Week 6 - Portfolio theory & asset pricing

Question 1

What happens when the correlation between the returns on 2 assets…
1. Corr(R1, R2) = 1
2. Corr(R1, R2) = -1?

Answer 1

1. The portfolio’s std dev increases linearly with E(R)
2. We can have a point that gives us a RISK-FREE portfolio. It is beneficial to DIVERSIFY (invest in both assets) - can reduce risk & increase E(R)

Question 2

What is the variance of a risk-free asset? And its covariance with another asset?

*Why we get diff. expected return on asset i? Different betas, different SENSITIVITIES to market risk

Answer 2

Variance = 0, Covariance = 0, Risk-free assets do NOT vary!

So the variance of portfolio just becomes variance of risky asset

Question 3

1. LINEAR equation of the capital allocation line (CAL) / capital market line (CML)
2. What is the gradient called? Axes of the graph?
3. What happens on the line above the risky asset/portfolio A?

Answer 3

1. E(Rp) = rrf + ( (E(RA) - rrf) / σRA ) σRp
   > Represents the risk-return opportunity set an investor can obtain by varying the weights of investing in the risk-free asset and risky asset/portfolio
   *y-intercept = risk-free rate, rrf
2. The gradient = Sharpe ratio. Larger Sharpe ratio is better
   Expected return vs Standard deviation
3. Not investing money with risk-free rate, but now BORROWING money with the risk-free rate and investing more than our wealth (own money + borrowed money) in the risky asset/portfolio.

Question 4

1. LINEAR equation of the capital allocation line (CAL) / capital market line (CML)
2. What is the gradient called? Axes of the graph?
3. What happens on the line above the risky asset/portfolio A?

Answer 4

1. E(Rp) = rrf + ( (E(RA) - rrf) / σRA ) σRp
   > Represents the risk-return opportunity set an investor can obtain by varying the weights of investing in the risk-free asset and risky asset/portfolio
   *y-intercept = risk-free rate, rrf
2. The gradient = Sharpe ratio. Larger Sharpe ratio is better
   Expected return vs Standard deviation
3. Not investing money with risk-free rate, but now BORROWING money with the risk-free rate and investing more than our wealth (own money + borrowed money) in the risky asset/portfolio.

Question 5

What should investors invest in according to CAPM?

Answer 5

Tangency portfolio MAXIMISES the Sharpe ratio (gradient of capital allocation line CAL), thus investors should only hold the tangency portfolio + the risk-free asset

When CAPM holds, ie. in equilibrium

Question 6

CAPM equation & what does it tell us?

Answer 6

CAPM tells us the expected return a stock should earn is based on its risk (beta)!

E(Ri) = rrf + βi [ E(RM) - rrf ]

• where βi is the market beta of asset i
• & [ E(RM) - rrf ] is the Market risk premium (MRP) = the excess return on the market portfolio, ie. In CAPM, you get rewarded for systematic, market risk b/c holding market portfolio rather than individual assets.

Beta = a stocks’ sensitivity to changes in the market portfolio (from TB C8.2)

Question 7

5 CAPM assumptions (answers from notes + 2019/workshop 3)

Answer 7

1. Investors can borrow/lend unlimited amounts at the SAME risk-free rate
2. Asset markets are frictionless and info is COSTLESS & available to all investors -> no transaction costs or taxes
3. Investors are rational MEAN-VARIANCE OPTIMISERS (only care about mean & variance)
4. Investors have HOMOGENEOUS EXPECTATIONS about securities, ie. expected returns & the covariance matrix of security returns {& std dev}
   ^strong assumption
   -> hence all investors get the same efficient frontier, & all hit the same tangency portfolio since same risk-free rate & MV optimisers
5. All investors are risk-averse & plan for ONE identical HOLDING PERIOD to maximise the expected utility of their end-of-period wealth.

Question 8

What is the beta of the market portfolio? Beta of the risk-free asset?

Answer 8

Market beta of the market = 1
Market beta of the risk-free asset = 0

Question 9

1. If CAPM holds, ie. in equilibrium, what happens to the security market line (SML)?
2. What is the gradient of the SML? Axes?

Answer 9

1. If CAPM holds, ie. in eqm, ALL assets/portfolios plot ON the security market line (SML)
   *Not in reality though
2. Gradient of SML = market risk premium (MRP)
   Expected return vs Beta

Question 10

What happens if a stock does not lie on the security market line (SML)? eg. If lies above SML?

Alpha

Answer 10

If the stock plots above the SML, it is underpriced and is generating an ABNORMAL/excess +VE RETURN (+ve alpha).

Alpha = the extra expected return on the stock

Trading implication: give this stock higher weight in your portfolio than it has in the market portfolio & benefit from earning higher E(R) w/o bearing the appropriate level of risk

Question 11

3 applications for CAPM

Answer 11

1. Dividend model to VALUE STOCKS
   - need to have a risk-adjusted discount rate to discount risky assets and find PV of future dividend stream b/c you can’t use risk-free rate to discount risky assets
2. Valuation of projects
   - Use CAPM expected return as the discount rate to find NPV and decide which project to invest in
3. As a benchmark model for portfolio selection

Question 12

2 problems of CAPM that erode our confidence in its accuracy

In other words, 2 factors other than beta explain expected returns -> contradict CAPM

Answer 12

1. Size anomaly
   - stocks by smaller firms tend to have higher E(r) than big stocks (holding beta constant)
2. Value anomaly
   - mean returns on stocks with high book-to-market tend to be larger than those on low book-to-market stocks (holding beta equal)