LG 4 8.3 Risk of a Portfolio

LG 3 LG 4 8.3 Risk of a Portfolio

In real-world situations, the risk of any single investment would not be viewed independently of other assets. New investments must be considered in light of their impact on the risk and return of an investor’s portfolio of assets. The finan-

efficient portfolio cial manager’s goal is to create an efficient portfolio, one that provides the max-

A portfolio that maximizes

imum return for a given level of risk. We therefore need a way to measure the

return for a given level of risk.

return and the standard deviation of a portfolio of assets. As part of that analysis, we will look at the statistical concept of correlation, which underlies the process of diversification that is used to develop an efficient portfolio.

PORTFOLIO RETURN AND STANDARD DEVIATION The return on a portfolio is a weighted average of the returns on the individual

assets from which it is formed. We can use Equation 8.5 to find the portfolio return, r p :

(8.5) j=1

r p = ( w 1 * r 1 )+( w 2 * r 2 )+Á+ ( w n * r n )= a w j * r j

where

w j = proportion of the portfolio’s total dollar value represented by asset j

r j = return on asset j

Of course, g n j=1 w j = 1 , which means that 100 percent of the portfolio’s assets

must be included in this computation.

Example 8.9 3 James purchases 100 shares of Wal-Mart at a price of $55 per share, so his total investment in Wal-Mart is $5,500. He also buys 100 shares of Cisco Systems at

$25 per share, so the total investment in Cisco stock is $2,500. Combining these two holdings, James’s total portfolio is worth $8,000. Of the total, 68.75% is invested in Wal-Mart ($5,500 , $8,000) and 31.25% is invested in Cisco Systems

($2,500 , $8,000). Thus, w 1 = 0.6875, w 2 = 0.3125, and w 1 + w 2 = 1.0 . The standard deviation of a portfolio’s returns is found by applying the for-

mula for the standard deviation of a single asset. Specifically, Equation 8.3 is used when the probabilities of the returns are known, and Equation 8.3a (from footnote 4) is applied when analysts use historical data to estimate the standard

PART 4

Risk and the Required Rate of Return

Example 8.10 3 Assume that we wish to determine the expected value and standard deviation of returns for portfolio XY, created by combining equal portions (50% each) of

assets X and Y. The forecasted returns of assets X and Y for each of the next 5 years (2013–2017) are given in columns 1 and 2, respectively, in part A of Table

8.6. In column 3, the weights of 50% for both assets X and Y along with their respective returns from columns 1 and 2 are substituted into Equation 8.5. Column 4 shows the results of the calculation—an expected portfolio return of 12% for each year, 2013 to 2017.

Furthermore, as shown in part B of Table 8.6, the expected value of these portfolio returns over the 5-year period is also 12% (calculated by using Equation 8.2a, in footnote 3). In part C of Table 8.6, portfolio XY’s standard deviation is calculated to be 0% (using Equation 8.3a, in footnote 4). This value should not be surprising because the portfolio return each year is the same— 12%. Portfolio returns do not vary through time.

TA B L E 8 . 6 Expected Return, Expected Value, and Standard Deviation of Returns for Portfolio XY

A. Expected Portfolio Returns

Forecasted return

Expected portfolio

Portfolio return calculation a return, r p Year

Asset X

Asset Y

B. Expected Value of Portfolio Returns, 2013–2017 b

C. Standard Deviation of Expected Portfolio Returns c

(12% - 12%) 2 + (12% - 12%) 2 + (12% - 12%) 2 + (12% - 12%) 2 s + (12% - 12%) 2

a Using Equation 8.5. b Using Equation 8.2a found in footnote 3.

c Using Equation 8.3a found in footnote 4.

