 ### IRR explained in detail

The internal rate of return on an investment or project is the "annualized effective compounded return rate" or rate of return that sets the net present value of all cash flows (both positive and negative) from the investment equal to zero. Equivalently, it is the discount rate at which the net present value of the future cash flows is equal to the initial investment, and it is also the discount rate at which the total present value of costs (negative cash flows) equals the total present value of the benefits (positive cash flows).

Speaking intuitively, IRR is designed to account for the time preference of money and investments. A given return on investment received at a given time is worth more than the same return received at a later time, so the latter would yield a lower IRR than the former, if all other factors are equal. A fixed income investment in which money is deposited once, interest on this deposit is paid to the investor at a specified interest rate every time period, and the original deposit neither increases nor decreases, would have an IRR equal to the specified interest rate. An investment which has the same total returns as the preceding investment, but delays returns for one or more time periods, would have a lower IRR.

Truly understanding what’s actually happening with IRR will give you a big advantage. Let’s walk through a detailed example of IRR and show you exactly what it does, step-by-step.

Source: Property Metrics

Suppose we are faced with the following series of cash flows: This is pretty straightforward. An investment of \$100,000 made today will be worth \$161,051 in 5 years. As shown the IRR calculated is 10%. Now let’s take a look under the hood to see exactly what’s happening to our investment in each of the 5 years: As shown above in year 1 the total amount we have invested is \$100,000 and there is no cash flow received. Since the 10% IRR in year 1 we receive is not paid out to us as an interim cash flow, it is instead added to our outstanding investment amount for year 2. That means in year 2 we no longer have \$100,000 invested, but rather we have \$100,000 + 10,000, or \$110,000 invested.

Now in year 2 this \$110,000 earns 10%, which equals \$11,000. Again, nothing is paid out in interim cash flows so our \$11,000 return is added to our outstanding internal investment amount for year 3. This process of increasing the outstanding “internal” investment amount continues all the way through the end of year 5 when we receive our lump sum return of \$161,051. Notice how this lump sum payment includes both the return of our original \$100,000 investment, plus the 10% return “on” our investment.

This is much more intuitive than the mathematical (and typical) explanation of IRR as “the discount rate that makes the net present value equal to zero.”

While technically correct, that doesn’t exactly help us all that much in understanding what IRR actually means. As shown above, the IRR is clearly the percentage rate earned on each dollar invested for each period it is invested. Once you break it out into its individual components and step through it period by period, this becomes easy to see.

## What IRR is Not

IRR can be a very helpful decision indicator for selecting an investment. However, there is one very important point that must be made about IRR: it doesn’t always equal the annual compound rate of return on an initial investment.

Let’s take an example to illustrate. Suppose we have the following series of cash flows that also generates a 10% IRR: In this example an investment of \$100,000 is made today and in exchange we receive \$15,000 every year for 5 years, plus we also sell the asset at the end of year 5 for \$69,475. The calculated IRR of 10% is exactly the same as our first example above. But let’s examine what’s happening under the hood in order to see why these are two very different investments: As shown above in year 1 our outstanding investment amount is \$100,000, which earns a return on investment of 10% or \$10,000. However, our total interim cash flow in year 1 is \$15,000, which is \$5,000 greater than our \$10,000 return “on” investment. That means in year 1 we get our \$10,000 return on investment, plus we also get \$5,000 of our original initial investment back.

Now, notice what happens to our outstanding internal investment in year 2. It decreases by \$5,000 since that is the amount of capital we recovered with the year 1 cash flow (the amount in excess of the return on portion). This process of decreasing the outstanding “internal” investment amount continues all the way through the end of year 5. Again, the reason why our outstanding initial investment decreases is because we are receiving more cash flow each year than is needed to earn the IRR for that year. This extra cash flow results in capital recovery, thus reducing the outstanding amount of capital we have remaining in the investment.

Why does this matter? Let’s take another look at the total cash flow columns in each of the above two charts. Notice that in our first example the total \$161,051 while in the second chart the total cash flow was only \$144,475. But wait a minute, I thought both of these investments had a 10% IRR?! Well, indeed they did both earn a 10% IRR, as we can see by revisiting the definition or IRR:

The Internal rate of return (IRR) for an investment is the percentage rate earned on each dollar invested for each period it is invested.

The internal rate of return measures the return on the outstanding “internal” investment amount remaining in an investment for each period it is invested. The outstanding internal investment, as demonstrated above, can increase or decrease over the holding period. It says nothing about what happens to capital taken out of the investment. And contrary to popular belief, the IRR does not always measure the return on your initial investment.

## The Myth of The Reinvestment Rate Assumption

One of the most commonly cited limitations of the IRR is the so called “reinvestment rate assumption.” In short, the reinvestment rate assumption says that the IRR assumes interim cash flows are reinvested at the IRR, which of course isn’t always feasible. The idea that the IRR assumes interim cash flows are reinvested is a major misconception that’s unfortunately still taught by many business school professors today.

As shown in the step-by-step approach above, the IRR makes no such assumption. The internal rate of return is a discounting calculation and makes no assumptions about what to do with periodic cash flows received along the way. It can’t because it’s a DISCOUNTING function, which moves money back in time, not forward.

This is not to say the the IRR doesn’t have some limitations, as shown in the examples above. It’s just to say that the “reinvestment rate assumption” is not among them. Should you take into account the yield you can earn on interim cash flows that you reinvest? Absolutely, and there have been various measures introduced over the years to turn the IRR into a measure of return on the initial investment. Some of the more popular approaches include the modified internal rate of return (MIRR), the capital accumulation method, and the external rate of return (ERR).