## session 3 - Olivier Brandouy

Figure 7.7 The Opportunity Set of the Debt and Equity Funds ... Figure 7.13 Capital Allocation Lines with .... and HP's Security Characteristic Line (SCL). (). () ().
SESSION 3 Optimal Risky Portfolios and Index Models (chap 7 & 8 BKM) Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS IAE de Paris, Master Finance, 2013-2013

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The Investment Decision Top-down process with 3 steps: 1. Capital allocation between the risky portfolio and riskfree asset 2. Asset allocation across broad asset classes 3. Security selection of individual assets within each asset class

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Diversification and Portfolio Risk Market risk ● Systematic or nondiversifiable Firm-specific risk ● Diversifiable or nonsystematic

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.2 Portfolio Diversification

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Covariance and Correlation Portfolio risk depends on the correlation between the returns of the assets in the portfolio ●

Covariance and the correlation coefficient provide a measure of the way returns of two assets vary ●

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Two-Security Portfolio: Return

wr

+ wEr E

rp

=

rP

= Portfolio Return

D

D

wD = Bond Weight rD

= Bond Return

wE = Equity Weight rE

= Equity Return

E (rp ) = wD E (rD ) + wE E (rE ) Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Two-Security Portfolio: Risk

σ = w σ + w σ + 2 wD wE Cov ( rD , rE ) 2 p

2 D

2 D

2 E

2 E

σ D2 = Variance of Security D σ = Variance of Security E 2 E

Cov ( rD , rE ) = Covariance of returns for

Security D and Security E Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Two-Security Portfolio: Risk Another way to express variance of the portfolio: ●

σ P2 = wD wD Cov(rD , rD ) + wE wE Cov (rE , rE ) + 2wD wE Cov (rD , rE )

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Covariance Cov(rD,rE) = ρ DEσ Dσ E ρ D,E = Correlation coefficient of returns σ D = Standard deviation of returns for Security D σ E = Standard deviation of returns for Security E Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Correlation Coefficients: Possible Values Range of values for ρ 1,2 + 1.0 >

ρ > -1.0

If ρ = 1.0, the securities are perfectly positively correlated If ρ = - 1.0, the securities are perfectly negatively correlated Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Correlation Coefficients When ρDE = 1, there is no diversification

σ P = wEσ E + wDσ D When ρDE = -1, a perfect hedge is possible

σD wE = = 1 − wD σD +σ E Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Three-Asset Portfolio E (rp ) = w1 E (r1 ) + w2 E (r2 ) + w3 E (r3 ) σ p2 = w12σ 12 + w22σ 22 + w32σ 32 + 2 w1w2σ 1, 2 + 2w1w3σ 1,3 + 2 w2 w3σ 2,3

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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The Minimum Variance Portfolio The minimum variance portfolio is the portfolio composed of the risky assets that has the smallest standard deviation, the portfolio with least risk. ●

When correlation is less than +1, the portfolio standard deviation may be smaller than that of either of the individual component assets. ●

When correlation is -1, the standard deviation of the minimum variance portfolio is zero. ●

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Correlation Effects The amount of possible risk reduction through diversification depends on the correlation. ● The risk reduction potential increases as the correlation approaches -1. ●

If ρ = +1.0, no risk reduction is possible. If ρ = 0, σP may be less than the standard deviation of either component asset. If ρ = -1.0, a riskless hedge is possible. Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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The Sharpe Ratio Maximize the slope of the CAL for any possible portfolio, P. ● The objective function is the slope: ●

SP =

E (rP ) − rf

σP

The slope is also the Sharpe ratio.

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.8 Determination of the Optimal Overall Portfolio

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.9 The Proportions of the Optimal Overall Portfolio

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Markowitz Portfolio Selection Model Security Selection ● The first step is to determine the riskreturn opportunities available. ● All portfolios that lie on the minimumvariance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.10 The Minimum-Variance Frontier of Risky Assets

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Markowitz Portfolio Selection Model We now search for the CAL with the highest reward-to-variability ratio

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Markowitz Portfolio Selection Model Everyone invests in P, regardless of their degree of risk aversion. ●

More risk averse investors put more in the risk-free asset. Less risk averse investors put more in P.

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Capital Allocation and the Separation Property The separation property tells us that the portfolio choice problem may be separated into two independent tasks ● Determination of the optimal risky portfolio is purely technical. ● Allocation of the complete portfolio to Tbills versus the risky portfolio depends on personal preference. ●

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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The Power of Diversification Remember:

n

n

σ 2P =∑ ∑ w i w j Cov ( r i , r j ) i=1 j =1

If we define the average variance and average covariance of the securities as:

n

σ2=

1 σ 2i ∑ n i =1 n

n

1 Cov= Cov (r i , r j ) ∑ ∑ n (n−1) j =1 j≠i i=1

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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The Power of Diversification We can then express portfolio variance as:

1 2 n −1 σ = σ + Cov n n 2 P

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes

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Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Optimal Portfolios and Nonnormal Returns Fat-tailed distributions can result in extreme values of VaR and ES and encourage smaller allocations to the risky portfolio. ●

If other portfolios provide sufficiently better VaR and ES values than the mean-variance efficient portfolio, we may prefer these when faced with fattailed distributions. ●

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Risk Pooling and the Insurance Principle Risk pooling: merging uncorrelated, risky projects as a means to reduce risk. ●

increases the scale of the risky investment by adding additional uncorrelated assets.

