1. 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.

2. 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)

3. 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

4. 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?

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

6. 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

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

8. 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

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

10. 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

11. 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

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

13. 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

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

15. 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

16. 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:

17. Figure 6.4 The Investment Opportunity Set

18. 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:

19. Table 6.4 Utility Levels for Various Positions in Risky Assets

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

21. Table 6.5 Spreadsheet Calculations of Indifference Curves

22. Figure 6.8 Finding the Optimal Complete Portfolio

23. 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

24. 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

25. 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

26. 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.

27. 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

28. 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

29. 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

30. 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.

31. Figure 7.2 Portfolio Diversification

32. 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)

33. 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)

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

35. 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

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

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

38. Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

39. Figure 7.3 Portfolio Expected Return

40. Figure 7.4 Portfolio Standard Deviation

41. Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

42. 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

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

44. 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

45. Figure 7.7 Debt and Equity Funds with the Optimal Risky Portfolio

46. Figure 7.8 Determination of the Optimal Overall Portfolio

47. Figure 7.9 The Proportions of the Optimal Complete Portfolio

48. 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

49. Figure 7.10 The Minimum-Variance Frontier of Risky Assets

50. 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

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

52. 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

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

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

55. 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

56. Table 7.4 Risk Reduction of Equally Weighted Portfolios

57. 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

58. 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

59. 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