Textbook Questions

5 Modern Portfolio Concepts

Learning Goals

After studying this chapter, you should be able to:

1. LG 1 Understand portfolio objectives and the procedures used to calculate portfolio return and standard deviation.

2. LG 2 Discuss the concepts of correlation and diversification and the key aspects of international diversification.

3. LG 3 Describe the components of risk and the use of beta to measure risk.

4. LG 4 Explain the capital asset pricing model (CAPM) conceptually, mathematically, and graphically.

5. LG 5 Review the traditional and modern approaches to portfolio management.

6. LG 6 Describe portfolio betas, the risk-return tradeoff, and reconciliation of the two approaches to portfolio management.

United Rentals Inc. (URI) rents construction and industrial equipment to contractors, businesses, governments, and individuals. The company specializes in heavy equipment such as earth-moving machines and forklifts. During the recession that began in 2007, many companies found that they did not have enough work to do to keep the machines they already owned running, so naturally the demand for rental equipment suffered. URI stock reached a 2007 peak of over $35 per share in May, but after that began a long slide, hitting bottom at $2.52 in March 2009.

That spring, the economy began to show signs of life, and URI stock surged, rising nearly 200% from its low point by August 2009. Heiko Ihle, a stock analyst for the Gabelli & Co. money management firm, issued a “buy” rating on URI despite the fact that the company had high leverage (meaning that it borrowed a lot of money to finance its operations). Ihle noted that URI stock had a high beta, meaning that it moved sharply when the broader market shifted.

Mr. Ihle’s recommendation proved to be a good one. From the end of August 2009 to the end of August 2014, the value of URI stock climbed almost 1,200% and was trading above $119 per share. Over that same period of time, the S&P 500 Index, a widely used indicator of the overall stock market, rose by a less dramatic 95%.

In this chapter we continue to explore the tradeoff between risk and return, and we’ll see that a stock’s beta—its sensitivity to movements in the overall stock market—has a big effect on both the stock’s risk and the return that it offers investors.

(Sources: Yahoo! Finance; “U.S. Hot Stocks: Legg Mason, JDA Software Active in Late Trading,” July 20, 2009; The Wall Street Journal Digital Network, http://online.wsj.com/article/BT-CO-20090720-713541.html.)

Principles of Portfolio Planning

1. LG 1

2. LG 2

Investors benefit from holding portfolios of investments rather than single investments. Without necessarily sacrificing returns, investors who hold portfolios can reduce risk. Surprisingly, the volatility of a portfolio may be less than the volatilities of the individual assets that make up the portfolio. In other words, when it comes to portfolios and risk, the whole is less than the sum of its parts!

A portfolio is a collection of investments assembled to meet one or more investment goals. Of course, different investors have different objectives for their portfolios. The primary goal of a  growth-oriented portfolio  is long-term price appreciation. An  income-oriented portfolio  is designed to produce regular dividends and interest payments.

Portfolio Objectives

Setting portfolio objectives involves definite tradeoffs, such as the tradeoff between risk and return or between potential price appreciation and income. How investors evaluate these tradeoffs will depend on their tax bracket, current income needs, and ability to bear risk. The key point is that portfolio objectives must be established before one begins to invest.

The ultimate goal of an investor is an  efficient portfolio , one that provides the highest return for a given risk level. Efficient portfolios aren’t necessarily easy to identify. Investors usually must search out investment alternatives to get the best combinations of risk and return.

Portfolio Return and Standard Deviation

The first step in forming a portfolio is to analyze the characteristics of the securities that an investor might include in the portfolio. Two of the most important characteristics to examine are the returns that each asset might be expected to earn and the uncertainty surrounding that expected return. As a starting point, we will examine historical data to see what returns stocks have earned in the past and how much those returns have fluctuated to get a feel for what the future might hold.

The portfolio return is calculated as a weighted average of returns on the assets (i.e., the investments) that make up the portfolio. You can calculate the portfolio return, rp, by using  Equation 5.1 . The portfolio return depends on the returns of each asset in the portfolio and on the fraction invested in each asset, wj.

PortfolioReturn=(Proportion of portfolios totaldollar valueinvested inasset 1×Returnon asset1)+(Proportion of portfolio's totaldollar valueinvested inasset 2×Returnon asset2)+...+(Proportion ofportfolio's totaldollar valueinvested inassetn×Returnon assetn)=n∑j=1(Proportion of portfolio's totaldollar valueinvested inassetj×Returnon assetj)Portfolio Return=(Proportion of portfolios total dollar value invested in asset 1×Return on asset 1)+(Proportion of portfolio's total dollar value invested in asset 2×Return on asset 2)+...+(Proportion of portfolio's total dollar value invested in asset n×Return on asset n)=∑j=1n(Proportion of portfolio's total dollar value invested in asset j×Return on asset j)Equation5.1

rp=(w1×r1)+(w2×r2)+...+(wn×rn)=n∑j=1(wj×rj)rp=(w1×r1)+(w2×r2)+...+(wn×rn)=∑j=1n(wj×rj)Equation5.1a

The fraction invested in each asset, wj, is also known as a portfolio weight because it indicates the weight that each asset receives in the portfolio. Of course, n∑j=1wj=1∑j=1nwj=1, which means that the sum of the portfolio weights must equal 100%. In other words, when you add up the fractions invested in all of the assets, that sum must equal 1.0.

Panel A of  Table 5.1  shows the historical annual returns on two stocks, International Business Machines Corp. (IBM) and Celgene Corp. (CELG), from 2005 through 2014. Over that period, IBM earned an average annual return of 9.0%, which is close to the average annual return on the U.S. stock market during the past century. In contrast, Celgene Corp. earned a spectacular 40.7% average annual return. Although Celgene may not repeat that kind of performance over the next decade, it is still instructive to examine the historical figures.

Excel@Investing

Table 5.1 Individual And Portfolio Returns And Standard Deviation Of Returns For International Business Machines (Ibm) And Celgene (Celg)

Source: End-of-year closing prices are obtained from Yahoo Finance and are adjusted for dividends and stock splits.

A. Individual and Portfolio Returns
	(1)	(2)	(3)	(4)
	Historical Returns *	Portfolio Weights	Portfolio Return
Year (t)	rIBM%	rCELG%	WIBM = 0.86	WCELG = 0.14	rp%
2005	−15.8%	144.3%	(0.86 × -15.8%) + (0.14 × 144.3%) =	6.6%
2006	19.8%	77.5%	(0.86 × 19.8%) + (0.14 × 77.5%) =	27.9%
2007	12.8%	−19.7%	(0.86 × 12.8%) + (0.14 × -19.7%) =	8.3%
2008	−20.8%	19.7%	(0.86 × -20.8%) + (0.14 × 19.7%) =	−15.1%
2009	58.6%	0.7%	(0.86 × 58.6%) + (0.14 × 0.7%) =	50.5%
2010	14.3%	6.2%	(0.86 × 14.3%) + (0.14 × 6.2%) =	13.1%
2011	27.4%	14.3%	(0.86 × 27.4%) + (0.14 × 14.3%) =	25.6%
2012	5.9%	16.1%	(0.86 × 5.9%) + (0.14 × 16.1%) =	7.3%
2013	−0.2%	115.3%	(0.86 × -0.2%) + (0.14 × 115.3%) =	16.0%
2014	−12.4%	32.4%	(0.86 × -12.4%) + (0.14 × 32.4%) =	−6.1%
Average Return	9.0%	40.7%			13.4%
B. Individual and Portfolio Standard Deviations
Standard Deviation Calculation for IBM: SIBM=   ⎷10∑t=1(rt−¯r)2n−1=√(−15.8%−9.0%)2+...+(−12.4%−9.0%)210−1=√5015.4%210−1=23.6%SIBM=∑t=110(rt−r¯)2n−1=(−15.8%−9.0%)2+...+(−12.4%−9.0%)210−1=5015.4%210−1=23.6%
Standard Deviation Calculation for CELG: SCELG=   ⎷10∑t=1(rt−¯r)2n−1=√(144.3%−40.7%)2+...+(32.4%−40.7%)210−1=√2,5913.3%210−1=53.7%SCELG=∑t=110(rt−r¯)2n−1=(144.3%−40.7%)2+...+(32.4%−40.7%)210−1=2,5913.3%210−1=53.7%
Standard Deviation Calculation for Portfolio: SP=   ⎷10∑t=1(rt−¯r)2n−1=√(6.6%−13.4%)2+...+(−6.1%−13.4%)210−1=√3045.8%210−1=18.4%SP=∑t=110(rt−r¯)2n−1=(6.6%−13.4%)2+...+(−6.1%−13.4%)210−1=3045.8%210−1=18.4%
* Annual rate of return is calculated based on end-of-year closing prices.

