Efficient divesification

Efficient Diversification
BUSFIN 1321: Investment Management
Taha Ahsin
University of Pittsburgh

Overview
• Asset Allocation
• Optimal Risky Portfolio
• Single-Index Stock Market

Asset Allocation with Two Risky Assets

Diversification and Portfolio Risk
• Stock returns will be subject to two sources
of risk:
▶ General economic conditions: business
cycle, inflation, interest rates, etc.
▶ Firm-specific factors: R&D, management,
etc.
• Diversification can reduce firm-specific risk
→ unique risk, nonsystematic risk,
diversifiable risk
• Risk that all firms are subject to cannot be
diversified away
→ market risk, systematic risk,
non-diversifiable risk
Portfolio risk as a function of the number of stocks in the portfolio
Goal of efficient diversification:
⇒ Construct portfolios that provide the lowest possible risk for any given level of expected return
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

Risk Pooling, Risk Sharing, and Time Diversification
• Risk pooling: Pooling together many independent sources of risk
Total payoff of insurance policies:Var (
n
∑
i=1
xi ) = nσ2
Average payoff of insurance policies:Var ( 1
n
n
∑
i=1
xi ) = σ2
n
• Risk sharing: Reducing exposure to any individual investment
Risk to individual investor:Var ( 1
n
n
∑
i=1
xi ) = σ2
n
• True diversification: Spreading exposure of a fixed amount across multiple sources of uncertainty
• Across time, requires an investor with a two-year horizon to put half as much per year than a one-year horizon
⇒ Halves exposure to the market
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

Covariance and Correlation
• Suppose you desire an optimal combination of two risky portfolios
▶ Desire portfolios with an inverse relationship in performance with one another
▶ This relationship is measured in the covariance
• Covariance: Measures the tendency of asset returns to vary in tandem
Cov (rS , rB ) =
n
∑
i=1
p(i)[rS (i) − E (rS )][rB (i) − E (rB )]
▶ Measures the deviations from the mean of two variables
▶ Sign and magnitude based on whether deviations move together
• Correlation coefficient: Measures degree of covariance of two variables on a scale of -1 to 1
Correlation coefficient = ρSB = Cov (rS , rB )
σS σB
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

The Three Rules of Two-Risky-Assets Portfolios
• The rate of return on a portfolio is the weighted average of returns on the component
securities, with the portfolio proportions as weights
rP = wB rB + wS rS
• 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 ) = wB E (rB ) + wS E (rS )
• The variance of the rate of return on a two-risky-assets portfolio is
σ2
P = (wB σB )2 + (wS σS )2 + 2(wB σB )(wS σS )ρBS
⇒ Portfolio variance depends on individual variance plus the covariance
⇒ Small or negative correlation will reduce the overall portoflio risk
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

The Risk-Return Trade-Off with Two-Risky-Assets Portfolios
• Analyst can present investors with the investment opportunity set
▶ Set of available portfolio risk-return combinations
▶ Shape will vary depending on correlation between assets
▶ Mean-variance criterion (MVC): Investors will prefer assets with higher mean and lower variance
E (rA) ≥ E (rB ) and σA ≤ σB
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

The Risk-Return Trade-Off with Two-Risky-Assets Portfolios
• Analyst can present investors with the investment opportunity set
• Portfolios below the upward sloping line should be rejected by MVC
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

The Optimal Risky Portfolio with a
Risk-Free Asset

The Optimal Risky Portfolio with a Risk-Free Asset
• Suppose we add T-bills as an additional investment (Risk-Free Asset)
• Can construct CALs using the risk-free asset and any portfolio on the opportunity set
• We can then find the CAL with the highest Sharpe ratio
• Optimal portfolio: The best combination of risky assets to be mixed with safe assets when
forming the complete portfolio
• Using calculus, the optimal portfolio weight is:
wB = [E (rB ) − rf ]σ2
S − [E (rS ) − rf ]σB σS ρBS
[E (rB ) − rf ]σ2
S + [E (rS ) − rf ]σ2
B − [E (rB ) − rf + E (rS ) − rf ]σB σS ρBS
wS = 1 − wB
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

