RISK & RETURN
When we were pricing the share of stock, we estimated that next year’s dividend would be $1.15. In actuality, the dividend that will be paid next year is a range of possible outcomes:
Distribution / AProbability / Outcome
0.25 / * / $1.00 / = / $0.25
0.50 / * / $1.15 / = / $0.58
0.25 / * / $1.30 / = / $0.33
1.00 / E(V)= mean / = / $1.15
The estimated dividend of $1.15 represents a weighted average of the possible outcomes, where each possible outcome is weighted by its probability of occurrence. The weighted average is known as the mean of the distribution, or the expected value in the parlance of finance. Why must the probabilities sum to 1.00?
If the possible outcomes are plotted versus their respective probabilities of occurrence, the resulting graph appears as
This is known as a point distribution since it represents three points of possible outcomes. In reality, there are a large number of possible outcomes, each with a specific probability of occurrence. Rather than try to enumerate every possible outcome and its probability of occurrence, each point in this distribution really represents a range of possible outcomes. Thus, a more appropriate representation of the distribution might look as follows:
Even this is a simplification. The true distribution of possible outcomes is continuous, although it would be virtually impossible to specify every possible dividend that might be paid and the probability of each. Nonetheless, the true distribution appears as on the previous page.
Now consider a second distribution of possible outcomes for next year’s dividend:
Distribution / BProbability / Outcome
0.10 / * / $0.80 / = / $0.08
0.20 / * / $1.00 / = / $0.20
0.40 / * / $1.15 / = / $0.46
0.20 / * / $1.30 / = / $0.26
0.10 / * / $1.50 / = / $0.15
E(V)= mean / = / $1.15
The expected value of this distribution is also $1.15 for next year’s dividend. If we plot Distribution B along with Distribution A, we obtain the following graph:
Your “best guess” as to the actual dividend that will be paid is $1.15 for both distributions. If you could choose between the two distributions, which one would you prefer?
Most individuals would prefer Distribution A over Distribution B since there is less variability about the expected value with Distribution A. The variability about the expected value is how we define “risk” in finance.
How do is the risk of a distribution of possible outcomes measure? Some would be inclined to define risk as the range of possible outcomes, where Distribution A ranges from $1.00 to $1.30 while Distribution B ranges from $0.80 to $1.50 in possible outcomes. However, if we give Distribution A a one-millionth probability of the dividend only being $0.80 and a one-in-a-million chance of the dividend being $1.50 then the range of the two distributions is identical.
A better measure of the variability of a distribution is the standard deviation:
The standard deviation of Distribution A would be calculated as
or a little over ten and one-half cents. The standard deviation not only reflects the range of possible outcomes, but also there probabilities.
What is the interpretation of the standard deviation? Recall from statistics that, for a normal distribution, the mean plus/minus one standard deviation represents approximately 68% of the area of the entire distribution. In the case of our expected dividend of $1.15 for next year, the mean ($1.15) minus one standard deviation ($0.106) yields $1.044 while the mean plus one standard deviation yields a value of $1.256. Thus, there is a 68% probability that the actual dividend will fall between $1.04 and $1.26 (rounding to the nearest penny). The mean plus/minus two standard deviations represents approximately 95% of the area under the curve of the distribution. There is a 96% probability that the actual dividend next year will be between $0.94 and $1.36 in value. What is the probability that the dividend will be above $1.36?
The standard deviation of Distribution B is $0.183 (check it at home). As we could see from both the numbers in the distribution as well as from the graphs, Distribution B has more variability than Distribution A as confirmed by the standard deviations.
Consider a third distribution of possible outcomes, Distribution C:
Distribution / CProbability / Outcome
0.25 / * / $1.10 / = / $0.275
0.50 / * / $1.25 / = / $0.625
0.25 / * / $1.40 / = / $0.350
E(V)= mean / = / $1.25
Distribution C is identical in shape as Distribution A but has a higher expected value.
The standard deviation of Distribution C is $0.106 like that of Distribution A. But if you had to choose between Distribution A and Distribution C, which would you prefer? Distribution C has a higher expected value and is therefore preferred to Distribution A. Does that mean that Distribution C will turn out to be better than Distribution A? Only if you could do it over and over again, then on average Distribution C would be better than Distribution A.
A better measure of risk is the coefficient of variation which is defined as
The coefficient of variation measures the variability relative to the expected value and thus expresses the standard deviation as a percentage of the expected value. For our three distributions, the coefficients of variation are
According to the coefficients of variation, Distribution B is the most risky with a variability of 15.9 cents per dollar of expected return. Distribution A is the second most risky with a variability of 9.2 cents per dollar of expected return. Distribution C is the least risky with a variability of only 8.5 cents per dollar of expected return. The coefficient of variation has ranked the three distributions in the same manner that risk-averse individuals would.
These three distributions were designed in such a manner that it is clear which distribution is preferred to another. In reality, it is generally the case that the distribution with the higher expected value also has the higher variability. This is why the coefficient of variation is a better measure than the standard deviation of the stand-alone risk of an individual project.
