The internal rate of return (IRR) is frequently used by corporations to compare and decide between capital projects, but it can also help you evaluate certain financial events in your own life, like lotteries 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. Read on to learn more about how this method is used.

IRR UsesAs we mentioned above, one of the uses of IRR is by corporations that wish to compare capital projects. For example, a corporation will evaluate an investment in a new plant versus an extension of 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 stock buyback, the analysis must show that the company's own stock is a better investment (has a higher IRR) than any other use of the funds for other capital projects, or than any acquisition candidate at current market prices.

Calculation ComplexitiesThe 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.
Calculating IRRThe simplest example of computing an IRR is by using the example of 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 a 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%.

The formula for IRR, using this example, is as follows:

• Where the initial payment (CF₁) is $200,000 (a positive inflow)
• Subsequent cash flows (CF ₂, CF ₃, CF N) are negative $1050 (negative because it is being paid out)
• Number of payments (N) is 30 years times 12 = 360 monthly payments
• Initial Investment is $200,000
• IRR is 4.8% divided by 12 (to equate to monthly payments) = 0.400%

Figure 1: The formula for calculating internal rate of return (IRR)

Power of CompoundingIRR 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 approximately the current 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.

Other IRR UsesIRR 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.

ConclusionAs the number of trading methodologies, mutual funds, alternative investment plans and stocks has been increasing 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.

In introductory finance courses, we are taught to calculate the standard deviation of the portfolio as a measure of risk, but part of this calculation is the covariance of these two, or more, stocks. So, before going into portfolio selections, understanding covariance is very important.



What Is Covariance?Covariance is a measure of how two variables move together. It measures if the two move in the same direction (a positive covariance) or in opposite directions (a negative covariance). In this article, the variables will usually be stock prices, but it can be anything.
In the stock market, there is a strong emphasis placed on reducing the amount of risk taken on for the same amount of return. When constructing a portfolio, an analyst will select stocks that will work well together. This usually means that these stocks do not move in the same direction. Covariance can tell how the stocks move together, but to determine the strength of the relationship, we need to look at the correlation.

Calculating CovarianceThe calculation for covariance of a stock starts with finding a list of previous prices. This is labeled as "historical prices" on most quote pages. Typically, the closing price for each day is used to find the return from one day to the next. Do this for both stocks, and build a list to begin the calculations.
For example:

Day	ABC Returns (%)	XYZ Returns (%)
1	1.1	3
2	1.7	4.2
3	2.1	4.9
4	1.4	4.1
5	0.2	2.5
Table 1: Daily returns for two stocks using the closing prices

From here, we need to calculate the average return for each stock: Now, it is a matter of taking the differences between ABC's return and ABC's average return, and multiplying it by the difference between XYZ's return and XYZ's average return. The last step is to divide the result by the sample size and subtract one. If it was the entire population, then you could just divide by the population size.
Represented by this equation:



 For example:
In this situation, we are using a sample, so we divide by the sample size (five) minus one.
You can see that the covariance between the two stock returns is 0.665, which means that they move in the same direction. When ABC had a high return, XYZ also had a high return.
Using Microsoft Excel
In Excel, you can easily find the covariance by using one the following functions:

= COVARIANCE.S()    for a sample
or
= COVARIANCE.P()    for a population
You will need to set up the two lists of returns in vertical columns, just like in Table 1. Then, when prompted, select each column. In Excel, each list is called an "array," and there should be two arrays inside the brackets, separated by a comma.
Meaning      
In the example, there is a positive covariance so the two stocks tend to move together. When one has a high return, the other tends to have a high return as well. If the result was negative, then the two stocks would tend to have opposite returns; when one had a positive return, the other would have a negative return.
Uses of Covariance
Finding that two stocks have a high or low covariance might not be a useful metric on its own, but the covariance can be used to calculate the correlation. The correlation will give a measurement between -1 and 1, and adds a strength value on how the stocks move together. If the correlation is 1, they move perfectly together, and if the correlation is -1, the stocks move perfectly in opposite directions. If the correlation is 0, then the two stocks move in random directions from each other. (To know more about correlation and portfolio management.

The covariance can also be used to find the standard deviation of a multi-stock portfolio. The standard deviation is the accepted calculation for risk, and is extremely important when selecting stocks. Typically, you would want to select stocks that move in opposite directions. If the chosen stocks work well together, then the risk will be lower given the same amount or potential return.
ConclusionCovariance is a common statistical calculation which can show how two stocks tend to move together. We can only use historical returns so there will never be complete certainty about the future. Also, it should not be used on its own. Instead, it can be used in other, more important, calculations such as correlation, or standard deviation

Managers	Average Annual Return	Beta
Manager A	10%	0.90
Manager B	14%	1.03
Manager C	15%	1.20

Now, you can compute the Treynor value for each:

T(market) = (.10-.05)/1 = .05
T(manager A) = (.10-.05)/0.90 = .056
T(manager B) = (.14-.05)/1.03 = .087
T(manager C) = (.15-.05)/1.20 = .083

The higher the Treynor measure, the better the portfolio. If you had been evaluating the portfolio manager (or portfolio) on performance alone, you may have inadvertently identified manager C as having yielded the best results. However, when considering the risks that each manager took to attain their respective returns, Manager B demonstrated the better outcome. In this case, all three managers performed better than the aggregate market.

