efficient diversification chapter 6. diversification and portfolio risk suppose your portfolio has...
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Diversification and Portfolio Risk
Suppose your portfolio has only 1 stock, how many sources of risk can affect your portfolio? Uncertainty at the market level Uncertainty at the firm level
Market risk Systematic or Nondiversifiable
Firm-specific risk Diversifiable or nonsystematic
If your portfolio is not diversified, the total risk of portfolio will have both market risk and specific risk
If it is diversified, the total risk has only market risk
Covariance and Correlation
Why the std (total risk) decreases when more stocks are added to the portfolio?
The std of a portfolio depends on both standard deviation of each stock in the portfolio and the correlation between them
Example: return distribution of stock and bond, and a portfolio consists of 60% stock and 40% bond
state Prob. stock (%) Bond (%) Portfolio
Recession 0.3 -11 16
Normal 0.4 13 6
Boom 0.3 27 -4
Covariance and Correlation What is the E(rs) and σs?
What is the E(rb) and σb?
What is the E(rp) and σp?
E(r) σ
Bond 6 7.75
Stock 10 14.92
Portfolio 8.4 5.92
Covariance and Correlation When combining the stocks into the portfolio, you get the average
return but the std is less than the average of the std of the 2 stocks in the portfolio
Why? The risk of a portfolio also depends on the correlation between 2 stocks How to measure the correlation between the 2 stocks Covariance and correlation
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Covariance and Correlation Prob rs E(rs) rb E(rb) P(rs- E(rs))(rb-
E(rb))
0.3 -11 10 16 6 -630.4 13 10 6 6 00.3 27 10 -4 6 -51
Cov (rs, rb) = -114-114 The covariance tells the direction of the relationship between the 2 assets, but
it does not tell the whether the relationship is weak or strong Corr(rs, rb) = Cov (rs, rb)/ σs σb = -114/(14.92*7.75) = -0.99
Covariance
1,2 = Correlation coefficient of returns
1,2 = Correlation coefficient of returns
Cov(r1r2) = 12Cov(r1r2) = 12
1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2
1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2
Correlation Coefficients: Possible Values
If If = 1.0, the securities would be = 1.0, the securities would be perfectly positively correlatedperfectly positively correlated
If If = - 1.0, the securities would be = - 1.0, the securities would be perfectly negatively correlatedperfectly negatively correlated
If If ρρ = 0, no correlation = 0, no correlation
Range of values for 1,2
-1.0 < < 1.0
Two Asset Portfolio St Dev – Stock and Bond
Deviation Standard Portfolio
Variance Portfolio
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Numerical Example: Bond and Stock
Returns
E(Bond) = 6% E(Stock) = 10%
Standard Deviation
Bond = 12% Stock = 25%
Correlation Coefficient
(Bonds and Stock) = 0
Numerical Example: Bond and Stock
Case 1: Weights Bond = .5 Stock = .5
What is the E(rp) and σp
E(rp) = 8%
σp = 13.87%
Average std = (25+12)/2 = 18.5 By combining stocks, get average return, but the risk is lower
than average
Numerical Example: Bond and Stock
Case 1: Weights Bond = .75 Stock = .25
What is the E(rp) and σp
E(rp) = 7%
σp = 10.96%
• By combining you get higher return than bond but lower risk than bond
• This is power of diversification
Two Asset Portfolio St Dev – Stock and Bond
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• Std of the portfolio is always smaller than the weighted average of the 2 std in the portfolio.
• Std of the portfolio is maximized when the correlation = 1, and is minimized when correlation = -1
•No diversification benefit when the correlation = 1
Extending to Include Riskless Asset The optimal combination becomes linear A single combination of risky and riskless
assets will dominate
Dominant CAL with a Risk-Free Investment (F)
CAL(O) dominates other lines -- it has the best risk/return or the largest slope
Slope = (E(R) - Rf) / E(RP) - Rf) / PE(RA) - Rf) /
Regardless of risk preferences combinations of O & F dominate
Extending Concepts to All Securities
The optimal combinations result in lowest level of risk for a given return
The optimal trade-off is described as the efficient frontier
These portfolios are dominant
Asset Allocation
“Asset allocation accounts for 94% of the differences in pension fund performance”
Identify investment opportunities (risk-return combinations)
Choose the optimal combination according to investor’s risk attitude
Optimal Portfolio Construction
Step 1: Using available risky securities (stocks) to construct efficient frontier.
Step 2: Find the optimal risky portfolio using risk-free asset
Step 3: Now We have a risk-return tradeoff, choose your most favorable asset allocation
The feasible set of portfolios represents all portfolios that can be constructed from a given set of stocks.
An efficient portfolio is one that offers: the most return for a given amount of risk, or
the least risk for a give amount of return.
The collection of efficient portfolios is called the efficient set or efficient frontier.
IB2 IB1
IA2IA1
Optimal PortfolioInvestor A
Optimal Portfolio
Investor B
Risk p
ExpectedReturn, rp
Optimal Portfolios
When a risk-free asset is added to the feasible set, investors can create portfolios that combine this asset with a portfolio of risky assets.
The straight line connecting rRF with M, the tangency point between the line and the old efficient set, becomes the new efficient frontier.
What impact does rRF have onthe efficient frontier?
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M Risk, p
Efficient Set with a Risk-Free Asset
The Capital MarketLine (CML):
New Efficient Set
..B
rM^
ExpectedReturn, rp
The Capital Market Line (CML) is all linear combinations of the risk-free asset and Portfolio M.
Portfolios below the CML are inferior.The CML defines the new efficient set.All investors will choose a portfolio on the
CML.
What is the Capital Market Line?
Disadvantages of the efficient frontier approach
The efficient frontier was introduced by Markowitz (1952) and later earned him a Nobel prize in 1990.
However, the approach involved too many inputs, calculations If a portfolio includes only 2 stocks, to calculate the variance of the
portfolio, how many variance and covariance you need?
If a portfolio includes only 3 stocks, to calculate the variance of the portfolio, how many variance and covariance you need?
If a portfolio includes only n stocks, to calculate the variance of the portfolio, how many variance and covariance you need?
n variances n(n-1)/2 covariances
Single index model
level firm at they uncertaint toduereturn ofcomponent :
levelmarket at they uncertaint toduereturn ofcomponent :
market the toistock of nessresponsive :
intercept :
market of premiumrisk :
istock of premiumrisk :
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Single index model
risk specific :
componentrisk systematic :
risk Total:
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When we diversify, all the specific risk will go away, the only risk left is systematic risk component
22221
2 .......... mnmp
Now, all we need is to estimate beta1, beta2, ...., beta n, and the variance of the market. No need to calculate n variance, n(n-1)/2 covariances as before
Estimate beta
Run a linear regression according to the index model, the slope is the beta
For simplicity, we assume beta is the measure for market risk Beta = 0 Beta = 1 Beta > 1 Beta < 1