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Risk and Return

correlation

CORRELATION

A statistical measure of the relationship between any two

Correlation is a statistical measure of the relationship between any two series of

series of numbers.

numbers. The numbers may represent data of any kind, from returns to test scores. If two series tend to vary in the same direction, they are positively corre-

positively correlated Describes two series that move

lated. If the series vary in opposite directions, they are negatively correlated. For

in the same direction.

example, suppose we gathered data on the retail price and weight of new cars. It is likely that we would find that larger cars cost more than smaller ones, so we

negatively correlated Describes two series that move

would say that among new cars weight and price are positively correlated. If we

in opposite directions.

also measured the fuel efficiency of these vehicles (as measured by the number of miles they can travel per gallon of gasoline), we would find that lighter cars are

correlation coefficient more fuel efficient than heavier cars. In that case, we would say that fuel

correlation between two series. economy and vehicle weight are negatively correlated. 5 The degree of correlation is measured by the correlation coefficient, which

A measure of the degree of

perfectly positively ranges from + 1 for perfectly positively correlated series to - 1 for perfectly nega- correlated

tively correlated series. These two extremes are depicted for series M and N in

Describes two positively correlated series that have a

Figure 8.4. The perfectly positively correlated series move exactly together

correlation coefficient of 1. +

without exception; the perfectly negatively correlated series move in exactly opposite directions.

perfectly negatively correlated Describes two negatively

DIVERSIFICATION

correlated series that have a correlation coefficient of 1. -

The concept of correlation is essential to developing an efficient portfolio. To reduce overall risk, it is best to diversify by combining, or adding to the portfolio, assets that have the lowest possible correlation. Combining assets that have a low correlation with each other can reduce the overall variability of a portfolio’s returns. Figure 8.5 (see page 324) shows the returns that two assets, F and G, earn over time. Both assets earn the same average or expected return, r, but note that when F’s return is above average, the return on G is below average and vice versa. In other words, returns on F and G are negatively correlated, and when these two assets are combined in a portfolio, the risk of that portfolio falls without reducing the average return (that is, the portfolio’s average return is also ). r

FIGURE 8.4

Perfectly Positively Correlated

Perfectly Negatively Correlated

Correlations The correlation between

N series M and series N

5. Note here that we are talking about general tendencies. For instance, a large hybrid SUV might have better fuel economy than a smaller sedan powered by a conventional gas engine. This does not change the fact that the general

PART 4

Risk and the Required Rate of Return

FIGURE 8.5

Portfolio of

Diversification

Asset F

Asset G

Assets F and G

Combining negatively

correlated assets to reduce, or diversify, risk

For risk-averse investors, this is very good news. They get rid of something that they don’t like (risk) without having to sacrifice what they do like (return). Even if assets are positively correlated, the lower the correlation between them, the greater the risk reduction that can be achieved through diversification.

uncorrelated Some assets are uncorrelated—that is, there is no interaction between their

Describes two series that lack

returns. Combining uncorrelated assets can reduce risk, not as effectively as

any interaction and therefore

combining negatively correlated assets but more effectively than combining pos-

have a correlation coefficient

itively correlated assets. The correlation coefficient for uncorrelated assets is

close to zero.

close to zero and acts as the midpoint between perfect positive and perfect nega- tive correlation.

The creation of a portfolio that combines two assets with perfectly positively correlated returns results in overall portfolio risk that at minimum equals that of the least risky asset and at maximum equals that of the most risky asset. However, a portfolio combining two assets with less than perfectly positive cor- relation can reduce total risk to a level below that of either of the components.

For example, assume that you buy stock in a company that manufactures machine tools. The business is very cyclical, so the stock will do well when the economy is expanding, and it will do poorly during a recession. If you bought shares in another machine-tool company, with sales positively correlated with those of your firm, the combined portfolio would still be cyclical and risk would not be reduced a great deal. Alternatively, however, you could buy stock in a dis- count retailer, whose sales are countercyclical. It typically performs worse during economic expansions than it does during recessions (when consumers are trying to save money on every purchase). A portfolio that contained both of these stocks might be less volatile than either stock on its own.

