1 / 45

Week 8 Lecture 8

Week 8 Lecture 8. Ross, Westerfield and Jordan 7e Chapter 13 Return, Risk and the Security Market Line. Last Lecture. Returns Holding Period Returns Averages: AM, GM Risk Variance Standard Deviation There is a reward for bearing risk Positive risk-return relationship Risk Premium

necia
Download Presentation

Week 8 Lecture 8

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Week 8Lecture 8 Ross, Westerfield and Jordan 7e Chapter 13 Return, Risk and the Security Market Line

  2. Last Lecture.. • Returns • Holding Period Returns • Averages: AM, GM • Risk • Variance • Standard Deviation • There is a reward for bearing risk • Positive risk-return relationship • Risk Premium • EMH: weak, semi-strong, strong

  3. Chapter 13 Outline • Expected Returns and Variances • Probabilities • Portfolios • Risk and Returns • The principle of diversification • Risk: Systematic and Unsystematic • The Security Market Line (SML) • Capital Asset Pricing Model (CAPM) • Reward to Risk Ratio

  4. Expected Returns • Consider an asset which has many possible future returns, returns that are not equally likely.. What is the average return? What is the expected return? • Average or Expected returns is based on the average of all possible future returns weighted by their probabilities • Suppose there are T possible returns, and that R1 has probability p1 of occurring, R2 has probability p2 , …, and RT has probability pT . Then:

  5. Example: Expected Returns • Suppose you have predicted the following returns for stocks C and T in three possible states of nature. What are the expected returns? State Probability C T Boom 0.3 15% 25% Normal 0.5 10% 20% Recession ??? 2% 1% • RC = 0.3(0.15) + 0.5(0.10) + 0.2(0.02) = 9.99% • RT = 0.3(0.25) + 0.5(0.20) + 0.2(0.01) = 17.7%

  6. Variance and Standard Deviation • Variance and standard deviation still measure the volatility of returns • Using unequal probabilities for the entire range of possibilities • Weighted average of squared deviations

  7. Example: Variance and Standard Deviation • Consider the previous example. What are the variance and standard deviation for each stock? • E(R)C = 9.9% and E(R)T = 17.7% • Stock C • 2 = 0.3(0.15-0.099)2 + 0.5(0.10-0.099)2 + 0.2(0.02-0.099)2 = = 0.3(0.051)2 + 0.5(0.001)2 + 0.2(-0.079)2 = = 0.3(0.002601) + 0.5(0.000001) + 0.2(0.006241) = = 0.0007803 + 0.0000005 + 0.0012482 = 0.002029 •  = √σ2 = √0.002029 = 0.045044 = 4.5% • Stock T • 2 = 0.3(0.25-0.177)2 + 0.5(0.20-0.177)2 + 0.2(0.01-0.177)2 = = 0.3(0.073)2 + 0.5(0.023)2 + 0.2(-0.167)2 = = 0.0015987 + 0.0002645 + 0.0055778 = 0.007441 •  = √σ2 = √0.007441 = 0.086261 = 8.63%

  8. Quick Quiz • Consider the following information: State Probability ABC, Inc. (%) Boom 0.25 15 Normal 0.50 8 Slowdown 0.15 4 Recession 0.10 -3 • What is the expected return? • What is the variance? (0.00267475) • What is the standard deviation?

  9. Portfolios • A portfolio is a collection of assets • An asset’s risk and return are important in how they affect the risk and return of the portfolio • The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets

  10. Example: Portfolio Weights • Suppose you have $15,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security? • $2000 of DCLK • $3000 of KO • $4000 of INTC • $6000 of KEI DCLK: 2/15 = 0.133 KO: 3/15 = 0.2 INTC: 4/15 = 0.267 KEI: 6/15 = 0.4

  11. Portfolio Expected Returns • The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio (example 13.3) • Step 1: calculate E(Rasset) based on probability of state • Step 2: calculate E(RP) based on weights of assets • You can also find the expected return by finding the portfolio return in each possible state and computing the expected value as we did with individual securities (ex.13.5) • Step 1: calculate E(RP) in each state, eg. boom or bust • Step 2: add the state returns weighted by each probability

  12. Example : E(RP) • Consider the following information State Probability(p) X Z Boom 0.25 15% 10% Normal 0.60 10% 9% Recession 0.15 5% 10% • What are the expected return for a portfolio with an investment of $6000 in asset X and $4000 in asset Z?