Suppose we want to calculate the return on a portfolio containing investments in both IBM and Celgene. The first step in that calculation is to determine how much of each stock to hold. In other words, we must to decide what weight each stock should receive in the portfolio. Let’s assume that we want to invest 86% of our money in IBM and 14% in CELG. What kind of return would such a portfolio earn?

We know that over this period, Celgene earned much higher returns than IBM, so intuitively we might expect that a portfolio containing both stocks would earn a return higher than IBM’s but lower than Celgene’s. Furthermore, because most (86%) of the portfolio is invested in IBM, you might guess that the portfolio’s return would be closer to IBM’s than to Celgene’s.

Columns 3 and 4 in Panel A show the portfolio’s return each year. The average annual return on this portfolio was 13.4% and as expected it is higher than the return on IBM and lower than the return on Celgene. By investing a little in Celgene, an investor could earn a higher return than would be possible by holding IBM stock in isolation.

What about the portfolio’s risk? To examine the risk of this portfolio, starting by measuring the risk of the stocks in the portfolio. Recall that one measure of an investment’s risk is the standard deviation of its returns. Panel B of  Table 5.1  applies the formula for standard deviation that we introduced earlier to calculate the standard deviation of returns on IBM and Celgene stock. Or, if you prefer, rather than using the formulas in  Table 5.1  to find the standard deviation of returns for IBM and CELG, you can construct an Excel spreadsheet to do the calculations, as shown below. The standard deviation of IBM’s returns is 23.6%, and for Celgene’s stock returns the standard deviation is 53.7%. Here again we see evidence of the tradeoff between risk and return. Celgene’s stock earned much higher returns than IBM’s stock, but Celgene returns fluctuate a great deal more as well.

Because Celgene’s returns are more volatile than IBM’s, you might expect that a portfolio containing both stocks would have a standard deviation that is higher than IBM’s but lower than Celgene’s. In fact, that’s not what happens. The final calculation in Panel B inserts the IBM-Celgene portfolio return data from column 4 in Panel A into the standard deviation formula to calculate the portfolio’s standard deviation. Panel B shows the surprising result that the portfolio’s returns are less volatile than are the returns of either stock in the portfolio! The portfolio’s standard deviation is just 18.4%. This is great news for investors. An investor who held only IBM shares would have earned an average return of only 9.0%, but to achieve that return the investor would have had to endure IBM’s 23.6% standard deviation. By selling a few IBM shares and using the proceeds to buy a few Celgene shares (resulting in the 0.86 and 0.14 portfolio weights shown in  Table 5.1 ), an investor could have simultaneously increased his or her return to 13.4% and reduced the standard deviation to 18.4%. In other words, the investor could have had more return and less risk at the same time. This means that an investor who owns nothing but IBM shares holds an inefficient portfolio—an alternative portfolio exists that has a better return-to-risk tradeoff. That’s the power of diversification. Next, we will see that the key factor in making this possible is a low correlation between IBM and Celgene returns.

Correlation and Diversification

Diversification involves the inclusion of a number of different investments in a portfolio, and it is an important aspect of creating an efficient portfolio. Underlying the intuitive appeal of diversification is the statistical concept of correlation. Effective portfolio planning requires an understanding of how correlation and diversification influence a portfolio’s risk.

Correlation

Correlation  is a statistical measure of the relationship between two series of numbers. If two series tend to move in the same direction, they are  positively correlated . For instance, if each day we record the number of hours of sunshine and the average daily temperature, we would expect those two series to display positive correlation. Days with more sunshine tend to be days with higher temperatures. If the series tend to move in opposite directions, they are  negatively correlated . For example, if each day we record the number of hours of sunshine and the amount of rainfall, we would expect those two series to display negative correlation because, on average, rainfall is lower on days with lots of sunshine. Finally, if two series bear no relationship to each other, then they are  uncorrelated . For example, we would probably expect no correlation between the number of hours of sunshine on a particular day and the change in the value of the U.S. dollar against other world currencies on the same day. There is no obvious connection between sunshine and world currency markets.

The degree of correlation—whether positive or negative—is measured by the  correlation coefficient , which is usually represented by the Greek symbol rho (r). It’s easy to use Excel to calculate the correlation coefficient between IBM and Celgene stock returns, as shown in the following spreadsheet.

Excel will quickly tell you that the correlation coefficient between IBM and Celgene during the 2005–2014 period was -0.43. The negative figure means that there was a tendency over this period for the two stocks to move in opposite directions. In other words, years in which IBM’s return was better than average tended to be years in which Celgene’s returns was worse than average, and vice versa. A negative correlation between two stocks is somewhat unusual because most stocks are affected in the same way by large, macroeconomic forces. In other words, most stocks tend to move in the same direction as the overall economy, which means that most stocks will display at least some positive correlation with each other.

Because IBM is a major provider of information technology services and Celgene is a biopharmaceutical manufacturer, it is not too surprising that the correlation between these two stocks is not strongly positive. The companies compete in entirely different industries, have different customers and suppliers, and operate within very different regulatory constraints; however, the relatively large (-0.43) magnitude of their negative correlation raises concerns and should cause us to question the validity of basing investment decisions on this correlation measure. Perhaps the sample period we are using to estimate this correlation is too short or is not truly representative of the investment performance of these two stocks. The 2005 to 2014 period that we are focusing on consists of just 10 yearly return observations, and during this particular period there were no fewer than three strong systematic market-wide events (i.e., a financial crisis, a Great Recession, and an economic recovery). Those sharp macroeconomic fluctuations tended to drive most securities’ returns up and down at the same time, which in turn leads to a positive correlation between most pairs of stocks, even when those stocks are drawn from different industries. Ten yearly observations is without question a small sample size, and it may be too small, at least in this case, to accurately capture a meaningful measure of correlation between IBM and Celgene. One way to address this concern is to increase the period of time over which the correlation is being measured and in this way increase the number of yearly observations. Alternatively, one could use monthly returns over the same 10-year period, thereby increasing the number of observations by a factor of 12.

What’s the Correlation?

For any pair of investments that we might want to study, the correlation coefficient ranges from +1.0 for  perfectly positively correlated  series to −1.0 for  perfectly negatively correlated  series.  Figure 5.1  illustrates these two extremes for two pairs of

Figure 5.1 The Correlations of Returns between Investments M and P and Investments M and N.

Investments M and P produce returns that are perfectly positively correlated and move exactly together. On the other hand, returns on investments M and N move in exactly opposite directions and are perfectly negatively correlated. In most cases, the correlation between any two investments will fall between these two extremes.

investments: M and P, and M and N. M and P represent the returns on two investments that move perfectly in sync, so they are perfectly positively correlated. In the real world it is extremely rare to find two investments that are perfectly correlated like this, but you could think of M and P as representing two companies that operate in the same industry, or even two mutual funds that invest in the same types of stocks. In contrast, returns on investments M and N move in exactly opposite directions and are perfectly negatively correlated. While these two extreme cases can be illustrative, the correlations between most asset returns exhibit some degree (ranging from high to low) of positive correlation. Negative correlation is the exception.

Diversification

As a general rule, the lower the correlation between any two assets, the greater the risk reduction that investors can achieve by combining those assets in a portfolio.  Figure 5.2  shows negatively correlated assets F and G, both having the same average return, ¯rr¯. The portfolio that contains both F and G has the same return, ¯rr¯, but has less risk (variability) than either of the individual assets because some of the fluctuations in asset F cancel out fluctuations in G. As a result, the combination of F and G is less volatile than either F or G alone. Even if assets are not negatively correlated, the lower the positive correlation between them, the lower the resulting risk.

Table 5.2  shows the average return and the standard deviation of returns for many combinations of IBM and Celgene stock. Columns 1 and 2 show the percentage of the portfolio invested in IBM and Celgene, respectively, and columns 3 and 4 show the portfolio average return and standard deviation. Notice that as you move from the top of the table to the bottom (i.e., from investing the entire portfolio in IBM to investing all of it in Celgene), the portfolio return goes up. That makes sense because as you move from top to bottom, the percentage invested in Celgene increases, and Celgene’s average return is higher than IBM’s. The general conclusion from column 3 is that when a portfolio contains two stocks, with one having a higher average return than the other, the portfolio’s return rises the more you invest in the stock with the higher return.

Figure 5.2 Combining Negatively Correlated Assets to Diversify Risk

Investments F and G earn the same return on average, ¯rr¯ but they are negatively correlated, so movements in F sometimes partially offset movements in G. As a result, a portfolio containing F and G (shown in the rightmost graph) exhibits less variability than the individual assets display on their own while earning the same return.

