Chapter 6
Efficient Diversification
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DIVERSIFICATION AND PORTFOLIO RISK
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Suppose your risky portfolio comprises only one stock: stock of Digital Computer Corporation.
What are the sources of risk affecting this “portfolio”?
We can identify two broad sources of uncertainty.
1) The risk from general economic conditions, like business cycles, inflation, interest rates,
exchange rates. None of these economic factors can be predicted with certainty.
2) Firm-specific influences: those factors that affect the firm without affecting other firms. For
example, the success of the firm in research and development, the management style, and the
management philosophy.
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Now consider adding another security to the risky portfolio.
If you invest half of your risky portfolio in ExxonMobil, leaving the other half in Digital, what
happens to portfolio risk?
Because the firm-specific influences on the two stocks are unrelated, this strategy should reduce
portfolio risk.
For example, when oil prices fall, hurting ExxonMobil, computer prices might rise, helping digital.
The two effects are offsetting, which stabilizes portfolio return.
But why stop at only two stocks?
Diversifying into many securities continues to reduce exposure to firm-specific factors, so portfolio
volatility should continue to fall.
Ultimately, however, there is no way to avoid all risk.
All securities are affected by common macroeconomic factors.
We cannot eliminate exposure to general economic risk, no matter how many stocks we hold.
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Figure 6.1 illustrates these concepts.
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When all risk is firm-specific, as in Figure 6.1A, diversification can reduce risk to arbitrarily low
levels.
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With all risk sources independent, and with investment spread across many securities, exposure to
any particular source of risk is negligible.
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The insurance principle: risk reduction by spreading exposure across many independent risk
sources.
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If you invest in many securities and their risks are independent, your risk becomes very small.
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When a common source of risk affects all firms, however, even extensive diversification cannot
eliminate all risk.
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In Figure 6.1B, portfolio standard deviation falls as the number of securities increases, but it is not
reduced to zero.
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Market risk: risk that is attributable to marketwide risk resources. This is the risk that remains after
diversification.
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This risk is also called: systematic risk, or nondiversifiable risk.
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The risk that can be eliminated by diversification is called firm-specific risk, diversifiable risk,
unique risk or nonsystematic risk.
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Figure 6.1 – Portfolio
risk as a function of the
number of stocks in the
portfolio
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ASSET ALLOCATION WITH TWO RISKY ASSETS
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We will study efficient diversification: we will construct portfolios that provide the lowest possible
risk for any given level of expected return.
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Portfolios of two risky assets are relatively easy to analyze, and they illustrate the principles and
considerations that apply to portfolios of many assets.
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So we will begin with a two–asset risky portfolio: we assume the two assets are a bond fund and a
stock fund.
Covariance and Correlation
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To optimally construct a portfolio from risky assets, we need to understand how the uncertainties of
asset returns interact.
A key determinant of portfolio risk is the extent to which the returns on the two assets vary either in
agreement or in opposition.
Portfolio risk depends on the covariance between the returns of the assets in the portfolio.
We will use a simple scenario analysis.
The scenario analysis in Spreadsheet 6.1 shows four possible scenarios for the economy: a
severe recession, a mild recession, normal growth, and a boom.
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Important to know:
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The performance of stocks follows the broad economy, returning, respectively, -37%, -11%, 14%,
and 30% in the four scenarios.
We assume bonds perform best in a mild recession, returning 15% (because falling interest rates
result in capital gains), and in the normally growth scenario, where their return is 8%.
They suffer from defaults in severe recession, resulting in a negative return, -9%, and from inflation
(which leads to higher nominal interest rates) in the boom scenario, where their return is -5%.
Notice that bonds outperform stocks in both the mild and severe recession scenarios.
In both normal growth and boom scenarios, stocks outperform bonds.
The expected return
on each fund equals the probability-weighted average of the outcomes in the
market
four scenarios.
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
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Important to know:
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The variance is the probability-weighted average of the squared deviations of the actual returns
form the expected return.
• The standard deviation is the square root of the variance.
Spreadsheet 6.2 – Variance and standard deviation of returns
• These values are computed in Spreadsheet 6.2.
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What about the risk and return characteristics of a portfolio made up from the stock and bond
funds?
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Consider a portfolio with 40% in stocks and 60% in bonds.
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See spreadsheet 6.3
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The portfolio return is the weighted average of the returns on each fund with weights equal to the
proportion of the portfolio invested in each fund.
Portfolio return in severe recession = (.40 x -37%) + (.60 x -9%) = -20.2%
Portfolio return in mild recession = (.40 x -11%) + (.60 x 15%) = 4.6%
Portfolio return in normal growth = (.40 x 14%) + (.60 x 8%) = 10.4%
Portfolio return in a boom = (.40 x 30%) + (.60 x -5%) = 9%
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We multiply the expected return in each scenario by the probability of that scenario, and we add
these.
(.05 x -20.2%) + (.25 x 4.6%) + (.4 x 10.4%) + (.3 x 9%)
= -1.01 + 1.15 + 4.16% + 2.7% = 7% = the expected return on the whole portfolio.
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Then we find the variance of the whole portfolio = 44.26.
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And we find the standard deviation of the whole portfolio = 6.65
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The standard deviation of the whole portfolio is less than the standard deviation of the pure stocks
portfolio.
