Unit 6: Risk and Return

Returns

Typically, the return on stocks has two components: dividends and capital gains. As a holder of some stocks, you own a part of the company and you have the right to benefit from the profitability of the company. The cash that you receive arising from the ownership is called a dividend. Even during periods where the company loses money, the companies may strategically decide to distribute dividends (as long as they have cash and sufficient retained earnings from the past) in order to satisfy the expectations of their shareholders and manage their position in the market. In addition to dividends, you will also have capital gains (losses) if the market price of your stocks has increased (decreased) since you purchased them.

Calculative Returns

The dividends represent the income component of your returns whereas capital gain/loss comes from changes in the market price of the shares. Note that the capital gain/loss is used for return calculations regardless of whether the shares are actually sold at that price (in which case the gain/loss is said to be realized) or not (unrealized gain/loss).

Percentage return is nothing but dividend yield plus capital gain yield, where dividend yield represents the percentage of income return.

In historical returns, in order to evaluate the performance of different investment alternatives over time we need to use some statistics that will summarize the data. We will use the arithmetic average (mean) to measure the expected value and the standard deviation to represent the volatility or dispersion around that expected value (footnote: some sources prefer to use the geometric average; however, we will use the more common arithmetic average in this book). e average return is calculated by summing the return over the years and then dividing by the number of years, T. Another important statistic is the standard deviation of returns that we will use to measure the volatility. e standard deviation of a series represents the dispersion or spread of the observations around the mean. A large standard deviation means that the observations tend to be widely dispersed around the mean and can easily change drastically from one period to the next. As a result, standard deviation is often used as a measure of risk for a given distribution. It is important to note that there can be other ways to calculate the dispersion or risk in returns as well, however, standard deviation (?) or variance (?2), of returns serves as a simple and basic risk measure. The equations for these calculations can be seen on page 159.

Risk-free return is the rate of return that investors require to invest in risk free investments in that environment. e short term T-Bill rate is usually used to measure it. e Risk premium is the return in excess of the risk-free rate that investors require to compensate for the risk of an investment.

The Efficiency of Financial Markets

After people have observed that prices in financial markets are almost constantly changing, many of them had questions about whether these markets were efficient. Our general statement of the Efficient Markets Hypothesis (EMH) is given below. In an (informationally) efficient market, prices fully reflect all available information. In such a market, it should not be possible for market participants to consistently earn excess returns, beyond the return appropriate to investments of that level of risk. Two important parts of the EMH definition that we should pay more attention to are “fully reflect” and “all available information”. The level of efficiency across different markets varies greatly, however, for developed markets the findings tend to indicate that markets tend to be generally weak form and even roughly semistrong form efficient. However, findings have shown that it is possible to find private information that would provide excess returns, yet not as frequently as imagined.

Risk and Return

When deciding on an investment, we should take into account not only their expected returns and risk levels, but also possible interactions/ co-movements with other securities. One way of estimating the expected return of a security that we have seen is by calculating the average return over some past period. However, other approaches that are more forward looking may also be used. Furthermore, we may also be interested in the comovement of certain securities that can be measured with covariance and correlation. Especially during certain periods such as announcements affecting a specific industry, the returns of stocks in the corresponding industry may show similar trends.

Return and Risk of Individual Securities

Suppose that there are three states of economy: depression, normal, boom with equally likely probability to happen. Considering two companies from the market, Compatible is the one growing with the general economic conditions whereas Divergent’s business is independent from macroeconomic conditions. Expected return of each company would be the average of different possibilities. e variance would be the weighted average of squared distances, and standard deviation would be the square root of the variance. The standard deviation is more useful for interpretation purposes as it is in the same units as the original and average values. Equations for these calculations can be seen on page 163 and 164.

Covariances and Correlations

e covariance between Compatible and Divergent is a measure of whether the returns tend to move in the same direction, in other words, do they tend to be above and below average at the same time or at different times? The first step in the calculation is to calculate their differences from their relative means, and multiply them with each other for each state of the economy. Second, taking a weighted average of these terms gives us the covariance. When both returns tend to be above or below their relative averages at the same time, then the covariance becomes positive, and negative when they tend to diverge. Equation for this calculation can be seen on page 164.

The product term will be:

(i) Positive , whenever both Compatible and Divergent returns are above or below the average at the same time (i.e., + times +, or – times -),

(ii) Negative , whenever Compatible and Divergent returns di er in being above or below their relative averages in the same period (i.e., + and -, or - and +).

Correlation is simply the covariance that has been standardized to be between -1 to +1, as a result, the sign of correlation is also determined by whether the covariance is positive or negative. A positive correlation means that the investments will tend to have high or low returns at the same time.

Since covariance is standardized by dividing by the standard deviations, the correlation always lies between +1 (perfect positive correlation) and -1 (perfect negative correlation). Zero correlation would mean there is no relation at all. Another important feature of this standardization is that the magnitude can be comparable: the values of correlation between X and Y, and correlation between Y and Z, can be compared to determine which one is more related.

