What is Risk?

Risk refers to the possibility of an unfavorable event occurring. The higher the risk, the greater the probability of an unfavorable event or the more unfavorable the event could be. The interaction of the probability of the unfavorable event and the degree of negativity associated with the event is critical to determining the risk. For instance, imagine that you are going to participate in a coin flip. It will cost you $1.00 to participate. If you flip a head, you get $1.01. If you flip a tail you get $0.99. Even though there is a fairly high probability of the unfavorable event (50% chance of tails), the outcome is so minor (you lose $0.01) that this would be a low risk event. Now, consider a slightly different coin flip. This time, instead of flipping the coin once, you will flip it 3 times. If you get 3 heads, you receive $10,000. If you flip 3 tails, you owe $10,000. Anything else you get your $1.00 back. Even though the probability of the bad outcome (there is a 12.5% chance of flipping 3 straight tails) is much smaller, this is a much riskier event due to the bad outcome being substantially worse.

In finance terms, our "unfavorable event" refers to earning less than expected. Any time we have a chance to earn less than expected on an investment opportunity we are exposed to risk. Note that this is a more strict definition than defining risk as the possibility of losing money. We want to be careful to think of risk as the possibility of earning less than expected instead of as the possibility of losing money.

Sources of Risk

• General Economic Conditions
  • Inflation
  • Recession
  • Interest Rates
• Company Specific Factors
  • Business Risk Factors
    • Volatility of Sales
    • Volatility of Input Prices
    • Strikes and other Labor Situations
    • Product Lines
    • Fixed vs Variable Costs
  • Financial Risk Factors
    • Degree of Debt Financing

Note that the above list is a sample of broad factors and not a specific list. For example, consider what happens when we have a large increase in oil and gasoline prices. One immediate impact is inflation. The higher energy prices are, by definition, inflation in energy, but it goes beyond that. Now it costs more for firms to distribute their products to suppliers which is likely to cause the inflation to spill over to other areas. As we search for alternative energy sources (like ethanol) we see corn prices rising. As corn is used to feed cattle, this means a likely increase in beef costs as well. Also, consumers are now spending more to fill up their gas tanks and more to buy a variety of food products so there is less money available to spend on entertainment and other goods/services. This could lead to a recession environment (I say could not will because there are always so many influences on the economy that this is just one of many factors impacting economic growth).

Diversification
Diversification refers to the concept that by holding a number of different securities from a spectrum of industries, we can negate the impact of company specific factors on our returns. We will come back to this issue (one of the most important concepts in finance) in more detail later in this chapter.

Expected Return and Standard Deviation of a Single Security

Expected Return
The expected return of a security is based on the probability distribution of returns. Specifically, the expected return is the probability of a specific state of nature occurring times the return under that state of nature summed across all possible states of nature. In formula terms, it is

• kbar represents the expected return of the stock
• P1 represents the probability of the first possible outcome (state of nature)
• k1 represents the return under the first outcome (state of nature)
• Pn represents the probability of the nth possible outcome (state of nature)
• kn represents the return under the nth outcome (state of nature)

Don't worry if the formula and definition seem intimidating, the process is relatively simple. Consider the following example. After researching Stock A we have determined that there are 3 possible outcomes for the next year (3 states of nature). The first possibility is the economy enters a recession causing the stock to have a return of -15%. The probability of this occurring is 20%. The second possibility is that the economy goes smoothly, but does not experience rapid growth causing the stock to rise offer a 10% return. The probability of this occurring is 50%. The third possibility is that the economy booms, causing the stock to provide a 35% rate of return. The probability of the economy booming is 30% (note that the probabilities must sum to 1.0).

What is the expected rate of return?

kbar = (.20)(-15%) + (.50)(10%) + (.30)(35%) =
kbar = -3% + 5% + 10.5% = 12.5%

Standard Deviation
The standard deviation measures the variability of possible returns and is represented by the Greek symbol sigma. The smaller the standard deviation, the more likely we are going to earn something "close" to our expected return. The greater the standard deviation, the greater the chance that we may earn something far more (good) or far less (bad) than our expected return. The formula for this is (remember that kbar is our symbol for expected return):

• sigma represents the standard deviation
• P1 represents the probability of the first outcome (state of nature)
• k1 represents the return under the first outcome (state of nature)
• kbar represent the expected return for the stock
• Pn represents the probability of the nth outcome (state of nature)
• kn represents the return under the nth outcome (state of nature)

Calculation Notes:

• It is easy to get confused with decimals and percents. The best way to do these calculations is to always leave the weights as decimals and the returns as a regular number. For instance, if you have a probability of 0.10 and a return of 15%, you would put the probability into your calculator as 0.10 and the return as 15.
• Be careful with your order of operations.
  • Do (k1 - kbar) first
  • Then square that
  • Then multiply by P1
  • Repeat for all n states of nature
  • Add them up
  • Finally, take the square root

Consider our previous example. What is the standard deviation for stock A?