Because this measure only uses systematic risk, it assumes that the investor already has an adequately diversified portfolio and, therefore, unsystematic risk (also known as diversifiable risk) is not considered. As a result, this performance measure should really only be used by investors who hold diversified portfolios.

Sharpe RatioThe Sharpe ratio is almost identical to the Treynor measure, except that the risk measure is the standard deviation of the portfolio instead of considering only the systematic risk, as represented by beta. Conceived by Bill Sharpe, this measure closely follows his work on the capital asset pricing model (CAPM) and by extension uses total risk to compare portfolios to the capital market line. (For related topics, check out Understanding The Sharpe Ratio and The Sharpe Ratio Can Oversimplify Risk.)

The Sharpe ratio can be easily defined as:

(Portfolio Return – Risk-Free Rate) / Standard Deviation

Using the Treynor example from above, and assuming that the S&P 500 had a standard deviation of 18% over a 10-year period, let's determine the Sharpe ratios for the following portfolio managers:

Manager	Annual Return	Portfolio Standard Deviation
Manager X	14%	0.11
Manager Y	17%	0.20
Manager Z	19%	0.27

S(market) = (.10-.05)/.18 = .278
S(manager X) = (.14-.05)/.11 = .818
S(manager Y) = (.17-.05)/.20 = .600
S(manager Z) = (.19-.05)/.27 = .519

Once again, we find that the best portfolio is not necessarily the one with the highest return. Instead, it's the one with the most superior risk-adjusted return, or in this case the fund headed by manager X.

Unlike the Treynor measure, the Sharpe ratio evaluates the portfolio manager on the basis of both rate of return and diversification (as it considers total portfolio risk as measured by standard deviation in its denominator). Therefore, the Sharpe ratio is more appropriate for well diversified portfolios, because it more accurately takes into account the risks of the portfolio.

Jensen MeasureLike the previous performance measures discussed, the Jensen measure is also based on CAPM. Named after its creator, Michael C. Jensen, the Jensen measure calculates the excess return that a portfolio generates over its expected return. This measure is also known as alpha. (For related reading, see Bettering Your Portfolio With Alpha And Beta.)

The Jensen ratio measures how much of the portfolio's rate of return is attributable to the manager's ability to deliver above-average returns, adjusted for market risk. The higher the ratio, the better the risk-adjusted returns. A portfolio with a consistently positive excess return will have a positive alpha, while a portfolio with a consistently negative excess return will have a negative alpha

The formula is broken down as follows:

Jensen's Alpha = Portfolio Return – Benchmark Portfolio Return

Where: Benchmark Return (CAPM) = Risk Free Rate of Return + Beta (Return of Market – Risk-Free Rate of Return)

So, if we once again assume a risk-free rate of 5% and a market return of 10%, what is the alpha for the following funds?

Manager	Average Annual Return	Beta
Manager D	11%	0.90
Manager E	15%	1.10
Manager F	15%	1.20

First, we calculate the portfolio's expected return:

ER(D)= .05 + 0.90 (.10-.05) = .0950 or 9.5% return
ER(E)= .05 + 1.10 (.10-.05) = .1050 or 10.50% return
ER(F)= .05 + 1.20 (.10-.05) = .1100 or 11% return

Then, we calculate the portfolio's alpha by subtracting the expected return of the portfolio from the actual return:

Alpha D = 11%- 9.5% = 2.5%
Alpha E = 15%- 10.5% = 4.5%
Alpha F = 15%- 11% = 4.0%

Which manager did best? Manager E did best because, although manager F had the same annual return, it was expected that manager E would yield a lower return because the portfolio's beta was significantly lower than that of portfolio F.

Of course, both rate of return and risk for securities (or portfolios) will vary by time period. The Jensen measure requires the use of a different risk-free rate of return for each time interval considered. So, let's say you wanted to evaluate the performance of a fund manager for a five-year period using annual intervals; you would have to also examine the fund's annual returns minus the risk free return for each year and relate it to the annual return on the market portfolio, minus the same risk free rate. Conversely, the Treynor and Sharpe ratios examine average returns for the total period under consideration for all variables in the formula (the portfolio, market and risk-free asset). Like the Treynor measure, however, Jensen's alpha calculates risk premiums in terms of beta (systematic, undiversifiable risk) and therefore assumes the portfolio is already adequately diversified. As a result, this ratio is best applied with diversified portfolios, like mutual funds.

Conclusion
Portfolio performance measures should be a key aspect of the investment decision process. These tools provide the necessary information for investors to assess how effectively their money has been invested (or may be invested). Remember, portfolio returns are only part of the story. Without evaluating risk-adjusted returns, an investor cannot possibly see the whole investment picture, which may inadvertently lead to clouded investment decisions.