CS代写 Efficient Diversification: Markowitz model

Efficient Diversification: Markowitz model
Source: Bodie, Kane and Marcus, Investments, 12 ed., McGraw-Hill, 2021

Investment Decision

• The investment decision is a top-down process
1. Capital allocation (risky versus risk-free)
2. Asset allocation
3. Security selection
• Optimal risky portfolio construction
• Efficient diversification
• Long-term vs. short-term investment horizons

Diversification and Portfolio Risk
• Market risk
• Attributable to marketwide risk sources
• Remains even after diversification
• Also called systematic or nondiversifiable risk
• Firm-specific risk
• Risk that can be eliminated by diversification
• Also called diversifiable or nonsystematic risk

Portfolio Risk as a Function of the Number of Stocks in the Portfolio
• Panel A: All risk is firm • Panel B: Some risk is specific systematic

Portfolio Diversification

Portfolios of Two Risky Assets
• Expected return
• Weighted average of expected returns on the component securities
• Portfolio risk
• Variance of the portfolio is a weighted sum of covariances, and each weight is the product of the portfolio proportions of the pair of assets

Portfolios of Two Risky Assets: Expected Return
Consider a portfolio made up of equity (stocks) and debt (bonds)…
where rP = rate of return on portfolio
wD = proportion invested in the bond fund wE = proportion invested in the stock fund rD = rate of return on the debt fund
rE = rate of return on the equity fund
rwrwr pDDEE
E (rp )  w D E (rD )  wE E (rE )

Portfolios of Two Risky Assets: Risk • Variance of rP
2 w22 w22 2w w Covr ,r  pDDEEDEDE
• Bond variance
• Equity variance
• Covariance of returns for bond and equity
Covr ,r  DE

Portfolios of Two Risky Assets: Covariance • Covariance of returns on bond and equity
Cov(r,rE)  D DEDE
􏰁 DE = Correlation coefficient of returns
􏰁 D = Standard deviation of bond returns 􏰁 E = Standard deviation of equity returns

Portfolios of Two Risky Assets: Correlation Coefficients 1 • Range of values for correlation coefficient
 1.0  1.0
• If  = 1.0perfectly positively correlated securities
• If  = 0the securities are uncorrelated
• If  = − 1.0perfectly negatively correlated securities

Portfolios of Two Risky Assets: Correlation Coefficients 2 • When ρDE = 1, there is no diversification
P wEE wDD
• When ρDE = −1, a perfect hedge is possible
wE  D 1wD D E

Portfolios of Two Risky Assets: Example — 50%/50% Split • Table 7.1 Descriptive statistics for two mutual funds
Expected return, E(r ) Standard deviation, σ Covariance, Cov(rD, rE) Correlation coefficient, ρDE
• Expected Return:
E(r)w E(r)wE(r) pDDEE
• Variance:
.508%.5013% 10.5% 2 w22 w22 2w w Covr ,r 
.502 122 .502 202 2.5.572172 P  172 13.23%

Portfolio Expected Return

Computation of Portfolio Variance from the Covariance Matrix

Portfolio Standard Deviation

Portfolio Expected Return as a Function of Standard Deviation

The Minimum-Variance Portfolio
• The minimum-variance portfolio has a standard deviation smaller than that of either of the individual component assets
• Risk reduction depends on the correlation:
• If  = +1.0, no risk reduction is possible
• If  = 0, σP may be less than the standard deviation of either component asset
• If  = -1.0, a riskless hedge is possible

The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs
Portfolio A
E(r )8.9% A
A 11.45% Portfolio B
E(r )9.5% B
B 11.70%

The Sharpe Ratio
• Objective is to find the weights wD and wE that result in the highest slope of the CAL
• Thus, our objective function is the Sharpe ratio:
Erp rf p

The Sharpe Ratio: Example
Portfolio A E(r )8.9%
A 11.45%
SAAf .34
 A Portfolio B
8.9%5% 11.45%
E(r )9.5% B
B 11.70%
SBB f .38
9.5%5%  B 11.70%

Debt and Equity Funds with the Optimal Risky Portfolio
Optimal Risky Portfolio
E(r )11% P
Er r Pf
SP  11%5%
14.2%  .42

Determination of the Optimal Complete Portfolio
Optimal Allocation to P A4
Er r Pf
 11%5% .7439 4(14.2%)2

The Proportions of the Optimal Complete Portfolio
Overall Portfolio
E(r)11% y.7439 P
P 14.2% rf 5%
E ( r )  y  E ( r )  (1  y )  r
Overall p f
 .7439 11%  .2561 5%
Overall .743914.2%10.56%
SOverall 9.46%5%.42 10.56%

Optimization Model 1
• Security selection
• Determine the risk-return opportunities available
– Minimum-variance frontier of risky assets
• All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations
– Efficient frontier of risky assets is the portion of the frontier that lies above the global minimum-variance portfolio

The Minimum-Variance Frontier of Risky Assets

Optimization Model 2
• Security selection (continued)
• Search for the CAL with the highest Shape ratio (that is, the steepest slope)
• Individual investor chooses the appropriate mix between the optimal risky portfolio P and T-bills
• Everyone invests in P, regardless of their degree of risk aversion
– More risk averse investors put less in P – Less risk averse investors put more in P

The Efficient Frontier of Risky Assets with the Optimal CAL

Optimization Model 3
• Capital allocation and the separation property
• 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 risk-free versus the risky portfolio depends on personal preference

Capital Allocation Lines with Various Portfolios from the Efficient Set

Optimization Model 4
• The power of diversification
2 wwCovr,r
• Assume we define the average variance and average covariance of the securities as:
1n 2 2
Cov  Covr,r
nn1 ij j1 i1

Optimization Model 5
• The power of diversification (continued)
• We can then express portfolio variance as
2 12 n1Cov
• Portfolio variance can be driven to zero if the average covariance is zero
• The risk of a highly diversified portfolio depends on the covariance of the returns of the component securities

Risk Reduction of Equally Weighted Portfolios
Universe Size n
Portfolio Weights w = 1/n (%)
Standard Deviation (%)
Reduction in σ
Standard Reduction Deviation inσ
50.00 35.36 22.36 20.41 15.81 15.08 11.18 10.91 5.00 4.98
14.64 1.95 0.73 0.27 0.02
50.00 8.17 41.83
36.06 0.70 35.36
33.91 0.20 33.71
32.79 0.06 32.73
21 100 101
4.76 1 0.99
31.86 0.00 31.86
16.67 10 9.09

Optimization Model 6
• Optimal portfolios and non-normal returns
• Fat-tailed distributions can result in extreme values of VaR and ES
– Practice way to estimate values of VaR and ES in the presence of fat tails is called bootstrapping
• If other portfolios provide sufficiently better VaR and ES values than the mean-variance efficient portfolio, we may prefer these when faced with fat-tailed distributions

Risk Pooling, Risk Sharing, and Time Diversification • Risk pooling vs. risk sharing
• Variance of average insurance policy payoff decreases with the number of policies
1n12 Var x  n2
• Variance of the total payoff becomes more uncertain
Var n x n2
  i1 

Time Diversification
• True diversification
• Requires holding fixed the total funds put at risk, and spreading the exposure across multiple sources of uncertainty

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