Risk and Rates of Return

The relationship between risk and return in complicated, but in general, there is a direct tradeoff. Total risk is defined as variability in returns and can be separated into systematic and unsystematic risk. An investor can reduce unsystematic risk by maintaining a diversified portfolio. This article is devoted to exploring the components of risk and discussing techniques for reducing, measuring and pricing risk and return.

Risk consists of uncertainty and the possible exposure to a negative result. The stock market is fraught with risks as well as rewards. Why would a stock investor be willing to forgo a risk-free investment such as a savings account that reliably earns 1-2 percent interest for a security that could lose all its value? A smart investor understands the tradeoff and how to estimate the return required in order to compensate for risk. Most investors are risk averse and would prefer a low-risk alternative. However, there is a certain return where the investor is willing to take on additional risk. The riskier the investment, the higher the expected return must be. The following diagram illustrates the pattern of the risk-return trade-off.

There are five critical components to understanding risk and return:

• Defining risk;
• Measuring systematic risk and return;
• Measuring unsystematic risk and return;
• Reducing risk; and return;
• Pricing risk and return.

Defining Risk

Risk comes in many shapes and sizes. There are countless risks to consider when assessing an investment. The following are examples of some common risks:

• Legal risk is the possibility that a company will be subject to unexpected litigation.
• Industry risk includes such factors as changing technology, foreign competitors, etc.
• Operational risk is the potential for internal processes or systems to fail.
• Currency risk is an issue for companies that do business in other countries. They are subject to exchange rates that could unexpectedly fluctuate and affect profits.
• Liquidity risk is the chance that no one will buy a security that needs to be sold. This is different from the price per share dropping to zero. Generally publicly traded stocks have low liquidity risks. Shares of private companies or assets such as houses can have high liquidity risks.

The proceeding risks are just a few of the myriad risk factors that can affect a security. There are certain risks that are particularly important when discussing the risk of a fixed-income investment such as a bond. Interest rate risk, inflation risk, and credit risk can significantly affect the return on a bond:

• Interest rate risk is the possibility that interest rates will change before you sell a bond;
• Inflation risk refers to unexpected changes in inflation; and
• Credit risk is the potential that the issuer will default on its debt obligation. If a bond is defaulted, an investor can lose interest as well as principle.

As a rule, bond prices fall when interest rates and inflation rise. Bonds are acutely sensitive because they have coupon payments that are fixed into the future and do not adjust for changes in interest rates or inflation. Therefore, when interest rates and inflation rise, the present value of the bond is worth less than its purchase price. In general, short-term bonds carry a lower interest rate and less inflation risk. They have a lower risk premium because the duration of time is brief. Credit risk is another important factor when choosing a bond. Credit ratings, which can help assess risk, are published by companies such as Moody's and Standard and Poor's. Interest coverage ratios (EBIT/interest expense) can also provide insight into credit risk. If a company's earnings do not have enough cushion to cover interest expense, then there is a high risk of default.

Stock prices fluctuate for a variety of reasons, but here we will focus on placing risk into two very distinct categories. Understanding the differences between these two categories is the basis for financial evaluation of risk. Some stock price fluctuations are due to factors that affect many securities in the market. For example, an economic downturn can cause many companies to experience unexpected losses. These kinds of risks are very different from random factors that affect only one firm. For example, a union strike at an automotive firm would only affect that specific company. The financial community has defined two categories of risk: systematic or market risk, and unsystematic or firm-specific risk.

Systematic risk includes economic and market factors that affect almost every company in the market. This risk cannot be eliminated by diversification. Interest rates and inflation are two kinds of risk that would fall into this category. Other examples of systematic risk include the country going to war or tax cuts approved by Congress.

Unsystematic risks are factors that only affect individual securities. Legal risk, industry risk, operational risk, currency risk, and liquidity risk fall into this category. Other examples of unsystematic risk include a fire in the main warehouse of a company, the CEO getting killed in an auto accident or a low-cost competitor entering the market. If you own only one stock, this kind of risk is very important. However, once you diversify your portfolio with several stocks, this unsystematic risk can be virtually eliminated. In financial markets, investors are only rewarded for bearing systematic risk, because this is the only kind of risk that cannot be eliminated through diversification.

Reducing Risk

In 1952, Harry Markowitz introduced an investment strategy called "portfolio diversification." He demonstrated how an investor can reduce risk and the overall standard deviation, or spread, of returns by creating a portfolio of securities.

