Economics 134a

1
Economics 134a
2
Ipod

• Ch. 8 deleted from the course
• Reminder ditto with Ch. 11.

3
(No Transcript)
4
Today

• Optimal Portfolios (taking securities prices,
  returns etc. as given)
• Tuesday Eric will lecture on CAPM equilibrium
  what will be like if everyone chooses portfolios
  optimally
• Analogy to microeconomics demand-supply
  analysis vs. general equilibrium

5
Portfolio Expected Return and Standard Deviation
6
Portfolios

• Define a portfolio consisting of proportion x of
  one and 1-x of the other (measured by dollar
  value)
• If xlt0 or xgt1, thats a short position in one of
  the securities
• What are the expectations and SD of its return?
• This depends on the expectation and SD of both
  stocks
• and also on their correlation

7
Analytically

• The expected return of a portfolio equals the
  average of the ERs of the securities in the
  portfolio
• Where the weights equal the portfolio weights x
  and 1-x
• However, The SD of a portfolio equals the average
  of the SDs of the securities in the portfolio
  ONLY in the case of perfect positive correlation
  (corr. coefficient 1)
• What happens in the general case? Depends on
  correlation

8
Perfect Positive Correlation
9
Riskless Portfolio

• In the case of perfect positive correlation,
  there exists a portfolio that is riskless. It
  involves a short sale of one of the securities

10
Perfect Negative Correlation
11
Riskless Portfolio

• In the case of perfect negative correlation,
  there exists a portfolio that is riskless. It
  involves positive holdings of both securities.

12
Examples

• The easiest examples for perfect positive and
  negative correlations are in the binomial case.
• This is the same as the example given below, but
  only 2 states.

13
Intermediate Case
14
How do you derive these curves?

• You can do it analytically (see the text), but
  its easier to understand this via numerical
  examples
• Go back to Supertech and Slowpoke.
• Figure out the payoffs of portfolios composed of
  100 - 0 Supertech-Slowpoke
• Then do 80-20, 60-40, 40-60, 20-80 and 0-100.
• This is easiest on a spreadsheet!
• You may also want to analyze portfolios with
  short positions

15

• Then compute means and standard deviations of
  portfolio returns.
• This will give you the analysis when return
  correlation is -0.16.
• Youd need another example for other correlations.

16
Lets do Supertech and Slowpoke ...

• Here are the data again
• What is the payoff of the portfolio in the red
  space?
• -.5(5) 1.5(-20) -32.5

17
Portfolio Payoffs
18

• Finally, compute means and standard deviations

19
Here are the portfolio means and SDs

• This looks like slide 15 intermediate case

20
Exercise

• repeat this using the following data

21

• Note that the returns in this case are positively
  linearly related.
• So the locus of portfolio means and standard
  deviations looks like slide 7

22
Another exercise
23

• In this case Slowpokes returns 60
  Supertechs returns (linear relation with
  negative coefficient).
• So the locus of expected return and SD looks like
  slide 8

24
Review

• Weve done portfolio theory when there are 2
  securities
• If the returns are perfectly positively
  correlated, the mean and SD of portfolios lie on
  the line connecting the 2 securities
• If they are negatively (or less than perfectly
  positively) correlated, you can get LESS RISK
  than would be implied by this line
• This is the case for diversification.

25
Many Securities
26
Mean-SD combinations of portfolios

• All payoffs are east of a (generally) curved line

27
Efficient portfolios

• Have payoffs that maximize expected return for
  each level of SD
• These are the payoffs on the upward-facing
  portion of the line
• Note, any of the securities individually may or
  may not be efficient. Usually not.

28

• A portfolio consisting of 100 Slowpoke is
  inefficient it has lower expected return and
  higher risk than a portfolio with 25 Supertech.
• Check this out using the table on slide 15
• Conclusion a risk-averse agent will always
  choose and efficient portfolio.
• Which one? Depends on how risk-averse he is.

29
Efficient portfolios
30
How can you tell when a portfolio is inefficient?

• Any portfolio that lies below the line connecting
  2 other portfolios must be inefficient.

31
However

• Portfolio payoffs that lie above the line may or
  may not be efficient

32
Risk-aversion

• You can represent attitudes toward risk (like any
  aspect of preferences) using indifference curves.

33
Steepness and risk-aversion

• Steep indifference curves mean an investor will
  sacrifice a lot of expected return to get a small
  decrease in risk
• Such an investor is very risk-averse
• Flatter indifference curve mean an investor is
  not very risk averse.

34
Important special case

• Risk-neutrality

35
Risk-free Asset

• For the most part, weve assumed there are no
  risk-free securities or portfolios
• Exception perfect positive or negative
  correlation.
• If there is a risk-free asset, and there also
  exist risk-free portfolios, they must have the
  same return.
• Otherwise arbitrage is possible

36
Risk-free Asset
37
Risk-free asset and Tangency Portfolio
38
Risk Preferences and Optimal Portfolios
39
Quick Notes

• All optimal portfolios now are weighted averages
  of a tangency portfolio and the risk-free
  asset.
• Very risk averse investors hold more of risk-free
  asset
• Less risk-averse investors hold more of the
  tangency portfolio and less of the riskless asset
• Maybe even a negative amount of the risk-free
  asset
• Margin account.
• Notice that portfolios located northeast of the
  tangency portfolio must be short the risk-free
  asset.

40
Optimal portfolios

• Regardless of risk-aversion, all risky assets
  are held in the same relative proportions.
• Optimal portfolios southwest of the tangency
  portfolio involve some lending at the risk-free
  rate
• smaller holdings of risky assets
• Such portfolios are appropriate for risk-averse
  investors.