Interpreting Expected Return and Standard Deviation

Expected return gives us an idea of how much we will make on the investment. Remember that it is really how much we WILL make, but how much we would make ON AVERAGE if we could repeat the holding period an infinite number of times. Think of a situation where you are asked to pick a number between 1 and 10. If you select the correct number you get $100 and if not you get nothing. Any one time that you try this, you will either receive $100 (if you are lucky) or $0. However, if you could repeat the exercise 100,000 times, you would find that you would make almost exactly $10 per time. It is critical to know expected values when selecting investments. For instance, if I told you I would sell you the opportunity to do this exercise for $9 per pick and you could pick as often as you wanted it would be an excellent opportunity. If I offered the same thing for $11, you would (hopefully) walk away. However, it is just as important to understand that the expected return is only an average return and not the return we WILL receive in any particular instance.

Now consider a similar exercise -- pick a number between 1 and 5. If you select the correct number, you get $50 and if not you get nothing. The expected value for this exercise is the same as the previous exercise ($10). So, imagine that I offered you the opportunity to participate in either one (pick from 10 numbers or pick from 5 numbers) for $8. You can only play once. Most people will now choose to pick from 5 numbers. Why? Because it has less risk (a lower standard deviation) and offers the same expected return. This is the concept of risk aversion. As a side note, if you still picked the 10-number game it is likely because the stakes are small (the entertainment value of the gamble outweighs the financial aspect). As the stakes increase, the vast majority of people will choose the 5-number game.

Moving away from our example, let's put this in finance terms. Consider two stocks. Stock A has an expected return of 10% and a standard deviation of 25%. Stock B has an expected return of 10% and a standard deviation of 30%. Which should you choose and why? (Answer to follow ...think about it first)

Now consider two other stocks. Stock C has an expected return of 7% and a standard deviation of 20%. Stock D has an expected return of 9% and a standard deviation of 28%. Which should you choose and why? (Again, spend some time thinking before reading the next paragraph)

In the first example, you should choose stock A and so should everyone else. Stock B offers us no additional compensation (expected return) to entice us to take the higher risk (standard deviation). Therefore, it is irrational in a risk-averse framework to invest in stock B. In the second example, you could choose stock C or stock D and someone else may make the same choice or the opposite choice. Here, the choice is based on your individual level of risk aversion. Stock D is riskier, but it also compensates us for that risk with a higher expected return. Is the compensation enough? That depends on the individual. For those that are less risk-averse, they require less additional compensation to take on the extra risk so they will likely take stock D. For those that are more risk averse, they will take stock C because the extra compensation is not enough for them to take the extra risk. Take a few moments and try to think of what factors impact YOUR level of risk aversion. One last thought -- remember that taking stock D does not mean you WILL earn a higher return, just that you will ON AVERAGE. If you always earned a higher return from stock D, then it wouldn't be riskier.

Expected Return and Standard Deviation of a Portfolio

Expected Return
The expected return for a portfolio is simply the weighted average of each stock held in the portfolio. The formula here is

• kbar represents the expected return for the portfolio
• W1 represents the weight (proportion of portfolio) of stock 1
• kbar1 represents the expected return for stock 1
• Wn represents the weight (proportion of portfolio) of stock n
• kbarn represents the expected return for stock n

Again, let's consider an example. What is the expected return of a portfolio made up of 60% Stock A and 40% Stock B when the expected return for Stock A is 10% and the expected return for Stock B is 20%?

kbar = (.60)(10%) + (.40)(20%) =
kbar = 6% + 8% = 14%

Standard Deviation
The standard deviation of a portfolio becomes more complicated. It depends not only on the standard deviation and weightings of each stock, but also on the correlation between pairs of stocks.

The CORRELATION between a pair of stocks measures how closely the returns for each stock are related. A negative correlation means that the price of one stock tends to fall while the other rises (prices/returns are inversely related). A positive correlation means that the price of one stock tends to rise while the other rises (prices/returns are positively related). Correlations can range from -1.0 to 1.0.