If two securities are positively correlated (i.e., move together when the market changes), there is no impact on risk. However, if two securities are negatively correlated (i.e., securities do not move together), the portfolio is considered diversified and risk is reduced. Gains from one security in the portfolio can offset losses from another, lessening the overall exposure to a negative return.

Portfolio diversification can only mitigate risk associated with unsystematic risk. As you increase the number of securities in your portfolio, unsystematic risk can be virtually eliminated. However, you cannot diversify away systematic risk because, by definition, it affects all companies. Investors are only compensated for systematic risk because unsystematic risk is expected to be diversified away.

In portfolios of thirty-plus randomly selected stock, unsystematic risk is virtually eliminated.

Measuring Unsystematic Risk & Return

In order to measure risk, we must first understand rates of return. Suppose you invested $1,000 in a security and then sold it for $1,200. What was your rate of return? The past rate of return can be measure by the equation:

Amt. Received -- Amt. Invested/Amt Invested = Rate of Return

$1,200 -- $1,000/$1000 = 12%

Your next question might be what the return will be if you reinvest that money in a new security. To determine future expectations, we can calculate the expected value of a security. This is derived from the mean value of the probability distribution of possible returns. A probability distribution simply lists the potential returns and their associated probabilities. To calculate the expected return, we determine the weighted average of all the possible returns (Pr), where the weights are the probabilities (K) associated with each return and there are n possible outcomes.

Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.

In practice, investors usually focus on three scenarios: a worst-case scenario, an expected-case scenario and a best-case scenario. Suppose a particular security A has a 10% chance of a-10% return (worst-case scenario), 75% chance of 5% return (expected-case scenario) and 15% chance of 15% return (best-case scenario). The expected return is:

0.1(-0.1) + 0.75(0.05) + 0.15(0.15) = 5%

Expected return alone cannot tell us which stock to invest in. Two securities may have the same expected return, but one may have a very wide spread, or variability of returns. A wide spread indicates the possibility of some very negative returns and is therefore more risky. Take the following two examples. Both investment A and investment B have the same expected return, but A has a higher spread or standard deviation. Investors would generally prefer investment B.

Standard deviation (s) is a tool that can help us evaluate the difference between investment A and investment B. The standard deviation measures the variability of the outcomes expressed in a probability distribution. Greater standard deviation means greater uncertainty and, therefore, greater risk. Standard deviation is expressed as the weighted probability average difference between an investment's possible returns and its expected return. Standard deviation is calculated using the formula:

Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.

Standard deviation is criticized because it scales in such a way that does not allow for direct comparison of more than one standard deviation. For example, take investment A with a 25% chance of a -5% return (worst-case scenario), 50% chance of 2% return (expected-case scenario), 25% chance of 9% return (best-case scenario), and an expected return of 2%. Compare this to investment B with a 25% chance of a -10% return (worst-case scenario), 50% chance of 4% return (expected-case scenario), 25% chance of 18% return (best-case scenario), and an expected return of 4%.

σ of A = [(0.25*(-0.05-0.02)^2)+(0.5*(0.02-0.02)^2)+(0.25*(0.09-0.02)^2)]^(1/2) = 4.9%

σ of B = [(0.25*(-0.1-0.04)^2)+(0.5*(0.04-0.04)^2)+(0.25*(0. 18-0.04)^2)]^(1/2) = 9.9%

Is security B really two times more risky than security A? No, standard deviations cannot be used for this kind of direct comparison.

The coefficient of variation (CV) eliminates the scale problem by dividing standard deviation by expected return. Essentially, it gives us risk per unit of return.

Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.

CV of security A = .049/.02 = 2.47

CV of security B = .099/.04 = 2.47

Using the CV method illustrates how the securities are actually equal in their riskiness.

Measuring unsystematic risk is important when you own just one security. However, since systematic risk can be virtually eliminated through diversification, we next need to focus on measuring systematic risk.

Measuring Systematic Risk & Return

Systematic risk remains even after diversification. To measure systematic risk, we look at how much a security moves when the market moves. This is called covariance. If a company's returns are more sensitive to changes in the market, then its risk is escalated. A company who moves in exact parallel is exactly as risky as the market. In the case that a company amplifies the markets returns, then the security is considered more risky. When a company moves in the same direction as the market, but not as much, it is less risky. Finally, a company that moves in the exact opposite direction of the market is called a hedge.