DIVERSIFICATION OF RISK
Since individuals are risk-averse, they are interested in decreasing the risk to which they are exposed. This is accomplished through diversification. While diversification will reduce the probability of any large losses being incurred, it will also, by definition, reduce the probability of any large gains being realized. Nonetheless, the risk-averse individual is willing to give up the large gains in order to avoid the large losses. This is a consequence of utility functions. Recall from microeconomics that an individual’s utility function is assumed to exhibit the fact that more is always preferred over less, but at a decreasing rate (the principle of declining marginal utility).
Risk-aversion is characterized by the fact that, for a gain or loss of equal size with equal probability of occurrence, the gain in utility from winning is less than the loss of utility from losing. If the two outcomes have equal probability of occurring, then the expected utility from taking the risk (pink line) is less than the utility of the expected value (blue line). The individual would be equally happy to have a risk-free (or certain) equivalent amount that is less than the expected value and avoid the uncertainty altogether. Thus, individuals are willing to pay money in order to avoid taking risk. Do individuals actually do this? This is what insurance is all about.
Insurance companies use the statistical law of large numbers to reduce their risks, whereas the individual is faced with an either/or situation – the individual either lives or dies; the individual’s house either burns or doesn’t burn down, etc.
When it comes to investments, however, the individual can use the law of large numbers and diversify his/her portfolio in order to reduce the overall risk of the portfolio. The total risk of an investment (variability, measured in either absolute terms with the standard deviation, or relative terms with the coefficient of variation) can be decomposed into two portions:
Systematic risk is the risk related to the economic system as a whole. That is, it affects all companies in the same manner (although not to the same extent). What’s an example of an economic event that is bad news for all companies? A rise in interest rates is bad for two reasons: first, any company that has any debt will have to pay more in interest expense; secondly, an increase in interest rates leads to a decrease in the present value of the companies’ stock prices. Variability has an upside as well, and a fall in interest rates is good for all companies. Systematic risk is not diversifiable, however; that is, it cannot be eliminated.
Unsystematic, or diversifiable (or firm-specific), risk is the part of the total variability that can be eliminated through diversification. What’s an example of unsystematic risk? The risk that a tornado destroys one of your manufacturing facilities is an example. It has nothing to do with management’s abilities, or the industry – it just happened. What’s an example of the upside risk of unsystematic risk? (A tornado destroys your competition’s plant!) The diversification occurs when you own stock in both companies. The loss sustained by having a manufacturing facility destroyed in one company is offset (in part, at least) by the gain that occurs to the other company as a result.
Contrary to what many think, however, you do not need to be wealthy to by well-diversified. Simply choosing your investments wisely can achieve significant diversification with a relatively few number of stocks. Random simulations of stock portfolios have been made to see what reduction in risk occurs through diversification. The results are in the following table:
Number of Stocks / Percentage of Unsystematicin Portfolio / Risk Remaining in Portfolio
1 / 100%
2 / 57%
4 / 32%
8 / 16%
16 / 8%
32 / 4%
64 / 2%
128 / 1%
* / *
* / *
Market / 0%
Why is only the unsystematic portion of the risk considered? Because there is nothing that can be done about the systematic risk.
So with only one stock in our portfolio, say General Motors, all of the unsystematic risk remains since we have not diversified at all. By adding one more stock, on average, we can eliminated almost one-half of the risk that is diversifiable. Obviously, this second stock is not Ford or Daimler-Chrysler. Rather, it is a stock such as American Airlines. Four stocks eliminates almost 70% of the diversifiable risk. So now our portfolio contains General Motors, American Airlines, and maybe IBM and Taco Cabana. Eight stocks will eliminate 84% of the nonsystematic risk. Note that it takes ever increasing numbers of stocks to reduce the remaining unsystematic risk. To totally eliminate it requires owning a little bit of everything, which is “the market”.
Diversification – The Two-Asset Portfolio
How exactly does diversification eliminate the unsystematic risk? Consider two investment alternatives:
If we plot these two assets in risk-return space, we get the following graph:
Notice that if investors are risk-averse, these two assets are priced appropriate since Asset 2, which is more risky, has a higher expected return to compensate for its higher risk.
Suppose we create a portfolio with these two assets where we put 50% of our money in Asset 1 and the other 50% in Asset 2. What do you think the expected return on our two-asset portfolio will be?
10% * .5 = 5%
14% * .5 = 7%
E(RP) = 12%
The two-asset portfolio has an expected return of 12%, which is just a weighted average of the returns of each of the assets in the portfolio. The standard deviation of the portfolio is not as simple.
where
x is the proportion invested in Asset 1
(1-x) is the proportion invested in Asset 2
r1,2 is the correlation between Asset 1 and Asset 2
Recall from statistics what a correlation coefficient represents. The correlation coefficient tells you how two assets are related to one another. A correlation coefficient ranges between two extremes: +1 (perfectly positively correlated) and –1 (perfectly negatively correlated).
