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## Week 8 Lecture 8

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**Week 8Lecture 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 • EMH: weak, semi-strong, strong**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**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:**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%**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**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%**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?**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**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**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**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?**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%**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%**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)**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%**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?**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%**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%**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**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**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.**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)**Table 13.7**More assets Less risk**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**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.**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**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**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**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**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**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**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 :**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)**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.**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**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**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**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)**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**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**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?**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