CAPM
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CAPM and Multifactor Models
Learning Objective • Understanding of: – CAPM model – Systematic and Unsystematic Risk – Beta – Security Market Line – Performance measures: Treynor ratio, Sharpe ratio and Jensen’s Alpha – Single Index Model – Arbitrage pricing theory and Multifactor models
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Capital Asset Pricing Model (CAPM) • Total risk on investment in a financial asset maybe categorized into: – Idiosyncratic/ diversifiable risk: Risk that can be diversified away by putting your money in a well-diversified portfolio. This include risk due to factors specific to a firm like management quality, market perception about the firm’s performance etc. – Systematic/Un-diversifiable risk: Even a well-diversified portfolio is exposed to changes in market variables impacting all the companies and therefore, this component cannot be diversified away. However, systematic risk of a portfolio would depend on how exposed it is to market variables.
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Capital Asset Pricing Model (CAPM) • Market value of an asset maybe arrived at by: – Finding present value (PV) of all future cash flows (cash inflow minus cash outflows) that the asset may generate – For instance, a property maybe priced at by finding present value of lease rentals (post all costs like Tax, maintenance costs etc.) that property may generate in future
• In case of financial assets like equity shares, future earnings would be the free cash flows (FCF) available to equity shareholders (free cash flows of the firm – interest payments to debt-holders – dividend on preference capital)
• But what would be the discount factor (i.e. expected rate of return of shareholders) to arrive at PV of FCF available to shareholders? • This is the precise question answered by the CAPM
• CAPM helps to arrive at an estimate of returns expected by shareholders on their investment in a company
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CAPM
• Market portfolio (say, BSE Sensex, NSE Nifty) is taken as a well-diversified portfolio with systematic risk of 1
• Any other portfolio is benchmarked to market portfolio using a parameter called Beta Therefore, if Beta of a portfolio is 1.3, it means it is 1.3 times as risky as marke portfolio and investors in this portfolio would expect a return of 1.3 times that generated by the market portfolio. • Expected return on a stock = Rf + Beta * (Rm – Rf) – where Rf = Risk-free return (return on government securities) – Rm = Return expected from the market portfolio – (Rm – Rf) = Market risk premium. • Expected return on a stock = Risk-free + Risk premium where, Risk-premium on the stock = Beta * Market risk-premium
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CAPM • Assumptions of CAPM – All investors have homogeneous expectations – Investors choose their risky asset proportions by maximizing the Sharpe ratio – Investors can borrow or lend unlimited amounts at the risk-free rate – The market is in equilibrium at all times
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CAPM and SML
• Expected returns computed by CAPM at different Beta levels can be expressed as Security Market Line. Securities lying above SML are giving excess return over and above what Beta Risk implies Security Market Line SML
Returns (Expected and Actual)
35%
Actual Returns
SML shows the expected return at a given level of Beta denotes the actual return given by various securities
30% 25% 20%
Actual Return < Expected i.e. Negative Jenson's Alpha. So SELL
Actual Return > Expected i.e. Positive Jenson's Alpha. So BUY
15% 10%
Rf 5%
0% 0
0.2
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0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Beta
7
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Performance Measures – Treynor Ratio
• Treynor Ratio: The ratio measures the excess return over the risk-free rate given by a portfolio with respect to its Beta • TR= (Rp – Rf) / Beta
where Rp = portfolio return, Rf = risk free return
• Higher the Treynor measure, the better the portfolio • This measure should be used only for well-diversified portfolio
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Performance Measures – Treynor Ratio
• Example Annual return from Dow Jones for 5 years is 8% annually. Avg. annual risk free return is 3.5%. Then compare three below mentioned performances A. Portfolio A with return of 10% and Beta of 1.1 B. Portfolio B with return of 12% and beta of 1.2 C. Portfolio C with return of 14% and beta of 1.5
• Solution TR of Market = (.08 – .035)/1 = .045 (Beta of market is 1) A. TR of A = (.1 – .035)/1.1 = .059 B. TR of B = (.12 – .035)/1.2 = .071 C. TR of C = (.14 – .035)/1.5 = .070
Though, all the portfolios have outperformed the market, B gives maximum return for the systematic risk taken, as determined by the Treynor Ratio
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Performance Measures – Sharpe Ratio
• Sharpe ratio: The ratio measures the excess return over the risk-free rate given by a portfolio with respect to its total risk (measured by standard deviation of returns). The ratio therefore can be used for single securities as well.