Incorporating Investor Risk-Aversion
• All investors will choose the optimal portfolio as their risky portfolio
• All investors will want to lie on the CAL because it returns the steepest reward for risk
• Allocation between the safe asset and the optimal portfolio will depend on risk aversion
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

The Efficient Frontier of Risky Assets
• With many risky assets, you can improve
investment opportunities
• Efficient frontier: Maximizing expected
return at each level of portfolio volatility
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

Finding Optimal Allocation
• Use Markowitz model to find efficient
frontier:
⇒ Choose a level of SD, maximize on E(r)
⇒ Or choose a level of E(r), minimize on SD
⇒ Additional possible constraints
▶ Short-sale restriction
▶ Min. dividend yield
▶ SRI and ESG constraints
• Find optimal portfolio → CAL with highest
Sharpe ratio
• Choose desired allocation on CAL
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

Separation Property
Separation Property: The property that implies portfolio choice can be separated into two
independent tasks: (1) determination of the optimal risky portfolio, which is a purely technical
problem, and (2) the personal choice of the best mix of the risky portfolio and the risk-free asset.
• Best risky portfolio is the same for all clients regardless of risk aversion
• Risk aversion only matters when selecting desired point on CAL
• Optimization is straight forward → More difficult to use accurate input data
• Managers compete on security analysis used to produce input estimates
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

A Single-Index Stock Market

Index Models
• Efficient frontier complicated by optimization across many assets
• In contrast, an index model focuses on one common systematic factor
• Index model: Model that relates a security’s returns to returns on both a broad market index
and idiosyncratic/firm-specific factors
• Estimates two components of risk for a security or portfolio factor
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

Motivating the Index Model
• Scatter diagram describes typical
relationship
• Slope reflects sensitivty to market conditions
⇒ Steeper line → more responsive
⇒ Deviations → firm-specific factors
Scatter diagram for Disney against the market index
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

Constructing the Index Model
• Excess return Ri = ri − rf , difference between return on security i and risk-free rate
• RM (t ) represents market excess return for month t
• Regression equation:
Ri (t ) = αi + βi RM (t ) + ei (t )
• Alpha (α): The expected excess return when market excess is zero
• Beta (β): The sensitivity of a security’s return to the return on the market index
⇒ Measures systematic risk i.e. the portion of risk common to the entire economy
⇒ Cyclical stock: higher-than-average sensitivity
⇒ Defensive stocks: Less sensitivity (β < 1)
• Residual (ei ): Component of return variance independent of the market factor
• Security characteristic line (SCL): Plot of predicted excess returns (line-of-best-fit)
• Total variance of return is sum of two components
Variance(Ri ) = Variance(αi + βi RM + ei ) = Variance(βi RM ) + Variance(ei )
Variance(Ri ) = Systematic risk + Firm risk
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

Statistical Interpretation
• Index model can be interpreted as a single-variable regression equation
• Regression line represents expectations of Ri given RM
→ Actual returns include a residual component ei
• R-squared: Measure of the relative importance of systematic variance:
R-squared = SystematicVariance
TotalVariance = β2
i σ2
M
β2
i σ2
M + σ2(ei )
• High R-squared: market dominates firm factors in stock return
• Betas exhibit a statistical property called mean reversion
▶ High-β securities tend to exhibit lower β in the future
▶ Can adjust beta estimate toward 1 for better predictive power
▶ Simple method to adjust: weighted average between sample average and 1
βadjusted = [(2/3) × bβ] + [(1/3) × 1]
Asset Allocation Optimal Risky Portfolio Single-Index Stock Market

Treynor-Black Model: Using Security Analysis with the Index Model
• Suppose you want to add security i to a passive portfolio (e.g. index fund M and T-bills)
⇒ Find highest Sharpe ratio between the index fund and the additional security
• First compute the following variable w0
i using an index model with excess returns Ri and RM
and SD for portfolio M equal σM
Ri = αi + βi RM + ei
w0
i = αi /σ2(ei )
RM /σ2
M
• Use w0
i to determine weight wi :
w ∗
i = w0
i
1 + w0
i (1 − βi )
• New Sharpe ratio determined by the information ratio αi
σ(ei ) :
S2
new − S2
M =
h αi
σ(ei )
i2
• Can find the optimal mix between active portfolio and passive portfolio