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ERIC WANG
Chapter6.pptx
6. Efficient Diversification
Instructor: Seongcheol Paeng
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6.1 Diversification and Portfolio Risk
When all risk is firm-specific, as in Figure 6.1A, diversification can reduce risk to low levels.
When a common source of risk affects all firms, however, even extensive diversification cannot eliminate all risk. In Figure 6.1B, portfolio standard deviation falls as the number of securities increases, but it is not reduced to zero.
The risk that remains even after diversification is called market risk, risk that is attributable to market-wide risk sources. Equivalent terms are systematic risk or non-diversifiable risk.
The risk that can be eliminated by diversification is called unique risk, firm-specific risk, nonsystematic risk, or diversifiable risk.
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6.1 Diversification and Portfolio Risk
Figure 6.2 shows the effect of portfolio diversification, using data on NYSE stocks.
The figure shows the average standard deviations of equally weighted portfolios constructed by selecting stocks at random as a function of the number of stocks in the portfolio.
On average, portfolio risk does fall with diversification, but the power of diversification to reduce risk is limited by common sources of risk.
International diversification may further reduce portfolio risk, but here too, global economic and political factors affecting all countries to various degrees will limit the extent of risk reduction.
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6.2 Asset Allocation with Two Risky Assets
Covariance and Correlation
The scenario analysis in Spreadsheet 6.1 posits four possible scenarios for the economy: a severe recession, a mild recession, normal growth, and a boom.
The last row of Spreadsheet 6.1 shows that the expected return of the stock fund is 10% and that of the bond fund is 5%.
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Spreadsheet 6.1 Capital market expectations for the stock and bond funds
6.2 Asset Allocation with Two Risky Assets
Covariance and Correlation
The variance is the probability-weighted average of the squared deviation of actual return from the expected return; the standard deviation is the square root of the variance.
These values are computed in Spreadsheet 6.2.
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Spreadsheet 6.2 Variance and standard deviation of returns
6.2 Asset Allocation with Two Risky Assets
Covariance and Correlation
Suppose we form a portfolio with 40% in stocks and 60% in bonds.
Notice that while the portfolio’s expected return is just the weighted average of the expected return of the two assets, the portfolio standard deviation is actually lower than that of either component fund.
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6.2 Asset Allocation with Two Risky Assets
Covariance and Correlation
(6.1)
The negative value for the covariance indicates that the two assets, on average, vary inversely; when one performs well, the other tends to perform poorly.
(6.2)
Correlation is a pure number and can range from −1 to +1. A correlation of −1 indicates that one asset’s return varies perfectly inversely with the other’s.
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6.2 Asset Allocation with Two Risky Assets
Covariance and Correlation
The correlation coefficient of = −.49 in Equation 6.2 confirms the tendency of the returns on the stock and bond funds to vary inversely.
In fact, a fraction of = .24 of the variance of stocks can be explained by the returns on bonds.
Equation 6.2 shows that whenever the covariance is called for in a calculation we can replace it with the following expression using the correlation coefficient:
(6.3)
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6.2 Asset Allocation with Two Risky Assets
Using Historical Data
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6.2 Asset Allocation with Two Risky Assets
The Three Rules of Two-Risky-Asset Portfolios
: bond fund, stock fund, : rate of return on a portfolio
Rule1: + (6.4)
Rule2: + (6.5)
Rule3: )() (6.6)
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6.2 Asset Allocation with Two Risky Assets
The Risk-Return Trade-Off with Two-Risky-Assets Portfolio
=5%; =8%; =10%; =19%; =.2
=40%; =60%
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6.2 Asset Allocation with Two Risky Assets
The Risk-Return Trade-Off with Two-Risky-Assets Portfolio
What the analyst can and must do is show investors the entire investment opportunity set.
investment opportunity set: Set of available portfolio risk-return combinations.
We find the investment opportunity set using Spreadsheet 6.5. Columns A and B set out several different proportions for investments in the stock and bond funds.
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Spreadsheet 6.5
6.2 Asset Allocation with Two Risky Assets
The Mean-Variance Criterion
These risk-return combinations are plotted in Figure 6.3.
Investors desire portfolios that lie to the “northwest” in Figure 6.3. These are portfolios with high expected returns (toward the “north” of the figure) and low volatility (to the “west”).
These preferences mean that we can compare portfolios using a mean-variance criterion in the following way:
Portfolio A is said to dominate portfolio B if all investors prefer A over B.
This will be the case if it has higher mean return and lower variance or standard deviation:
and
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Figure 6.3 The investment opportunity set with the stock and bond funds
6.2 Asset Allocation with Two Risky Assets
The Mean-Variance Criterion
For example, the stock fund in Figure 6.3 dominates portfolio Z; the stock fund has higher expected return and lower volatility.