  13. Example : E(Rp) continued..(1) • Weight X = 0.6, Weight Z = 0.4 • First way of calculating E(RP): • Step 1: Calculate the expected return of each asset based on each probability of state occurring: E(RX) = (0.25x0.15) + (0.6x0.1) + (0.15x0.05) = 10.5% E(RZ) = (0.25x0.1) + (0.6x0.09) + (0.15x0.1) = 9.4% Boom Normal Recession • Step 2: Calculate the E(RP) based on the weights of each asset: • E(RP) = 0.6x10.5% + 0.4x9.4% = 10.06%

  14. Example : E(RP) continued..(2) • Weight X = 0.6, Weight Z = 0.4 • Second way to calculate E(RP): • Step 1: Calculate the E(RP) in each state based on each asset weight: E(RP)Boom = (0.6 x 0.15) + (0.4 x 0.10) = 13% E(RP)Normal = (0.6 x 0.10) + (0.4 x 0.09) = 9.6% E(RP)Recession = (0.6 x 0.05) + (0.4 x 0.10) = 7% • Step 2: Calculate total E(RP) using probabilities as weights: E(RP) = pB x E(RB) + pN x E(RN) + pR x E(RR) Boom Normal Recession E(RP) = (0.25x13%) + (0.6x9.6%) + (0.15x7%) = 3.25% + 5.756% + 1.05% = 10.06%

  15. Portfolio Variance with Probabilities • 1) Compute the portfolio return for each state, boom, bust.. etc. (step 1):E(RPstate) = w1R1 + w2R2 • 2) Compute the expected portfolio return using probabilities as for a single asset (step 2): E(RP) = p1 x E(Rstate1) + p2 x E(Rstate2) + p3 x E(Rstate3) • 3) This E(RP) becomes the mean • 4) Compute the deviations of each state from the mean, then square the deviation: [E(RPstate)-E(RP)]2 • 5) Multiply the squared deviation with probability of each state, then sum: ∑ (pstate x [E(RPstate)-E(RP)]2)

  16. Example: Variance & SD • Variance: ∑ (pstate x [E(RPstate)-E(RP)]2) • Portfolio return in each state (boom, normal, recession) and Two-asset (X and Z) total portfolio return (slide 13) • E(Rp)boom = 13% • E(Rp)normal = 9.6% • E(Rp)recession = 7% • E(Rp) = 10.06% • Variance: Var = 0.25(0.13-0.1006)2 + 0.6(0.096-0.1006)2 + 0.15(0.07-0.1006)2 = Var = 0.00021609 + 0.000012696 + 0.000140454 = 0.00036924 SD = √0.00036924 = 0.019215619 = 1.92%

  17. Example 2: E(R), Variance & SD • Consider the following information: • Invest 50% of your money in Asset A State Probability A B portfolio Boom 0.4 30% -5% 12.5% Bust 0.6 -10% 25% 7.5% • What are the expected return and standard deviation for each asset? • What are the expected return and standard deviation for the portfolio?