6.65 < 18.63
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The standard deviation of the whole portfolio is less than the standard deviation of the portfolio of
pure bonds.
6.65 < 8.27
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Why?
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The low risk of the portfolio is due to the inverse relationship between the performances of the
stock and bond funds.
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In a mild recession, stocks do poorly (-11% return), but this is offset by the large positive return
(+15% return) on the bond fund.
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In the boom scenario, bonds do poorly (-5% return), but stocks do very well (+30% return).
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Portfolio risk is reduced because variations in the returns of the two assets are generally offsetting.
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Spreadsheet 6.3 – Performance of a portfolio invested in the stock and bond funds
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How can we measure the tendency of the returns on the two assets to vary in agreement or in
opposition to each other?
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The two statistics that provide this measure are the covariance and the correlation coefficient.
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We start in Spreadsheet 6.4 with the deviation of the return on each fund from its expected value
(columns C and D).
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In column E, we multiply the stock fund’s deviation from its mean by the bond fund’s deviation from
its mean.
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The product will be positive if both asset returns exceed their respective means or if both are less
than their respective means.
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The product will be negative if one asset has a return that exceeds its mean, and the other asset
has a return that is less than its mean.
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Row 4 shows that in a mild recession, the stock fund return is less than its expected value by 21%,
while the bond fund return exceeds its expected value by 10%.
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Therefore, the product of the two deviations is -21 x 10 = -210, as reported in column E.
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The product of deviations is negative because one asset (stocks) performs poorly while the other
(bonds) performs well.
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Spreadsheet 6.4 – Covariance between the returns of the stock and bond funds
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The probability –weighted average of the products is called covariance.
Covariance: measures the average tendency of the asset returns to vary in agreement.
The formula for the covariance of the returns on the stock and bond funds is given in the equation
below.
Each particular scenario in this equation is labeled or “indexed” by i.
In general, i ranges from scenario 1 to n.
Here, n = 4.
The probability of each scenario is denoted p(i).
The covariance of the stock and bond fund is computed in cell F7 of Spreadsheet 6.4.
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.
The unit of covariance is percent square.
It is difficult to interpret the magnitude of covariance.
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An easier statistic to interpret is the correlation coefficient.
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Correlation coefficient: the covariance divided by the product of the standard deviations of the
returns on each fund.
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We denote the correlation coefficient by the Greek letter rho, ρ.
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The Correlation coefficient can range from -1 to +1.
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The Correlation coefficient of -1 indicates a perfect inverse correlation.
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The Correlation coefficient of +1 indicates a perfect positive correlation.
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A Correlation coefficient of zero indicates that the returns of the two assets are unrelated.
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The correlation coefficient of ρSB = -.49 confirms the tendency of the returns on the stock and bond
funds to vary inversely.
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Using Historical Data
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We can estimate risk and return using historical data.
The idea is that variability and covariability change slowly over time.
Thus, if we estimate these statistics from recent data, our estimates will provide useful predictions
for the near future.
Example
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Consider the 10 annual returns for the two mutual funds presented in the following spreadsheet.
This sample period is shorter than most analysts would use.
But we will pretend that it is adequate to estimate returns and risk with acceptable precision.
In practice, analysts would use higher-frequency data (e.g., monthly or even daily data).
The spreadsheet starts with the returns on the stock fund and the bond fund (columns B and C).
We use Excel to find average returns, standard deviation, covariance, and correlation coefficient
(see rows 18-21).
We also confirm (in cell F14) that covariance is the average value of the cross-product of the two
asset’s deviations from their mean returns.
The correlation coefficient between stock and bond returns in this example is low but
positive(0.20).
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The Three Rules of Two-Risky-Assets Portfolios
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Suppose a proportion denoted by wB is invested in the bond fund.
Suppose the remainder, 1 – wB, denoted by wS, is invested in the stock fund.
The properties of the portfolio are determined by the following three rules governing combinations
of random variables.
Rule 1: The rate of return on a portfolio is the weighted average of returns on the component
securities, with the portfolio proportions as weights.
rP = w B rB + w S rS
Rule 2: The expected rate of return on a portfolio is similarly the weighted average of the expected
returns on the component securities, with the portfolio proportions as weights.
E(rp) = wBE(rB) + wSE(rS)
Rule 3: The variance of the rate of return on a two-risky-assets portfolio is
σ2P = (wBσB)2 + (wSσS)2 + 2(wBσB)(wSσS)ρBS
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ρBS is the correlation coefficient between the returns on the stock and bond funds.
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The variance of a portfolio is the sum of the contributions of the component-security variances plus
a term that involves the correlation coefficient between the returns on the component securities.
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Let us look at the last term.
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When the correlation between the component securities is small or negative, there will be a greater
tendency for returns on the two assets to offset each other.
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This will reduce portfolio risk.
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The equation tells us that portfolio variance is lower when the correlation coefficient is lower.
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There are large potential gains from diversification.
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The Risk-Return Trade-Off with Two-Risky-Assets Portfolios
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Suppose an investor estimates the following input data:
E(rB) = 5%; 𝜎𝐵 = 8%; E(rS) = 10%; 𝜎𝑆 = 19%; ρBS = .2
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Currently, all funds are invested in the bond fund.
