The polynomial defining the IRR may have more than one real, non-negative root. What is the rationale for picking the smallest?
• Inconsistency in recommendations based on IRR and NPV: The smallest real non-negative root may be positive (suggesting a good investment), with a negative NPV (suggesting a bad investment).
• No IRR: All roots may be complex.

Berk and DeMarzo (2014) present the example in Table 1 as an illustration of the third pitfall. They imagine an author who receives an advance of $750 hundred thousands, sacrifices an income of $500 hundred thousand in each year of writing a book, and, finally, receives a royalty of one million dollars upon publication. The roots of the polynomial defining the NPV are -1.71196 + 0.78662 j, -1.71196 - 0.78662 j, 0.04529 + 0.30308 j, 0.04529 - 0.30308 j. All of these roots are complex; none satisfy the definition of the IRR.

Table 1: A Non-Standard Investment

Year	Revenue
0	750
1	-500
2	-500
3	-500
4	1,000

5.0 Issues for Multiple Interest Rate Analysis

Osborne, in his 2014 book, extends his 2010 analysis of the NPV to consider the first and second pitfall above. Nowhere do I know of is an Osborne expression for the NPV derived for an example in which the third pitfall arises.

The idea that the pitfalls above for the use of the IRR might be a problem for multiple interest rate analysis was suggested to me anonymously. On even hours, I do not see this. Why should I care about how many roots there are in an Osborne expression for the NPV, their sign, or even if they are complex?

On the other hand, I wonder about how non-standard investments relate to the theory of production. I know that an example can be constructed, in which the price of a used machine becomes negative before it becomes positive. Can the varying efficiency of the machine result in a non-standard investment? After all, the cash flow, in such an example of joint production, is the sum of the price of the conventional output of the machine and the price of the one-year older machine. Even when the latter is negative, the sum need not be negative. But, perhaps, it can be in some examples.

Not all techniques in models with joint production, of the production of commodities by means of commodities, can be represented as dated labor flows. I guess one can still talk about NPVs. Can one formulate an algorithm, based on NPVs, for the choice of technique? How would certain annoying possibilities, such as cycling be accounted for? Can one always formulate an Osborne expression for the NPV? Do properties of multiple interest rates have implications for, for example, a truncation rule in a model of fixed capital? Perhaps a non-standard investment, for a fixed capital example and one pitfall noted above, always has a cost-minimizing truncation in which the pitfall does not arise. Or perhaps the opposite is true.

Anyway, I think some issues could support further research relating models of production in economics and finance theory. Maybe one obtains, at least, a translation of terms.

Appendix: Technical Terminology

See body of post for definitions.

• Flow Input, Point Output
• Investment
• Investment Project
• Internal Rate of Return (IRR)
• Net Present Value (NPV)
• Non Standard Investment
• Osborne Expression (for NPV)
• Point-Input, Flow Output model
• Standard Investment

References

• Jonathan Berk and Peter DeMarzo (2014). Corporate Finance, 3rd edition. Boston: Pearson Education
• Michael Osborne (2010). A resolution to the NPV-IRR debate? Quarterly Review of Economics and Finance, V. 50, Iss. 2: 234-239.
• Michael Osborne (2014). Multiple Interest Rate Analysis: Theory and Applications. New York: Palgrave Macmillan
• Robert Vienneau (2016). The choice of technique with multiple and complex interest rates, DRAFT.