Excel@Investing

Table 5.2 Portfolio Returns And Standard Deviations For International Business Machines (Ibm) And Celgene (Celg)

(1)	(2)	(3)	(4)
Portfolio Weights	Portfolio Return	Portfolio Standard Deviation
WIBM	WCELG	¯rIBM=9.0%r¯IBM=9.0%	¯rCELG=40.7%r¯CELG=40.7%
1.0	0.0	(1.0 × 9.0%) + (0.0 × 40.7%) = 9.0%	23.6%
0.9	0.1	(0.9 × 9.0%) + (0.1 × 40.7%) = 12.1%	19.6%
0.8	0.2	(0.8 × 9.0%) + (0.2 × 40.7%) = 15.3%	17.3%
0.7	0.3	(0.7 × 9.0%) + (0.3 × 40.7%) = 18.5%	17.5%
0.6	0.4	(0.6 × 9.0%) + (0.4 × 40.7%) = 21.7%	20.0%
0.5	0.5	(0.5 × 9.0%) + (0.5 × 40.7%) = 24.8%	24.3%
0.4	0.6	(0.4 × 9.0%) + (0.6 × 40.7%) = 28.0%	29.4%
0.3	0.7	(0.3 × 9.0%) + (0.7 × 40.7%) = 31.2%	35.1%
0.2	0.8	(0.2 × 9.0%) + (0.8 × 40.7%) = 34.3%	41.1%
0.1	0.9	(0.1 × 9.0%) + (0.9 × 40.7%) = 37.5%	47.3%
0.0	1.0	(0.0 × 9.0%) + (1.0 × 40.7%) = 40.7%	53.7%
Example: Calculation of the Standard Deviation for the Equally Weighted Portfolio
SIBM=23.6%SIBM=23.6%
SCELG=53.7%SCELG=53.7%
PIBM,CELG=−0.43PIBM,CELG=−0.43
SP=√W2iS2i+W2jS2j+2WjWjpi,jSiSjSP=W2iS2i+W2jS2j+2WjWjpi,jSiSj
SP=√0.52×23.6%2+0.52×53.7%2+2(0.5×0.5×−0.43×23.6%×53.7%)=24.3%SP=0.52×23.6%2+0.52×53.7%2+2(0.5×0.5×−0.43×23.6%×53.7%)=24.3%

Column 4 shows the standard deviation of returns for different portfolios of IBM and Celgene. Here again we see a surprising result. A portfolio invested entirely in IBM has a standard deviation of 23.6%. Intuitively, it might seem that reducing the investment in IBM slightly and increasing the investment in Celgene would increase the portfolio’s standard deviation because Celgene stock is so much more volatile than IBM stock. However, the opposite is true, at least up to a point. The portfolio standard deviation initially falls as the percentage invested in Celgene rises. Eventually, however, increasing the amount invested in Celgene does increase the portfolio’s standard deviation. So the general conclusion from column 4 is that when a portfolio contains two stocks, with one having a higher standard deviation than the other, the portfolio’s standard deviation may rise or fall the more you invest in the stock with the higher standard deviation.

Figure 5.3  illustrates the two lessons emerging from  Table 5.2 . The curve plots the return (y-axis) and standard deviation (x-axis) for each portfolio listed in  Table 5.2 . As the portfolio composition moves from 100% IBM to a mix of IBM and Celgene, the portfolio return rises, but the standard deviation initially falls. Therefore, portfolios of IBM and Celgene trace out a backward-bending arc. Clearly no investor should place all of his or her money in IBM because the investor could earn a higher return with a lower standard deviation by holding at least some stock in Celgene. However, investors who want to earn the highest possible returns, and who therefore will invest heavily in Celgene, have to accept a higher standard deviation.

Figure 5.3 Portfolios of IBM and Celgene

Because the returns of IBM and Celgene are not highly correlated, investors who hold only IBM shares can simultaneously increase the portfolio return and reduce its standard deviation by holding at least some Celgene shares. At some point, however, investing more in Celgene does increase the portfolio volatility while also increasing its expected return.

Excel@Investing

The relationship between IBM and Celgene is obviously a special case, so let’s look at the more general patterns that investors encounter in the markets.  Table 5.3  presents the projected returns from three assets—X, Y, and Z—in each of the next five years (2018–2022).  Table 5.3  also shows the average return that we expect each asset to earn over the five-year period and the standard deviation of each asset’s returns. Asset X has an average return of 12% and a standard deviation of 3.2%. Assets Y and Z each have an average return of 16% and a standard deviation of 6.3%. Thus, we can view asset X as having a low-return, low-risk profile while assets Y and Z are high-return, high-risk stocks. The returns of assets X and Y are perfectly negatively correlated—they move in exactly opposite directions over time. The returns of assets X and Z are perfectly positively correlated—they move in precisely the same direction.

Investor Facts

Not an Ideal [Cor]relationship During periods of economic uncertainty and high stock market volatility, the correlation between different assets tends to rise. In the summer of 2011, with concerns about European economies rocking markets, the correlation between the different sectors of the S&P 500 stock index reached 97.2%, the highest correlation since the 2008 financial crisis. During these periods, macroeconomic news announcements have a larger impact on stock returns than individual events at specific companies, so the correlation between stocks rises and the benefits of diversification fall.

(Source: Charles Rotblut, “AAII Investor Update: Highly Correlated Stock Returns Are Temporary,” September 16, 2011,  http://seekingalpha .com/article/294067-aaii-investor-update-highly-correlated-stock-returns-are-temporary .)

Portfolio XY (shown in  Table 5.3 ) is constructed by investing 2323 in asset X and 1313 in asset Y. The average return on this portfolio, 13.3%, is a weighted average of the average returns of assets X and Y (23×12%+13×16%)(23×12%+13×16%). To calculate the portfolio’s standard deviation, use the equation shown in  Table 5.2  with a value of −1.0 for the correlation between X and Y. Notice that portfolio XY generates a predictable 13.3% return every year. In other words, the portfolio is risk-free and has a standard deviation of 0.

Now consider portfolio XZ, which is created by investing 2323 in asset X and 1313 in asset Z. Like portfolio XY, portfolio XZ has an expected return of 13.3%. Notice, however, that portfolio XZ does not provide a risk-free return. Its return fluctuates from year to year, and its standard deviation is 4.2%.

To summarize, the two portfolios, XY and XZ, have identical average returns, but they differ in terms of risk. The reason for that difference is correlation. Movements in X are offset by movements in Y, so by combining the two assets in a portfolio, the investor can reduce or eliminate risk. Assets X

Table 5.3 Expected Returns And Standard Deviations For Assets X, Y, And Z And Portfolios XY And XZ

	Portfolio’s Projected Returns
	Asset’s Projected Returns	E(rxy)	E(rxz)
Year (t)	E(rX%)	E(rY%)	E(rZ%)	[2/3 × E(rX%) + 1/3 × E(rY%)]	[2/3 × E(rX%) + 1/3 × E(rZ%)]
2018	8.0%	24.0%	8.0%	13.3%	8.0%
2019	10.0%	20.0%	12.0%	13.3%	10.7%
2020	12.0%	16.0%	16.0%	13.3%	13.3%
2021	14.0%	12.0%	20.0%	13.3%	16.0%
2022	16.0%	8.0%	24.0%	13.3%	18.7%
Average Return	12.0%	16.0%	16.0%	13.3%	13.3%
Standard Deviation	3.2%	6.3%	6.3%	0.0%	4.2%

and Z move together, so movements in one cannot offset movements in the other, and the standard deviation of portfolio XZ cannot be reduced below the standard deviation of asset X.

Excel@Investing

Figure 5.4  illustrates how the relation between a portfolio’s expected return and standard deviation depends on the correlation between the assets in the portfolio. The black line illustrates a case like portfolio XY where the correlation coefficient is −1.0. In that case, it is possible to combine two risky assets in just the right proportions so that the portfolio return is completely predictable (i.e., has no risk). Notice that in this situation, it would be very unwise for an investor to hold an undiversified position in the least risky asset. By holding a portfolio of assets rather than just one, the investor moves up and to the left along the black line to earn a higher return while taking less risk. Beyond some point, however, increasing the investment in the more risky asset pushes both the portfolio return and risk higher, so the investor’s portfolio moves up and to the right along the second segment of the black line.

The red line in  Figure 5.4  illustrates a situation like portfolio XZ in which the correlation coefficient is +1.0. In that instance, when an investor decreases his or her investment in the low-risk asset to hold more of the high-risk asset, the portfolio’s expected return rises, but so does its standard deviation. The investor moves up and to

Figure 5.4 Risk and Return for Combinations of Two Assets with Various Correlation Coefficients

This graph illustrates how a low-return, low-risk asset can be combined with a high-return, high-risk asset in a portfolio, and how the performance of that portfolio depends on the correlation between the two assets. In general, as an investor shifts the portfolio weight from the low-return to the high-return investment, the portfolio return will rise. But the portfolio’s standard deviation may rise or fall depending on the correlation. In general, the lower the correlation, the greater the risk reduction that can be achieved through diversification.

the right along the red line. An investor might choose to invest in both assets, but making that decision is a matter of one’s risk tolerance, and not all investors will make that choice. In other words, when the correlation between two assets is −1.0, diversifying is definitely the right move, but when the correlation is +1.0, whether to diversify or not is less obvious.