But the investor is considering a portfolio invested 40% in stock and 60% in bonds.
Using Rule 2, the expected return of this portfolio is
E(rP) = (.4x10%) + (.6x5%) = 7%
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This represents an increase of 2% compared to a bond-only investment.
Using Rule 3, the portfolio standard deviation is
𝜎𝑃 =
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.4x19 2 + .6x8 2 + 2 .4x19 x .6x8 x. 2 = 9.76%
When the investor invests 40% of his portfolio in stocks, he increases his return and he increases
his risk.
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Example
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Suppose we invest 85% in bonds and only 15% in stocks.
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This portfolio has an expected return greater than bonds.
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(.85 x 5%) + (.15 x 10%) = 5.75%.
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5.75% > 5%
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This portfolio has a standard deviation less than bonds.
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We find the portfolio variance:
(.85 x 8)2 + (.15 x 19)2 + 2(.85 x 8)(.15 x 19) x .2 = 62.1
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The portfolio standard deviation is 62.1 = 7.88
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This is less than the standard deviation of either bonds or stocks alone.
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7.88 < 19 (standard deviation of stocks alone)
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7.88 < 8 (standard deviation of bonds alone)
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Taking on a more volatile asset (stocks) actually reduces portfolio risk!
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This shows the power of diversification.
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We find investment proportions that will reduce portfolio risk even further.
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The risk-minimizing proportions are 90.8% in bonds and 9.2% in stocks.
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With these proportions the portfolio standard deviation will be 7.80, and portfolio’s expected return
will be 5.46%.
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Is this portfolio preferable to the one with 15% in the stock fund?
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That depends on how the investor trades off risk against return because the portfolio with the lower
variance also has a lower expected return.
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What the analyst can and must do is show investors the entire investment opportunity set.
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The investment opportunity set: the set of all attainable combinations of risk and return offered by
portfolios formed using the available assets in differing proportions.
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We find the investment opportunity set using Spreadsheet 6.5.
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Columns A and B set out several different proportions for investment in stock and bond funds.
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The next columns present the portfolio expected return and standard deviation corresponding to
each allocation.
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These risk-return combinations are plotted in Figure 6.3.
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Spreadsheet 6.5 – The investment opportunity set with the stock and bond funds
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Figure 6.3 – The investment opportunity set with the stock and bond funds
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The Mean-Variance Criterion
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Investors desire portfolios that lie to the “northwest” in Figure 6.3.
When we move to the “northwest” this means less risk and more expected returns.
The more we move upwards and to the left, the better.
This means more return and less risk.
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:
E(rA) ≥ E(rB) and A ≤ B
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Graphically, when we plot the expected return and standard deviation of each portfolio in Figure 6.3,
portfolio A will lie to the northwest of B.
Given a choice between portfolios A and B, all investors would choose A.
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 be rejected. These are
inefficient.
Any portfolio on the downward-sloping portion of the curve is “dominated” by the portfolio that lies
directly above it on the upward-sloping portion of the curve.
The best choice among the portfolios on the upward-sloping portion of the curve is not very obvious
because, in this region, higher expected return is accompanied by greater risk.
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So far we have assumed that the correlation coefficient between stock returns and bond returns is
0.2. (ρBS = .2)
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We know that diversification is less effective when correlations are higher.
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What would be the implications of perfect positive correlation between bonds and stocks?
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The equation for portfolio variance is σ2P = (wBσB)2 + (wSσS)2 + 2(wBσB)(wSσS)ρBS
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When ρBS = 1,
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With perfect positive correlation, the portfolio standard deviation is a weighted average of the
component security standard deviations.
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In this (and only this) circumstance, there are no gains from diversification.
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Both the portfolio mean and the standard deviation are simple weighted averages.
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Figure 6.4 shows the opportunity set with perfect positive correlation – a straight line through
component securities.
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No portfolio can be discarded as inefficient in this case, and the choice among portfolios depends
on risk aversion.
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Perfect positive correlation is the only case in which there is no benefit from diversification.
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Figure 6.4 – Investment opportunity sets for bonds and stocks with various correlation coefficients
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Figure 6.4A – Investment opportunity sets for bonds and stocks with various correlation coefficients
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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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Our analysis has ranged from very attractive diversification benefits (ρBS < 0) to no benefits at all
(ρBS = 1).
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For ρBS within this range, benefits will be in between.
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A realistic correlation coefficient between stocks and bonds is .2.
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This is based on historical experience.
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ρBS = .2 is a lot better for diversification than perfect positive correlation and a bit worse than zero
correlation.
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Spreadsheet 6.6 – Investment opportunity set for stocks and bonds with various correlation coefficients
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As we vary the correlation coefficient between stocks and bonds, the portfolio standard deviation
changes.
The more correlation between stocks and bonds, the higher the portfolio standard deviation.
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Negative correlation between a pair of assets is also possible.
When correlation is negative, there will be even greater diversification benefits.
Let us start with the extreme.
With a perfect negative correlation, ρBS = -1.
