Optimal Risky Portfolio

Question 1

What is Optimizing Portfolio Risk?

Answer 1

• Will not include risk-free security

* Will maximize investor’s utility

Question 2

What is the Calculation Process?

Answer 2

1. Estimate the statistics for prospective security returns.
2. Start with arbitrary portfolio weights.
3. Calculate portfolio expected return and standard deviation.
4. Calculate utility score for portfolio.
5. Change portfolio weights to maximize utility score.

Question 3

What are the results?

Answer 3

• More risk-averse investors’ portfolios should have lower standard deviation and expected returns.
• Less risk-averse investors’ portfolios should have higher standard deviation and expected returns.

Question 4

What are the two methods of Optimal Risky Portfolio with a Risk-Free Security?

Answer 4

◦Mutual fund separation

◦Analysis of all securities at once

Question 5

What is Mutual fund separation?

Answer 5

It refers to the division of investment capital between risk-free and risky assets.

Question 6

How do you determine mutual fund separation?

Answer 6

• Calculate expected return and standard deviation of arbitrary portfolio.
• Calculate reward-to-volatility ratio (or slope of capital allocation line):
• Find portfolio weights that maximize slope of capital allocation line.
• Optimize risk allocation

Question 7

How do you use Portfolio Using All

Securities process?

Answer 7

Same process as optimizing portfolio without risk-free securities:
•Estimate statistics for prospective security returns.
•Calculate portfolio expected return and standard deviation.
•Calculate utility score.
•Maximize utility score (by changing weights).

Question 8

What are some Issues Related to Portfolios?

Answer 8

• Constraints on the portfolio
• Buy-and-hold vs. rebalanced portfolios
• Rebalancing frequency
• Ex-ante vs. ex-post
• Naive vs. efficient diversification

Question 9

How can you Estimate the Inputs?

Answer 9

•Using the Markowitz model ◦Expected returns
◦Covariance between returns
•Estimating numbers ◦Use historical data.
◦Adjust historical estimates using common sense, insight about the future.

Question 10

What is Market Equilibrium?

Answer 10

• Investors combine securities into portfolios to maximize reward-to-volatility ratio.
• Mutual fund separation shows optimal portfolio will have highest ratio. ◦Frontier shape depends on security return statistics.

Question 11

Provide a Frictionless Market Example

Answer 11

• Investors with identical expectations will use same statistics for optimization.
• Investors will include all available risky securities in portfolio.

Question 12

What is asset pricing?

Answer 12

•Risky portfolio weights depend on expected returns and standard deviations. ◦If expected return is high compared to risk, give more weight.
◦When demand increases, security price rises, and

• Risk premiums on securities will be proportional to their risk (in equilibrium).
• Security’s risk is proportional to its covariance with market portfolio returns.

These arguments constitute the capital asset pricing model (CAPM).

Question 13

What is the Capital Asset Pricing Model?

Answer 13

Security’s risk in a market portfolio is proportional to covariance of its returns with those of the market portfolio.
•Cov(ri, rM) is measure of said security’s risk to the portfolio.
•Investors care about risk only in portfolio context, not in totality.
◦What matters is how much is being contributed to their portfolio (i.e., its covariance).

Question 14

What is beta?

Answer 14

• Measures risk of particular security, portfolio, and so on
• Gauges risk a security/portfolio adds to a diversified portfolio
• Can be estimated as Cov(ri, rM) divided by Var(rM)◦Historical betas found by running regression between ri and rM

Question 15

What is the beta of a risk free security?

Answer 15

•Beta of risk-free security: βf = 0◦Contributes no risk when added to market portfolio

Question 16

What is the beta of the market portfolio?

Answer 16

•Beta of market portfolio: βM = 1◦Adds all of its risk to portfolio

Question 17

What are some observations between beta and correclatoin?

Answer 17

•Beta is affected by standard deviation and correlation between security and market.
◦Security with high volatility but low market correlation: low beta
◦Security with low volatility but high market correlation: high beta
•Beta and correlation are not equivalent.
•Beta is composite of correlation and volatility of security returns.
•Risk-free security and market portfolio have 100 percent correlation.
◦Movement in returns is due to changes in market portfolio.

Question 18

What are some Issues with CAPM?

Answer 18

•Objections: ◦Security returns affected by forces other than market.
◦Issue of whether or not it is a theory.
•Alternatives: ◦Arbitrage pricing theory
◦Model that includes market, size, and “value-ness”
It remains popular despite these issues due to its simplicity.