Portfolios that lie below the minimum-variance portfolio in the figure can therefore be rejected out of hand as inefficient.
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Figure 6.3 The investment opportunity set with the stock and bond funds
6.2 Asset Allocation with Two Risky Assets
The Mean-Variance Criterion
Figure 6.4 shows the opportunity set with perfect positive correlation—a straight line through the component securities.
No portfolio can be discarded as inefficient in this case, and the choice among portfolios depends only on risk aversion.
Perfect positive correlation is the only case in which there is no benefit from diversification.
Whenever ρ < 1, the portfolio standard deviation is less than the weighted average of the standard deviations of the component securities.
Therefore, there are benefits to diversification whenever asset returns are less than perfectly positively correlated.
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Figure 6.4 Investment opportunity sets for bonds and stocks with various correlation coefficients
6.2 Asset Allocation with Two Risky Assets
The Mean-Variance Criterion
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Spreadsheet 6.6 Investment opportunity set for stocks and bonds with various correlation coefficients
6.3 The Optimal Risky Portfolio with a Risk-Free Asset
When we add the risk-free asset to a stock-plus-bond risky portfolio, the resulting opportunity set is the straight line that we called the CAL (capital allocation line).
We now consider various CALs constructed from risk-free bills and a variety of possible risky portfolios, each formed by combining the stock and bond funds in alternative proportions.
We start in Figure 6.5 with the opportunity set of risky assets constructed only from the bond and stock funds. The lowest-variance risky portfolio is labeled MIN (denoting the minimum-variance portfolio).
is drawn through it and shows the risk-return tradeoff with various positions in T-bills and portfolio MIN.
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Figure 6.5 The opportunity set of stocks, bonds, and a risk-free asset with two capital allocation lines
6.3 The Optimal Risky Portfolio with a Risk-Free Asset
Spreadsheet 6.6 (see bottom panel of column E) shows that portfolio MIN’s expected return is 5.46% and its standard deviation (SD) is 7.80%. Portfolio A (row 10 in Spreadsheet 6.6) offers an expected return of 6% with an SD of 8.07%.
(6.8)
; (6.9)
optimal risky portfolio: The best combination of risky assets to be mixed with safe assets to form the complete portfolio.
. (6.10)
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Figure 6.5 The opportunity set of stocks, bonds, and a risk-free asset with two capital allocation lines
6.3 The Optimal Risky Portfolio with a Risk-Free Asset
Figure 6.6 clearly shows the improvement in the risk-return trade-off obtained with .
To find the composition of the optimal risky portfolio, O, we search for weights in the stock and bond funds that maximize the portfolio’s Sharpe ratio.
Using the risk premiums (expected excess return over the risk-free rate) of the stock and bond funds, their standard deviations, and the correlation between their returns in Equation 6.10, we find that the weights of the optimal portfolio are = .568 and = .432.
Equations 6.5, 6.6, and 6.8 imply that E() = 7.16%, = 10.15%, and therefore the Sharpe ratio of the optimal portfolio (the slope of its CAL) is
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Figure 6.6 The optimal capital allocation line with bonds, stocks, and T-bills
6.3 The Optimal Risky Portfolio with a Risk-Free Asset
Figure 6.7 shows one possible choice for the preferred complete portfolio, C. The investor places 55% of wealth in portfolio O and 45% in Treasury bills. The rate of return and volatility of the portfolio are
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Figure 6.7 The complete portfolio
6.3 The Optimal Risky Portfolio with a Risk-Free Asset
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Figure 6.8 The composition of the complete portfolio: The solution to the asset allocation problem
Figure 6.8 depicts the overall asset allocation.
The allocation reflects considerations of both efficient diversification (the construction of the optimal risky portfolio, O) and risk aversion (the allocation of funds between the risk-free asset and the risky portfolio O to form the complete portfolio, C).
6.4 Efficient Diversification with Many Risky Asset
The Efficient Frontier of Risky Assets
Now we can continue to take other points (each representing portfolios) from these three curves and further combine them into new portfolios, thus shifting the opportunity set even farther to the northwest.
You can see that this process would work even better with more stocks.
Moreover, the boundary or “envelope” of all the curves thus developed will lie quite away from the individual stocks in the northwesterly direction, as shown in Figure 6.10.
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Figure 6.9 Portfolios constructed with three stocks (A, B, and C)
6.4 Efficient Diversification with Many Risky Asset
The Efficient Frontier of Risky Assets
The analytical technique to derive the efficient set of risky assets was developed by Harry Markowitz in 1951 and ultimately earned him the Nobel Prize in Economics. We sketch his approach here.
