__Internal Rate of Return__

__Internal Rate of Return__

Internal rate of return is another important concept of cash flow analysis. It pertains specifically to the entire cash flow stream associated with an investment, not to a partial stream such as a cash flow at a single period. The streams to which this concept is applied typically have both negative and positive elements: the negative flows correspond to the payments that must be made; the positive flows to payments received. A simple example is a process of investing in a certificate of deposit for a fixed period of 1 year. Here there are two cash flow elements: the initial deposit or payment (a negative flow) and the final redemption (a positive flow).

Given a cash flow stream (x_{0}, x_{1},………, x_{n}) associated with an investment, we write the present value formula as

If the investment that corresponds to this stream is constructed from a series of deposits and withdrawals from a constant ideal bank at interest rate *r*, then from the main theorem on the present value in the previous section, PV would be zero. The idea behind the internal rate of return is to turn the procedure around. Given a cash flow stream, we write the expression for present value and the value of *r* that renders this present value equal to zero.

That value is called the internal rate of return because it is the interest rate implied by the internal structure of the cash flow stream. The idea can be applied to any series of cash flows.

The preliminary formal definition of the internal rate of return (IRR) is as follows:

*Internal rate of return** : Let (x _{0}, x_{1}, …….x_{n}) be a cash flow stream. Then the internal rate of return of this section is a number r satisfying the equation *

*0 = x _{0} + x_{1}/(1+r) +x_{2}/(1+r)^{2} +……………….x_{n}/(1+r)^{n}*

Equivalently, it is a number *r* satisfying 1/(1+*r*) =c [i.e., *r= *(1/c) – 1] , where *c *satisfies the polynomial equation

*0 = x _{0} + x_{1}c +x_{2}c^{2} +……………….x_{n}c^{n}*

We call this a preliminary definition because there may be ambiguity in the solution of the polynomial equation of degree *n.*

IRR can also be *defined* as a discount rate at which net present value is equal to zero. In other words, t is a rate of return which equates investment output or cash outflow to present value or cash inflow from an investment.

IRR is determined entirely by the cash flows of the stream. This is the reason why it is called the *internal* rate of return; it is defined internally without reference to the external financial world. It is the rate that an ideal bank would have to apply to generate the given stream from an initial balance of zero.

As we have seen that the internal rate of return is a polynomial equation in *c* of degree *n*, which does not, in general, have an analytic solution. However, it is almost always easy to solve the equation with a computer. Form algebraic theory it is known that such an equation always has at least one root, and may have as many as *n *roots, but some or all of these roots may be complex numbers. Fortunately, the most common form of investment, where there is an initial cash outlay followed by several positive flows, leads to a unique positive solution. Hence the internal rate of return is then well defined and relatively easy to calculate. The formal statement of the existence of the positive root embodies the main result concerning the internal rate of return.

x**Main theorem of internal rate of return**Suppose the cash flow stream (x_{0}, x_{1}, …….x_{n}) has x_{0}< 0 and_{k }≤ 0 for all k, k = 1, 2, ……n. with at least one term being strictly positive. Then there is a unique positive root to the equation

*0 = x _{0} + x_{1}c +x_{2}c^{2} +……………….x_{n}c^{n}*

* Furthermore, if > 0 (meaning that the total amount returned exceeds the initial investment), then the corresponding internal rate of return r = (1/c) – 1 is positive.*

** **If some or all solutions to the equations for internal rate of return are complex, the interpretation of these values is not simple. In general, it is reasonable to select the solution that has the largest real part and use that real part to determine the internal rate of return. In practice, however, this is not often a serious issue, since suitable real roots typically exist.

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