• sigma represents the standard deviation of the portfolio
• WA represents the weight (proportion of portfolio) of stock A
• sigmaA represent the standard deviation of stock A
• WB represents the weight (proportion of portfolio) of stock B
• CorrAB represents the correlation between the returns of stocks A and B

Again, lets work through an example. Consider a two-stock portfolio in which 60% of your money is invested in stock A and 40% of your money is invested in stock B. Stock A has a standard deviation of 50% and stock B has a standard deviation of 70%. The correlation between the returns for stock A and stock B are 0.30. You want to find the standard deviation of this portfolio.

Note that in this example, the standard deviation of the portfolio is LESS than the standard deviation of either stock separately. This illustrates the concept of diversification. As long as the correlation is less than 1.0 (which it will be for any two stocks), the risk of the portfolio is less than the average risk of the two securities which make up the portfolio (and sometimes -- like here -- even less than the lowest risk stock in the portfolio).

Diversifiable and Non-Diversifiable Risk

Remember earlier we discussed the possibility of lowering our firm-specific risk by holding a number of stocks from a wide range of industries? This concept of diversification allows us to greatly reduce our risk by holding a portfolio. Diversifiable risk refers to risk factors that are isolated towards one particular firm or industry. For example, if a drug manufacturer gets hit with a lawsuit related to one of the drugs it produces, that is likely to have an isolated impact. It will not have any effect on the stock prices of auto manufacturers, grocery retailers, banks, etc. Once we have enough stocks in our portfolio, bad news for any one of them will have a small impact on our overall portfolio as long as that bad news is contained to that one firm (in other words, as long as it comes from a diversifiable risk factor). By holding approximately 25-50 stocks, we can eliminate most of our diversifiable risk. While a portfolio of 10 stocks has much less risk than a portfolio of 5 stocks, a portfolio of 100 stocks offers very little risk reduction compared to a much smaller 50-stock portfolio.

Does this mean that there is we have eliminated all of our risk when we hold a 50-stock portfolio? No, we are still subject to general economic risk that affects all securities. This leftover risk is referred to as Non-diversifiable risk

Given the relative ease with which investors can virtually eliminate their firm-specific (diversifiable) risk, for most investors the level of non-diversifiable (market) risk associated with an investment becomes more important. It is important to note that while market risk impacts all stocks, it does not impact all stocks equally. Therefore, we need to a tool to measure how sensitive a stock is to the overall market. This tool is known as BETA. Standard deviation measures total risk (diversifiable risk + market risk) for a security, while beta measures the degree of market (non-diversifiable) risk. We don't have a risk measurement for diversifiable risk.

Beta

In addition to serving as a measure of market risk, Beta tells us how a particular stock moves in relation to the rest of the stock market as a whole.

• Beta of Stock A
• SigmaA represents the standard deviation of stock A
• Corr represents the correlation between stock A and the overall market
• SigmaMkt represents the standard deviation of the overall market

Consider the following example. Stock A has a standard deviation of 60% while the overall stock market has a standard deviation of 25%. Assuming that the correlation between Stock A and the overall market is 0.30, what is the beta of Stock A?

Beta = [(60)(.30)]/(25) = 18/25 = .72

What is the Market?
The market refers to a portfolio of all investment assets (stocks, bonds, gold , art, etc.). However, in more practical terms, the market usually refers to the stock market and can be measured by a market index (such as the S&P 500 or Dow Jones Industrial Average).

How do we Interpret Beta?

• Betas > 1.0 implies higher than average risk
• Betas = 1.0 implies average risk
• Betas < 1.0 imply less than average risk
• Most betas range between 0.4 and 1.8 (there are many outside this range, but the majority of stocks fall in this range)

Go back to our example where we calculated the beta for stock A. By itself, stock A is much riskier than the overall market as determined by its standard deviation. However, when we consider it as part of an overall portfolio its risk is much lower (less than average) due to the fact that it has a relatively low correlation to the overall market. The riskiness of stock A depends on whether we plan to use it as a stand alone investment or as part of a portfolio.

Standard Deviation vs Beta

At this point, we have introduced two risk measurements. The first is standard deviation and the second is beta. In some cases, these two risk measurements will tell a different story. For instance, stock A may have a standard deviation of 30% and a beta of 0.8 while stock B may have a standard deviation of 25% and a beta of 1.3. Which stock is riskier? The answer is depends on the specific situation. Because each risk measurement is measuring a different type of risk (standard deviation measures total risk while beta measures market risk), we need to think of situations where each is appropriate.