The blue line in  Figure 5.4  illustrates an intermediate case in which the correlation coefficient is between -1.0 and +1.0. This is what investors encounter in real markets most of the time—assets are neither perfectly negatively correlated nor perfectly positively correlated. When the correlation coefficient is between the extremes, portfolios of two assets lie along an arc (i.e., the blue line). When two assets have very low correlation, that arc may bend back upon itself, as was the case with IBM and Celgene. When the correlation is higher, but still below 1.0, the arc merely curves up and to the right. Even then, the benefits of diversification are better than when the correlation is 1.0, meaning that portfolios along the blue arc earn higher returns for the same risk compared to portfolios along the red line.

International Diversification

Diversification is clearly a primary consideration when constructing an investment portfolio. As noted earlier, many opportunities for international diversification are now available. Here we consider three aspects of international diversification: effectiveness, methods, and costs.

Investor Facts

Culture and Correlation In finance we usually think about economic factors that cause the returns of different stocks to be more or less correlated. But a recent study suggests cultural influences matter too. In some cultures, behavioral norms are stronger and society’s tolerance for deviations in those norms tends to be low. Researchers found that in these countries, stock returns were more highly correlated than those in countries with less rigid social norms. The study’s authors suggest that in “tighter” cultures, investors are more likely to buy or sell the same stocks at the same time, and that leads to highly correlated stock returns.

(Source: Cheol S. Eun, Lingling Wang, and Steven C. Xiao, “Culture and R2,” Journal of Financial Economics, 2015, Vol. 115, pp. 283–303.)

Effectiveness of International Diversification

Investing internationally offers greater diversification than investing only domestically. That is true for U.S. investors as well as for investors in countries with capital markets that offer much more limited diversification opportunities than are available in the United States. Broadly speaking, the diversification benefits from investing internationally come from two sources. The first source is that returns in different markets around the world do not move exactly in sync. In other words, the correlation between markets is less than +1.0. As you have already seen, the lower the correlation is between investments, the larger are the benefits from diversification. Unfortunately, as globalization has brought about greater integration of markets (both financial markets and markets for goods and services) around the world, the correlation in returns across national markets has risen. This trend reduces the benefit of international diversification.

However, the second source of the benefits of international diversification has been on the rise for many years. Over time, the number of stock markets around the world has been increasing. For example, at the beginning of the 20th century fewer than 40 countries in the world had active stock markets, but by the end of the century the number of stock markets had more than doubled. Just as someone who invests only in domestic stocks will generally have a more diversified portfolio if there are more stocks in the portfolio, so it is for investors who can diversify across many stock markets around the world rather than just a few.

On net, there is little question that it benefits investors to diversify internationally, even if the rising correlation across markets (especially the larger, more developed markets) limits these benefits to an extent. Next, we discuss how investors can access international markets to diversify their portfolios.

Investor Facts

U.S. Fund Fees a Bargain U.S. investors benefit from a large, competitive mutual fund industry. That industry offers investors a vast array of diversification opportunities, and the fees that U.S. mutual funds charge are a bargain compared to those of the rest of the world. A recent study finds that the total cost to invest in an average U.S. mutual fund is about 1.04% per year of the amount invested. The country with the next most affordable mutual fund costs is Australia at 1.41% per year, followed by France (1.64%), Germany (1.79%), and Italy and Switzerland (1.94%). Mutual fund costs in the United Kingdom are 2.21%, more than double the costs in the United States.

(Source: “Mutual Fund Fees Around the World,” Review of Financial Studies Vol. 22, No. 3, 1279–1310.)

Methods of International Diversification

Later in this text we will examine a wide range of alternatives for international portfolio diversification. We will see that investors can make investments in bonds and other debt instruments in U.S. dollars or in foreign currencies—either directly or via foreign mutual funds. Foreign currency investment, however, brings currency exchange risk. Investors can hedge this risk with contracts such as currency forwards, futures, and options. Even if there is little or no currency exchange risk, investing abroad is generally less convenient, more expensive, and riskier than investing domestically. When making direct investments abroad, you must know what you’re doing. You should have a clear idea of the benefits being sought and enough time to monitor foreign markets.

U.S. investors can capture at least some of the benefits of international diversification without having to send money abroad. Investors can buy stock of foreign companies listed on U.S. exchanges. Many foreign issuers, both corporate and government, sell their bonds (called Yankee bonds) in the United States. The stocks of more than 2,000 foreign companies, from more than 60 countries, trade in the United States in the form of American depositary shares (ADSs). Finally, international mutual funds provide foreign investment opportunities.

You might wonder whether it is possible to achieve the benefits of international diversification by investing in a portfolio of U.S.-based multinational corporations. The answer is yes and no. Yes, a portfolio of U.S. multinationals is more diversified than a portfolio of wholly domestic firms. Multinationals generate revenues, costs, and profits in many markets and currencies, so when one part of the world is doing poorly, another part may be doing well.

Investors who invest only in U.S.-based multinationals will still not enjoy the full benefits of international diversification. That’s because a disproportionate share of the revenues and costs generated by these firms is still in the United States. Thus, to fully realize the benefits of international diversification, it is necessary to invest in firms located outside the United States.

Costs of International Diversification

You can find greater returns overseas than in the United States, and you can reduce a portfolio’s risk by including foreign investments. Still, you should not jump to the conclusion that it is wise to invest all of your money in overseas assets. A successful global investment strategy depends on many things, just as a purely domestic strategy does. The percentage of your portfolio that you should allocate to foreign investments depends on your overall investment goals and risk preferences. Many investment advisers suggest allocations to foreign investments of about 20% to 30%, with two-thirds of this allocation in established foreign markets and the other one-third in emerging markets.

In general, investing directly in foreign-currency-denominated instruments is very costly. Unless you have hundreds of thousands of dollars to invest, the transaction costs of buying securities directly on foreign markets will tend to be high. A less costly approach to international diversification is to invest in international mutual funds, which offer diversified foreign investments and the professional expertise of fund managers. You could also purchase ADSs to make foreign investments in individual stocks. With either mutual funds or ADSs, you can obtain international diversification along with low cost, convenience, transactions in U.S. dollars, and protection under U.S. security laws.

Concepts in Review

Answers available at  http://www.pearsonhighered.com/smart

1. 5.1 What is an efficient portfolio, and what role should such a portfolio play in investing?

2. 5.2 How do you calculate the return and standard deviation of a portfolio? Compare the calculation of a portfolio’s standard deviation to that for a single asset.

3. 5.3 What is correlation, and why is it important with respect to portfolio returns? Describe the characteristics of returns that are (a) positively correlated, (b) negatively correlated, and (c) uncorrelated. Differentiate between perfect positive correlation and perfect negative correlation.

4. 5.4 What is diversification? How does the diversification of risk affect the risk of the portfolio compared to the risk of the individual assets it contains?

5. 5.5 Discuss how the correlation between asset returns affects the risk and return behavior of the resulting portfolio. Describe the potential range of risk and return when the correlation between two assets is (a) perfectly positive, (b) uncorrelated, and (c) perfectly negative.

6. 5.6 What benefit, if any, does international diversification offer the individual investor? Compare and contrast the methods of achieving international diversification by investing abroad versus investing domestically.

The Capital Asset Pricing Model

1. LG 3

2. LG 4

Intuitively we would expect that any risky investment should offer a return that exceeds what investors can earn on a risk-free investment. In other words, the return that investors expect to earn on a risky asset equals the risk-free rate plus a risk premium. But what determines the magnitude of the risk premium? In the previous section we learned that investors can reduce or eliminate many types of risk simply by diversifying their portfolios, a process that is neither particularly time consuming nor expensive. However, diversification can’t eliminate risk entirely. Therefore, from an investor’s perspective, the most worrisome risk is undiversifiable risk—the risk that can’t be eliminated through diversification. The more undiversifiable risk that a particular investment entails, the higher the risk premium it must offer to attract investors.

That logic provides the underpinning for a theory that links return and risk for all assets. The theory is called the capital asset pricing model, or the CAPM. The CAPM says that the expected return on a risky asset equals the risk-free rate plus a risk premium, and the risk premium depends on how much of the asset’s risk is undiversifiable. In this section, we introduce the concept of undiversifiable risk, and we explain how the CAPM quantifies that risk and links it to investment returns.

Components of Risk

The risk of an investment consists of two components: diversifiable and undiversifiable risk.  Diversifiable risk , sometimes called unsystematic risk, results from factors that are firm-specific, such as whether a new product succeeds or fails, the performance of senior managers, or a firm’s relationships with its customers and suppliers. Unsystematic risk is the portion of an investment’s risk that can be eliminated through diversification.  Undiversifiable risk , also called systematic risk or  market risk , is the inescapable portion of an investment’s risk. In other words, it’s the risk that remains even if a portfolio is well diversified. Systematic risk is associated with broad forces such as economic growth, inflation, interest rates, and political events that affect all investments and therefore are not unique to any single investment. The sum of undiversifiable risk and diversifiable risk is called  total risk .

