## And why firms use it as a measure of investment performance…

Note: The concepts discussed in this article are meant for those who have studied finance at some point

If you’ve studied finance at any point in your life, you would have come across the concept of Internal Rate of Return (“IRR”) – the rate at which the present value of cash inflows equals the present value of cash outflows. While it may be easy to learn the technical definition of IRR, many people struggle to explain what it actually means and apply it.

To unravel the mysteries of the IRR, however, let’s take the help of another return concept – Compounded Annual Growth Rate (“CAGR”).

### Compounded Annual Growth Rate

CAGR is the most popular return concept for measuring investment performance. The CAGR is that single rate which an investment earns year-on-year over the investment period, even though actual returns may vary each year. For instance, you make a Rs. 1,000 investment today which performs as follows over the next 4 years:

 Time End of Year Value Yearly Return 1 1,500 50.00% 2 3,000 100.00% 3 1,500 -50.00% 4 2,000 33.33%

Given your 4-year investment period, the CAGR will be that single rate at which Rs. 1,000 grows consistently every year to yield Rs. 2,000 at the end of 4 years. In our scenario, that rate comes to 18.92%, even though the yearly returns were significantly different.

Therefore, one needs the following 3 inputs in order to calculate the CAGR:

• Beginning Investment Value;
• Ending Investment Value; and
• Time period.

### Internal Rate of Return

As mentioned earlier, IRR is the rate at which the present value of cash inflows equals the present value of cash outflows. That is, it is the rate at which the Net Present Value of cash flows is equal to 0.

In corporate finance, the concept of IRR is often used to analyse projects with cash flows. The decision criteria for accepting or rejecting projects based on IRR starts with an assumption regarding the required rate of return. Once you have determined the required rate of return, then:

 If… Decision IRR > Required Rate Accept IRR < Required Rate Reject

We have been taught that one of the problems with using the IRR is the assumption that cash flows received during the life of the project are reinvested at the IRR rate – an assumption which may not be realistic. In fact, many investment firms declare their performance in terms of IRR which may not truly reflect the actual performance of the fund. We’ll see why later in the article.

Now let’s get back to our two return concepts and tie them together for a better understanding with the help of the following case studies.

### Case 1: No interim cash flows

Case 1 deals with a situation where you make an investment in Year 0 and receive a single cash flow at the end of the investment period, say Year 4. The cash flows are depicted below: #### CAGR

This is a simple problem and our 3 inputs into the CAGR calculation are as follows:

• Beginning Investment Value – Rs. 1,000
• Ending Investment Value – Rs. 2,000
• Time Period – 4 Years

Based on these inputs, the CAGR comes to 18.92%. Now let’s calculate the IRR of this investment.

#### IRR

Calculating the IRR is an iteration problem and can be guessed through trial and error. This can be done with the help of a financial calculator or the IRR function in excel. The inputs to be plugged in are:

• Initial cash outflow;
• Interim cash flows; and
• Ending cash inflow.

In our case, the beginning investment can be considered as the initial cash outflow. The ending investment value can be considered as the cash inflow in the terminal year. Notice that there are no interim cash flows. This yields an IRR of 18.92%.

Did you notice something? The CAGR and IRR are the same in this case.

### Case 2: With interim cash flows

In this case, we do away with the assumption of no interim cash flows and assume that we receive Rs. 200 in Year 1, Rs. 500 in Year 2, Rs. 400 in Year 3 and Rs. 900 in Year 4. Notice that the total cash inflows are the same as Case 1, i.e. Rs. 2,000. #### CAGR

Calculating the CAGR in this scenario is not as simple as in our earlier example. Notice that the inputs required for calculating the CAGR do not provide for interim cash flows. Even though our total cash flows from this investment is Rs. 2,000 (200+500+400+900), we cannot take this as our ending investment value.

Why? Because the cash flows we have received in years 1,2 and 3 would be reinvested to earn some return. Accordingly, we need to figure out what that rate is (called the reinvestment rate) in order to calculate the value of these interim cash flows at the end of year 4. This is what we get if we assume a reinvestment rate of 5% p.a. Now this problem becomes the same as in Case 1 with the following inputs:

• Beginning Investment Value – Rs. 1,000
• Ending Investment Value – Rs. 2,102
• Time Period – 4 Years

The CAGR comes to about 20.42% based on the above.

Note that even though the total cash flow during the investment period was Rs.2,000, the CAGR calculated in Case 2 was slightly higher than that calculated in Case 1. This is simply because of the reinvestment of the interim cash flows.

#### IRR

Using the same methodology discussed earlier, the IRR in this scenario comes to 27.41%, much higher than the CAGR we had calculated.

Why is there such a difference you might wonder. The difference is on account of the assumption regarding the reinvestment of interim cash flows. The IRR assumes that the interim cash flows of Rs. 200, 400 and 500 are all reinvested at the rate of 27.41%. It is because of this assumption that the IRR is higher than the CAGR which had assumed the interim cash flows be reinvested at 5%.

Pictorially, we can depict what is happening as follows: In other words, we can say that the main difference between IRR and CAGR is the assumption regarding the reinvestment rate of interim cash flows.

### Will the IRR always be greater than the CAGR?

Of course not. it all depends on the reinvestment rate. We can summarize the following relation between the CAGR and IRR:

 If… Then… Reinvestment rate of interim cash flows in CAGR is greater than IRR CAGR > IRR Reinvestment rate of interim cash flows in CAGR is equal to IRR CAGR = IRR Reinvestment rate of interim cash flows in CAGR is less than IRR CAGR < IRR

This takes us back to our original question – why do investment firms declare results in terms of the IRR and not CAGR. Probably because there is no need to make an explicit assumption regarding the reinvestment rate in case of IRR. The assumption is already built-in in its calculation.

This is why investors need to take results in IRR with a pinch of salt. Because the reinvestment rate is not stated explicitly in case of IRR, many investors overlook it. Not questioning this assumption may lead to investors accepting investment results which appear better than they actually are or would be under realistic assumptions.