  18. Example 2: E(R), Var & SD…asset E(Rasset) = (pstate1 x Rasset) + (pstate2 x Rasset) Var = ∑[pstate x (Rasset – E(Rasset)2] • Asset A: E(RA) = 0.4(0.30) + 0.6(-0.10) = 6% • Variance(A) = 0.4(0.30-0.06)2 + 0.6(-0.10-0.06)2 = 0.02304 + 0.01536 = 0.0384 • Std. Dev.(A) = √0.0384 = 19.6% • Asset B: E(RB) = 0.4(-0.05) + 0.6(0.25) = 13% • Variance(B) = 0.4(-0.05-0.13)2 + 0.6(0.25-0.13)2 = 0.01296+0.00864 = 0.0216 • Std. Dev.(B) = √ 0.0216 = 14.7%

  19. Example 2: E(R), Var & SD…portf. • Calculate the Expected return of portfolio in each state E(Rp)state = (wassetA x RA state) + (wassetB x RB state) • E(Rp)boom = 0.5(0.30) + 0.5(-0.05) = 12.5% • E(Rp)bust = 0.5(-0.10) + 0.5(0.25) = 7.5% • Then the overall Expected portfolio return E(Rp) = (pstate1 x E(Rp)state1) + (pstate2 x E(Rp)state2) • E(Rp) = 0.4(0.125) + 0.6(0.075) = 9.5% • Then the Variance of the portfolio Varp = ∑[pstate x (E(Rp)state – E(Rp))2] • Varp = 0.4(0.125 - 0.095)2 + 0.6(0.075 - 0.095)2 = 0.00036 + 0.00024 = 0.0006 • Then the Standard Deviation of the portfolio • SD = √0.0006 = 0.02449 = 2.45%

  20. Risk and Portfolio Theory • Risk Averse Investors: require a higher average return to take on a higher risk • Portfolio Theory Assumption: • Investors prefer the portfolio with the highest expected return for a given variance, or, the lowest variance for a given expected return • Expected returns and Variances of Portfolios derived from historical returns, variances, and covariances of individual assets in portfolio

  21. Covariance and Correlation Coefficient • Covariance is an absolute measure of the degree to which two variables move together over time relative to their individual mean. • Correlation Coefficient, ρ, is a standardised measure of the relationship between the two variables, ranging between -1.00 to +1.00

  22. Portfolio Variance and Standard Deviation for a 2-Asset Portfolio • In order to reduce the overall risk, it is best to have assets with low positive or negative correlation (covariance) • The smaller is the covariance between the assets, the smaller will be the portfolio’s variance.

  23. Example: Risk of 2-Asset Portfolio • Consider these two assets that have equal weights of 0.50 in the portfolio, and with the following returns and standard deviation: E(R1) = 30%, and σ1 = 0.20 E(R2) = 15% and σ2 = 0.12 Corr Coeff = 0.1 σp2 = (0.5)2(0.2)2 + (0.5)2(0.12)2 + 2(0.5)(0.5)(0.2)(0.12)(0.1) = = 0.01 + 0.0036 + 0.0012 = 0.0148 σp=√0.0148 = 0.1216 = 12.16% (lower risk for 22.5% portfolio return)

  24. Table 13.7 More assets Less risk

  25. Diversification • The Principle of Diversification :states that spreading an investment across many assets will eliminate some but not all of the risk. • Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns • Size of risk reduction depends on covariances between assets in the portfolio • However, there is a minimum level of risk that cannot be diversified away and that is the systematic portion

  26. Two Types of Risk • Systematic or Non-Diversifiable Risk • That portion of an asset’s risk attributed to the market factors that affect all firms and cannot be eliminated through the process of diversification. • Unsystematic or Diversifiable Risk • That portion of an asset’s risk which is firm specific and can be eliminated through the process of diversification.