Example 8.11 3 Table 8.7 presents the forecasted returns from three different assets—X, Y, and Z—over the next 5 years, along with their expected values and standard devia-

tions. Each of the assets has an expected return of 12% and a standard deviation of 3.16%. The assets therefore have equal return and equal risk. The return pat- terns of assets X and Y are perfectly negatively correlated. When X enjoys its highest return, Y experiences its lowest return, and vice versa. The returns of assets X and Z are perfectly positively correlated. They move in precisely the

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Risk and Return

Forecasted Returns, Expected Values, and Standard Deviations TA B L E 8 . 7 for Assets X, Y, and Z and Portfolios XY and XZ

Assets

Portfolios XY a XZ b

Year

(50% X ⴙ 50% Y) (50% X ⴙ 50% Z)

16 8 16 12 16 Statistics: c

Expected value

3.16% a Portfolio XY, which consists of 50 percent of asset X and 50 percent of asset Y, illustrates perfect negative correlation because these two return

Standard deviation d 3.16%

streams behave in completely opposite fashion over the 5-year period. Its return values shown here were calculated in part A of Table 8.6. b Portfolio XZ, which consists of 50 percent of asset X and 50 percent of asset Z, illustrates perfect positive correlation because these two return streams behave identically over the 5-year period. Its return values were calculated by using the same method demonstrated for portfolio XY in part A of Table 8.6.

c Because the probabilities associated with the returns are not given, the general equations, Equation 8.2a in footnote 3 and Equation 8.3a in footnote 4, were used to calculate expected values and standard deviations, respectively. Calculation of the expected value and standard deviation

for portfolio XY is demonstrated in parts B and C, respectively, of Table 8.6. d The portfolio standard deviations can be directly calculated from the standard deviations of the component assets with the following formula:

s r p = 2w 2 1 s 2 1 + w 2 2 s 2 2 + 2 w 1 w 2 c 1,2 s 1 s 2

where w 1 and w 2 are the proportions of component assets 1 and 2, s 1 and s 2 are the standard deviations of component assets 1 and 2, and c 1,2 is the correlation coefficient between the returns of component assets 1 and 2.

returns for X and Z are identical.) 6 Now let’s consider what happens when we

combine these assets in different ways to form portfolios. Portfolio XY Portfolio XY (shown in Table 8.7) is created by combining equal

portions of assets X and Y, the perfectly negatively correlated assets. (Calculation of portfolio XY’s annual returns, the expected portfolio return, and the standard deviation of returns was demonstrated in Table 8.6 on page 322.) The risk in this portfolio, as reflected by its standard deviation, is reduced to 0%, whereas the expected return remains at 12%. Thus, the combination results in the complete elimination of risk because in each and every year the portfolio earns a 12%

return. 7 Whenever assets are perfectly negatively correlated, some combination of the two assets exists such that the resulting portfolio’s returns are risk free.

Portfolio XZ Portfolio XZ (shown in Table 8.7) is created by combining equal portions of assets X and Z, the perfectly positively correlated assets. Individually, assets X and Z have the same standard deviation, 3.16%, and because they

6. Identical return streams are used in this example to permit clear illustration of the concepts, but it is not necessary for return streams to be identical for them to be perfectly positively correlated. Any return streams that move exactly together—regardless of the relative magnitude of the returns—are perfectly positively correlated.

7. Perfect negative correlation means that the ups and downs experienced by one asset are exactly offset by move-

PART 4

Risk and the Required Rate of Return

always move together, combining them in a portfolio does nothing to reduce risk—the portfolio standard deviation is also 3.16%. As was the case with port- folio XY, the expected return of portfolio XZ is 12%. Because both of these port- folios provide the same expected return, but portfolio XY achieves that expected return with no risk, portfolio XY is clearly preferred by risk-averse investors over portfolio XZ.

CORRELATION, DIVERSIFICATION, RISK, AND RETURN In general, the lower the correlation between asset returns, the greater the risk

reduction that investors can achieve by diversifying. The following example illus- trates how correlation influences the risk of a portfolio but not the portfolio’s expected return.