Portfolio Diversification
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Transcript of Portfolio Diversification
Portfolio DiversificationPortfolio DiversificationChapters 7 and 8Investments (BKM)
“Don’t put all of your eggs in one basket”
Systematic and specific Systematic and specific riskriskWhat would be the source of risk
of ADIB?1.Systematic risk: general
economy conditions (business cycle, inflation, interest rates, and exchange rates)
2.Specific risk: firm-specific risk What is the risk that we can
reduce?
Portfolio Risk as a Function of the Portfolio Risk as a Function of the Number of Stocks in the PortfolioNumber of Stocks in the Portfolio
Diversiable vs. Diversiable vs. nondiversiable risknondiversiable riskWe cannot eliminate the risk that
comes from common sourcesRisk cannot be reduced to zero by
diversifying our portfolioThe remaining component is:
market risk, or systematic risk, or nondiversifiable risk
The risk that can be eliminatated by diversification is unique risk, or firm-specific risk, or nonsystematic risk, or diversiable risk
Portfolio DiversificationPortfolio DiversificationNYSE stocksEqually-weighted portfolios randomly selected
The power of diversification is limited by systematic risk
W1 = Proportion of funds in Security 1W2 = Proportion of funds in Security 2r1 = return on Security 1r2 = return on Security 2E(): expected return
rp = W1r1 + W2r2
E(rp) = W1E(r1 ) + W2E(r2 )1
n
1iiw
Two-Security Portfolio: Two-Security Portfolio: ReturnReturn
p2 = w1
212 + w2
222 + 2W1W2 Cov(r1r2)
12 = Variance of Security 1
22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for Security 1 and Security 2
Two-Security Portfolio: Two-Security Portfolio: RiskRisk
Risk and returnRisk and returnThe expected return of the portfolio
is a weighted average of the expected returns of the assets that form the portfolio. The weight is the proportion invested in each asset
The variance of the portfolio is not a weighted average of the individual asset variances
The variance is reduced if the covariance term is negative
1,2 = Correlation coefficient of returns
Cov(r1r2) = 1,212
1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2
CovarianceCovariance
ExerciseExerciseCalculate the expected return and the variance of the portfolio that consists of 40% of debt and the remaining in equity
Range of values for 1,2
+ 1.0 > > -1.0
If = 1.0, the securities would be perfectly positively correlated
If = - 1.0, the securities would be perfectly negatively correlated
If then the variance is reduced
Correlation Coefficients: Correlation Coefficients: Possible ValuesPossible Values
Correlation Correlation It is always better to add to your
portfolios assets with lower or, even better, negative correlation with your existing positions
Portfolios of less than perfectly correlated assets always offer better risk-return opportunities than the individual component securities on their own
The lower the correlation between the assets, the greater the gain of diversification
Portfolio variancePortfolio variance
2p = W1
212 + W2
212
+ 2W1W2
rp = W1r1 + W2r2 + W3r3
Cov(r1r2)
+ W323
2
Cov(r1r3)+ 2W1W3
Cov(r2r3)+ 2W2W3
Three-Security PortfolioThree-Security Portfolio
E(rp) = W1E(r1) + W2E(r2) + W3E(r3 )
Correlation and varianceCorrelation and variance
Portfolio Expected Return as a Function Portfolio Expected Return as a Function of Investment Proportionsof Investment Proportions
Portfolio Standard Deviation as a Portfolio Standard Deviation as a Function of Investment ProportionsFunction of Investment Proportions
Portfolio risk and returnPortfolio risk and returnW1 and W2 can be <0 or >1 (short
sell)Portfolio standard deviation
decreases and then increasesWhere is the minimum-variance
portfolio?How much is the variance of the
minimum-variance portfolio? Compare it the variance of the two assets
Portfolio Expected Return as a function Portfolio Expected Return as a function of Standard Deviation of Standard Deviation
Portfolio opportunity set: possible combinations of the two assets
The Opportunity Set of the Debt and The Opportunity Set of the Debt and Equity Funds and Two Feasible CALsEquity Funds and Two Feasible CALs