Suppose we look at two assets’ returns in different periods and plot each period as a point. In addition let’s run a linear regression line. One possible graph and best-fit (linear regression) line might look like this:
Do you recall what the square of the correlation coefficient (R2, or coefficient of determination) interpretation is? It tells you the percentage of the variation in Asset 2 that is explained by the variation in Asset 1. In this case, every point falls on the straight regression line. Thus, the two assets are perfectly correlated. Because the slope of the line is positive, they are perfectly positively correlated, or the correlation coefficient is +1. The square of of +1 is 1 or 100%. Thus, if you know what change in Asset 1 occurs, you know exactly what the change in Asset 2 will be.
Suppose the scatter plot of asset returns and the regression line look like this:
The correlation between the two assets is still positive, but not perfectly positive. Given the change in Asset 1 we can make a prediction about what Asset 2 will do, but we won’t be able to estimate its exact value. As may be seen on the graph, the actual outcome of Asset 2 may be higher or lower than our estimate.
Suppose the data points and regression line have the following appearance:
In this case, the two assets have no (zero) correlation to one another. Given the change in Asset 1, it is not possible to even predict the direction of the change in Asset 2, let alone the actual value. Asset 2 is as likely to move in the opposite direction of Asset 1 as it is to move in the same direction.
What kind of correlation is this?
Notice that it is not perfect negative correlation because not all of the observation points fall one the regression line.
As in the case of perfect positive correlation, the square (R2) of the correlation coefficient of –1 is still 1 or 100% explanatory power.
Returning to our two-asset portfolio, let’s calculate the standard deviation of the portfolio when one-half of our money is invested in Asset 1 the other half is invested in Asset 2. The one variable in the equation for the standard deviation of a two-asset portfolio that we do not already know is the correlation coefficient. Assume that the correlation between the two assets is perfectly positive, so that the correlation coefficient is +1. Then the standard deviation of the portfolio is
In this case, the standard deviation is just a weighted average of the standard deviations of the assets in the portfolio, but this is only because they are perfectly positively correlated. Our graph now looks like
Note that the risk/return plot of our 50/50 portfolio falls on a straight line between the two assets. In fact, we can place ourselves anywhere on the line. How?
The portfolio we have created is not an example of diversification in the financial sense because the perfect positive correlation between the two assets results in a linear (and proportional) reduction in return for a reduction in risk. Diversification only occurs when two assets are less than perfectly positively correlated.
Suppose that Asset 1 and Asset 2 are positively correlated, but not perfectly positively correlated. Assume that the correlation coefficient is +0.6 so that the standard deviation of our 50/50 portfolio is now
The expected return on our portfolio is still 12%, but now the risk is less than the average of the two assets. If the proportions invested in each asset are varied, we will trace our a risk-return line that looks like the following:
The effect becomes more pronounced as the correlation decreases. For zero correlation,
A negative correlation of –0.6 has the following standard deviation and shape:
Notice that our 50/50 portfolio now has a higher return (12%) and less risk than the least risky of the two assets, Asset 1.
The best diversification would occur if we could combine two assets that are perfectly negatively correlated.
If the proportions invested in Asset 1 and Asset 2 were chosen properly, all of the risk of the portfolio could be eliminated.
Diversification can be summarized as follows: As long as two assets are less than perfectly positively correlated, there is a chance that a loss in one asset will be offset by a gain in the other asset. As the correlation between the two assets decreases, the probability of a loss being offset by a gain increases. What is the probability of a loss being offset by a gain if two assets are perfectly negatively correlated?
The Capital Asset Pricing Model
If all risky assets are plotted in terms of their risk/expected return on a graph, it would appear as
Since these assets are not perfectly positively correlated with one another, combining them into portfolios yields curves like we saw before. In fact, portfolios can be combined into larger portfolios that are not perfectly positively correlated with one another in order to achieve even greater diversification. The universe of risky assets, and portfolios of risky assets, that an individual can invest in is called the “Attainable Set” since investors can put themselves anywhere within that set simply by choosing the appropriate assets and allocating their funds in proper proportions.
From the investment opportunities available within the Attainable Set, however, a rational investor will only consider those investments that make up the northwest edge of the attainable set. This portion of Attainable Set is referred to as the “Efficient Frontier”. The portfolios on the Efficient frontier “dominate” all other portfolios since they provide the same or higher expected return and/or the same or less risk than other investment opportunities.
The objective of individuals is assumed to be to maximize expected utility. This does not necessarily mean maximize wealth, since there is value placed on non-monetary things as well, such as leisure time, time with family, etc. In the area of finance and investing, however, the objective of maximizing wealth is taken as a given subject to individual tradeoffs between risk and return (i.e., individual risk-aversion). Each individual possesses a set of parallel indifference curves, each curve corresponding to a specific level of utility. An individual would choose their investment portfolio by finding that portfolio that is just tangent to their highest level of utility indifference curve:
Individual A, therefore, would choose portfolio A on the Efficient Frontier since this maximizes their expected utility, while Individual B (who is more risk averse) would choose portfolio B to maximize their expected level of utility.