Context is everything. Victory over an obstacle needs to have appropriate awards. A win that costs you more than you were willing to pay is known as a Pyrrhic victory – named after the Greek King Pyrrhus who defeated the Romans around 280 B.C. but did so at such a great cost that the Greek King quipped that any more victories would ruin him.

In the stock market, similar situations happen all the time. You may succeed in turning the tide on a position and making it profitable, but at such a great cost that you took away from all your other positions to do so and ended up losing out on all the other gains you would’ve had.

It’s hard to know if you’ve done well managing your portfolio or not. You may think that simply beating the market averages for the year, a difficult enough feat, constitutes success, but you’d be mistaken. A gambler who bets everything on a roll of the dice and comes up a winner would certainly qualify as beating the markets, but he did so at tremendous risk. Risk that no investor would be willing to take.

Understanding risk is critical to being a profitable investor over the long haul. Our measurement of success therefore comes from two parts – percentage gains, and risk assumed.

**Analyzing Risk**

When you compare your performance to the market average, you generally pick an index that matches your portfolio set-up as closely as possible. It could be anything from the S&P 500 to the Russel 2000 or anything in-between. You need a benchmark to compare not only your performance to, but your risk profile as well.

Risk can be broken up into two parts – standard deviation and beta. Standard deviation measures the variability of an investments return with respect to the expected return of the investment. The smaller the deviation, the less risky, and the greater the deviation, the more risky it is. Actual return is within one standard deviation 68% of the time and within two standard deviations 95% of the time.

The second aspect of risk is beta. This measures to volatility of the investment relative to the market average. The S&P 500 is the most common representative and comes with an inherent beta of 1. An investment with a beta of 2 would mean that for every 1% change in the S&P 500, the investment should move 2%. If it had a beta of 0.50, then it would only move 0.50%.

So how does this all work together?

Let’s say that the S&P 500 has an expected return of 12% with a standard deviation of 4%. We already know its beta is 1. You pick an investment with an expected return of 18%, a standard deviation of 9%, and a beta of 2.

From this information, we know that the S&P 500 will most likely return between 8% and 16% while your investment will likely return between 9% and 27%. It’s clear that your investment is going to be far more volatile then the index but to truly make a risk comparison, we’ll need more information.

**The Risk Formulas**

Risk can be assessed using several formulas that can tell you whether or not you’ve beaten the market on a fair risk basis or not. These formulas all use a combination of expected return, beta, and standard deviation.

**Sharpe Ratio**

The Sharpe ratio is also called the reward-to-variability ratio. To understand how its used, you would need to compare the Sharpe ratios of both your portfolio and the market portfolio. If the Sharpe ratio for your portfolio is higher than the market portfolio’s ratio, then it means that your portfolio has beaten the market.

**Sr = Rm – Rf / σp**

Sr = Sharpe Ratio

Rm = Arithmetic mean of the portfolio being evaluated

Rf = Risk-free rate

σp = Standard deviation of the portfolio being evaluated

We can calculate arithmetic mean by simply adding up the past history of a portfolio’s performance and dividing by the number of periods used. For example, if you used 3 years of history of a portfolio that returned 7%, 9%, and 5%, than you would add them up and divide by 3 to give you a mean score of 7%.

The risk free-rate is the rate an investor could get for taking no risk. This is generally compared to the 10-year yield on treasuries. Right now, because rates are at historic lows, investing in stocks is less risky than usual. When interest rates rise, overall risk in the market will as well.

**Treynor Ratio**

While the Sharpe ratio tells you whether or not you beat the market, the Treynor ratio gives you a sense of how much you underperformed or outperformed the benchmark index. This ratio is also known as the reward-to-volatility ratio.

**Tr = Re – Rf / βp**

Tr = Treynor Ratio

Re = Expected return of the portfolio being evaluated

Rf = Risk-free rate

βp = Beta of the portfolio being evaluated

Let’s use a real world example to see how the Treynor ratio works. The S&P 500 averaged 20% returns over the past 3 years while the T. Rowe Price Blue Chip Growth Fund averaged about 24.5% and has a beta of 1.15. Finally, we’ll use a risk free rate of 3%.

S&P 500 = 20% – 3% / 1 = 17%

T.Rowe Fund = 24.5% – 3% / 1.15 = 18.7%

The results show that the Blue Chip Growth Fund has successfully beaten the market using the Treynor ratio. (http://finance.yahoo.com/q/rk?s=TRBCX+Risk)

**Jensen’s Alpha**

This is the most common ratio used to determine alpha. This is the figure that investors use to determine what a portfolio actually returned versus what it was expected to return. This figure can tell you the performance of a fund management team and let you know who’s actually beating the market and who’s not.

**αp = Re – [Rf + (Rm – Rf)βp]**

αp = Jensen’s Alpha

Re= Expected return of portfolio being evaluated

Rf = Risk-free rate

Rm = Expected return of market portfolio

βp = Beta of portfolio being evaluated

Like the other ratios on this list, Jensen’s alpha needs to be compared to a benchmark in order to give the results context. The result is expressed as a percentage, so if you calculate alpha of 3%, then you know that the portfolio outperformed the market by 3%. Positive results are indicative of out-performance while negative results indicate under-performance.

**Application In The Real World**

Most mutual fund’s will actually list the funds risk ratios in comparison to a benchmark index so you don’t have to manually do the calculations yourself. Understanding what they mean can make it much easier to see what fund actually beats the market on a consistent basis and what fund continually under-performs the averages.

You can apply these technique to your own investment portfolio as well. If you’ve been investing for several years, you can calculate your standard deviation, beta, and average return fairly easily. Keeping track of your own performance will tell you how well you’re managing your own investments. If you find that you’re struggling to keep up with the market, you might consider trusting management to professionals.

Hedge funds are an interesting animal on Wall Street when discussing risk analysis and alpha. While many hedge funds align fairly well to the market and can be compared using the S&P 500, many more pride themselves on being uncorrelated to the market allowing for consistent returns no matter what the market does.

The Sharpe ratio is one of the most common ratios used to analyze hedge funds. All you’ll need is a historical reference of past returns and you can analyze the Sharpe ratio of any hedge fund and see how much risk it assumes.

Successful investors must take into account the kind of risk they’re taking on to accomplish their returns. Anyone can toss the dice and hope for the best, but tenured investors know that consistency is the key to a long and profitable investment experience. Just because you beat the market from a reurn standpoint, doesn’t mean you can rely on that method to last. Measuring risk using the Sharpe, Treynor, and Jensen’s alpha ratio will help you analyze your own performance as well as that of other professionals.

**Fariba RonnasiCEO, Elite Wealth Management**

*Full Disclosures: **http://elitewm.com/disclosures/*

*This article is not intended as investment advice. Elite Wealth Management or its subsidiaries may hold long or short positions in the companies mentioned through stocks, options or other securities.*