• SM= (Rp – Rf) / σ , where Rp = portfolio return, Rf = risk free return, σ = Std. deviation of returns
• The higher the Sharpe measure, the better performer the portfolio/security as compared to others
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Performance Measures – Sharpe Ratio • Example: Annual return from Dow Jones for 5 years is 8% annually. Avg. annual risk free return is 3.5% and Market returns had 12% std deviation. Then compare performance of these portfolios: A. Portfolio A with return of 10% and 9% std deviation B. Portfolio B with return of 12% and 12% std deviation C. Portfolio C with return of 14% and 15% std deviation
• Solution: Using Sharpe ratio S.R. of Market = (.08 – .035)/.12 = .375 A. S.R. of A = (.1 – .035)/.09 = .65 B. S.R. of B = (.12 – .035)/.12 = .71 C. S.R. of C = (.14 – .035)/.15 = .7 Portfolio C has given maximum return but as per Sharpe measure, B is a better performer. However, all the portfolios have outperformed the market.
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Performance Measures – Jensen’s Alpha • Jensen’s Alpha: This measure determines portfolio’s return over and above returns expected by CAPM. • α = Rp – Rc , where Rp = Actual portfolio return, Rc = Expected return (CAPM) = Rf + Beta*(Rm – Rf) • Positive alpha (portfolio with positive excess return) is always preferred over negative alpha
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Performance Measures – Jensen’s Alpha • Example: Annual return from Dow Jones for 5 years is 8%. Risk free rate is 3.5% and market returns had 12% std deviation. Then compare three below mentioned performances a. Portfolio A with actual return of 10% and 0.9 beta b. Portfolio B with actual return of 12% and 1.2 beta c. Portfolio C with return of 12% and 1.3 beta
• Solution: a. α of A = .10 – ( .035+.9*(.08 –.035)) = .058 = 5.8% b. α of B = .12 – ( .035+1.2*(.08 –.035)) = .064 = 6.4% c. α of C = .12 – ( .035+1.3*(.08 –.035)) = .060 = 6% Portfolio B has the maximum alpha
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Single Index Model (SIM) • The model simply assumes that statistically return of a security maybe represented as a linear function of a single economic/market variable like GDP growth, interest rates etc. • Rit = θi + δiIt+εit , where Rit is the return on a security i at time t εit is the error term and It is the economic variable • If It is set as the market return premium, Rm – r, SIM will differ from CAPM
• In case of CAPM, we run a regression between excess returns given by a security over risk-free and excess returns given by Market above risk-free i.e. regression between (Ri – Rf) and (Rm – Rf). However, in SIM, we don’t run regression on excess return but total return. • For any security, the Total Variance of return = δ x variance of index + variance of specific risk
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Multifactor Models and APT
• CAPM assumes that excess returns by market and excess return on a security are correlated. Arbitrage Pricing Theory (APT) is a model which states that return on a security is related to its exposure to multiple factors. • Ri = ai + ∑bj*Fj + error
where, Ri = Return on a security “i” Bj = Sensitivity of return of security “i” to jth Factor Fj is the value of the jth factor
• Such a model can be fitted by running a multiple regression between return on a security and values of different market variables
• For example, Return on ABC share = 5% + 1.63*Change in Inflation rate – 0.67*Change in USD/INR rate + 2.87*Annual GDP growth rate
• If we consider only a single factor (excess market return over the risk-free), then APT is equivalent to CAPM.