Total risk=Undiversifiable risk+Diversifiable riskTotal risk = Undiversifiable risk + Diversifiable riskEquation5.2

Any careful investor can reduce or virtually eliminate diversifiable risk by holding a diversified portfolio of securities. Studies have shown that investors can eliminate most diversifiable risk by carefully selecting a portfolio of as few as two or three dozen securities, and most investors hold many more securities than that through investments such as mutual funds and pension funds. Because it is relatively easy to eliminate unsystematic risk through diversification, there is no reason for investors to expect a reward (i.e., higher returns) for bearing this kind of risk. Investors who fail to diversify are simply bearing more risk than they have to without getting a reward for doing so.

But no matter how many securities are in a portfolio, some systematic risk will remain. Remember, undiversifiable risk refers to the broad forces that tend to affect most stocks simultaneously, such as whether the economy is booming or in recession. Some stocks are more sensitive to these forces than others. For example, companies that produce luxury goods tend to do very well when the economy is surging, but when a recession hits, these companies struggle to find customers. On the other hand, some stocks are relatively insulated from swings in the business cycle. Companies that produce food and other basic necessities do not see their revenues and profits rise and fall sharply with the ups and downs of the economy.

This discussion implies that systematic risk varies from one stock to another, and stocks with greater systematic risk must offer higher returns to attract investors. To identify these stocks, we need a way to measure the undiversifiable risk associated with any particular stock. The CAPM provides just such a measure called the stock’s beta.

Beta: A Measure of Undiversifiable Risk

During the past 50 years, the finance discipline has developed much theory on the measurement of risk and its use in assessing returns. The two key components of this theory are beta, which is a measure of systematic risk, and the capital asset pricing model, which links an investment’s beta to its return.

First we will look at  beta , a number that quantifies undiversifiable risk. A security’s beta indicates how the security’s return responds to fluctuations in market returns, which is why market risk is synonymous with undiversifiable risk. The more sensitive the return of a security is to changes in market returns, the higher that security’s beta. When we speak of returns on the overall market, what we have in mind is something like the return on a broad portfolio of stocks or on a stock index. Analysts commonly use changes in the value of the Standard & Poor’s 500 Index or some other broad stock index to measure market returns. To calculate a security’s beta, you gather historical returns on the security and on the overall market to see how they relate to each other. You don’t have to calculate betas yourself; you can easily obtain them for actively traded securities from a variety of published and online sources. But you should understand how betas are derived, how to interpret them, and how to apply them to portfolios.

Deriving Beta

We can demonstrate graphically the relationship between a security’s return and the market return.  Figure 5.5  plots the relationship between the returns of two securities, United Parcel Service, Inc. (UPS) and FedEx Corporation (FDX), and

Figure 5.5 Graphical Derivation of Beta for Securities C and D

Betas can be derived graphically by plotting the coordinates for the market return and security return at various points in time and using statistical techniques to fit the “characteristic line” to the data points. The slope of the characteristic line is beta. For FedEx the beta is about 1.12, and for UPS the beta is about 0.76.

the market return measured as the return on the S&P 500 (GSPC). The return data necessary to plot the relationships shown in  Figure 5.5  are easily obtained from numerous online financial websites. In this case, we obtained historical closing prices from Yahoo! Finance by entering the security ticker symbols and downloading the end-of-year historical prices to a spreadsheet. In a spreadsheet, we used the end-of-year security prices to calculate the following annual returns (in the case of the S&P 500, we calculated annual returns based on the index level):

We calculated annual returns by dividing the current year-end price by the previous year-end price and subtracting 1. (Note: the prices we are using here have been adjusted in a way that accounts for dividends, meaning that the percentage price change includes both the stock’s capital gain or loss as well as its dividend payments.) For example, the annual return of 30.3% for UPS in 2010 is ($64.28 ÷, $49.35) − 1. Notice that we are simply letting the previous year’s closing price be the present value, the current year’s closing price be the future value, the one-year time interval be the number of periods, and then solving for the interest rate.

In  Figure 5.5  we plot coordinates showing the annual return on each stock (on the y-axis) and the S&P 500 (on the x-axis) in each year. Each green circle shows the return earned by UPS and the S&P 500 in a particular year, and each blue diamond shows the return on FedEx and the S&P 500 in a particular year. For example, the blue diamond and the green circle diamond in the upper right quadrant of the figure show that in 2013 FedEx earned a return of about 60%, UPS earned a return of roughly 45%, and the overall market’s return was about 30% (you can verify these numbers in the previous table).

With the data points plotted, we used Excel to insert a trendline (also called the characteristic line) that best fit the coordinates for each stock. The green line in  Figure 5.5  goes through the middle of the green circles (the coordinates for UPS and S&P 500 returns), so it shows the general relation between the return on the S&P 500 and the return on UPS. Similarly, the blue line is the line that best fits the blue diamonds (the coordinates for FedEx and S&P 500 returns), and it shows the relation between FedEx and S&P 500 returns. Remember that the equation for any straight line takes the form y = mx + b, where m represents the slope of the line, or the relation between x and y.  Figure 5.5  shows the equation for each trendline, and the slope for each line is the beta for that stock. For UPS, the equation for the characteristic line is y = 0.7629x + 1.3763, so the beta of UPS stock is 0.7639. The beta for FedEx is higher at 1.118. Because FedEx has a higher beta than UPS, we would say that FedEx stock is more sensitive to movements in the overall market. This also means that FedEx has more systematic risk, so overall we conclude the FedEx is a riskier investment than UPS.

Interpreting Beta

By definition, the beta for the overall market is equal to 1.0 (i.e., the market moves in a one-to-one relationship with itself). That also implies that the beta of the “average” stock is 1.0. All other betas are viewed in relation to this value.  Table 5.4  shows some selected beta values and their associated interpretations. As you can see, an investment’s beta can, in principle, be positive or negative, although nearly all investments have positive betas. The positive or negative sign preceding the beta number merely indicates whether the stock’s return moves in the same direction as the general market (positive beta) or in the opposite direction (negative beta).

Table 5.4 Selected Betas And Associated Interpretations

Beta	Comment	Interpretation
	Move in same direction as the market
0.0		Unaffected by market movement
	Move in opposite direction of the market

Most stocks have betas that fall between 0.50 and 1.75. The return of a stock that is half as responsive as the market (b = 0.5) will, on average, change by ½ of 1% for each 1% change in the return of the market portfolio. A stock that is twice as responsive as the market (b = 2) will, on average, experience a 2% change in its return for each 1% change in the return of the market portfolio. Listed here, for illustration purposes, are the actual betas for some popular stocks, as reported on Yahoo! Finance on February 27, 2015:

Stock	Beta	Stock	Beta
Amazon.com Inc.	1.27	Int’l Business Machines Corp.	0.88
Molson Coors Brewing Co.	1.45	Goldman Sachs Inc.	1.64
Bank of America Corp.	1.33	Microsoft Corp.	0.79
Procter & Gamble Co.	0.93	Nike Inc.	0.51
Walt Disney Co.	1.02	Celgene Corp.	1.84
eBay Inc.	0.82	Qualcomm Inc.	1.2
ExxonMobil Corp.	1.16	Sempra Energy	0.39
The Gap Inc.	1.63	Walmart Stores Inc.	0.29
Ford Motor Co.	0.76	Xerox Corp.	1.34
INTEL Corp.	0.95	YAHOO! Inc.	1.27

How to Estimate a Beta

Applying Beta

Individual investors will find beta useful. It can help in assessing the risk of a particular investment and in understanding the impact the market can have on the return expected from a share of stock. In short, beta reveals how a security responds to market forces. For example, if the market is expected to experience a 10% increase in its rate of return over the next period, we would expect a stock with a beta of 1.5 to experience an increase in return of about 15% (1.5 × 10%).

For stocks with positive betas, increases in market returns result in increases in security returns. Unfortunately, decreases in market returns are translated into decreasing security returns. In the preceding example, if the market is expected to experience a 10% decrease in its rate of return, then a stock with a beta of 1.5 should experience a 15% decrease in its return. Because the stock has a beta greater than 1.0, it is riskier than an average stock and will tend to experience dramatic swings when the overall market moves.

Stocks that have betas less than 1.0 are, of course, less responsive to changing returns in the market and are therefore less risky. For example, a stock with a beta of 0.50 will increase or decrease its return by about half that of the market as a whole. Thus, if the market return went down by 8%, such a stock’s return would probably experience only about a 4% (0.50 × 8%) decline.

Investor Facts

Which Beta? Working with betas is not an exact science. A researcher recently found that by browsing through 16 financial websites, one could find estimates of beta for the same company (Walt Disney) ranging from 0.72 to 1.39. If you try to estimate betas on your own, you will find that your estimates will vary depending on how much historical data you use in your analysis and the frequency with which returns are measured.