The graph that connects all the northwesternmost portfolios is called the efficient frontier of risky assets.
It represents the set of portfolios that offers the highest possible expected rate of return for each level of portfolio standard deviation.
These portfolios may be viewed as efficiently diversified. One such frontier is shown in Figure 6.10.
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Figure 6.10 The efficient frontier of risky assets and individual assets
6.4 Efficient Diversification with Many Risky Asset
The Efficient Frontier of Risky Assets
The three ways to draw the efficient frontier are (1) maximize the risk premium for any level of SD; (2) minimize the SD for any level of risk premium; and (3) maximize the Sharpe ratio for any level of SD (or risk premium).
Choosing the Optimal Risky Portfolio
The second step of the optimization plan involves the risk-free asset. Using the current risk-free rate, we search for the capital allocation line with the highest Sharpe ratio (the steepest slope), as shown in Figures 6.5 and 6.6.
The CAL formed from the optimal risky portfolio (O) will be tangent to the efficient frontier of risky assets discussed above. This CAL dominates all feasible CALs.
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6.4 Efficient Diversification with Many Risky Asset
The Preferred Complete Portfolio and a Separation Property
Finally, in the third step, each investor chooses the appropriate mix between the optimal risky portfolio (O) and T-bills, exactly as in Figure 6.7.
A portfolio manager will offer the same risky portfolio (O) to all clients, no matter what their degrees of risk aversion.
Risk aversion comes into play only when clients select their desired point on the CAL.
Regardless of risk aversion, all clients will use portfolio O as the optimal risky investment vehicle.
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Figure 6.7 The complete portfolio
6.4 Efficient Diversification with Many Risky Asset
The Preferred Complete Portfolio and a Separation Property
This result is called a separation property, introduced by James Tobin (1958), the 1983 Nobel Laureate for Economics: Its name reflects the fact that portfolio choice can be separated into two independent tasks.
The first task, to determine the optimal risky portfolio (O), is purely technical. Given the input data, the best risky portfolio is the same for all clients regardless of risk aversion.
The second task, construction of the complete portfolio from bills and portfolio O, is personal and depends on risk aversion. Here the client is the decision maker.
When different managers use different input data, they will develop different efficient frontiers and offer different “optimal” portfolios.
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6.4 Efficient Diversification with Many Risky Asset
Constructing the Optimal Risky Portfolio: An Illustration
To illustrate how the optimal risky portfolio might be constructed, suppose an analyst wished to construct an efficiently diversified global portfolio using the stock market indices of six countries.
The top panel of Table 6.1 shows the input list.
Examination of the table shows the U.S. index portfolio has the highest Sharpe ratio. China and Japan have the lowest, and the correlation of France and Germany with the U.S. is high.
Panel B shows the efficient frontier developed as follows: First we generate the global minimum-variance portfolio G by minimizing the SD with just the feasibility constraint, and then we find portfolio O by maximizing the Sharpe ratio subject only to the same constraint.
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6.4 Efficient Diversification with Many Risky Asset
Constructing the Optimal Risky Portfolio: An Illustration
A short sale is the sale of an asset or stock the seller does not own. It is generally a transaction in which an investor sells borrowed securities in anticipation of a price decline; the seller is then required to return an equal number of shares at some point in the future.
Observe that the SD of the global minimum-variance portfolio of 10.94% is far lower than that of the lowest-variance country (the U.K.), which has an SD of 14.93%.
Moreover, the Sharpe ratio of this portfolio is higher than that of all countries but the U.S! Still, even this portfolio will be rejected in favor of the highest Sharpe-ratio portfolio.
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6.4 Efficient Diversification with Many Risky Asset
Constructing the Optimal Risky Portfolio: An Illustration
Portfolio O attains a Sharpe ratio of .4477, compared to the U.S. ratio of .4013, a significant improvement that can be verified from the CAL shown in Panel C.
The points in Panel C are selected to have the same SD as those on the efficient frontier portfolios, so the risk premium for each equals the SD times the Sharpe ratio of portfolio O.
Panel D shows the efficient frontier when an additional constraint is applied to each portfolio, namely, that all weights must be nonnegative.
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6.4 Efficient Diversification with Many Risky Asset
Constructing the Optimal Risky Portfolio: An Illustration
Take a look at the two frontiers in Figure 6.11. The no-short-sale frontier is clearly inferior on both ends.
This is because both very low-return and very high-return frontier portfolios will typically entail short positions.
