Week 6 - Equity Premium and Asset Allocation

Question 1

What is equity premium?

Answer 1

The equity premium is the difference between the average returns on stock market index and the risk free rate

• estimated to be about 6% during the 100 years
• Stocks have a higher return on average than bonds
• however in a given year, stocks may have much lower returns than bonds

Question 2

What is risk aversion?

Answer 2

• risk aversion: demanding compensation for bearing risk
• The term risk-averse describes the investor who chooses the preservation of capital over the potential for a higher-than-average return
• the return pattern for stocks and bonds is consistent with risk aversion
• -> stocks have higher average returns and higher standard deviations historically…explanation: investors demand higher returns on average for stocks to offset the higher variation in the returns

Question 3

What is the equity premium puzzle?

Answer 3

• many economists think the 6% equity premium is too big under standard rational model of investor preference and market risk
• yes, economists agree that stocks are riskier
• but not risky enough to justify 6% per year for the level of risk aversion observed in the experiments

Question 4

Explanations of Equity Premium

Answer 4

• survivorship bias:
• -> we only looked at the US market which has been the best performing market over the past 100 years
• -> if you look at the whole world, than the premium is not that big
• maybe we are measuring risk incorrectly or incompletely
• it is hard to estimate future expected returns with historical realized returns
• investor sentiment and psychology

Question 5

Rethinking the equity risk premium

Answer 5

• long-term government bonds have gained 11.5% a year on average over the past three decades, beating the 10.8% increase in the S&P500
• on the other hand, many people claim “stocks are dead”
• retail investors seem to agree (they have been pulling money out of stock mutual funds every year since 2008)
• general consensus of academics and CFO’s is that the ERP is perhaps closer to 3-4%

Question 6

Are stocks less risky in the long-run?

Answer 6

• annualized return volatility decreases with investment horizon. BUT, what matters is your wealth at the end of your investment horizon, not the annualized return
• short fall risk decreases with investment horizon
• but this ignores the size of potential losses, which for some of the possible outcomes amount to complete ruin
• if you end up with a loss at the end of the investment horizon, the magnitude of the loss tends to be bigger, when the horizon is longer
• a better way to quantify the risk of a long-term investment is the market price of insuring it against short fall
• -> the cost of such long-term insurance premium is very high and increases with horizon
• -> it costs 20% of the initial value of portfolio to insure against shortfall risk over 10 year
• -> a 25 year policy would costs 30% of the initial portfolio value

Question 7

Diversify across time…

Answer 7

adding risks

Question 8

Diversify across assets…

Answer 8

dividing risks

Question 9

Expected portfolio return

Answer 9

E(rp) = wE(r1)+(1-w)E(r2)

Question 10

Variance of a two-risky asset portfolio

Answer 10

variance = w^2variance(r1)+(1-w)^2variance(r2)+2w(1-w)Covariance(r1,r2)

Question 11

Correlation between r1,r2

Answer 11

=cov(r1,r2)/SD(r1)SD(r2)

Question 12

Covariance r1, r2

Answer 12

=correlation(r1,r2)SD(r1)SD(r2)

Question 13

Diversification between two risky assets

Answer 13

• the portfolio has the same expected return as the individual stocks but it also has a smaller variance
• the portfolio variance will always be lower as long as the stocks are not perfectly correlated
• if correlation is sufficiently low we can find a portfolio with lower variance than either of the assets

Question 14

How is the mean-variance boundary formed?

Answer 14

• we picked the expected return we want, and then choose the weights of the portfolio so that the variance of the portfolio is minimized
• -> min portfolio variance; subject to E(rp) = K

Question 15

What does mean-variance portfolio theory tell us?

Answer 15

• the MV portfolio theory says that any investor will choose the optimal portfolio from the set of portfolios that:
• -> maximize expected return for a given level of risk
• -> minimize risk for a given level of expected return

Question 16

How do you find the minimum variance portfolio?

Answer 16

Suppose you want to minimize the variance of a portfolio with many securities already in it…

1. Find two securities already in the portfolio with different covariances with the portfolio
2. add a little weight to the security with a lower cov(ri,rp)…subtract a little from the security with the higher covariance
3. the portfolio variance is a little lower. Repeat

The variance of the portfolio will be minimized when all the securities have the same covariance with the portfolio
cov(r1,rp) = cov(r2,rp) = … = cov (rn, rp)

• an asset’s influence on a portfolio’s variance primarily depends on how it covaries with the other assets in the portfolio

Question 17

How do you find the tangency portfolio?

Answer 17

1. form a portfolio using all the risky securities
2. find two securities already in the portfolio with different risk premium to covariance ratios:
   E(ri)-rf/cov(ri,rp)
3. Add a little weight to the security with a higher ratio, and subtract a little from the security with the lower ratio
4. Keep repeating steps 1-3

Question 18

Finding the tangency portfolio

Answer 18

E(r1)- rf/cov(r1,rT) = E(r2) - rf/cov(r2,rT)…

- the ratio of risk premium to covariance is the same for all assets

Question 19

Implications

Answer 19

Suppose the following is true:
- everyone is risk averse
- we all agree on the same values for the expected returns, variances, and covariances of the securities
Result: everyone will hold some combination of the tangency portfolio and diskless asset - the more risk averse you are, the higher your percentage of the diskless asset
- if people have different inputs, they will perceive different tangency portfolios and thus different mean-variance frontiers

Question 20

How does investor risk aversion factor into portfolio selection?

Answer 20

• a portfolio manager will offer the same risky portfolio to ALL clients regardless of risk aversion
• the clients risk aversion comes into play in capital allocation (risky vs. risk-free), not in determining mix off risky assets

Question 21

How could different efficient frontiers be obtained by portfolio managers?

Answer 21

• different constraints

- different inputs used

Question 22

CAL vs. CML

Answer 22

The CML is a special case of the CAL, where the CML is tangent to the efficient frontier. Because all investors will hold the same risky portfolio, this tangent portfolio is the market portfolio.