(Source: “Betas Used by Professors: A Survey with 2,500 Answers.” Working paper, IESE Business School, University of Navarra, May 2009.)

Here are some important points to remember about beta:

· Beta measures the undiversifiable (or market) risk of a security.

· The beta for the market as a whole, and for the average stock, is 1.0.

· In theory, stocks may have positive or negative betas, but most stocks have positive betas.

· Stocks with betas greater than 1.0 are more responsive than average to market fluctuations and therefore are more risky than average. Stocks with betas less than 1.0 are less risky than the average stock.

Famous Failures in Finance Bulging Betas

Ford Motor Company has always been considered a cyclical stock whose fortunes rise and fall with the state of the economy. Ford’s beta was as high as 2.80 during the financial crisis, which hit auto manufacturers particularly hard and resulted in the bankruptcy of Ford’s major competitor, General Motors. Bank of America, another firm in an industry hit hard by the recession, had a beta of 1.96 during the crisis, indicating that it too was extremely sensitive to movements in the overall economy. Notice that both Ford and Bank of America have lower betas now than they did during the last recession.

The CAPM: Using Beta to Estimate Return

Intuitively, we expect riskier investments to provide higher returns than less risky investments. If beta measures the risk of a stock, then stocks with higher betas should earn higher returns, on average, than stocks with lower betas. About 50 years ago, finance professors William F. Sharpe and John Lintner developed a model that uses beta to formally link the notions of risk and return. Called the  capital asset pricing model (CAPM) , it attempts to quantify the relation between risk and return for different investments. It also provides a mechanism whereby investors can assess the impact of a proposed security investment on their portfolio’s risk and return. The CAPM predicts that a stock’s expected return depends on three things: the risk-free rate, the expected return on the overall market, and the stock’s beta.

The Equation

With beta, b, as the measure of undiversifiable risk, the capital asset pricing model defines the expected return on an investment as follows.

Expected returnon investmentj=Risk-freerate+[Beta forinvestmentj×(Expected marketreturn−Risk-freerate)]Expected return on investment j=Risk-free rate+[Beta for investment j×(Expected market return −Risk-free rate)]Equation5.3

rj=rrf+[bj×(rm−rrf)]rj=rrf+[bj×(rm−rrf)]Equation5.3a

where

rj=the expected return on investment j, given its risk as measured by betarrf=the risk-free rate of return; the return that can be earned on a risk-free investmentbj=beta coefficient, or index of undiversifiable risk for investment jrm=the expected market return; the average return on all securities(typicallymeasured by the average return on all securities in the Standard & Poor's 500Composite Index or some other broad stock market index)rj=the expected return on investment j, given its risk as measured by betarrf=the risk-free rate of return; the return that can be earned on a risk-free investmentbj=beta coefficient, or index of undiversifiable risk for investment jrm=the expected market return; the average return on all securities (typically measured by the average return on all securities in the Standard & Poor's 500 Composite Index or some other broad stock market index)

The CAPM can be divided into two parts: (1) the risk-free rate of return, rrf, and (2) the risk premium, bj×(rm−rrf)bj×(rm−rrf). The risk premium is the return investors require beyond the risk-free rate to compensate for the investment’s undiversifiable risk as measured by beta. The equation shows that as beta increases, the stock’s risk premium increases, thereby causing the expected return to increase. 1

1  Note that we are using the terms expected return and required return interchangeably here. Investors require investments to earn a return that is sufficient compensation based on the investment’s risk, and in equilibrium, the return that they require and the return that they expect to earn are the same.

Example

We can demonstrate use of the CAPM with the following example. Assume you are thinking about investing in Bank of America stock, which has a beta of 1.33. At the time you are making your investment decision, the risk-free rate (rrf) is 2% and the expected market return (rm) is 8%. Substituting these data into the CAPM equation,  Equation 5.3a , we get:

r=2%+1.33(8%−2%)=10%r=2%+1.33(8%−2%)=10%

You should therefore expect—indeed, require—a 10% return on this investment as compensation for the risk you have to assume, given the security’s beta of 1.33.

If the beta were lower, say, 1.0, the required return would be lower. In fact, in this case the required return on the stock is the same as the expected (or required) return on the market.

r=2%+1.0(8%−2%)=8%r=2%+1.0(8%−2%)=8%

If the beta were higher, say 2.0, the required return would be higher:

r=2%+2.0(8%−2%)=14%r=2%+2.0(8%−2%)=14%

Clearly, the CAPM reflects the positive tradeoff between risk and return: The higher the risk (beta), the higher the risk premium, and therefore the higher the required return.

The Graph: The Security Market Line

Figure 5.6  depicts the CAPM graphically. The line in the figure is called the  security market line (SML) , and it shows the expected return (y-axis) for any security given its beta (x-axis). For each level of undiversifiable risk (beta), the SML shows the return that the investor should expect to earn in the marketplace.

We can plot the CAPM by simply calculating the required return for a variety of betas. For example, as we saw earlier, using a 2% risk-free rate and an 8% market return, the required return is 10% when the beta is 1.33. Increase the beta to 2.0, and the required return equals 14% [2% + 2.0(8% - 2%)]. Similarly, we can find the required return for a number of betas and end up with the following combinations of risk (beta) and required return.

Risk (beta)	Required Return
0.0	2%
0.5	5%
1.0	8%
1.5	11%
2.0	14%
2.5	17%

Plotting these values on a graph (with beta on the horizontal axis and required return on the vertical axis) would yield a straight line like the one in  Figure 5.6 . It is clear from the SML that as risk (beta) increases, so do the risk premium and required return, and vice versa.

Some Closing Comments

The capital asset pricing model generally relies on historical data in the sense that the value of beta used in the model is typically based on calculations using historical returns. A company’s risk profile may change at any time as the company moves in and out of different lines of business, issues or retires debt, or takes

Figure 5.6 The Security Market Line (SML)

The security market line clearly depicts the tradeoff between risk and return. At a beta of 0, the required return is the risk-free rate of 2%. At a beta of 1.0, the required return is the market return of 8%. Given these data, the required return on an investment with a beta of 2.0 is 14% and its risk premium is 12% (14% – 2%).

other actions that affect the risk of its common stock. Therefore, betas estimated from historical data may or may not accurately reflect how the company’s stock will perform relative to the overall market in the future. Therefore, the required returns specified by the model can be viewed only as rough approximations. Analysts who use betas commonly make subjective adjustments to the historically determined betas based on other information that they possess.

Despite its limitations, the CAPM provides a useful conceptual framework for evaluating and linking risk and return. Its simplicity and practical appeal cause beta and CAPM to remain important tools for investors who seek to measure risk and link it to required returns in security markets. The CAPM also sees widespread use in corporate finance. Before they spend large sums of money on big investment projects, companies need to know what returns their shareholders require. Many surveys show that the primary method that companies use to determine the required rate of return on their stock is the CAPM.

Concepts in Review

Answers available at  http://www.pearsonhighered.com/smart

1. 5.7 Briefly define and give examples of each of the following components of total risk. Which type of risk matters, and why?

a. Diversifiable risk

b. Undiversifiable risk

2. 5.8 Explain what is meant by beta. What type of risk does beta measure? What is the market return? How is the interpretation of beta related to the market return?

3. 5.9 What range of values does beta typically exhibit? Are positive or negative betas more common? Explain.

4. 5.10 What is the capital asset pricing model (CAPM)? What role does beta play in the model? What is the risk premium? How is the security market line (SML) related to the CAPM?

5. 5.11 Is the CAPM a predictive model? Why do beta and the CAPM remain important to investors?

Traditional Versus Modern Portfolio Management

1. LG 5

2. LG 6

Individual and institutional investors currently use two approaches to plan and construct their portfolios. The traditional approach refers to the less quantitative methods that investors have been using since the evolution of the public securities markets. Modern portfolio theory (MPT) is a more mathematical approach that relies on quantitative analysis to guide investment decisions.

The Traditional Approach

Traditional portfolio management  emphasizes balancing the portfolio by assembling a wide variety of stocks and/or bonds. The typical emphasis is interindustry diversification. This produces a portfolio with securities of companies from a broad range of industries. Investors construct traditional portfolios using security analysis techniques that we will discuss later.

Table 5.5  presents some of the industry groupings and the percentages invested in them by a typical mutual fund that is managed by professionals using the traditional approach. This fund, American Funds’ Growth Fund of America (AGTHX), is an open-end mutual fund with a net asset value of $145.2 billion as of December 31, 2014. Its objective is to invest in a wide range of companies that appear to offer superior opportunities for growth of capital. The Growth Fund of America holds shares of more than 280 different companies and short-term securities issued from a wide range of industries. The AGTHX fund is most heavily invested in information technology, representing 21.7% of the portfolio. The consumer discretionary and health care industries represent 17.9% and 17.8% of the fund’s investment, respectively.