We use beta (or β) to measure this movement with the market. A beta is calculated by using a statistical tool called linear regression to measure how much the return on a security covaries with the market return. A beta of 1 indicates average risk. The S&P 500 tends to rise and fall with the same percentages of the market, and it has a beta close to 1. A beta greater than 1 indicates a security that has high sensitivity to market swings and is more volatile/risky. A security that is not very sensitive to market changes will have a beta under 1 and is considered less risky. A beta is a critical component in pricing risk in the capital asset pricing model, which will be described next.

Pricing Risk & Return

As stated previously, most investors are risk averse and expect to be compensated for taking on risk. We know that unsystematic risk can be mitigated through diversification, so we just need a model for pricing systematic risk. This is where the capital asset pricing model (CAPM) comes into play. CAPM is the most widely accepted model for pricing risk. This model assumes the securities markets are competitive and efficient. It also assumes that these markets are composed of rational, risk-averse investors whose goal is to maximize returns.

CAPM considers a linear relationship between required rate of return and risk. The required rate of return (Kj) is equal to the risk-free rate (Krf) plus some risk premium:

K_j = K_rf + Risk Premium

It prices risk using a security market line (SML). Theoretically, every security should fall on the SML. If the security is above the line, that indicates that the security is underpriced. If the security falls below the line, it is overpriced.

The formula is expressed as:

Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.

Kj indicates the required rate of return for an investor. Krf is the risk-free rate. A three-month treasury bond is a good approximation for Krf because these bonds are backed by the United States government and have a β close to 0. Some researchers in the financial community believe the duration of the risk-free security should match the duration of the cash flows. This logic would suggest using long-term treasury bonds. However, for our purposes, we will stick with a three-month treasure bond., βj is the beta of the security. As discussed previously, the beta measures how sensitive the stock is to changes in the overall market. Km is the return on market index. The S&P 500 index (β close to 1) is a good approximation of the market return.

Suppose a three-month treasury bond rate is 5%, the return on the S&P 500 index is 10%, and Company X has a beta of 1.5. According to CAPM, the required rate of return is

(0.05) + 1.5(0.1-0.05) = 12.5%

There has been widespread criticism of CAPM. Many in the financial community believe additional risk factors should be taken into account. In addition, CAPM is based on expectations, and the beta is derived from historical financial data. A company's historical data might not reflect investors' expectations about future risk. Some researchers point to the dot-com bubble of the late 1990s as a period when investors lost sight of the risk-return tradeoff. Valuations were inflated, and the high-tech stocks of the late 1990s fell way below the SML. The market correction brought these overpriced securities back in line.

As a result of the doubts raised by CAPM, alternate theories have surfaced. Steven Ross's arbitrage pricing theory (ATP) is one that warrants discussion. The risk-free rate is the starting point for this model as well. However, ATP breaks systematic risk, or the risk premium part of the CAPM equation, into several factors. It assumes some securities will be more sensitive to certain factors than other securities. For example, interest rates will affect some firms more than others. A mortgage lending company would be more sensitive to interest rate changes than a health care company, for example. Factors can include inflation, industrial activity, and interest rate spreads, among others.

K_j = K_rf + b₁(K_{factor 1}-K_rf)+ b₂ (K_{factor 2}-K_rf +…

Where

Kj indicates the required rate of return

Krf is the risk-free rate

b1 is sensitivity to factor 1

Kfactor 1 is the return on factor 1

This model may provide a more thorough analysis of risk; however, it is much harder to apply. ATP requires several layers of complex statistics and requires investors to identify each factor to which the security is sensitive. ATP has more statistical noise inherent in its model. CAPM appeals to many investors because it is simpler and more theoretically appealing. Both CAPM and ATP agree on the basic concepts that (a) investors require compensation for taking on additional risk and (b) are concerned with risk that cannot be eliminated through diversification.

Conclusion

The primary objective of investors is to capture the highest possible return on their investment. Ideally, investors will maximize their returns while minimizing risk. Portfolio diversification allows investors to minimize a portion of overall risk. This article has illustrated how investors are compensated with higher rates of return for taking on risk that cannot be eliminated through diversification. If investors are unwilling to accept risk, then they should expect a lower return. This risk-return trade-off has been translated into models for pricing risk.