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Questions
• Two stocks X and Y have delivered returns of 15% and 20% respectively over the las year. The stocks had standard deviation of 9% and 15% respectively. The marke return was 10% and std dev was 10%. If the risk free rate is 5%, according to Sharpe ratio A. A is a better performer than B and the market B. B is a better performer than A and the market C. Both A and B performed equally better than the market D. Both A and B performed equally worse than the market
• A stock X has a beta of 1.2 and gave a return of 20% returns last year. If the marke gave a return of 15% and the risk free rate is 5%, The Jensen alpha for X is A. 4% B. 5% C. 3% D. 2%
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Questions • Which of the following is/are true assumptions to the CAPM? A. Investors have a limited capacity to borrow and lend at the risk-free rate B. Information is freely and universally available to all the investors C. Investors are risk takers and seek to maximize their wealth D. Returns are normally distributed
• A stock X that has beta of 1.2 pays a return of 20%. If the market premium is 10% what will be the return on stock Y if its beta is 1.35? A. 21% B. 22.5% C. 22% D. 21.5%
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Questions • The SIM model of a stock X was found to be RXt = 10% + 1.5(Rm – Rf) + error . If the variance of the market last year was 15% and variance due to the unsystematic risk of the stock was 25%, what is the overall variance of the returns from the stock? A. 47.5% B. 50.5% C. 45% D. 48%
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Answers • SR of A = (.15 - .05) /0.09 = 1.11, SR of B = 1.00, SR of market = 0.5 – A. A is a better performer than B and the market • Jensen Alpha = 0.20 - (0.05 + 1.2(0.15-0.05) = 0.03 = 3% – C. 3% • Information is freely and universally available to all the investors – B. Information is freely and universally available to all the investors • X = rf + 1.2 * 10% gives rf = 8 and Y = 8 + 1.35*10 = 21.5% – D. 21.5%
• Total variance of the stock = delta X market variance + specific risk variance = 1.5x15% + 25% = 47.5% – A. 47.5%
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End of CAPM and Multifactor Models
CAPM and Multifactor Models
Learning Objective • Understanding of: – CAPM model – Systematic and Unsystematic Risk – Beta – Security Market Line – Performance measures: Treynor ratio, Sharpe ratio and Jensen’s Alpha – Single Index Model – Arbitrage pricing theory and Multifactor models
Neev Knowledge Management – Pristine
2
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Capital Asset Pricing Model (CAPM) • Total risk on investment in a financial asset maybe categorized into: – Idiosyncratic/ diversifiable risk: Risk that can be diversified away by putting your money in a well-diversified portfolio. This include risk due to factors specific to a firm like management quality, market perception about the firm’s performance etc. – Systematic/Un-diversifiable risk: Even a well-diversified portfolio is exposed to changes in market variables impacting all the companies and therefore, this component cannot be diversified away. However, systematic risk of a portfolio would depend on how exposed it is to market variables.
Neev Knowledge Management – Pristine
3
www.edupristine.co
Capital Asset Pricing Model (CAPM) • Market value of an asset maybe arrived at by: – Finding present value (PV) of all future cash flows (cash inflow minus cash outflows) that the asset may generate – For instance, a property maybe priced at by finding present value of lease rentals (post all costs like Tax, maintenance costs etc.) that property may generate in future
• In case of financial assets like equity shares, future earnings would be the free cash flows (FCF) available to equity shareholders (free cash flows of the firm – interest payments to debt-holders – dividend on preference capital)
• But what would be the discount factor (i.e. expected rate of return of shareholders) to arrive at PV of FCF available to shareholders? • This is the precise question answered by the CAPM
• CAPM helps to arrive at an estimate of returns expected by shareholders on their investment in a company
Neev Knowledge Management – Pristine
