If P(x) and Q(x) are polynomial functions, the function
is a rational function. In other words, a rational function is just a ratio of two polynomials. Unlike linear and polynomial functions, the domain of a rational function need not be the whole real line. We are in trouble when the denominator polynomial is zero, i.e., when Q(x) = 0. Loosely speaking, a rational function “goes to infinity” near the roots of the denominator polynomial.
Example 2.6 Consider the rational function
The numerator polynomial has no real root; in fact, the function graph shown in Fig. 2.9 does not cross the horizontal axis anywhere. The denominator polynomial has roots −2, 1, and 5, and the function behavior is critical near these roots, where the function goes to ±∞.
Example 2.7 (Internal rate of return) Let us consider a more interesting example from finance, involving polynomial and rational functions. In analyzing an investment, we often deal with a sequence of periodic cash flows Ct t = 0, 1, …, T. A positive cash flow at time t means that the investor receives some money, whereas a negative cash flow is, from her viewpoint, a payment. We deal with integer-valued time instants t, which are actually integer multiples of a basic time period, which could be a month or a year. To fix ideas, say that the basic time period is one year. Then, C0 is an immediate cash flow, C1 is a cash flow occurring in one year, etc. Quite often, C0 is the initial capital outlay to invest in a project, whereas Ct, t = 1, …, T, are the net cash inflows from the investment (they are not necessarily positive, though). A typical question is whether the project is worth financing. Arguably, if the straightforward sum of cash flows is negative, we are about to loose money. However, we already know from Section 1.2.3 that the time value of money should also be taken into account: Cash flows should be discounted using a discount rate r. The sum of discounted cash flows is the net present value (NPV) of the investment and is given by the following function of the discount rate:
Fig. 2.9 Graph of a rational function.
This function is actually a rational function of r, as by a straightforward manipulation we could recast it as the ratio of two polynomials:
In practice, this manipulation is of little use, as we shall see immediately.
According to financial theory, provided that we are able to estimate cash flows and to select a suitable discount rate accounting for risk, we should select investments with a positive NPV. As a numerical example, consider a project that requires $100 to be started, and will pay $20, $40, and $60 at the end of the first, second, and third years, respectively. Note that the sum of the three cash inflows is $120, which is larger than the initial cash outflow, but is the project really a good deal? That depends on the discount rate r. If we choose r = 10%, then NPV is
which suggests giving up the project. The decision depends critically on the chosen discount rate. The theory of corporate finance gives us some clue about this choice, but it is always wise to carry out a sensitivity analysis to investigate the impact on NPV of changes in the discount rate as well as in the predicted cash flows.
It is natural to wonder for which critical value of the discount rate the NPV of a cash flow sequence turns out to be zero, since this value draws the line between acceptance or rejection of an investment proposal. The critical discount rate is called internal rate of return (IRR) and is found by solving the following equation:
The IRR is sometimes used as an alternative tool to analyze investments. If we rely on IRR, then we should select an investment such that the IRR is larger than some required rate of return. This benchmark rate of return could be associated with an alternative project, or it could be a rate of return high enough that we are willing to take the risk of investing in the project. But how can we find the IRR? Although the NPV is a rational function of r, it is much better to transform it into a more manageable form. By the change of variable
we may transform the equation, which involves a rational function, into a polynomial form:
Luckily, very efficient numerical procedures to find roots a polynomial are included in many commercially available software packages. All we have to do is solve for y and transform the solution back to discount rates using r = (1 − y)/y. The IRR of the above cash flow stream is 8.21%. This also means that the NPV is negative for r > 8.21%, but positive for r < 8.21%. The larger the required rate of return, the larger the chance that the project is rejected.
However, we should see a potential trouble: A polynomial equation might have several positive real roots. In such a case, which is the correct IRR? Indeed, this is why the NPV criterion is typically considered a better one. One fortunate case is when only the first cash flow is negative, i.e., C0 < 0 and Ct > 0 for t > 0; for such a cash flow sequence, it can be shown that there is a unique real and positive IRR. In complex projects, with multiple capital outlays in time, there is no guarantee that only the first cash flow is negative. Nevertheless, there are quite relevant kinds of investment in which the condition applies. One such example are bonds. A bond is a financial instrument that is used by governments and corporations to finance their activities. When an investor buys a bond, she is basically lending money to the bond issuer. The face value F of the bond is the amount of money that is loaned to the bond issuer and will be repaid to the investor at a future time instant known as bond maturity, plus some interest. Typically, the bond issuer also promises periodic payments during the life of the bond; these payments are called coupons.9 For instance, if the face value is F = $1000 and the coupon rate is 6% per year, the coupons would be $60 each year, and the last cash flow would be $1060. Often, coupon payments are semiannual (every 6 months). In principle, bonds can be purchased from the issuer on primary markets, but they are typically traded on secondary markets at prices depending on many factors including prevailing interest rates. In bond investing, C0 is the price of the bond, the cash flows for t = 1, …, T − 1 correspond to coupon payments, Ct = C, and the last cash flow includes the face value of the bond, CT = C + F. In bond valuation and management, the IRR is referred to as yield to maturity.
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