In practice, each investor has a different threshold for risk. Investors tend to fall on a spectrum of risk aversion. On one side, there is the aggressive investor. The portfolio for an aggressive investor has a higher mix of equity securities (rather than debt securities) and looks to maximize growth returns (rather than investment income). The time horizon for these investments is generally long term (e.g., five or more years). The aggressive investor is not concerned with short-term volatility and tends to have a good understanding of the financial markets. On the other end of the spectrum is the conservative investor. The most conservative investments are those where the return is known prior to making an investment (e.g., savings accounts, bonds, other debt securities). These types of investments provide for some protection of principle with set income. The conservative investor forgoes higher returns for security.

Due to the shortcomings of the CAPM model as well as alternative pricing models, investors and financial managers should not rely on these theories alone to price risk. Nothing can replace the sound judgment and vigilance of an educated investor.

Terms & Concepts

Arbitrage Pricing Theory (APT): An alternative to the capital asset pricing model that breaks risk into multiple factors. The risk premium is calculated for each factor that affects the company.

Beta: Measure of a security's market risk determined by how a security's returns vary with market conditions. Denoted by symbol β.

Capital Asset Pricing Model (CAPM): Model of the relationship between expected risk and expected return, where the return on a security is equal to the risk-free rate plus a risk premium.

Coefficient of Variation: Measures the relative spread of a security by dividing the standard deviation by the expected return.

Correlation Coefficient: Measure of the extent that two variables move together.

Diversification: Investment strategy designed to reduce risk by acquiring several securities; also called "portfolio diversification."

Expected Return: Weighted average of possible returns of a security, where weights are probabilities associated with each return.

Investment Risk: Relates to the potential to earn a low or negative return on an investment.

Probability Distribution: Listing of probabilities and their associated outcomes.

Security Market Line (SML): Gives the expected rate of return of a security based on its risk.

Standard Deviation: Measure of the variability of outcomes expressed in a probability distribution. Denoted by symbol s.

Systematic Risk: Economic and market factors that affect most firms and that cannot be eliminated through diversification; also called "market risk."

Risk Aversion: Natural human dislike of taking risks and tendency to avoid additional risk.

Risk-Free Rate: Yield on risk-free investment. A three-month treasury bill can be used as an approximation.

Risk Premium: Reflects systematic risk as measured by beta.

Risk-Return Trade-off: Risk-averse investors require higher rates of return to induce them to invest in higher risk securities.

Unsystematic Risk: Risk associated with random events that affect specific securities. This risk can be virtually eliminated through diversification. Also called "firm-specific risk."

Bibliography

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Brigham, E. F., & Houston, J. F. (2003). Fundamentals of Financial Management. Cincinnati, OH: South-Western College Publishing.

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The capital asset pricing model. (1998). New England Economic Review, 44-45. Retrieved April 1, 2007, from EBSCO Online Database Academic Search Premier. http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=1293510&sit=ehost-live

Risk and return. (1991). Economist, 318(7692), 72-74. Retrieved April 1, 2007, from EBSCO Online Database Academic Search Premier. http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=11715833&site=ehost-live

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Suggested Reading

Cochrane. (1999). New facts in finance. Economic Perspectives 23 (3), 36-59. Retrieved April 1, 2007, from EBSCO Online Database Academic Search Premier. http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=2267526&site=ehost-live

Daníelsson, J. (2011). Financial risk forecasting: The theory and practice of forecasting market risk, with implementation in R and Matlab. Chichester, England: John Wiley. Retrieved November 19, 2013 from EBSCO online database eBook Academic Collection (EBSCOhost). http://search.ebscohost.com/login.aspx?direct=true&db=e000xna&AN=391323&site=ehost-live

Fama, E. F., & French, K. R. (2004). The capital asset pricing model: Theory and evidence. Journal of Economic Perspectives, 18 (3), 25-46. Retrieved April 1, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=14774675&site=ehost-live

Jagannathan, R., & McGrattan, E. R. (1995). The CAPM debate. Quarterly Review, 19(4), 2-18. Retrieved April 1, 2007, from EBSCO Online Database Academic Search Premier. http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=9602011014&site=ehost-live

Treynor, J. L. (1993). In defense of the CAPM. Financial Analysts Journal, 49(3), 11-13. Retrieved on April 1, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=6936586&site=ehost-live