4
www.edupristine.co
CAPM
• Market portfolio (say, BSE Sensex, NSE Nifty) is taken as a well-diversified portfolio with systematic risk of 1
• Any other portfolio is benchmarked to market portfolio using a parameter called Beta Therefore, if Beta of a portfolio is 1.3, it means it is 1.3 times as risky as marke portfolio and investors in this portfolio would expect a return of 1.3 times that generated by the market portfolio. • Expected return on a stock = Rf + Beta * (Rm – Rf) – where Rf = Risk-free return (return on government securities) – Rm = Return expected from the market portfolio – (Rm – Rf) = Market risk premium. • Expected return on a stock = Risk-free + Risk premium where, Risk-premium on the stock = Beta * Market risk-premium
Neev Knowledge Management – Pristine
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CAPM • Assumptions of CAPM – All investors have homogeneous expectations – Investors choose their risky asset proportions by maximizing the Sharpe ratio – Investors can borrow or lend unlimited amounts at the risk-free rate – The market is in equilibrium at all times
Neev Knowledge Management – Pristine
6
www.edupristine.co
CAPM and SML
• Expected returns computed by CAPM at different Beta levels can be expressed as Security Market Line. Securities lying above SML are giving excess return over and above what Beta Risk implies Security Market Line SML
Returns (Expected and Actual)
35%
Actual Returns
SML shows the expected return at a given level of Beta denotes the actual return given by various securities
30% 25% 20%
Actual Return < Expected i.e. Negative Jenson's Alpha. So SELL
Actual Return > Expected i.e. Positive Jenson's Alpha. So BUY
15% 10%
Rf 5%
0% 0
0.2
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0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Beta
7
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Performance Measures – Treynor Ratio
• Treynor Ratio: The ratio measures the excess return over the risk-free rate given by a portfolio with respect to its Beta • TR= (Rp – Rf) / Beta
where Rp = portfolio return, Rf = risk free return
• Higher the Treynor measure, the better the portfolio • This measure should be used only for well-diversified portfolio
Neev Knowledge Management – Pristine
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Performance Measures – Treynor Ratio
• Example Annual return from Dow Jones for 5 years is 8% annually. Avg. annual risk free return is 3.5%. Then compare three below mentioned performances A. Portfolio A with return of 10% and Beta of 1.1 B. Portfolio B with return of 12% and beta of 1.2 C. Portfolio C with return of 14% and beta of 1.5
• Solution TR of Market = (.08 – .035)/1 = .045 (Beta of market is 1) A. TR of A = (.1 – .035)/1.1 = .059 B. TR of B = (.12 – .035)/1.2 = .071 C. TR of C = (.14 – .035)/1.5 = .070
Though, all the portfolios have outperformed the market, B gives maximum return for the systematic risk taken, as determined by the Treynor Ratio
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Performance Measures – Sharpe Ratio
• Sharpe ratio: The ratio measures the excess return over the risk-free rate given by a portfolio with respect to its total risk (measured by standard deviation of returns). The ratio therefore can be used for single securities as well.
• SM= (Rp – Rf) / σ , where Rp = portfolio return, Rf = risk free return, σ = Std. deviation of returns
• The higher the Sharpe measure, the better performer the portfolio/security as compared to others
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Performance Measures – Sharpe Ratio • Example: Annual return from Dow Jones for 5 years is 8% annually. Avg. annual risk free return is 3.5% and Market returns had 12% std deviation. Then compare performance of these portfolios: A. Portfolio A with return of 10% and 9% std deviation B. Portfolio B with return of 12% and 12% std deviation C. Portfolio C with return of 14% and 15% std deviation
• Solution: Using Sharpe ratio S.R. of Market = (.08 – .035)/.12 = .375 A. S.R. of A = (.1 – .035)/.09 = .65 B. S.R. of B = (.12 – .035)/.12 = .71 C. S.R. of C = (.14 – .035)/.15 = .7 Portfolio C has given maximum return but as per Sharpe measure, B is a better performer. However, all the portfolios have outperformed the market.