The insurance principle: risk increases less than proportionally to the number of policies insured when the policies are uncorrelated ●

Sharpe ratio increases

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Risk Sharing As risky assets are added to the portfolio, a portion of the pool is sold to maintain a risky portfolio of fixed size. ●

Risk sharing combined with risk pooling is the key to the insurance industry. ●

True diversification means spreading a portfolio of fixed size across many assets, not merely adding more risky bets to an ever-growing risky portfolio. ●

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Investment for the Long Run Long Term Strategy

Short Term Strategy

Invest in the risky portfolio for 2 years.

Invest in the risky portfolio for 1 year and in the riskfree asset for the second year.

Long-term strategy is riskier. Risk can be reduced by selling some of the risky assets in year 2. “Time diversification” is not true diversification.

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

Index Models

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS IAE de Paris, Master Finance, 2013-2013

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Advantages of the Single Index Model Reduces the number of inputs for diversification ●

Easier for security analysts to specialize

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Single Factor Model

ri = E (ri ) + βi m + ei βi = response of an individual security’s return to the common factor, m. Beta measures systematic risk. m = a common macroeconomic factor that affects all security returns. The S&P 500 is often used as a proxy for m. ei = firm-specific surprises Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Single-Index Model ●

Regression Equation:

( ) ( ) ( ) R t = α + β R t + e t i i i M i Expected return-beta relationship: E ( Ri ) = α i + β i E ( RM )

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Single-Index Model Risk and covariance: ●

Variance = Systematic risk and Firm-specific risk: 2 2 2 2 σ = β σ + σ (e ) i betasi x market M Covariance = product of indexi risk:

Cov(ri , rj ) = βi β jσ M2

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Single-Index Model • Correlation = product of correlations with the market index βi β jσ M2 βiσ M2 β jσ M2 Corr (ri , rj ) = = = Corr (ri , rM ) xCorr (r j , rM ) σ iσ j σ iσ M σ jσ M

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Index Model and Diversification Variance of the equally weighted portfolio of firm-specific components: n

2

1 1 ̄2 2 σ ( e P )=∑ ( ) σ (e i )= σ (e ) n i=1 n 2

When n gets large, σ2(ep) becomes negligible and firm specific risk is diversified away.

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient βp

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

Figure 8.2 Excess Returns on HP and S&P 500

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Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 8.3 Scatter Diagram of HP, the S&P 500, and HP’s Security Characteristic Line (SCL)

RHP ( t ) = α HP + β HP RS & P 500 ( t ) + eHP ( t )

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Table 8.1 Interpretation Correlation of HP with the S&P 500 is 0.7238. The model explains about 52% of the variation in HP. HP’s alpha is 0.86% per month(10.32% annually) but it is not statistically significant. HP’s beta is 2.0348, but the 95% confidence interval is 1.43 to 2.53.

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Figure 8.4 Excess Returns on Portfolio Assets

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Alpha and Security Analysis 1. Use macroeconomic analysis to estimate the risk premium and risk of the market index. 2. Use statistical analysis to estimate the beta coefficients of all securities and their residual variances, σ2 (ei).

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Alpha and Security Analysis 3. Establish the expected return of each security absent any contribution from security analysis. 4. Use security analysis to develop private forecasts of the expected returns for each security.

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Single-Index Model Input List 1) Risk premium on the S&P 500 portfolio 2) Estimate of the SD of the S&P 500 portfolio 3) n sets of estimates of ● Beta coefficient ● Stock residual variances ● Alpha values

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Optimal Risky Portfolio of the Single-Index Model Maximize the Sharpe ratio – Expected return, SD, and Sharpe ratio: n+1 n+1 E ( R P )=α P +E (R M )βP =∑ w i α i +E ( R M ) ∑ w i βi i =1

2 P

2 M

2

1/2

[ ( 2

σ P =[ β σ +σ (e p ) ] = σ M

n+1

2

)

i =1 n+1

1/2

∑ wi βi +∑ w i σ (e i ) i =1

i=1

2

2

]

E ( RP) S P= σP Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Optimal Risky Portfolio of the Single-Index Model Combination of: ●

Active portfolio denoted by A

Market-index portfolio, the passive portfolio denoted by M

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

Optimal Risky Portfolio of the Single-Index Model

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Modification of active portfolio position:

0 w A w*A = 1 + (1 − β A ) wA0

When

β A = 1, w = w * A

0 A

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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The Information Ratio The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):

[ ]

αA S =S + σ( e A) 2 P

2 M

2

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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The Information Ratio The contribution of the active portfolio depends on the ratio of its alpha to its residual standard deviation. The information ratio measures the extra return we can obtain from security analysis.

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Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance Matrix

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Table 8.2 Portfolios from the SingleIndex and Full-Covariance Models

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Is the Index Model Inferior to the Full-Covariance Model? Full Markowitz model may be better in principle, but ●

Using the full-covariance matrix invokes estimation risk of thousands of terms. Cumulative errors may result in a portfolio that is actually inferior to that derived from the single-index model. – The single-index model is practical and decentralizes macro and security analysis.

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Beta Book: Industry Version of the Index Model Use 60 most recent months of price data

Use S&P 500 as proxy for M

Compute total returns that ignore dividends

Estimate index model without excess returns:

r = a + brm + e

*

Olivier BRANDOUY, based on INVESTMENTS | BODIE, KANE, MARCUS

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Beta Book: Industry Version of the Index Model The average beta over all securities is 1. Thus, our best forecast of the beta would be that it is 1. ●

Also, firms may become more “typical” as they age, causing their betas to approach 1. ●