Risk and Return
Investors purchase financial assets such as shares of stock because they desire to increase their wealth, i.e., earn a positive rate of return on their investments. The future, however, is uncertain; investors do not know what rate of return their investments will realize.
In finance, we assume that individuals base their decisions on what they expect to happen and their assessment of how likely it is that what actually occurs will be close to what they expected to happen. When evaluating potential investments in financial assets, these two dimensions of the decision making process are called expected return and risk.
The concepts presented in this paper include the development of measures of expected return and risk on an indivdual financial asset and on a portfolio of financial assets, the principle of diversification, and the Captial Asset Pricing Model (CAPM).
Expected Return
The future is uncertain. Investors do not know with certainty whether the economy will be growing rapidly or be in recession. As such, they do not know what rate of return their investments will yield. Therefore, they base their decisions on their expectations concerning the future.
The expected rate of return on a stock represents the mean of a probability distribution of possible future returns on the stock. The table below provides a probability distribution for the returns on stocks A and B.
State / Probability / Return onStock A / Return on
Stock B
1 / 20% / 5% / 50%
2 / 30% / 10% / 30%
3 / 30% / 15% / 10%
3 / 20% / 20% / -10%
In this probability distribution, there are four possible states of the world one period into the future. For example, state 1 may correspond to a recession. A probability is assigned to each state. The probability reflects how likely it is that the state will occur. The sum of the probabilities must equal 100%, indicating that something must happen. The last two columns present the returns or outcomes for stocks A and B that will occur in the four states.
Given a probability distribution of returns, the expected return can be calculated using the following equation:
where
· E[R] = the expected return on the stock,
· N = the number of states,
· pi = the probability of state i, and
· Ri = the return on the stock in state i.
Expected Return on Stocks A and BStock A
Stock B
So we see that Stock B offers a higher expected return than Stock A. However, that is only part of the story; we haven't yet considered risk.
Measures of Risk - Variance and Standard Deviation
Risk reflects the chance that the actual return on an investment may be very different than the expected return. One way to measure risk is to calculate the variance and standard deviation of the distribution of returns.
Consider the probability distribution for the returns on stocks A and B provided below.
State / Probability / Return onStock A / Return on
Stock B
1 / 20% / 5% / 50%
2 / 30% / 10% / 30%
3 / 30% / 15% / 10%
3 / 20% / 20% / -10%
The expected returns on stocks A and B were calculated on the Expected Return page. The expected return on Stock A was found to be 12.5% and the expected return on Stock B was found to be 20%.
Given an asset's expected return, its variance can be calculated using the following equation:
where
· N = the number of states,
· pi = the probability of state i,
· Ri = the return on the stock in state i, and
· E[R] = the expected return on the stock.
The standard deviation is calculated as the positive square root of the variance.
Variance and Standard Deviation on Stocks A and BNote: E[RA] = 12.5% and E[RB] = 20%
Stock A
Stock B
Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's. This, however, is only part of the picture because most investors choose to hold securities as part of a diversified portfolio.
Portfolio Risk and Return
Most investors do not hold stocks in isolation. Instead, they choose to hold a portfolio of several stocks. When this is the case, a portion of an individual stock's risk can be eliminated, i.e., diversified away. This principle is presented on the Diversification page. First, the computation of the expected return, variance, and standard deviation of a portfolio must be illustrated.
Once again, we will be using the probability distribution for the returns on stocks A and B.
State / Probability / Return onStock A / Return on
Stock B
1 / 20% / 5% / 50%
2 / 30% / 10% / 30%
3 / 30% / 15% / 10%
3 / 20% / 20% / -10%
From the Expected Return and Measures of Risk pages we know that the expected return on Stock A is 12.5%, the expected return on Stock B is 20%, the variance on Stock A is .00263, the variance on Stock B is .04200, the standard deviation on Stock S is 5.12%, and the standard deviation on Stock B is 20.49%.
Portfolio Expected Return
The Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio. The weights reflect the proportion of the portfolio invested in the stocks. This can be expressed as follows:
where
· E[Rp] = the expected return on the portfolio,
· N = the number of stocks in the portfolio,
· wi = the proportion of the portfolio invested in stock i, and
· E[Ri] = the expected return on stock i.
For a portfolio consisting of two assets, the above equation can be expressed as
Expected Return on a Portfolio of Stocks A and BNote: E[RA] = 12.5% and E[RB] = 20%
Portfolio consisting of 50% Stock A and 50% Stock B
Portfolio consisting of 75% Stock A and 25% Stock B
Portfolio Variance and Standard Deviation
The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together. Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient.
The Covariance between the returns on two stocks can be calculated using the following equation:
where
· s12 = the covariance between the returns on stocks 1 and 2,
· N = the number of states,
· pi = the probability of state i,
· R1i = the return on stock 1 in state i,
· E[R1] = the expected return on stock 1,
· R2i = the return on stock 2 in state i, and
· E[R2] = the expected return on stock 2.
The Correlation Coefficient between the returns on two stocks can be calculated using the following equation:
where
· r12 = the correlation coefficient between the returns on stocks 1 and 2,
· s12 = the covariance between the returns on stocks 1 and 2,
· s1 = the standard deviation on stock 1, and
· s2 = the standard deviation on stock 2.
