# 1.21 The Internal Rate of Return for Multiple Cash Flows

Well, that was pretty easy; we had only one cash outflow (PV) and one inflow (FV). What if we had multiple cash inflows? Our equation would look something like the following:

NPV = the discounted values of CF_{1} + CF_{2} + CF_{3} +…. Less initial outlay

= CF1/(1+R)^{1+ }CF2/(1+R)^{2}+ CF3/1+R)^{3 }Less initial outlay = NPV

Due to our having here *multiple* unknowns in the denominator, we cannot solve for R using a formula such as above (see Section #1.20), which had only one unknown, i.e., the value of the singular denominator. However, we can guess and guess – until we hit upon the correct answer.

Let’s get back to “Example 3.” We can agree that we are looking for a discount rate that will cause the NPV to equal zero. In Example 3, we had a positive NPV, so we will have to guess at a *higher* discount rate in order to *lower* the NPV, until it hits zero. So, let’s solve for that, by trial and error, *iteratively*, until we get it. We now have “Example #4.”

**Example #4:** – Annuity Cash Flows: One can easily imagine that, for the example (#3) above, there exists a THEORETICAL dollar solution wherein the NPV is *forced* to equal, i.e., where the present value of the project’s expected net inflows equals the project’s cost. This is true even though the firm’s true cost of capital discount rate is different.

In order to discover this other rate, let’s call it again the “Internal Rate of Return,” or IRR, we must try out several discount rates until we find the correct zero solution; it is an “iterative” or *trial and error* process. There is no direct mathematical solution to this. (You may use the tables….)

Be aware, once again, that in order for the solution to be zero, which is *less* than $1,258.80, we must use a *higher* discount rate than that used above. (Remember the basic rule: rates go up, present values go down.) Let’s try 10%, 15%, 20%, 25%. No good? NPV still not zero?

To make your life easy (for the moment), let us try a discount rate of 28%. The rest of the problem is as before (i.e., as in example #3).

**Given:**

- Same as Example 3 (except we won’t use the cost of capital discount rate!).
*k*= 28% (guessed at – after several failed tries)

**Solution:**

NPV = (PV _{annuity factor} _{for 5yrs @ 28%} for $1,000) – ($2,532)

= [($1,000) (2.532)] – $2,532

= 2,532 – 2,532 = 0

The IRR is 28%!

**Conclusion:**

If the discount rate (cost of capital) is less than the IRR, the NPV *must* be positive!