Optimal risky portfolioOptimal risky portfolioWe should find the weights that
give the highest slope of the Capital Allocation Line (CAL)
The objective function is the slope (Sharpe ratio: reward-to-volatility) of the CAL:
1W WGiven that
)()()(
)(
21
2211
rEWrEWrE
rrES
p
p
fpp
Optimal risky portfolioOptimal risky portfolioIn the case of two risky assets,
the weights of the optimal risky portfolio are:
21
2122
21
2122
11
1
),(])()([])([
),(])([])([
1
2
WW
rrCOVrrErrErrE
rrCOVrrErrEW
fff
ff
Optimal complete Optimal complete portfolioportfolioThe optimal complete portfolio is
formed once the optimal risky portfolio is set
The optimal complete portfolio consists of the optimal risky portfolio and the T-bills
Given the risk aversion A, the proportion invested in the risky portfolio is
pA
rfrpEy
2
)(
Determination of the Optimal Overall Determination of the Optimal Overall PortfolioPortfolio
Steps to form the optimal Steps to form the optimal complete portfoliocomplete portfolio1. Specify the return characteristics
of all securities (expected returns, variances, covariance)
2. Establish the risky portfolio, P (characteristics of P)
3. Allocate funds between risky and the risk-free asset (calculate the proportion invested in each asset)
Do exercise p 222 BKM (concept check 3)
Portfolio selection model: Portfolio selection model: MarkowitzMarkowitzGeneralize the portfolio
construction model to many risky securities and a risk-free asset
First step: determine the minimum-variance frontier: the minimum variance portfolio for any targeted expected return
The Minimum-Variance Frontier of Risky The Minimum-Variance Frontier of Risky AssetsAssets
Minimum-variance Minimum-variance portfolioportfolioThe bottom part of the efficient
frontier is inefficient. Why?Portfolios with the same risk have
different expected returnsSecond step: introduce the risk-
free asset and search for CAL with the highest reward-to-volatility ratio
Find the tangent CAL to the efficient frontier
Optimal complete Optimal complete portfolioportfolioLast step: choose between the
optimal risky portfolio and the risk-free asset
Risk Reduction of Equally Weighted Portfolios Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universesin Correlated and Uncorrelated Universes
ri = E(Ri) + ßiF + eßi = index of a securities’ particular
return to the factorF= some macro factor; in this case F
is unanticipated movement; F is commonly related to security returns
Assumption: a broad market index like the S&P500 is the common factor
Single Factor ModelSingle Factor Model
(ri - rf) = i + ßi(rm - rf) + ei
Risk Prem Market Risk Prem or Index Risk Prem
i= the stock’s expected return if the market’s excess return is zero
ßi(rm - rf) = the component of return due to
movements in the market index
(rm - rf) = 0
ei = firm specific component, not due to market
movements
Single Index Model: Security Single Index Model: Security Market Line (SML)Market Line (SML)
Let: Ri = (ri - rf)
Rm = (rm - rf)
Risk premiumformat
Ri = i + ßi(Rm) + ei
Risk Premium FormatRisk Premium Format
Market or systematic risk: risk related to the macro economic factor or market index.
Unsystematic or firm specific risk: risk not related to the macro factor or market index.
Total risk = Systematic + Unsystematic
Components of RiskComponents of Risk
i2 = i
2 m2 + 2(ei)
where;
i2 = total variance
i2 m
2 = systematic variance
2(ei) = unsystematic variance
Measuring Components of Measuring Components of RiskRisk
Total Risk = Systematic Risk + Unsystematic Risk
Systematic Risk/Total Risk = 2
ßi2
m2 / 2 = 2
Covariance =product of betas*market index risk
Correlation=product of correlations with the market index
Examining Percentage of Examining Percentage of VarianceVariance
Mjiji rrCOV 2),(
jMiMij *
Portfolio construction and Portfolio construction and the single-index modelthe single-index modelOnce we have estimated the SML
for all assets, the securities that will be chosen are those that have the highest Alphas (α)
Positive-Alpha securities are underpriced: long position
Negative-Alpha securities are overpriced: short position
Efficient Frontiers with the Index Model Efficient Frontiers with the Index Model and Full-Covariance Matrixand Full-Covariance Matrix