Capital Allocation to Risky Assets Chapter Six Capital Allocation to Risky Assets Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter Overview Risk aversion and its estimation Two-step process of portfolio construction Composition of risky portfolio Capital allocation between risky and risk-free assets Passive strategies and the capital market line (CML)

Risk and Risk Aversion Speculation Gamble Taking considerable risk for a commensurate gain Parties have heterogeneous expectations Bet on an uncertain outcome for enjoyment Parties assign the same probabilities to the possible outcomes

Risk and Risk Aversion Utility Values Investors are willing to consider: Risk-free assets Speculative positions with positive risk premiums Portfolio attractiveness increases with expected return and decreases with risk What happens when return increases with risk?

Table 6.1 Available Risky Portfolios Each portfolio receives a utility score to assess the investor’s risk/return trade off

Risk Aversion and Utility Values Utility Function U = Utility E(r) = Expected return on the asset or portfolio A = Coefficient of risk aversion σ2 = Variance of returns ½ = A scaling factor

Table 6.2 Utility Scores of Portfolios with Varying Degrees of Risk Aversion

Estimating Risk Aversion Use questionnaires Observe individuals’ decisions when confronted with risk – Prospect Theory Observe how much people are willing to pay to avoid risk – options market

Estimating Risk Aversion Mean-Variance (M-V) Criterion Portfolio A dominates portfolio B if: and

Capital Allocation Across Risky and Risk-Free Portfolios Asset Allocation The choice among broad asset classes that represents a very important part of portfolio construction The simplest way to control risk is to manipulate the fraction of the portfolio invested in risk-free assets versus the portion invested in the risky assets

Basic Asset Allocation Example Total market value $300,000 Risk-free money market fund $90,000 Equities $113,400 Bonds (long-term) $96,600 Total risk assets $210,000

Basic Asset Allocation Example Let y = Weight of the risky portfolio, P, in the complete portfolio (1-y) = Weight of risk-free assets

The Risk-Free Asset Only the government can issue default-free securities A security is risk-free in real terms only if its price is indexed and maturity is equal to investor’s holding period T-bills viewed as “the” risk-free asset Money market funds also considered risk-free in practice

Figure 6.3 Spread Between 3-Month CD and T-bill Rates

Portfolios of One Risky Asset and a Risk-Free Asset It’s possible to create a complete portfolio by splitting investment funds between safe and risky assets Let y = Portion allocated to the risky portfolio, P (1 - y) = Portion to be invested in risk-free asset, F

One Risky Asset and a Risk-Free Asset: Example rf = 7% E(rp) = 15% rf = 0% p = 22% The expected return on the complete portfolio: The risk of the complete portfolio:

Figure 6.4 The Investment Opportunity Set

Risk Tolerance and Asset Allocation The investor must choose one optimal portfolio, C, from the set of feasible choices Expected return of the complete portfolio: Variance:

Table 6.4 Utility Levels for Various Positions in Risky Assets

Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y

Table 6.5 Spreadsheet Calculations of Indifference Curves

Figure 6.8 Finding the Optimal Complete Portfolio

Passive Strategies: The Capital Market Line The passive strategy avoids any direct or indirect security analysis Supply and demand forces may make such a strategy a reasonable choice for many investors A natural candidate for a passively held risky asset would be a well-diversified portfolio of common stocks such as the S&P 500

Passive Strategies: The Capital Market Line The Capital Market Line (CML) Is a capital allocation line formed investment in two passive portfolios: Virtually risk-free short-term T-bills (or a money market fund) Fund of common stocks that mimics a broad market index

Passive Strategies: The Capital Market Line From 1926 to 2012, the passive risky portfolio offered an average risk premium of 8.1% with a standard deviation of 20.48%, resulting in a reward-to-volatility ratio of .40

Optimal Risky Portfolios Chapter Seven Optimal Risky Portfolios Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter Overview The investment decision: Optimal risky portfolio Capital allocation (risky vs. risk-free) Asset allocation (construction of the risky portfolio) Security selection Optimal risky portfolio The Markowitz portfolio optimization model Long- vs. short-term investing

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

Diversification and Portfolio Risk Market risk Risk attributable to marketwide risk sources and remains even after extensive diversification Also call systematic or nondiversifiable Firm-specific risk Risk that can be eliminated by diversification Also called diversifiable or nonsystematic

Figure 7.1 Portfolio Risk and the Number of Stocks in the Portfolio Panel A: All risk is firm specific. Panel B: Some risk is systematic or marketwide.

Figure 7.2 Portfolio Diversification

Portfolios of Two Risky Assets Portfolio risk (variance) 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 move together (covary)

Portfolios of Two Risky Assets: Return Portfolio return: rp = wDrD + wErE wD = Bond weight rD = Bond return wE = Equity weight rE = Equity return E(rp) = wD E(rD) + wEE(rE)

Portfolios of Two Risky Assets: Risk Portfolio variance: = Bond variance = Equity variance = Covariance of returns for bond and equity

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

Portfolios of Two Risky Assets: Correlation Coefficients Range of values for 1,2 - 1.0 > r > +1.0 If r = 1.0, the securities are perfectly positively correlated If r = - 1.0, the securities are perfectly negatively correlated

Portfolios of Two Risky Assets: Correlation Coefficients When ρDE = 1, there is no diversification When ρDE = -1, a perfect hedge is possible

Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

Figure 7.3 Portfolio Expected Return

Figure 7.4 Portfolio Standard Deviation

Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

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 The amount of possible risk reduction through diversification depends on the correlation: If r = +1.0, no risk reduction is possible If r = 0, σP may be less than the standard deviation of either component asset If r = -1.0, a riskless hedge is possible

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

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

Figure 7.7 Debt and Equity Funds with the Optimal Risky Portfolio

Figure 7.8 Determination of the Optimal Overall Portfolio

Figure 7.9 The Proportions of the Optimal Complete Portfolio

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

Figure 7.10 The Minimum-Variance Frontier of Risky Assets

Markowitz Portfolio Optimization Model Search for the CAL with the highest reward-to-variability ratio 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

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

Markowitz Portfolio Optimization Model 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

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

Markowitz Portfolio Optimization Model The Power of Diversification Remember: If we define the average variance and average covariance of the securities as:

Markowitz Portfolio Optimization Model The Power of Diversification We can then express portfolio variance as Portfolio variance can be driven to zero if the average covariance is zero (only firm specific risk) The irreducible risk of a diversified portfolio depends on the covariance of the returns, which is a function of the systematic factors in the economy

Table 7.4 Risk Reduction of Equally Weighted Portfolios

Markowitz Portfolio Optimization Model 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 fat-tailed distributions

Risk Pooling and the Insurance Principle Merging uncorrelated, risky projects as a means to reduce risk It 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 when the policies are uncorrelated Sharpe ratio increases

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