Without short sales, we cannot achieve lower or higher risk premiums than are offered by these portfolios.
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Figure 6.11 Efficient frontier and CAL from Table 6.1
6.5 A Single-Index Stock Market
We started this chapter with the distinction between systematic and firm-specific risk.
Index models are statistical models designed to estimate these two components of risk for a particular security or portfolio.
The first to use an index model to explain the benefits of diversification was another Nobel Prize winner, William F. Sharpe (1963).
The popularity of index models is due to their practicality.
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6.5 A Single-Index Stock Market
Index Model
+ + (6.11)
Let us use to denote the excess return on a security, that is, the rate of return in excess of the risk-free rate: = -
is the excess return on a broad market index (the S&P 500 is commonly used for this purpose), so variation in this term reflects the influence of economywide or macroeconomic events that generally affect all stocks to greater or lesser degrees.
The security’s beta, , is the typical response of that particular stock’s excess return to changes in the market index’s excess return.
The term in Equation 6.11 represents the impact of firm-specific or residual risk. The expected value of is zero, as the impact of unexpected events must average out to zero.
The term in Equation 6.11 is not a risk measure. Instead, represents the expected return on the stock beyond any return induced by movements in the market index. This term is called the security alpha.
A positive alpha is attractive to investors and suggests an underpriced security: Among securities with identical sensitivity (beta) to the market index, securities with higher alpha values will offer higher expected returns.
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6.5 A Single-Index Stock Market
Index Model
Because the firm-specific component of the stock return is uncorrelated with the market return, we can write the variance of the excess return of the stock as
Variance () = Variance ( + + )
= Variance ( ) + Variance ()
= + ()
= Systematic risk + Firm-specific risk (6.12)
Therefore, the total variance of the rate of return of each security is a sum of two components:
1. The variance attributable to the uncertainty of the entire market. This variance depends on both the variance of , denoted , and the beta of the stock on .
2. The variance of the firm-specific return, , which is independent of market performance.
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6.5 A Single-Index Stock Market
Statistical and Graphical Representation
Figure 6.12, which shows a scatter diagram of 12 observations of Dana’s excess return paired against the excess return of the market index.
Regression analysis uses a sample of historical returns to estimate the coefficients (alpha and beta) of the index model.
The algorithm finds the regression line, shown in Figure 6.12, that minimizes the sum of the squared deviations around it.
Hence, we say the regression line “best fits” the data in the scatter diagram. The line is called the security characteristic line (SCL).
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Figure 6.12 Scatter diagram for Dana Computer Corp.
6.5 A Single-Index Stock Market
Statistical and Graphical Representation
The regression line does not represent actual returns; points on the scatter diagram almost never lie exactly on the regression line.
Rather, the line represents average tendencies; it shows the expectation of given the market excess return, .
The algebraic representation of the regression line is
E(|) = + (6.13)
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Figure 6.12 Scatter diagram for Dana Computer Corp.
6.5 A Single-Index Stock Market
Statistical and Graphical Representation
The dispersion of the scatter of actual returns about the regression line is determined by the residual variance ().
The magnitude of firm-specific risk varies across securities. One way to measure the relative importance of systematic risk is to measure the ratio of systematic variance to total variance.
(6.14)
where ρ is the correlation coefficient between and .
At the extreme, when the correlation coefficient is either 1 or −1, the security return is fully explained by the market return and there are no firm-specific effects.
When the correlation coefficient is small (in absolute value terms), the market factor plays a relatively unimportant part in explaining the variance of the asset, and firm-specific factors dominate.
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6.5 A Single-Index Stock Market
Diversification in a Single-Index Security Market
It is quite different with firm-specific risk. Consider a portfolio of n securities with weights, (where = 1), in securities with nonsystematic risk, . The nonsystematic portion of the portfolio return is
Because the firm-specific terms, , are uncorrelated, the portfolio nonsystematic variance is the weighted sum of the individual firm-specific variances:
(6.15)
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Assignments
Problem Sets (Paraphrase with your own words.)
1. Explain Covariance and Correlation in Asset Allocation with Two Risky Assets.
2. Explain the Mean-Variance Criterion.
3. Explain optimal risky portfolio and draw the graph.
4. Figure 6.7 shows one possible choice for the preferred complete portfolio, C. The investor places 43% of wealth in portfolio O and 57% in Treasury bills. The rate of return and volatility of the portfolio are
5. Explain the Efficient Frontier of Risky Assets, Choosing the Optimal Risky Portfolio, and the Preferred Complete Portfolio and a Separation Property.
6. Explain the Single Index Model.
Deadline: 7/10 (Friday)
Submit it via email to [email protected]
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