The variance of portfolio P is
σ2P = (wBσB)2 + (wSσS)2 + 2(wBσB)(wSσS)ρBS
σ2P = (wBσB)2 + (wSσS)2 + 2(wBσB)(wSσS)(-1)
σ2P = (wBσB)2 + (wSσS)2 ‒ 2(wBσB)(wSσS)
σ2P = (wBσB – wSσS)2
𝜎𝑃 = 𝐴𝐵𝑆[𝑤𝐵𝜎B – 𝑤𝑆𝜎S]
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ABS = absolute value
With perfect negative correlation, the benefits from diversification stretch to the limit.
The equation above yields the proportions that will reduce portfolio standard deviation all the way to zero.
With our data, this will happen when wB = 70.37% and ws = 29.63%.
This will expose us to zero risk.
When wB = 70.37% and ws = 29.63%, portfolio standard deviation = 0 and expected return of the
portfolio = 6.48%.
But investing 29.63% in stocks, will increase the portfolio expected return from 5% to 6.48%.
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THE OPTIMAL RISKY PORTFOLIO WITH A RISK-FREE ASSET
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Now we expand the asset allocation problem to include a risk-free asset.
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When choosing their capital allocation between risky and risk-free portfolios, investors naturally will
want to work with the risky portfolio that offers the greatest reward for assuming risk.
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We will continue to use the input data from Spreadsheet 6.5.
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Suppose that we are still confined to the risky bond fund and the risky stock fund.
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But we can also invest in T-bills yielding 3%.
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When we add the risk-free asset to a portfolio constructed from stocks and bonds, the resulting
opportunity set is the straight line that we called the CAL (capital allocation line) in Chapter 5.
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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.
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Figure 6.5 – The opportunity set of stocks, bonds, and a risk-free asset with two capital allocation lines
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We start in Figure 6.5 with the opportunity set of risky assets constructed only from the bond and
stock funds. (This is the red curve.)
The lowest-variance risky portfolio is labeled MIN (denoting the minimum-variance portfolio).
CALMIN is drawn through it and shows the risk-return trade-off with various positions in T-bills and
portfolio MIN.
It is immediately evident from the figure that we could do better (i.e., obtain a higher Sharpe ratio)
by using portfolio A instead of MIN as the risky portfolio.
CALA dominates CALMIN in the sense that if offers a higher expected return for any level of
volatility.
The slope of the CAL corresponding to a risky portfolio, P, is the Sharpe ratio of that portfolio, that
is, the ratio of its expected excess return to its standard deviation:
E rp − rf
SP =
σP
With a T-bill rate of 3%, we can find and compare the Sharpe ratios for portfolios A and MIN.
Portfolio A has an investment in the stock fund of 0.2, and an investment in the bond fund of 0.8.
It offers an expected return of 6% with an SD of 8.07% (see row 10 of Spreadsheet 6.6).
Portfolio MIN has an expected return of 5.46 and an SD of 7.8. (See spreadsheet 6.6, E19 and
E20).
The Sharpe ratios of these two portfolios are:
5.46 − 3
6−3
SMIN = 7.80 =.32
SA = 8.07 =.37
Portfolio A has a higher Sharpe ratio.
Portfolio A offers an improvement in the risk-return trade-off, when compared to portfolio MIN.
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But why stop at portfolio A?
We can continue to rotate the CAL upward until it reaches the ultimate point of tangency with the
investment opportunity test.
This produces the CAL with the highest feasible Sharpe ratio.
Therefore, the tangency portfolio in Figure 6.6 is the optimal risky portfolio (denoted portfolio O) to
mix with T-bills.
In general, the optimal risky portfolio is one that generates the steepest CAL.
Optimal risky portfolio: the best combination of risky assets to be mixed with the safe asset when
forming the complete portfolio.
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.
With only two risky assets, we can solve for the optimal portfolio weights using the following
formula:
This equation can be obtained by calculus.
We set the derivative of the Sharpe ratio with respect to wB = 0, to find the maximum Sharpe ratio.
Then we solve for wB. This is the optimal wB.
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Figure 6.6 – The optimal capital allocation line with bonds, stocks, and T-bills
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We find that the weights of the optimal portfolio are wB(O) = .568 and wS(O) = .432.
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Using these weights, we find that E(rO) = 7.16%, σO = 10.15%.
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Therefore, the Sharpe ratio of the optimal portfolio (the slope of its CAL) is
SO=
E rO – rf
σO
=
7.16 − 3
= .41
10.15
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.41 is higher than the Sharpe ratios provided by either the bond or stock portfolios alone.
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The Sharpe ratio of the bond portfolio =
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5−3 2
= = .25
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10 − 3
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The Sharpe ratio of the stock portfolio = 19 = 19 = .368
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In the last chapter, we saw that the preferred complete portfolio formed from a risky portfolio and
risk-free asset depends on risk aversion.
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More risk-averse investors prefer low risk portfolios despite the lower expected return.
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More risk-tolerant investors choose higher risk, higher-expected-return portfolios.
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Both types of investors, however, will choose portfolio O as their risky portfolio as it provides the
highest return per unit of risk.
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It has the steepest capital allocation line.
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Investors will differ only in their allocation of investment funds between portfolio O and the risk-free
asset.
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Figure 6.7 shows one possible choice for the preferred complete portfolio, C.
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The investor places 55% of wealth in portfolio O and 45% in Treasury bills.