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Performance Measures – Jensen’s Alpha • Jensen’s Alpha: This measure determines portfolio’s return over and above returns expected by CAPM. • α = Rp – Rc , where Rp = Actual portfolio return, Rc = Expected return (CAPM) = Rf + Beta*(Rm – Rf) • Positive alpha (portfolio with positive excess return) is always preferred over negative alpha
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Performance Measures – Jensen’s Alpha • Example: Annual return from Dow Jones for 5 years is 8%. Risk free rate is 3.5% and market returns had 12% std deviation. Then compare three below mentioned performances a. Portfolio A with actual return of 10% and 0.9 beta b. Portfolio B with actual return of 12% and 1.2 beta c. Portfolio C with return of 12% and 1.3 beta
• Solution: a. α of A = .10 – ( .035+.9*(.08 –.035)) = .058 = 5.8% b. α of B = .12 – ( .035+1.2*(.08 –.035)) = .064 = 6.4% c. α of C = .12 – ( .035+1.3*(.08 –.035)) = .060 = 6% Portfolio B has the maximum alpha
Neev Knowledge Management – Pristine
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Single Index Model (SIM) • The model simply assumes that statistically return of a security maybe represented as a linear function of a single economic/market variable like GDP growth, interest rates etc. • Rit = θi + δiIt+εit , where Rit is the return on a security i at time t εit is the error term and It is the economic variable • If It is set as the market return premium, Rm – r, SIM will differ from CAPM
• In case of CAPM, we run a regression between excess returns given by a security over risk-free and excess returns given by Market above risk-free i.e. regression between (Ri – Rf) and (Rm – Rf). However, in SIM, we don’t run regression on excess return but total return. • For any security, the Total Variance of return = δ x variance of index + variance of specific risk
Neev Knowledge Management – Pristine
14
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Multifactor Models and APT
• CAPM assumes that excess returns by market and excess return on a security are correlated. Arbitrage Pricing Theory (APT) is a model which states that return on a security is related to its exposure to multiple factors. • Ri = ai + ∑bj*Fj + error
where, Ri = Return on a security “i” Bj = Sensitivity of return of security “i” to jth Factor Fj is the value of the jth factor
• Such a model can be fitted by running a multiple regression between return on a security and values of different market variables
• For example, Return on ABC share = 5% + 1.63*Change in Inflation rate – 0.67*Change in USD/INR rate + 2.87*Annual GDP growth rate
• If we consider only a single factor (excess market return over the risk-free), then APT is equivalent to CAPM.
Neev Knowledge Management – Pristine
15
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Questions
• Two stocks X and Y have delivered returns of 15% and 20% respectively over the las year. The stocks had standard deviation of 9% and 15% respectively. The marke return was 10% and std dev was 10%. If the risk free rate is 5%, according to Sharpe ratio A. A is a better performer than B and the market B. B is a better performer than A and the market C. Both A and B performed equally better than the market D. Both A and B performed equally worse than the market
• A stock X has a beta of 1.2 and gave a return of 20% returns last year. If the marke gave a return of 15% and the risk free rate is 5%, The Jensen alpha for X is A. 4% B. 5% C. 3% D. 2%
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Questions • Which of the following is/are true assumptions to the CAPM? A. Investors have a limited capacity to borrow and lend at the risk-free rate B. Information is freely and universally available to all the investors C. Investors are risk takers and seek to maximize their wealth D. Returns are normally distributed
• A stock X that has beta of 1.2 pays a return of 20%. If the market premium is 10% what will be the return on stock Y if its beta is 1.35? A. 21% B. 22.5% C. 22% D. 21.5%
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Questions • The SIM model of a stock X was found to be RXt = 10% + 1.5(Rm – Rf) + error . If the variance of the market last year was 15% and variance due to the unsystematic risk of the stock was 25%, what is the overall variance of the returns from the stock? A. 47.5% B. 50.5% C. 45% D. 48%
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Answers • SR of A = (.15 - .05) /0.09 = 1.11, SR of B = 1.00, SR of market = 0.5 – A. A is a better performer than B and the market • Jensen Alpha = 0.20 - (0.05 + 1.2(0.15-0.05) = 0.03 = 3% – C. 3% • Information is freely and universally available to all the investors – B. Information is freely and universally available to all the investors • X = rf + 1.2 * 10% gives rf = 8 and Y = 8 + 1.35*10 = 21.5% – D. 21.5%
• Total variance of the stock = delta X market variance + specific risk variance = 1.5x15% + 25% = 47.5% – A. 47.5%
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End of CAPM and Multifactor Models