Table 5.5 The Growth Fund Of America (Agthx) Investments In Select Industry Groups As Of December 31, 2014

(Source: Data from The Growth Fund of America, Class A Shares, Quarterly Fund Fact Sheet, December 31, 2014.)

The Growth Fund of America appears to adhere to the traditional approach to portfolio management. Its total portfolio value is $145.2 billion, of which 80.8% ($117.3 billion) is U.S. equities, 10.2% ($14.8 billion) is non-U.S. equities, 0.2% ($290.4 million) is U.S. bonds, and 8.8% ($12.8 billion) is cash & equivalents.

Sector Breakdown	Percentage
Information technology	21.7%
Consumer discretionary	17.9%
Health care	17.8%
Industrials	9.6%
Financials	8.2%
Energy	7.7%
Consumer staples	4.6%
Materials	2.8%
Telecommunication services	0.6%
Utilities	0.1%

Analyzing the stock position of the Growth Fund of America, which accounts for 91% of the fund’s assets, we observe the traditional approach to portfolio management at work. This fund holds numerous stocks from a broad cross-section of the universe of available stocks. The stocks are a mix of large and small companies. The fund’s largest individual holding is Amazon.com Inc., which accounts for 3.7% of the portfolio. Google Inc., the world’s do-everything search engine, ranks second, at 3.3%. The third largest holding, 2.3%, is Gilead Sciences. Although many of the fund’s stocks are those of large, recognizable companies, its portfolio does include stocks of smaller, less recognizable firms.

Those who manage traditional portfolios tend to invest in well-known companies for three reasons. First, fund managers and investors may believe that investing in well-known companies is less risky than investing in lesser-known firms. Second, the securities of large firms are more liquid and are available in large quantities. Third, institutional investors prefer successful, well-known companies because it is easier to convince clients to invest in them. Called window dressing, this practice of loading up a portfolio with successful, well-known stocks makes it easier for institutional investors to sell their services.

Investor Facts

Watch Thy Neighbor’s Portfolio A new study finds that the portfolios held by mutual fund managers who live near each other (e.g., in the same zip code) are more similar than portfolios held by managers whose residences are farther apart (e.g., in the same city but not in the same zip code).

(Source: Veronica K. Pool, Noah Stoffman, and Scott E. Yonker, “The People in Your Neighborhood: Social Interactions and Mutual Fund Portfolios,” Journal of Finance, forthcoming.)

One tendency often attributed to institutional investors during recent years is that of “herding”—investing in securities similar to those held by their competitors. These institutional investors effectively mimic the actions of their competitors. In the case of The Growth Fund of America, for example, its managers would buy stocks in companies that are held by other large, growth-oriented mutual funds. While we don’t know for certain why The Growth Fund of America’s managers bought specific stocks, it is clear that most funds with similar objectives hold many of the same well-known stocks.

Modern Portfolio Theory

During the 1950s, Harry Markowitz, a trained mathematician, first developed the theories that form the basis of modern portfolio theory. In the years since Markowitz’s pioneering work, many other scholars and investment experts have contributed to the theory.  Modern portfolio theory (MPT)  uses several basic statistical measures to develop a portfolio plan. Portfolios formed using MPT principles estimate the average returns, standard deviations, and correlations among many combinations of investments to find an optimal portfolio. According to MPT, the maximum benefits of diversification occur when investors find securities that are relatively uncorrelated and put those securities together in a portfolio. Two important aspects of MPT are the efficient frontier and portfolio betas.

Watch Your Behavior

Don’t Be Underdiversified Many research studies have found that investors tend to be underdiversified, holding too few stocks in their portfolios. Investors tend to invest too heavily in companies that are familiar to them, such as local companies. Underdiversification results in inefficient portfolios that perform worse, earning lower returns (by as much as 3% annually according to one study) and experiencing higher volatility compared to well-diversified portfolios.

The Efficient Frontier

At any point in time, you are faced with hundreds of investments from which to choose. You can form any number of possible portfolios. In fact, using only a few different assets, you could create an unlimited number of portfolios by changing the proportion of each asset in the portfolio.

If we were to create all possible portfolios, calculate the return and risk of each, and plot each risk-return combination on a graph, we would have the feasible, or attainable, set of possible portfolios. This set is represented by the shaded area in  Figure 5.7 . It is the area bounded by ABYOZCDEF. As defined earlier, an efficient portfolio is a portfolio that provides the highest return for

Figure 5.7 The Feasible, or Attainable, Set and the Efficient Frontier

The feasible, or attainable, set (shaded area) represents the risk-return combinations attainable with all possible portfolios; the efficient frontier is the locus of all efficient portfolios. The point O, where the investor’s highest possible indifference curve is tangent to the efficient frontier, is the optimal portfolio. It represents the highest level of satisfaction the investor can achieve given the available set of portfolios.

a given level of risk. For example, let’s compare portfolio T to portfolios B and Y shown in  Figure 5.7 . Portfolio Y appears preferable to portfolio T because it has a higher return for the same level of risk. Portfolio B also “dominates” portfolio T because it has lower risk for the same level of return.

The boundary BYOZC of the feasible set of portfolios represents all efficient portfolios—those portfolios that provide the best tradeoff between risk and return. This boundary is the  efficient frontier . All portfolios on the efficient frontier are preferable to all other portfolios in the feasible set. Any portfolios that would fall to the left of the efficient frontier are not available for investment because they fall outside of the attainable set. For example, anyone would love to buy an investment with an extremely high return and no risk at all, but no such investment exists. Portfolios that fall to the right of the efficient frontier are not desirable because their risk-return tradeoffs are inferior to those of portfolios on the efficient frontier.

We can, in theory, use the efficient frontier to find the highest level of satisfaction the investor can achieve given the available set of portfolios. To do this, we would plot on the graph an investor’s indifference curves. These curves indicate, for a given level of utility (satisfaction), the set of risk-return combinations about which an investor would be indifferent. These curves, labeled I1, I2, and I3 in  Figure 5.7 , reflect increasing satisfaction as we move from I1 to I2 to I3. The optimal portfolio, O, is the point at which indifference curve I2 meets the efficient frontier. The investor cannot achieve the higher utility provided by I3 because there is no investment available that offers a combination of risk and return falling on the curve I3.

If we introduced a risk-free investment-paying return rf into  Figure 5.7 , we could eventually derive the equation for the capital asset pricing model introduced previously. Rather than focus further on theory, let’s shift our attention to the more practical aspects of the efficient frontier and its extensions.

Portfolio Betas

As we have noted, investors strive to diversify their portfolios by including a variety of noncomplementary investments that allow investors to reduce risk while meeting their return objectives. Remember that investments embody two basic types of risk: (1) diversifiable risk, the risk unique to a particular investment, and (2) undiversifiable risk, the risk possessed, at least to some degree, by every investment.

A great deal of research has been conducted on the topic of risk as it relates to security investments. The results show that, in general, to earn a higher return, you must bear more risk. Just as important, however, are research results showing that the positive relation between risk and return holds only for undiversifiable risk. High levels of diversifiable risk do not result in correspondingly high levels of return. Because there is no reward for bearing diversifiable risk, investors should minimize this form of risk by diversifying the portfolio so that only undiversifiable risk remains.

Risk Diversification

As we’ve seen, diversification minimizes diversifiable risk by offsetting the below-average return on one investment with the above-average return on another. Minimizing diversifiable risk through careful selection of investments requires that the investments chosen for the portfolio come from a wide range of industries.

To better understand how diversification benefits investors, let’s examine what happens when we begin with a single asset (security) in a portfolio and then expand the portfolio by randomly selecting additional securities. Using the standard deviation, sp, to measure the portfolio’s total risk, we can depict the behavior of the total portfolio risk as more securities are added in  Figure 5.8 . As we add securities to the portfolio (x-axis), the total portfolio risk (y-axis) declines because of the effects of diversification, but there is a limit to how much risk reduction investors can achieve.

Figure 5.8 Portfolio Risk and Diversification

As more securities are combined to create a portfolio, the total risk of the portfolio (measured by its standard deviation, sp) declines. The portion of the risk eliminated is the diversifiable risk; the remaining portion is the undiversifiable, or relevant, risk.

On average, most of the risk-reduction benefits of diversification can be gained by forming portfolios containing two or three dozen carefully selected securities, but our recommendation is to hold 40 or more securities to achieve efficient diversification. This suggestion tends to support the popularity of investment in mutual funds.

Because any investor can create a portfolio of assets that will eliminate virtually all diversifiable risk, the only  relevant risk  is that which is undiversifiable. You must therefore be concerned solely with undiversifiable risk. The measurement of undiversifiable risk is thus of primary importance.

Calculating Portfolio Betas

As we saw earlier, beta measures the undiversifiable, or relevant, risk of a security. The beta for the market is equal to 1.0. Securities with betas greater than 1.0 are more risky than the market, and those with betas less than 1.0 are less risky than the market. The beta for the risk-free asset is 0.