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The rate of return and volatility of the portfolio are:
E(rC) = 3% x .45 + 7.16% x .55 = 5.29%
𝜎C = .55 x 10.15% = 5.58%
Figure 6.7 – The complete portfolio
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Because the portfolio O is a mix of the bond fund and stock fund with weights is 56.8% and 43.2%,
the overall asset allocation of the complete portfolio is as follows:
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Figure 6.8 depicts the overall asset allocation.
The allocation reflects:
1. Efficient diversification: the construction of the optimal risky portfolio, O.
2. Risk aversion: the allocation of funds between the risk-free asset and the risky portfolio O, to
form the complete portfolio, C.
Figure 6.8 – The composition of the complete portfolio: The solution to the asset allocation problem
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EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS
The Efficient Frontier of Risky Assets
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To get a sense of how additional risky assets can improve investment opportunities, look at Figure
6.9.
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Points A, B, and C represent the expected returns and standard deviations of three stocks.
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The curve passing through A and B shows the risk-return combinations of portfolios formed from
those two stocks.
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The curve passing through B and C shows portfolios formed from these two stocks.
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Now observe point E on the AB curve and point F on the BC curve.
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These points represent two portfolios chosen from the set of AB and BC combinations.
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The curve that passes through E and F in turn represents portfolios constructed from portfolio E
and F.
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Because E and F are themselves constructed from A, B, and C, this curve shows some of the
portfolios constructed from these three stocks.
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Notice that curve EF extends the investment opportunity set to the northwest, which is the desired
direction.
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Investing in A, B, and C (3 stocks) can increase the return and reduce the standard deviation
compared to investing in two stocks only.
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Figure 6.9 – Portfolios constructed with three stocks (A, B, and C)
• Along the EF curve (the black curve), we are investing in stocks A, B, and C.
• Notice: the black curve goes higher and more to the left than the two red curves.
• This shows the benefits of diversification.
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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.
The analytical technique to derive the efficient set of risky assets was developed by Harry
Markowitz and is often referred to as the Markowitz model.
We sketch his approach here.
First, we determine the risk-return opportunity set.
The aim is to construct from the universe of available securities the northwestern-most portfolios in
terms of expected return and standard deviation.
The input data are the expected returns and standard deviations of each asset in the universe,
along with the correlation coefficients between each pair of assets.
The plot that connects all the northwestern-most 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 Panel A of Figure 6.10.
Efficient frontier: graph representing a set of portfolios that maximizes expected returns at each
level of portfolio volatility.
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Figure 6.10 – The efficient frontier of
risky assets and individual assets. In
Panel B, we maximize E(r) for any
choice of standard deviation. In
Panel C, we minimize standard
deviation for each choice of E(r).
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There are two equivalent ways to produce the efficient frontier.
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We will show these in Panels B and C of Figure 6.10.
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The first method: to maximize the expected return for any level of standard deviation.
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For the first method, maximizing the expected return for any level of risk, we choose a target value for
standard deviation, for example, SD = 12%, and search for the portfolio with the highest possible
expected return consistent with this level of volatility.
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So, we give our optimization software an assignment to maximize expected return subject to two
constraints:
(i) The feasibility constraint: the portfolio weights must sum to 1. Any legitimate portfolio must
have weights that sum to 100%.
(ii) The portfolio SD must match the targeted value, σ = 12%.
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The optimization software searches over all portfolios with σ = 12% and finds the highest feasible
portfolio on the vertical line drawn at σ = 12%.
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This is the portfolio with the highest expected return.
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That portfolio is located at the top of the vertical arrow that rises from SD = 12%.
•
The horizontal tracer line from this point to the vertical axis indicates that for this portfolio, E(r) = 10%.
•
You now have one point on the efficient frontier.
•
Repeat for other target levels of volatility and you will find other points on the frontier.
•
We repeat this for SD = 6, 8, 10, 12, 14, for example.
•
When we “connect the dots” as we have done on Panel B, we can draw a frontier like that in Figure
6.10.
43
•
The second method is to minimize the volatility for any level of expected return.
•
Here, you need to draw a few horizontal lines above the global minimum-variance portfolio G.
•
Portfolios lying below G are inefficient because they offer a lower risk premium and higher variance
than G.
•
In Panel C, we have drawn the first horizontal line at an expected return of 10%.
•
Now the software’s assignment is to minimize the SD subject to the feasibility constraint.
•
We require that E(r) = 10%.
•
So, we seek the portfolio that is farthest to the left along the horizontal line drawn at a level of 10%
•
This is the portfolio with the lowest SD consistent with an expected return of 10%.
•
This portfolio must be at σ = 12%.
•
Repeat tis approach using other expected returns, and you will find other points along the efficient
frontier.
•
We repeat this for different expected returns like 8%, 9%, 10%, 11%, 13% and 14%.
•
Again, we connect the dots as we have done in Panel C, and we will have the frontier of Figure
6.10.
44
•
•
•
•
In practice, various additional constraints may preclude an investor from choosing portfolios on the
efficient frontier.
An institution can be prohibited from taking short positions in any asset.
In this situation, the portfolio manager must add constraints to the optimization program that rule
out negative (short) positions.
