The internal rate of return (IRR) is frequently used by companies to analyze profit centers and decide between capital projects. But this budgeting metric can also help you evaluate certain financial events in your own life, like mortgages and investments.

The IRR is the interest rate (also known as the discount rate) that will bring a series of cash flows (positive and negative) to a net present value (NPV) of zero (or to the current value of cash invested). Using IRR to obtain net present value is known as the discounted cash flow method of financial analysis.

## IRR Uses

As we mentioned above, IRR is a key tool in corporate finance. For example, a corporation will evaluate investing in a new plant versus extending an existing plant based on the IRR of each project. In such a case, each new capital project must produce an IRR that is higher than the company's cost of capital. Once this hurdle is surpassed, the project with the highest IRR would be the wiser investment, all other things being equal (including risk).

IRR is also useful for corporations in evaluating stock buyback programs. Clearly, if a company allocates a substantial amount to a repurchasing its shares, the analysis must show that the company's own stock is a better investment—that is, has a higher IRR—than any other use of the funds, such as creating new outlets or acquiring other companies.

## IRR Calculation Complexities

The IRR formula can be very complex depending on the timing and variances in cash flow amounts. Without a computer or financial calculator, IRR can only be computed by trial and error.

One of the disadvantages of using IRR is that all cash flows are assumed to be reinvested at the same discount rate, although in the real world these rates will fluctuate, particularly with longer-term projects. IRR can be useful, however, when comparing projects of equal risk, rather than as a fixed return projection.

The general formula for IRR that includes net present value is:

$\begin{aligned} 0 &= CF_0 + \frac{CF_1}{(1 + IRR)} + \frac{CF_2}{(1 + IRR)^2} + \dotso + \frac{CF_n}{(1 + IRR)^n} \\ &= NPV = \sum^N_{n = 0} \frac{CF_n}{(1 + IRR)^n} \\ &\textbf{where:}\\ &CF_0=\text{Initial investment/outlay}\\ &CF_1, CF_2, \dotso, CF_n=\text{Cash flows}\\ &n=\text{Each period}\\ &N=\text{Holding period}\\ &NPV=\text{Net present value}\\ &IRR=\text{Internal rate of return}\\ \end{aligned}$

## An Example of an IRR Calculation

The simplest example of computing an IRR is by taking one from everyday life: a mortgage with even payments. Assume an initial mortgage amount of $200,000 and monthly payments of $1,050 for 30 years. The IRR (or implied interest rate) on this loan annually is 4.8%.

Because the stream of payments is equal and spaced at even intervals, an alternative approach is to discount these payments at a 4.8% interest rate, which will produce a net present value of $200,000. Alternatively, if the payments are raised to, say $1,100, the IRR of that loan will rise to 5.2%.

Here's how the above formula for IRR works using this example:

- The initial payment (CF
_{1}) is $200,000 (a positive inflow) - Subsequent cash flows (CF
_{2}, CF_{3}, CF_{n}) are negative $1,050 (negative because it is being paid out) - Number of payments (N) is 30 years x 12 = 360 monthly payments
- Initial Investment is $200,000
- IRR is 4.8% divided by 12 (to equate to monthly payments) = 0.400%

## IRR and the Power of Compounding

IRR is also useful in demonstrating the power of compounding. For example, if you invest $50 every month in the stock market over a 10-year period, that money would turn into $7,764 at the end of the 10 years with a 5% IRR, which is more than the current 10-year Treasury (risk-free) rate.

In other words, to get a future value of $7,764 with monthly payments of $50 per month for 10 years, the IRR that will bring that flow of payments to a net present value of zero is 5%.

Compare this investment strategy to investing a lump-sum amount: to get the same future value of $7,764 with an IRR of 5%, you would have to invest $4,714 today, in contrast to the $6,000 invested in the $50-per-month plan. So, one way of comparing lump-sum investments versus payments over time is to use the IRR.

## IRR and Investment Returns

IRR analysis can be useful in dozens of ways. For example, when the lottery amounts are announced, did you know that a $100 million pot is not actually $100 million? It is a series of payments that will eventually lead to a payout of $100 million but does not equate to a net present value of $100 million.

In some cases, advertised payouts or prizes are simply a total of $100 million over a number of years, with no assumed discount rate. In almost all cases where a prize winner is given an option of a lump-sum payment versus payments over a long period of time, the lump-sum payment will be the better alternative.

Another common use of IRR is in the computation of portfolio, mutual fund or individual stock returns. In most cases, the advertised return will include the assumption that any cash dividends are reinvested in the portfolio or stock. Therefore, it is important to scrutinize the assumptions when comparing returns of various investments.

What if you don't want to reinvest dividends, but need them as income when paid? And if dividends are not assumed to be reinvested, are they paid out or are they left in cash? What is the assumed return on the cash? IRR and other assumptions are particularly important on instruments like whole life insurance policies and annuities, where the cash flows can become complex. Recognizing the differences in the assumptions is the only way to compare products accurately.

## The Bottom Line

As the number of trading methodologies, alternative investment plans, and financial asset classes has increased exponentially over the last few years, it is important to be aware of IRR and how the assumed discount rate can alter results, sometimes dramatically.

Many accounting software programs now include an IRR calculator, as do Excel and other programs. A handy alternative for some is the good old HP 12c financial calculator, which will fit in a pocket or briefcase.