Efficient Diversification

Risk Premiums and Risk Aversion Degree to which investors are unwilling to accept uncertainty Risk aversion If T-Bill denotes the risk-free rate, rf, and variance, σp2 , denotes volatility of the portfolio returns then: The risk premium of a portfolio is:

Risk Premiums and Risk Aversion To quantify the degree of risk aversion with parameter A: Or:

The Sharpe (Reward-to-Volatility) Measure

ASSET ALLOCATION ACROSS RISKY AND RISK-FREE PORTFOLIOS

Allocating Capital Possible to split investment funds between safe and risky assets Risk free asset: T-bills Risky asset: stock (or a portfolio)

Allocating Capital Issues Examine risk vs return tradeoff Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

The Risky Asset: Example Total portfolio value = $300,000 Risk-free value = $90,000 Risky (Vanguard and Fidelity) = $210,000 Vanguard (V) = 54% Fidelity (F) = 46%

The Risky Asset: Example Vanguard 113,400/300,000 = 0.378 Fidelity 96,600/300,000 = 0.322 Portfolio P 210,000/300,000 = 0.700 Risk-Free Assets F 90,000/300,000 = 0.300 Portfolio C 300,000/300,000 = 1.000

Calculating the Expected Return: Example rf = 7% srf = 0% E(rp) = 15% sp = 22% y = % in p (1-y) = % in rf

Expected Returns for Combinations E(rc) = yE(rp) + (1 - y)rf rc = complete or combined portfolio For example, y = .75 E(rc) = .75(.15) + .25(.07) = .13 or 13%

Variance on the Possible Combined Portfolios = 0, then s p c = Since rf y s s

Combinations Without Leverage = .75(.22) = .165 or 16.5% If y = .75, then = 1(.22) = .22 or 22% If y = 1 = 0(.22) = .00 or 0% If y = 0 s s s

Using Leverage with Capital Allocation Line Borrow at the Risk-Free Rate and invest in stock Using 50% Leverage rc = (-.5) (.07) + (1.5) (.15) = .19 sc = (1.5) (.22) = .33

Investment Opportunity Set with a Risk-Free Investment

Risk Aversion and Allocation Greater levels of risk aversion lead to larger proportions of the risk free rate Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

ASSET ALLOCATION WITH TWO RISKY ASSETS

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 returns on two assets to vary either in tandem or in opposition

Two-Asset Portfolio Return: Stock and Bond

Covariance and Correlation Coefficient

Correlation Coefficients: Possible Values Range of values for r 1,2 -1.0 < r < 1.0 If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated

Two-Asset Portfolio Standard Deviation: Stock and Bond

Two-Risky-Asset Portfolio Rate of return on the portfolio: Expected rate of return on the portfolio:

Two-Risky-Asset Portfolio Variance of the rate of return on the portfolio:

Numerical Example: Bond and Stock Returns Bond = 6% Stock = 10% Standard Deviation Bond = 12% Stock = 25% Weights Bond = .5 Stock = .5 Correlation Coefficient (Bonds and Stock) = 0

Numerical Example: Bond and Stock Returns Return = 8% .5(6) + .5 (10) Standard Deviation = 13.87% [(.5)2 (12)2 + (.5)2 (25)2 + … 2 (.5) (.5) (12) (25) (0)] ½ [192.25] ½ = 13.87

Investment Opportunity Set for Stocks and Bonds

Investment Opportunity Set for Stocks and Bonds with Various Correlations

THE OPTIMAL RISKY PORTFOLIO WITH A RISK-FREE ASSET

Extending to Include Riskless Asset The optimal combination becomes linear A single combination of risky and riskless assets will dominate

Opportunity Set Using Stocks and Bonds and Two Capital Allocation Lines

Dominant CAL with a Risk-Free Investment (F) CAL(O) dominates other lines -- it has the best risk/return ratio or the largest slope Slope =

Optimal Capital Allocation Line for Bonds, Stocks and T-Bills

The Complete Portfolio

The Complete Portfolio – Solution to the Asset Allocation Problem

EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS

Extending Concepts to All Securities The optimal combinations result in lowest level of risk for a given return Markowitz Portfolio Theory a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.

Portfolios Constructed from Three Stocks A, B and C

The Efficient Frontier of Risky Assets and Individual Assets