Here we add more constraints to the optimization program: all asset weights in the optimal portfolio
must be greater than or equal to zero.
•
•
•
Some clients may want to ensure a minimum dividend yield.
In this case, input data must include a set of expected dividend yields.
The optimization program is given the constraint that the expected portfolio dividend yield will
equal to or exceed the desired level.
•
•
Another common constraint pertains to socially responsible investing (SRI) which rules out
investments in firms engaged in businesses deemed by the investor to be objectionable, for
example, tobacco or arms productions.
SRI would constrain portfolio weights on such firms to be zero.
•
•
•
In principle, portfolio managers can tailor an efficient frontier to meet any particular objective.
Any constraint carries a price tag.
An efficient frontier subject to additional constraints will offer a lower Sharpe ratio.
45
Choosing the Optimal Risky Portfolio
•
There is another way to find the best risky portfolio.
•
We know that that optimal portfolio is the one that maximizes the Sharpe ratio.
•
Rather than solving for the entire efficient frontier, we can proceed directly to determining the
optimal portfolio.
•
We ask our optimization program to maximize the Sharpe ratio subject only to the feasibility
constraint (that portfolio weights sum to 1).
•
The portfolio providing the highest possible Sharpe ratio is the optimal portfolio O.
46
The Preferred Complete Portfolio and a Separation Property
•
Finally, each investor chooses the appropriate mix between the optimal risky portfolio (O) and Tbills, exactly as in Figure 6.7.
•
A portfolio manager will offer some 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 points on the CAL.
•
This result is called a separation property, introduced by James Tobin (1958).
•
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).
•
This is purely technical.
•
Given the input data, the best risky portfolio is the same for all clients regardless of risk aversion.
•
The second task: the construction of the complete portfolio from Treasury bills and portfolio O.
•
This is personal and depends on risk aversion.
•
Here the client is the decision maker.
47
•
In practice, optimal risky portfolios for different clients may vary because of constraints on short
sales, dividend yield, tax considerations, or other client preferences.
•
Our analysis, suggests that a few portfolios may be sufficient to serve the demands of a wide
range of investors.
•
We see here the theoretical basis of the mutual fund industry.
•
The (computerized) optimization technique is the easiest part of portfolio construction.
•
When managers use different input data, they will develop different efficient frontiers and offer
different “optimal” portfolios.
•
Therefore, the real arena of the competition among portfolio managers is in the sophisticated
security analysis that produces the input estimates.
•
The rule of GIGO, garbage in-garbage out applies to portfolio selection.
•
If security analysis is good, analysts can find the real optimal portfolio, O.
•
If security analysis is bad or “garbage”, analysts cannot find the real optimal portfolio.
•
The portfolio that seems optimal is not really optimal.
•
Their output is “garbage”.
•
When security analysis is poor, it is better to use a passive strategy: to invest in a market index.
48
Constructing the Optimal Risky Portfolio: An Illustration
•
Suppose an analyst wished to construct an efficiently diversified global portfolio using the stock
market indices of six countries.
•
Panel A of Table 6.1 shows the input list.
•
The values for standard deviations and the correlation matrix are estimated from historical data.
•
Forecasts of risk premiums are generated from fundamental analysis.
•
Examination of the table shows the U.S. index portfolio has the highest Sharpe ratio (.4013).
•
Given the high Sharpe ratio for the United States in this period, one might be tempted to conclude
that U.S. investors would not have benefited much from international diversification.
•
But even in this sample period, we will see that diversification is beneficial.
•
Panel B shows the efficient frontier developed.
•
First, we generate the global minimum-variance portfolio G by minimizing the SD with just the
feasibility constraint.
•
Then we find portfolio O by maximizing the Sharpe ratio subject only to the same constraint.
•
To fill out the curve, we choose more risk premiums.
•
For each risk premium, we minimize portfolio volatility.
•
Now we have 13 points to draw the graph in Figure 6.11.
•
One of these points is the maximum-Sharpe-ratio portfolio, O.
49
Table 6.1 - Efficient frontiers for international diversification with and without short sales and CAL
with short sales
50
51
52
Figure 6.11 – Efficient frontier and CAL from Table 6.1
53
•
The results are striking.
•
•
•
•
•
The U.S. stock market index has a Sharpe ratio of .4013.
The U.S. stock market index has the highest Sharpe ratio among all the stock indices.
Portfolio O has a Sharpe ratio of .4477.
.4477 > .4013
It is better to invest in portfolio O, than in the U.S. stock market index.
•
The stock market index with the lowest standard deviation is the index of the United Kingdom. It is
.1495.
Look at the minimum variance portfolio, G: it has a standard deviation of .1094.
.1094 < .1493
10.94% < 14.93%
The SD of G is < the SD of the British stock market index.
When you combine the stock market indices of six countries you can reduce the variance and the
standard deviation.
Portfolio G is formed by taking short positions in the stock indices of Germany and France.
This is probably because the stock indices of Germany and France are highly correlated with the
stock indices of U.S.A. and U.K.
Portfolio G has a relatively large position in the relatively low-risk United Kingdom.
•
•
•
•
•
•
54
•
Many institutional investors are prohibited from taking short positions.
•
Individuals may be averse to large short positions.