  27. Figure 13.1

  28. Total Risk • Total risk = systematic risk + unsystematic 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

  29. Systematic Risk Principle • There is a reward for bearing risk • There is not a reward for bearing risk unnecessarily • The expected return on a risky asset depends only on that asset’s systematic risk since unsystematic risk can be diversified away

  30. Measuring Systematic Risk = β • We use the beta coefficient to measure systematic risk • Beta measures the responsiveness of a security to movements in the market. • Market beta βm= 1 • Therefore if: • βA= 1, the asset has the same systematic risk as the overall market • βA < 1 implies the asset has less systematic risk than the overall market • βA > 1 implies the asset has more systematic risk than the overall market

  31. Estimation of Beta • Two ways: • Based on the formula calculate the covariance of the asset with the market, calculate the variance of the market, then divide the two • Slope function in excel • What is the market? • The index

  32. The Capital Asset Pricing Model (CAPM) • The capital asset pricing model defines the relationship between risk and return • If we know an asset’s systematic risk, we can use the CAPM to determine its expected return • This is true whether we are talking about financial assets or physical assets

  33. CAPM E(RA) = Rf + A(E(RM) – Rf) • Where: • E(RA) = expected return on asset A • Rf = risk free rate • A = beta of asset A • E(RM) = expected return on the market • Note: E(RM) – Rf = Market Risk Premium

  34. Example - CAPM • If the beta for IBM is 1.15, the risk-free rate is 5%, and the expected return on market is 12%, what is the required rate of return for IBM? • Applying the CAPM :

  35. Example - CAPM • Consider the betas for each of the assets given earlier. If the risk-free rate is 2.13% and the market risk premium is 8.6%, what is the expected return for each? E(RA) = Rf + A(E(RM) – Rf)

  36. Security Market Line • The security market line (SML) is the graphical representation of CAPM • Shows the relationship between systematic risk and expected return • Positive slope • The higher the risk, the higher the return • According to the CAPM, all stocks must lie on the SML, otherwise they would be under or over-priced.

  37. Security Market Line (SML) Asset expectedreturn (E (Ri)) SML A - undervalued E (RA) = E (RM) – Rf E (RB) B - overvalued E (RM) market Rf Assetbeta (i) M= 1.0 A B

  38. Reward to Risk Ratio • SML slope = Reward to Risk Ratio = Market Risk Premium • In equilibrium, all assets and portfolios must have the same reward-to-risk ratio and they all must equal the reward-to-risk ratio for the market • If not, assets are undervalued or overvalued

  39. Reward-to-Risk Ratio: Example • If RM=12%, Rf = 6% • Slope = (E(RM) – Rf)/M = market risk premium • = (12% - 6%)/1 = 6% • If asset A has E(RA) = 15%, βA = 1.3, and asset B has E(RB) = 10%, βB = 0.8 • Asset B offers insufficient reward for its level of risk, so B is relatively overvalued compared to A, or A is relatively undervalued

  40. CAPM and Beta of Portfolio • If w1, w2, …, wn, are the proportions of the portfolio invested in n assets 1, 2, …, n, the beta of a portfolio (P) can be written: • Example: If 30% of a portfolio is invested in asset 1 and the balance in asset 2, and asset 1’s beta=1.7 while asset 2’s beta =1.2, what is the beta of the portfolio (P)

  41. Example: Portfolio Beta • Consider our previous four securities and their betas: Security Weight Beta DCLK .133 2.685 KO .2 0.195 INTC .167 2.161 KEI .4 2.434 • What is the portfolio beta? • 0.133(2.685) + 0.2(0.195) + 0.167(2.161) + 0.4(2.434) = 1.731

  42. Factors Affecting E(R) E(RA) = Rf + A(E(RM) – Rf) • Pure time value of money – measured by the risk-free rate Rf • Reward for bearing systematic risk – measured by the market risk premium E(RM) – Rf • Amount of systematic risk – measured by beta β

  43. Quick Quiz • What is the difference between systematic and unsystematic risk? • What type of risk is relevant for determining the expected return? • Consider an asset with a beta of 1.2, a risk-free rate of 5% and a market return of 13%. • What is the reward-to-risk ratio in equilibrium? • What is the expected return on the asset?

  44. Lecture 8 - Summary • Expected returns, variances and standard deviation • Using probabilities • Using historical returns • For a single asset and for a portfolio • The principle of diversification • Systematic and Unsystematic risk • SML and CAPM • Reward to Risk ratio

  45. End Lecture 8

More Related