• Single Security and/or Poorly Diversified Portfolio -- If you are going to place your entire investment into a single security or a poorly diversified portfolio, then STANDARD DEVIATION is the appropriate risk measurement. In this situation, you have not diversified away the majority of the firm-specific risk, so you need to include it in your analysis. Standard deviation does this because it includes both sources (market and firm-specific) of risk.
• Adding a Security to a Well-Diversified Portfolio -- If you own a well-diversified portfolio and you are planning to add a single security to that portfolio, then the firm-specific risk of the security you are adding is not relevant. The reason it is not relevant is because it will be one of many stocks in the large portfolio and the firm-specific risk will be diversified away. What matters is how that stock moves with the overall market. Since we measure this market risk with beta, our appropriate risk measurement here is BETA.
• Choosing Between 2 (or more) Well-Diversified Portfolios -- If you are choosing between two or more portfolios that are each well-diversified, then you can use either STANDARD DEVIATION OR BETA as your risk measurement. The reason for this is that at this point, the firm-specific risk is already diversified away so that your total risk and market risk should be essentially the same. Thus, whichever portfolio has the higher standard deviation should also have the higher beta (if not, you know the portfolios are not well-diversified).

Beta and Required Return

Capital Asset Pricing Model (CAPM)
During the late-60's and early-70's some finance professors developed the Capital Asset Pricing Model (CAPM). One of the key components of this model is the Security Market Line (SML) which states that the required rate of return for a stock is dependent on the beta of that stock. While technically, the SML is a subset of the larger model (CAPM), in practice the two terms are typically used interchangeably. Thus, think of them as the same basic model. Specifically, the SML states that

• where ka is the required return for stock A,
• kRF is the risk-free rate of interest (often approximated by the yield on 10-year Treasury Bond),
• Ba is the beta for stock A,
• km is the expected return on the market (often approximated by the S&P 500).

Let's calculate the required return for a particular stock. Stock B has a beta of 1.4. The expected return on the market is 11% and the 3-month TBill rate is 5%. Based on this, we can calculate the required return for stock B as follows --

k = 5% + (1.4)(11% - 5%) =
k = 5% + (1.4)(6%) =
k = 5% + 8.4% = 13.4%

Important Implications of the CAPM/SML

• According to the SML, high beta stocks should, on average, earn higher returns than low beta stocks
• According to the SML, the only factor that should cause consistent differences in returns across stocks is beta.
• When interest rates rise, required returns should increase causing stock prices to decline (assuming all-else-equal)
• When investors become more risk-averse, the risk premium (km - kRF) should increase which will increase required returns and (all-else-equal) cause stock prices to decline.

Under equilibrium conditions, the required return should equal the expected return. Let's consider this for a minute. In our example above, we estimated that the required return for stock B should be 13.4%. What would happen if the expected return (kbar) for this stock was 16%?

• We could earn more than we need to compensate us for our risk by buying this stock, so we do so ==>
• Everyone else sees the same situation, so they are also buying stock B ==>
• Since everyone is trying to buy the same stock, the increased demand causes the price to increase ==>
• While the price is going up, the company itself is not changing. This means everyone is paying more for the same set of expected cash flows ==>
• Since we are paying more for the same set of cash flows, our expected return is declining ==>
• Once the expected return falls back in line to match the 13.4% required return, the price will stabilize and we will be back in equilibrium.

For practice in understanding this concept, think through what would happen if the required return was 13.4% and the expected return was 9%. Also, think about what types of things may push us out of equilibrium and why this is relevant for explaining stock price movements.

Empirical Findings of the SML
While the CAPM/SML met with a lot of early success and became the standard for estimating required returns in the field of finance very quickly, it ran into problems in the 1990's and is deemed less reliable at the moment. There have been some alternative models introduced since then, however there has not been a new standard-bearer to take its place. As of right now, the SML is still commonly used in practice, however there is growing focus on the alternatives. My view is that it is critical for you to know and understand the basic premise of the CAPM and SML (that role of market risk in explaining returns) but also to be aware of some of the problems (listed below).

• While market returns play a major role in explaining returns of individual stocks, Beta doesn't do a very good job of explaining future returns. In other words, after controlling for other factors, it is not clear that high beta stocks actually outperform
• Small firms seem to earn higher returns than can be explained by beta
• Firms with a low market-to-book ratio (MV/BV) tend to earn higher returns than can be explained by beta
• Firms that have been top performers in the past 6-12 months tend to earn higher returns in the following 6-12 months than can be explained by beta

Again, the above flaws do not mean that the CAPM/SML are useless. They provide a simplified framework for understanding how market risk relates to returns. However, recognize that they are not perfect models and the process of understanding stock returns is more complex than the SML indicates. As our knowledge increases, better models will likely evolve.