•
Panel D shows the efficient frontier when an additional constraint is applied to each portfolio,
namely, that all weights must be nonnegative.
•
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.
•
Therefore, the short-sale constrained efficient frontier diverges further from the unconstrainted
frontier for extreme-risk premiums.
55
A SINGLE-INDEX STOCK MARKET
•
Index models: statistical models designed to estimate systematic risk and firm-specific risk for a
particular security or portfolio.
•
The first to use an index to explain the benefits of diversification was Nobel Prize winner, William F.
Sharpe (1963).
•
The popularity of index models is due to their practicality.
•
Index models are simple.
•
The index model asserts that one common systematic factor is responsible for all the covariability
of stock returns, with all other variability due to firm-specific factors.
•
This assumption dramatically simplifies the analysis.
56
•
The intuition that motivates the index model can be seen in Figure 6.12.
•
We begin with a historical sample of paired observations of excess returns on the market index
and a particular security, let’s say shares in Disney.
•
In Figure 6.12, we have 60 pairs of monthly excess returns, one for each month in a five-year
sample.
•
Each dot represents the pair of returns in one particular month.
•
In July 2018, Disney’s excess return was 9.15% while the market’s was 3.16%.
•
To describe the typical relation between the return on Disney’ and the return on the market index,
we fit a straight line through the scatter diagram in Figure 6.12.
•
It is clear from this “line of best fit” that there is a positive relation between Disney’s return and the
market’s.
•
This is evidence for the importance of broad market conditions on the performance of Disney’s
stocks.
•
The slope of the line reflects the sensitivity of Disney’s return to market conditions.
•
A steeper line would imply that its rate of return is more responsive to the market return.
•
On the other hand, the scatter diagram also shows that market conditions are not the entire story.
•
If returns perfectly tracked those of the market, then all return pairs would lie exactly on the line.
•
The scatter of points around the line is evidence that firm-specific events also have a significant
impact on Disney’s return.
57
Figure 6.12 – Scatter diagram for Disney against the market index, and Disney’s security market line
(Monthly data, January 2014 – December 2018)
58
•
•
•
•
•
•
•
•
•
How might we determine the line of best fit?
We use Ri to denote an excess return,
The market index, M, has an excess return of RM = rM – rf.
Disney’s excess return is RDisney = rDisney – rf.
We estimate the line using a single-variable linear regression.
We regress Disney’s excess return on the excess return of the index, RM.
Excess return: rate of return in excess of the risk-free rate.
For any stock i, we denote the excess return in month t by Ri(t) and the market index excess return by
RM(t).
Then the index model can be written as the following regression equation:
Ri(t) = αi + βiRM(t) + ei(t)
•
•
•
•
•
•
•
•
The intercept of this equation is security i’s alpha (denoted by the Greek letter αi).
Alpha (αi): the security’s expected return when the market excess return is zero.
It is the vertical intercept in Figure 6.12.
We can think of alpha as the expected return on the stock in excess of the T-bill rate beyond any return
induced by movements in the broad market.
The slope of the line in Figure 6.12 is called the security’s beta coefficient, βi.
Beta (βi): the amount by which the security excess return tends to increase or decrease for every 1%
increase, or decrease in the excess return on the index.
Beta measures the security’s sensitivity to marketwide economic shocks.
Beta is a natural measure of systematic risk.
59
•
The term ei(t) is the zero-mean, firm-specific surprise in the security return in month t.
•
It is called the residual.
•
The greater the residuals (positive or negative), the wider is the scatter of returns around the
straight line in Figure 6.12.
•
This scatter reflects the impact of firm-specific risk.
•
We also call it residual risk.
•
Both systematic risk and residual risk contribute to the volatility of returns.
60
•
When we estimate the regression line for the scatter diagram in Figure 6.12, we obtain the
following estimates:
Rdisney(t) = .23% + 1.046RM (t) + ei(t)
•
This regression line “best fits” the data in the scatter diagram.
•
We call this line Disney’s security characteristic line (SCL).
•
Security characteristic line (SCL): plot of a security’s predicted excess return given the excess
return of the market.
•
The SCL tells us that on average, Disney’s stock rose an additional 1.046% for every additional
1% return in the stock market index.
•
The average value of the residual is zero.
•
The intercept of the equation is positive.
•
For this sample period, in a month where the market had an excess return of zero, Disney’s
excess return would be predicted to be .23%.
•
Investors naturally will be attracted to stocks with positive values of alpha.
•
Positive values of alpha imply higher average excess returns without the “cost” of any additional
exposure to market risk.
•
For example, α may be large if you think a security is underpriced and therefore offers an attractive
expected return.
61
•
The average beta of all stocks in the economy is 1.
•
The average response of a stock changes in a market index composed of all stocks must be 1-for-1.
•
The beta of the market index is, by definition, 1: The index obviously responds 1-for-1 to changes in
itself.
•
Cyclical stocks: have higher-than-average sensitivity to the broad economy and therefore have
betas greater than 1.
•
Cyclical stocks: stocks whose prices follow the ups and downs of the business cycle.
•
The price of these shares goes up during periods of economic expansion, and the price falls during
downturns.
•
These are also called aggressive stocks.
•
Defensive stocks: stocks with betas that are less than 1.
•
The returns of these stocks respond less than 1-for-1 to market returns.
