Return risk and the Security market line quantfinancejobs

1. Chapter 13 Return risk and the Security market linehttp://www.quantfinancejobs.com

2. 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)

3. 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

4. Expected Returns (2) • RA = 6% risk premium = 6 – 3.2=2.8% • RB = 7% risk premium = 7 – 3.2=3.8% • If the risk-free rate = 3.2%, what is the risk premium for each stock?

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

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

7. 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

8. 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 WA=.25 • $9,000 of B WB=.45 • $5,000 of C WC=.25 • $1,000 of D WD=.05

9. 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

10. 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) = 11.77

11. 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

12. Portfolio Variance (2) • Consider the following information Invest 60% of your money in Asset A • State Probability A B • Boom .5 70% 10% • Recession .5 -20% 30% • What is the expected return and standard deviation for each asset? • What is the expected return and standard deviation for the portfolio?

13. Solution: • 1.

14. 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?

15. Solution: • Var=1058.1 (1458+32-431.87)

16. Different Correlation Coefficients (1)

17. Different Correlation Coefficients (2)

18. Different Correlation Coefficients(3)

19. Possible Relationships between Two Stocks

20. 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.

21. Diversification (2)

22. 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

23. 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

24. 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

25. 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.

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

27. 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

28. 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

29. Portfolio Diversification (1)

30. Portfolio Diversification (2)

31. 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

32. 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

33. 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

34. 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

35. High and Low Betas

36. Portfolio Betas • Consider the previous example with the following four securities • Security Weight Beta • A .133 3.69 • B .2 0.64 • C .267 1.64 • D .4 1.79 • What is the portfolio beta? 1.773

37. 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

38. 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%.

39. Portfolio Expected Returns and Betas Rf

40. 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?

41. 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

42. 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

43. Security Market Line (2)

44. 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

45. 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?

46. 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