0. Return risk and the Security market line
Chapter 13
Return risk and the Security market line

1. Chapter Outline Expected Returns and Variances of a portfolio
Announcements, Surprises, and Expected Returns
Risk: Systematic and Unsystematic
Diversification and Portfolio Risk
Systematic Risk and Beta
The Security Market Line (SML)

2. Expected Returns (1)
Expected returns are based on the probabilities of possible outcomes
Expected means average if the process is repeated many times
Expected return = return on a risky asset expected in the future

3. Expected Returns (2) Probability Expected return Stock A Stock B Boom
0.2
20%
15%
Normal
0.4
10% 8%
Recession
-5% 2%
RA =
RB =
If the risk-free rate = 3.2%, what is the risk premium for each stock?

4. Variance and Standard Deviation (1)
Unequal probabilities can be used for the entire range of possibilities
Weighted average of squared deviations

5. Variance and Standard Deviation (2)
Consider the previous example. What is the variance and standard deviation for each stock?
Stock A
Stock B

6. Portfolio = a group of assets held by an investor
Portfolios
The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets
Portfolio = a group of assets held by an investor
Portfolio weights = Percentage of a portfolio’s total value in a particular asset

7. Portfolio Weights
Suppose you have $ 20,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security?
$5,000 of A
$9,000 of B
$5,000 of C
$1,000 of D

8. Portfolio Expected Returns (1)
The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio
You can also find the expected return by finding the portfolio return in each possible state and computing the expected value

9. Expected Portfolio Returns (2)
Consider the portfolio weights computed previously. If the individual stocks have the following expected returns, what is the expected return for the portfolio?
A: 19.65%
B: 8.96%
C: 9.67%
D: 8.13%
E(RP) =

10. Portfolio Variance (1) Steps:
Compute the portfolio return for each state: RP = w1R1 + w2R2 + … + wnRn
Compute the expected portfolio return using the same formula as for an individual asset
Compute the portfolio variance and standard deviation using the same formulas as for an individual asset

11. Portfolio Variance (2) Consider the following information
Invest 60% of your money in Asset A
State Probability A B
Boom % 10%
Recession % 30%
What is the expected return and standard deviation for each asset?
What is the expected return and standard deviation for the portfolio?

12. Solution:

13. Another Way to Calculate Portfolio Variance
Portfolio variance can also be calculated using the following formula:
Correlation is a statistical measure of how 2 assets move in relation to each other
If the correlation between stocks A and B = -1, what is the standard deviation of the portfolio?

14. Solution:

15. Different Correlation Coefficients (1)

16. Different Correlation Coefficients (2)

17. Different Correlation Coefficients(3)

18. Possible Relationships between Two Stocks

19. Diversification (1)
There are benefits to diversification whenever the correlation between two stocks is less than perfect (p < 1.0)
If two stocks are perfectly positively correlated, then there is simply a risk- return trade-off between the two securities.

20. Diversification (2)

21. Expected vs. Unexpected Returns
Expected return from a stock is the part of return that shareholders in the market predict (expect)
The unexpected return (uncertain, risky part):
At any point in time, the unexpected return can be either positive or negative
Over time, the average of the unexpected component is zero
Total return = Expected return + Unexpected return

22. Announcements and News
Announcements and news contain both an expected component and a surprise component
It is the surprise component that affects a stock’s price and therefore its return
Announcement = Expected part + Surprise

23. Systematic Risk Risk factors that affect a large number of assets
Also known as non-diversifiable risk or market risk
Examples: changes in GDP, inflation, interest rates, general economic conditions

24. Unsystematic Risk Risk factors that affect a limited number of assets
Also known as diversifiable risk and asset-specific risk
Includes such events as labor strikes, shortages.

25. Returns Unexpected return = systematic portion + unsystematic portion
Total return can be expressed as follows:
Total Return = expected return + systematic portion + unsystematic portion

26. Effect of Diversification
Portfolio diversification is the investment in several different asset classes or sectors
Diversification is not just holding a lot of assets
Principle of diversification = spreading an investment across a number of assets eliminates some, but not all of the risk

27. The Principle of Diversification
Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns
Reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another
There is a minimum level of risk that cannot be diversified away and that is the systematic portion

28. Portfolio Diversification (1)

29. Portfolio Diversification (2)

30. Diversifiable (Unsystematic) Risk
The risk that can be eliminated by combining assets into a portfolio
If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away
The market will not compensate investors for assuming unnecessary risk

31. Total Risk
The standard deviation of returns is a measure of total risk
For well diversified portfolios, unsystematic risk is very small
Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk

32. Systematic Risk Principle
There is a reward for bearing risk
There is no reward for bearing risk unnecessarily
The expected return (and the risk premium) on a risky asset depends only on that asset’s systematic risk since unsystematic risk can be diversified away

33. Measuring Systematic Risk
Beta (β) is a measure of systematic risk
Interpreting beta:
β = 1 implies the asset has the same systematic risk as the overall market
β < 1 implies the asset has less systematic risk than the overall market
β > 1 implies the asset has more systematic risk than the overall market

34. High and Low Betas

35. Total vs. Systematic Risk
Consider the following information:
Standard Deviation Beta
Security A 20%
Security B 30%
Which security has more total risk?
Which security has more systematic risk?
Which security should have the higher expected return?

36. Solution:

37. Portfolio Betas
Consider the previous example with the following four securities
Security Weight Beta A B C D
What is the portfolio beta?

38. Beta and the Risk Premium
The higher the beta, the greater the risk premium should be
The relationship between the risk premium and beta can be graphically interpreted and allows to estimate the expected return

39. Consider a portfolio consisting of asset A and a risk-free asset
Consider a portfolio consisting of asset A and a risk-free asset. Expected return on asset A is 20%, it has a beta = 1.6. Risk- free rate = 8%.

40. Portfolio Expected Returns and BetasRf

41. Reward-to-Risk Ratio:
The reward-to-risk ratio is the slope of the line illustrated in the previous slide
Slope = (E(RA) – Rf) / (A – 0)
Reward-to-risk ratio =
If an asset has a reward-to-risk ratio = 8?
If an asset has a reward-to-risk ratio = 7?

42. The Fundamental Result
The reward-to-risk ratio must be the same for all assets in the market
If one asset has twice as much systematic risk as another asset, its risk premium is twice as large

43. Security Market Line (1)
The security market line (SML) is the representation of market equilibrium
The slope of the SML is the reward-to- risk ratio: (E(RM) – Rf) / M
The beta for the market is always equal to one, the slope can be rewritten
Slope = E(RM) – Rf = market risk premium

44. Security Market Line (2)

45. The Capital Asset Pricing Model (CAPM)
The capital asset pricing model defines the relationship between risk and return
E(RA) = Rf + A(E(RM) – Rf)
If we know an asset’s systematic risk, we can use the CAPM to determine its expected return

46. CAPM
Consider the betas for each of the assets given earlier. If the risk-free rate is 4.5% and the market risk premium is 8.5%, what is the expected return for each?
Security
Beta
Expected Return A
3.6 B .7 C
1.7 D
1.9

47. Factors Affecting Expected Return
Time value of money – measured by the risk-free rate
Reward for bearing systematic risk – measured by the market risk premium
Amount of systematic risk – measured by beta