•
Defensive stocks: stocks that provide stable earnings and consistent returns, even during an
economic downturn.
62
•
The index model is mostly descriptive.
•
The index model separates the realized rate of return on a security into macro (systematic) and
micro (firm-specific) components.
•
The excess rate of return on each security is the sum of three components:
63
•
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 (Ri) = Variance (αi + βiRM + ei)
= Variance (βiRM) + Variance (ei)
= β2i σ 2M + σ2 (ei)
= Systematic risk + Firm - specific risk
•
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 RM and the beta of the stock.
2. The variance of the firm-specific return, ei.
64
STATISTICAL INTERPRETATION OF THE SINGLE INDEX MODEL
•
•
•
•
•
The regression line does not represent actual returns.
Points on the scatter diagram almost never lie exactly on the regression line.
The line represents average tendencies.
It shows the expectation of RDisney given the market excess return, RM.
The algebraic representation of the regression line is:
•
This reads: The expectation of RDisney given a value of RM equals the intercept plus the slope
coefficient times the value of RM.
Because the regression line represents expectations, and these expectations may not be realized
(as Figure 6.12 shows).
The actual returns also include a residual, ei, reflecting the firm-specific component of return.
This surprise is measured by the vertical distance between the point of the scatter diagram and the
regression line.
For example, in July 2018, when the market excess return was 3.16%, we would have predicted
Disney’s excess return to be .23% + 1.046 x 3.16% = 3.53%.
In fact, its actual excess return was 9.15%, resulting in a large positive residual of 5.62% as shown
in Figure 6.12.
Disney must have had very good firm-specific news in that month.
•
•
•
•
•
•
65
•
•
The dispersion of the scatter of actual returns about the regression line is measured by the variance of
the residuals, σ2(e).
The magnitude of this 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.
This is called the R-square of the regression line:
•
•
R-square is the ratio of systematic variance to total variance.
R-square is the proportion of total variance that can be attributed to market fluctuations.
•
•
R-square can take values that range from 0 to 1.
A high R-square tells us that systematic factors dominate firm-specific factors in determining the return
on the stock.
A low R-square tells us that firm-specific factors dominate the systematic factors in determining the
return on the stock.
•
•
•
•
Variance (Ri) = Systematic risk + Firm - specific risk.
•
Total variance = systematic variance + variance of the residuals [ σ2(e) ]
•
Variance of the residuals [ σ2(e) ] represents firm-specific risk.
66
•
R-square tells us how good is the “fit” along the regression line.
•
When the “fit” is good, R-square is high.
•
This means the size of the vertical distances from the regression line is small (σ2(e) is small).
•
This means firm-specific risk is a small portion of total risk.
•
And systematic risk is a large portion of total risk.
•
When the “fit” is bad, R-square is small.
•
This means the size of the vertical distances from the regression is large (σ2(e) is big).
•
This means firm-specific risk is a large portion of total risk.
•
And systematic risk is a small portion of total risk.
67
Example
•
We enter 12 months of hypothetical returns for two stocks and the market index, as well as the
risk-free return.
•
We calculate excess returns for the market index, for stock ABC, and for stock XYZ.
•
We regress the excess returns of stock ABC on the excess returns of the market index.
•
We regress the excess returns of stock XYZ on the excess returns of the market index.
•
For stock ABC the regression results are:
R-square = .252
Alpha = .34%
Beta = .525
•
The equation is RABC(t) = .34% + .525 RM (t) + ei(t)
•
ABC has low systematic risk, with a beta of only .525.
•
Its R-square is .252, indicating that only 25.2% of the variance in its returns was driven by
movements in the market index. (Systematic risk / Total risk = .252)
•
ABC has little systematic risk.
•
ABC has a lot of firm-specific risk.
68
69
Learning From the Index Model
•
Table 6.2 shows index model regressions for a sample of almost 20 large companies.
•
Notice that the average beta of the firms is 1.093, not so far from the expected value of 1.0.
•
The model seems to provide reasonable estimates of systematic risk.
•
As one would expect, firms with high exposure to the state of the macroeconomy (e.g., U.S. Steel,
Marathon Oil, and Amazon) have betas greater than 1.
•
Firms whose business is less sensitive to the macroeconomy (e.g., Newmont Mining, Campbell
Soup, and Starbucks) have betas less than 1.
•
The t-statistic for beta is statistically significant.
•
We can reject the hypothesis that the true beta is zero.
•
We conclude that β ≠ 0.
•
There is strong evidence that returns on these stocks are driven in part by the performance of the
broad market.
•
The alpha estimates were not statistically significant.
•
We would not be able to reject the hypothesis that the true values of alpha were zero.
•
We conclude that α = 0
70
•
The column titled Residual Standard Deviation is the standard deviation of the residual terms, e,
and is our measure of firm-specific risk.
•
Firm-specific risk is high, averaging 6.5% per month.
•
This reminds us of the importance of diversification.
•
The high levels of firm-specific risk are reflected as well in the R-square of these regressions.
•
R-square is 0.253 on the average.
•
R-square is low.
•
This means systematic risk is low.
•
This means firm-specific risk is a relatively large portion of total risk.
•
Therefore, we need to diversify our portfolios.
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