Covariance and Correlation Coefficent between the Returns on Stocks A and BNote: E[RA] = 12.5%, E[RB] = 20%, sA = 5.12%, and sB = 20.49%.
Using either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows:
The standard deviation on the porfolio equals the positive square root of the the variance.
Variance and Standard Deviation on a Portfolio of Stocks A and BNote: E[RA] = 12.5%, E[RB] = 20%, sA = 5.12%, sB = 20.49%, and rAB = -1.
Portfolio consisting of 50% Stock A and 50% Stock B
Portfolio consisting of 75% Stock A and 25% Stock B
Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A. This is the essence of Diversification, by forming portfolios some of the risk inherent in the individual stocks can be eliminated.
Capital Asset Pricing Model (CAPM)
Because investors are risk averse, they will choose to hold a portfolio of securities to take advantage of the benefits of Diversification. Therefore, when they are deciding whether or not to invest in a particular stock, they want to know how the stock will contribute to the risk and expected return of their portfolios.
The standard deviation of an individual stock does not indicate how that stock will contribute to the risk and return of a diversified portfolio. Thus, another measure of risk is needed; a measure of a security's systematic risk. This measure is provided by the Capital Asset Pricing Model (CAPM).
Systematic and Unsystematic RiskAn asset's total risk consists of both systematic and unsystematic risk.
Systematic risk, which is also called market risk or undiversifiable risk, is the portion of an asset's risk that cannot be eliminated via diversification. The systematic risk indicates how including a particular asset in a diversified portfolio will contribute to the riskiness of the portfolio
Unsystematic risk, which is also called firm-specific or diversifiable risk, is the portion of an asset's total risk that can be eliminated by including the security as part of a diversifiable portfolio.
The Capital Asset Pricing Model (CAPM) provides an expression which relates the expected return on an asset to its systematic risk. The relationship is known as the Security Market Line (SML) equation and the measure of systematic risk in the CAPM is called Beta.
The Security Market Line (SML)
The SML equation is expressed as follows:
where
· E[Ri] = the expected return on asset i,
· Rf = the risk-free rate,
· E[Rm] = the expected return on the market portfolio,
· bi = the Beta on asset i, and
· E[Rm] - Rf = the market risk premium.
The graph below depicts the SML. Note that the slope of the SML is equal to (E[Rm] - Rf) which is the market risk premium and that the SML intercepts the y-axis at the risk-free rate.
In capital market equilibrium, the required return on an asset must equal its expected return. Thus, the SML equation can also be used to determine an asset's required return given its Beta.
The Beta (Bi)
The beta for a stock is defined as follows:
where
· sim = the Covariance between the returns on asset i and the market portfolio and
· s2m = the Variance of the market portfolio.
Note that, by definition, the beta of the market portfolio equals 1 and the beta of the risk-free asset equals 0.
An asset's systematic risk, therefore, depends upon its covariance with the market portfolio. The market portfolio is the most diversified portfolio possible as it consists of every asset in the economy held according to its market portfolio weight.
Example Problems1. Find the expected return on a stock given that the risk-free rate is 6%, the expected return on the market portfolio is 12%, and the beta of the stock is 2.
2. Find the beta on a stock given that its expected return is 16%, the risk-free rate is 4%, and the expected return on the market portfolio is 12%.
Critical assumptions of CAPM
The CAPM is simple and elegant. Consider the many assumptions that underlie the model. Are they valid?
· Zero transaction costs. The CAPM assumes trading is costless so investments are priced to all fall on the capital market line. If not, some investments would hover below and above the line -- with transaction costs discouraging obvious swaps. But we know that many investments (such as acquiring a small business) involve significant transaction costs. Perhaps the capital market line is really a band whose width reflects trading costs.
· Zero taxes. The CAPM assumes investment trading is tax-free and returns are unaffected by taxes. Yet we know this to be false: (1) many investment transactions are subject to capital gains taxes, thus adding transaction costs; (2) taxes reduce expected returns for many investors, thus affecting their pricing of investments; (3) different returns (dividends versus capital gains, taxable versus tax-deferred) are taxed differently, thus inducing investors to choose portfolios with tax-favored assets; (4) different investors (individuals versus pension plans) are taxed differently, thus leading to different pricing of the same assets.
· Homogeneous investor expectations. The CAPM assumes invests have the same beliefs about expected returns and risks of available investments. But we know that there is massive trading of stocks and bonds by investors with different expectations. We also know that investors have different risk preferences. Again, it may be that the capital market line is a fuzzy amalgamation of many different investors' capital market lines.
· Available risk-free assets. The CAPM assumes the existence of zero-risk securities, of various maturities and sufficient quantities to allow for portfolio risk adjustments. But we know even Treasury bills have various risks: reinvestment risk -- investors may have investment horizons beyond the T-bill maturity date; inflation risk -- fixed returns may be devalued by future inflation; currency risk -- the purchasing power of fixed returns may diminish compared to that of other currencies. (Even if investors could sell assets short -- by selling an asset she does not own, and buying it back later, thus profiting from price declines -- this method of reducing portfolio risk has costs and assumes unlimited short-selling ability.)