Return and Risk of Portfolios

The aim of investors is to increase returns and decrease the risk. However, as we have seen, this is not easy as risk and return are not independent of each other. In order to reach this aim, the investor should consider:

(i) Expected return of individual securities, and how each would contribute to the portfolio return,

(ii) Standard deviation of individual securities, correlation between securities, and how these would impact the standard deviation of the portfolio.

Expected return of a portfolio is simply the weighted average of the returns of securities. The variance of a portfolio is a function of the individual security return variances and of the covariance between them. As can be observed from the formula between. There is a positive relationship with both the variances and covariance. Equations for these calculations can be seen on page 166.

Effect of Diversification on a Portfolio

In case returns are perfectly positively correlated the standard deviation of portfolio is just the weighted average of risk levels of single securities, and hence there is no benefit from diversification. As long as is ? less than 1, there will always be a diversification benefit to investing in a portfolio. Remembering that the correlation lies between -1 and +1, the benefit would be eliminated at +1, whereas the maximum diversification benefit would be reached at -1. The results are the same for portfolios with more than two securities; as long as correlations between pairs of securities are less than 1, the investor would always have a portfolio with a lower standard deviation compared to the weighted average of the securities.

Efficient Set for Two Securities

For a risk-averse investor as higher expected return is desirable (a good) while higher standard deviation is undesirable (a bad) the indifference curves in this case would be as shown in the figure and the investment decision would be made at the point that the highest indifference curve is tangent to the feasible set.

Efficient Set for Multiple Securities

In real life investors are faced with many securities, however, the fundamentals of what we have learned from two securities are roughly generalizable to this environment as well.

The expected return for a portfolio composed of N assets would be calculated as a weighted average of each asset’s return. Expected return and Variance of an N asset portfolio can be calculated with the equations given on page 171.

Expected and Unexpected Returns, Systematic and Unsystematic Risk

Unexpected returns are the result of new unexpected information arriving to the market during the period which changes some of the factors we believe are pertinent to the valuation of a firm. In any given period, unexpected returns may be negative or positive, however, on average they should be zero. is implies that actual returns should be equal to expected returns on average, otherwise there would be a bias in expectations that should not be able to persist in a market with rational participants. The risky part of returns (unexpected returns) can be broken down into a systematic risk and unsystematic risk components. Both of these risks are the result of new unexpected information arriving to the market and being incorporated into prices. The difference is that, some of these risks are relevant to all (or almost all) securities while others are relevant only to one security (or a limited, small group of securities). The importance of the separation of risks into systematic and unsystematic is that unsystematic risks may be eliminated at little or no cost with proper diversification. As unsystematic risk can be eliminated as shown below, then there should be no compensation for taking on this type of risk. Market, portfolio or systematic risk is the uncertainty inherent to the market that cannot be controllable. On the other hand, the diversifiable, unique or unsystematic risk is the uncertainty that is related to the invested asset. Therefore, unsystematic risk can be reduced through diversification whereas systematic risk cannot.

Combining Risky Securities with Risk-Free Assets

Investors may want to combine a risky security with some lower risk or even risk-free securities to change the characteristics of the portfolio. Recall that the risk-free return is the return on securities that always yield their expected returns, regardless of the economic environment. The risk-free return is usually approximated with the returns from very short-term government securities.

Efficient Set with Multiple Securities and Risk-Free Asset

The Capital Market Line (CML) is the line that connects the risk-free asset with the market portfolio, where the line is just tangent to the efficient frontier on an expected return/ standard deviation graph. e CML depicts the tradeoff between risk and return for diversified (efficient) portfolios. relationship between the expected returns of efficient portfolios subject to standard deviation of a market portfolio and risk-free rate. The CML equation is given on page 176.

Measuring Risk of a Security: Beta

In order to describe the risk-return relationship for individual securities, we need a risk measure, which measures the relevant (systematic) risk as opposed to the total risk measured by the standard deviation. The relevant risk measure that is used is called the beta coefficient (or just beta for short) and measures the comovement of a securities returns with the market return, which is assumed to have average risk. Beta is estimated by standardizing the covariance between the security “i” and market portfolio with the variance of the market with the equation given on page 176. In addition to being able to measure only the relevant (systematic) risk of securities, beta has another very nice property. The beta of a portfolio of securities is simply a weighted average of their individual betas. So, for a portfolio of N securities, the portfolio beta would be calculated in a similar way to calculating the portfolio expected returns.

The Security Market Line (SML) and Capital Asset Pricing Model (CAPM)

The SML relationship, also referred to as the CAPM, should hold for both individual securities (including riskfree) and portfolios of securities. On SML, we can denote not only portfolios, but also single securities as well. Besides, SML is the relationship between expected returns and beta, not standard deviation. In this regard, CML deals with portfolio risk whereas SML deals with systematic risk.

Capital-Asset-Pricing Model is a model describing the relationship between the systematic risk of a security, namely beta, and expected returns. As long as the market risk premium is positive, higher beta value bring higher returns.

The difference between Security Market Line and Capital Market Line is that: on CML we are considering a portfolio that is composed of a risky assets plus a risk-free asset, and all the efficient points on this line denotes the expected return and standard deviation of this combination.