The  portfolio beta, bp , is merely the weighted average of the betas of the individual assets in the portfolio. You can easily calculate a portfolio’s beta by using the betas of the component assets. To find the portfolio beta, bp, calculate a weighted average of the betas of the individual stocks in the portfolio, where the weights represent the percentage of the portfolio’s value invested in each security, as shown in  Equation 5.4 .

Portfolio beta=(Proportion of portfolio's totaldollar valuein asset 1×Betaforasset 1)+(Proportion of portfolio'stotal dollar value in asset 2×Beta for asset 2)+...+(Proportion of portfolio's total dollar value in assetn×Beta for assetn)=n∑j=1(Proportion ofportfolio's totaldollar valuein assetj×Beta for assetj)Portfolio beta =(Proportion of portfolio's total dollar value in asset 1×Beta for asset 1)+(Proportion of portfolio's total dollar value in asset 2×Beta for asset 2)+...+(Proportion of portfolio's total dollar value in asset n×Beta for asset n)=∑j=1n(Proportion of portfolio's total dollar value in asset j×Beta for asset j)Equation5.4

bp=(w1×b1)+(w2×b2)+...+(wn×bn)=n∑j=1(wj×bj)bp=(w1×b1)+(w2×b2)+...+(wn×bn)=∑j=1n(wj×bj)Equation5.4a

Of course, n∑j=1wj=1,∑j=1nwj=1, which means that 100% of the portfolio’s assets must be included in this computation.

Portfolio betas are interpreted in exactly the same way as individual asset betas. They indicate the degree of responsiveness of the portfolio’s return to changes in the market return. For example, when the market return increases by 10%, a portfolio with a beta of 0.75 will experience a 7.5% increase in its return (0.75 × 10%). A portfolio with a beta of 1.25 will experience a 12.5% increase in its return (1.25 × 10%). Low-beta portfolios are less responsive, and therefore less risky, than high-beta portfolios.

To demonstrate, consider the Austin Fund, a large investment company that wishes to assess the risk of two portfolios, V and W. Both portfolios contain five assets, with the proportions and betas shown in  Table 5.6 . We can calculate the betas for portfolios V and W, bv and bw, by substituting the appropriate data from the table into  Equation 5.4 , as follows.

Table 5.6 Austin Fund’S Portfolios V And W

	Portfolio V	Portfolio W
Asset	Proportion	Beta	Proportion	Beta
1	0.10	1.65	0.10	0.80
2	0.30	1.00	0.10	1.00
3	0.20	1.30	0.20	0.65
4	0.20	1.10	0.10	0.75
5	0.20	1.25	0.50	1.05
Total	1.00		1.00

bv=(0.10×1.65)+(0.30×1.00)+(0.20×1.30)+(0.20×1.10)+(0.20×1.25)=0.165+0.300+0.260+0.220+0.250=1.195≈1.20––––––––––bw=(0.10×0.80)+(0.10×1.00)+(0.20×0.65)+(0.10×0.75)+(0.50×1.05)=0.080+0.100+0.130+0.075+0.525=0.91––––––––––bv=(0.10×1.65)+(0.30×1.00)+(0.20×1.30)+(0.20×1.10)+(0.20×1.25)=0.165+0.300+0.260+0.220+0.250=1.195≈1.20__bw=(0.10×0.80)+(0.10×1.00)+(0.20×0.65)+(0.10×0.75)+(0.50×1.05)=0.080+0.100+0.130+0.075+0.525=0.91__

Portfolio V’s beta is 1.20, and portfolio W’s is 0.91. These values make sense because portfolio V contains relatively high-beta assets and portfolio W contains relatively low-beta assets. Clearly, portfolio V’s returns are more responsive to changes in market returns—and therefore more risky—than portfolio W’s.

Interpreting Portfolio Betas

If a portfolio has a beta of 1.0, the portfolio experiences changes in its rate of return equal to changes in the market’s rate of return. The 1.0 beta portfolio would tend to experience a 10% increase in return if the stock market as a whole experienced a 10% increase in return. Conversely, if the market return fell by 6%, the return on the 1.0 beta portfolio would also fall by 6%.

Table 5.7  lists the expected returns for three portfolio betas in two situations: an increase in market return of 10% and a decrease in market return of 10%. The portfolio with a beta of 2.0 moves twice as much (on average) as the market does. When the market return increases by 10%, the portfolio return increases by 20%. When the market return declines by 10%, the portfolio’s return will fall by 20%. This portfolio would be considered a high-risk, high-return portfolio.

The middle, 0.5 beta portfolio is considered a low-risk, low-return portfolio. This would be a conservative portfolio for investors who wish to maintain a low-risk investment posture. The 0.5 beta portfolio is half as volatile as the market.

A portfolio with a beta of -1.0 moves in the opposite direction from the market. A bearish investor would probably want to own a negative-beta portfolio because this

Table 5.7 Portfolio Betas And Associated Changes In Returns

Portfolio Beta	Changes in Market Return (%)	Change in Expected Portfolio Return (%)
+ 2.0	+ 10.0%	+ 20.0%
	− 10.0%	− 20.0%
+ 0.5	+ 10.0%	+ 5.0%
	− 10.0%	− 5.0%
− 1.0	+ 10.0%	− 10.0%
	− 10.0%	+ 10.0%

Figure 5.9 The Portfolio Risk-Return Tradeoff

As the risk of an investment portfolio increases from 0, the return provided should increase above the risk-free rate, rf. Portfolios A and B offer returns commensurate with their risk, portfolio C provides a high return at a low-risk level, and portfolio D provides a low return for high risk. Portfolio C is highly desirable; portfolio D should be avoided.

type of investment tends to rise in value when the stock market declines, and vice versa. Finding securities with negative betas is difficult, however. Most securities have positive betas because they tend to experience return movements in the same direction as changes in the stock market.

The Risk-Return Tradeoff: Some Closing Comments

Another valuable outgrowth of modern portfolio theory is the specific link between undiversifiable risk and investment returns. The basic premise is that an investor must have a portfolio of relatively risky investments to earn a relatively high rate of return. That relationship is illustrated in  Figure 5.9 . The upward-sloping line shows the  risk-return tradeoff . The point where the risk-return line crosses the return axis is called the  risk-free rate, rf . This is the return an investor can earn on a risk-free investment such as a U.S. Treasury bill or an insured money market deposit account.

As we proceed upward along the risk-return tradeoff line, portfolios of risky investments appear, as depicted by four investment portfolios, A through D. Portfolios A and B are investment opportunities that provide a level of return commensurate with their respective risk levels. Portfolio C provides a high return at a relatively low risk level—and therefore would be an excellent investment. Portfolio D, in contrast, offers high risk but low return—an investment to avoid.

Reconciling the Traditional Approach and MPT

We have reviewed two fairly different approaches to portfolio management: the traditional approach and MPT. The question that naturally arises is which technique should you use? There is no definite answer; the question must be resolved by the judgment of each investor. However, we can offer a few useful ideas.

The average individual investor does not have the resources and the mathematical acumen to implement a total MPT portfolio strategy. But most individual investors can extract and use ideas from both the traditional and MPT approaches. The traditional approach stresses security selection, which we will discuss later in this text. It also emphasizes diversification of the portfolio across industry lines. MPT stresses reducing correlations between securities within the portfolio. This approach calls for diversification to minimize diversifiable risk. Thus, diversification must be accomplished to ensure satisfactory performance with either strategy. Also, beta is a useful tool for determining the level of a portfolio’s undiversifiable risk and should be part of the decision-making process.

We recommend the following portfolio management policy, which uses aspects of both approaches:

· Determine how much risk you are willing to bear.

· Seek diversification among types of securities and across industry lines, and pay attention to how the return from one security is related to that from another.

· Consider how a security responds to the market, and use beta in diversifying your portfolio to keep the portfolio in line with your acceptable risk level.

· Evaluate alternative portfolios to make sure that the portfolio selected provides the highest return for the acceptable level of risk.

Concepts in Review

Answers available at  http://www.pearsonhighered.com/smart

1. 5.12 Describe traditional portfolio management. Give three reasons why traditional portfolio managers like to invest in well-established companies.

2. 5.13 What is modern portfolio theory (MPT)? What is the feasible or attainable set of all possible portfolios? How is it derived for a given group of investments?

3. 5.14 What is the efficient frontier? How is it related to the attainable set of all possible portfolios? How can it be used with an investor’s utility function to find the optimal portfolio?

4. 5.15 Define and differentiate among the diversifiable, undiversifiable, and total risk of a portfolio. Which is considered the relevant risk? How is it measured?

5. 5.16 Define beta. How can you find the beta of a portfolio when you know the beta for each of the assets included within it?

6. 5.17 Explain how you can reconcile the traditional and modern portfolio approaches.