Portfolio Diversification Modern Portfolio Theory

Types of Risk  Total risk when a security is held alone –e.g. standard deviation and variance  Diversifiable Risk –Unique risk, Firm-specific risk, nonsystematic risk  Nondiversifiable Risk –Market risk, Systematic risk  In a “well-diversified” portfolio, the only risk from a security that remains is the systematic risk.

Sample Statistics  Expected return E(r A ) =  s=1 to n p (s) r A(s) Where s is the possible state, p (s) is the probability that state s will occur, r A(s) is the return on asset A when state s occurs  Variance and standard deviation  A 2 =  s=1 to n p(s) [r A (s) - E(r A )] 2  A = square root of  A 2  If ex ante probabilities are not available –Use historic arithmetic average as a proxy for E(r) –Use historic sample standard deviation as a proxy for standard deviation

Effectiveness of Diversification  Covariance between two securities  AB =  s=1 to n p(s) [r A (s) - E(r A )] [r B (s) - E(r B )] –Use historic sample covariance as a proxy  Correlation coefficient between two securities –   =  AB / (  A  B ) –   must be between -1 and 1 –Special cases of     = 1   = 0   = -1  When does diversification work?

Return and Risk of a Portfolio  Return of a Portfolio  r =  i=1 to k w i r i –where k is the number of securities in the portfolio  Variance of a Portfolio   2 =  i=1 to k  j=1 to k w i w j  ij  i  j

Examples of Diversification  The 2-asset Case  Multiple assets and diversification  Portfolio Risk versus Security Risk

Portfolio: An Ex Ante Example Economyp(s)r A (s)r B (s) Boom0.4030%-5% Bust %25%  E(r A ) = 6%, E(r B ) = 13%   A = 19.60%,  B = 14.70%  Covariance between assets A and B: –  A,B = 0.40 x ( ) x ( ) x ( ) x ( ) =  Correlation Coefficient –  A,B = / (0.196 x 0.147) = -1

Portfolio Example (continued)  Portfolio weights –50% in Asset A and 50% in Asset B  E(r) on portfolio = 0.50 x x 0.13 = 9.5%  Portfolio variance = w Y 2  Y 2 + w Z 2  Z w Y w Z  Y,Z = x x x 0.5 x 0.5 x =  Standard Deviation = 2.45%

Portfolio Example (concluded)  Portfolio weights –put 3/7 in asset A and 4/7 in asset B: –An alternative approach, compute portfolio’s return in each scenario Economyp (s) r A(s) r B(s) r portfolio (s) Boom0.4030%-5%10% Bust %25%10% E(r portfolio ) = 10%  portfolio = 0%

The Risky Portfolio: Asset Allocation  Investment Opportunity Set –Different risk-return combinations created using different portfolio weights Even 2 assets can provide an infinite number of combinations!  Minimum Risk Portfolio  Efficient Set (Efficient Frontier) –Portfolios that represent the best risk-return combinations

Investment Opportunity Set: -1 <  < 1 Minimum Variance Portfolio Efficient Set * * * * * *

Investment Opportunity Set:  = 1 Minimum Variance Portfolio Efficient Set

Investment Opportunity Set:  = -1 Minimum Variance Portfolio Efficient Set * * * * *

The Complete Portfolio: Asset Allocation  The Risk-free Asset  The Optimal Risky Portfolio  The Optimal Capital Allocation Line (CAL)  Choosing the complete portfolio

The Optimal Capital Allocation Line Optimal Risky Portfolio * * * * * * Rf = 8% Optimal CAL

Asset Allocation: A Summary  Constructing the Optimal CAL –Construct the Investment Opportunity Set –Identify the Efficient Frontier –Identify the Optimal Risky Portfolio  Choose the Complete Portfolio –Allocate investment between the risk-free asset and the Optimal Risky Portfolio –Choose a point on the Optimal CAL

The Separation Property  Construction of the Optimal CAL is independent of an investor’s risk preference  An investor’s risk preference only affects his choice of the complete portfolio

The Complete Portfolio: An Example  The Optimal Risky Portfolio (P): Assetwt in PE(r) A % B % –E(r P ) =.3765 x 10% x 17% = %  The Risk-free Asset: r f = 8%  Investor John puts 45% in the risk-free asset and 55% in the optimal risky portfolio (P): –E(r C ) =.45 x 8% +.55 x % = 11.5%

The Complete Portfolio Example (continued)  For John’s portfolio (45% in the risk-free asset): –How much of Assets A and B does he own? wt in A =.3765 x.55 =.207 wt in B =.6235 x.55 =.343 wt in risk-free asset =.450  Investor Adam puts 90% in the risk-free asset: –E(r C ) =.90 x 8% +.10 x % = 8.64% wt in A =.3765 x.10 =.038 wt in B =.6235 x.10 =.062 wt in risk-free asset =.900

Asset Weights  Wt of assets in the (Optimal) Risky Portfolio: –Depends only on the assets’ expected return, variances, and covariances  Wt of the (Optimal) Risky Portfolio in the Complete Portfolio: –investor’s risk preference  Wt of assets in the Complete Portfolio: –wt of assets in the Risky Portfolio * wt of the Risky Portfolio in the Complete Portfolio

Asset Pricing Models  The Single-Factor Model –R i = E(R i ) +  i M + e i E(R i ) is the expected excess return on stock i M is the unexpected change in the factor e i is an unexpected firm-specific event

The Market Index Model  Regression and the Market Index Model –Ri = ai + biRM + ei Intercept ( ai): Abnormal return Slope (bi): Sensitivity to the Market Residual (ei): Unexpected firm-specific event RM (X - independent): Excess return on the Market Ri (Y - dependent): Excess return on stock i

Variance in the Market Model  Total Variance  2 i =  i 2  2 M +  2 (e i ) Systematic Variance + Firm-specific Variance  Proportion explained by the Market Model  2 =  i 2  2 M /  2 i Systematic variance / Total variance

Scatter Diagram of the Market Model  =  =  =