Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
FINS5513 Lecture 3 CAPM and SIM 2❑ 3.1 General Equilibrium: Derivation of the CAPM ➢ Capital Asset Pricing Model (CAPM) Assumptions ➢ The Market Portfolio and Capital Market Line ➢ Derivation of the CAPM ❑ 3.2 Interpreting the CAPM ➢ Beta ➢ Systematic and Unsystematic Risk ➢ Security Market Line ➢ Applications and Extensions of the CAPM ❑ 3.3 The Single Index Model (SIM) ➢ Limitations of the CAPM ➢ SIM ➢ Alpha ➢ SIM Risk Measures Lecture Outline 3.1 General Equilibrium: Derivation of the CAPM FINS5513 4Capital Asset Pricing Model (CAPM) Assumptions FINS5513 5❑ Markowitz established modern portfolio theory through important contributions in 1952 and 1959 ❑ Based on this work, Sharpe, Lintner, Mossin and others published papers between 1964- 1966 which collectively became known as the Capital Asset Pricing Model (CAPM) ❑ CAPM is a model for deriving expected returns on risky assets under equilibrium conditions ❑ CAPM is derived under a general equilibrium framework.The assumptions under which the CAPM is derived simplify the world with regard to: ➢ Individual behaviour ➢ Market structure ❑ The CAPM assumptions are often considered to be stylised and not reflective of reality Capital Asset Pricing Model (CAPM) Evolution 6❑ Individual behaviour ➢ Investors are rational, mean-variance optimisers (as per Markowitz) ➢ Investors are price takers - no investor is large enough to influence equilibrium prices ➢ Investors common planning horizon is a single period ➢ Investors have homogeneous expectations on the statistical properties of all assets (i.e. same expected returns and covariances and all relevant information is publicly available) ❑ Market structure ➢ Investors can borrow and lend at a common risk-free rate with no borrowing constraints ➢ All assets are publicly held and traded on public exchanges ➢ Perfect capital markets - there are no financial frictions such as short selling constraints, transaction costs, taxes etc ❑ Under these assumptions, all investors derive the same efficient frontier, CAL, and ∗ CAPM Assumptions 7The Market Portfolio and Capital Market Line FINS5513 8❑ The foundations of the CAPM come from Markowitz and separation theorem: ➢ Sharpe began with the question “What if everyone is optimising a la Markowitz?” ❑ Under separation theorem, EVERY rational investor invests along the CAL, regardless of risk aversion – because all portfolio combinations on the CAL have the highest Sharpe ratio ❑ Therefore, in market equilibrium all investors hold the same optimal risky portfolio ∗ ❑ If, in equilibrium, all investors are holding the same ∗, this must be the market portfolio , and must comprise all assets ➢ The weight of each asset in is the asset’s market value divided by the total value of ➢ Every investor holds some portion of this market portfolio ❑ Since ∗ is the market portfolio , also has the highest possible Sharpe ratio Separation Theorem Implications 9❑ The rational way to increase return (and risk) is to invest more in (rather than deviating from and buying risky assets in different weightings to their weightings in ) ❑ Since every investor invests in , the common CAL associated with is called the Capital Market Line (CML) The Capital Market Line ❑ The CML is equivalent to the optimal CAL. It is the aggregation of EVERY investors’ CAL which are all the same ❑ is equivalent to ∗. It is the aggregation of EVERY investor’s optimal risky portfolio ∗ which are all the same ❑ Every investor invests along the CML. Movements along the CML represent different allocations between and the risk-free asset 10 ❑ An investor can implement an entirely passive strategy requiring no security analysis ❑ Requires two assets: ➢ Risk-free: short-term T-Bills or money market mutual fund ➢ Risky: mutual fund or an ETF tracking a broad-based market index (eg the S&P500) ❑ Then, draw a line joining the two assets. This represents is a practical version of the CML ❑ According to BKM Table 6.7, the U.S. equity market returned 11.72% with a standard deviation of 20.36% (assume for simplicity this represents the Market portfolio although it is only a proxy) and 1-month T-bills (the risk-free asset) returned 3.38% between 1926 and 2018. a) Draw the Capital Market Line. What is the slope of this line and what does it represent? b) Aggressive Investor A and Conservative Investor C target maximum portfolio risk of 25% and 14% respectively. For investors A and C: calculate their allocation to the equity market (∗) and T-bills (1 − ∗), the expected return on their complete portfolio (∗), the Sharpe ratio of their complete portfolio, and their implied risk aversion coefficient . Example: A Practical CML 11 ➢ The slope of the CML is the market’s Sharpe ratio Risk Premium = 0.1172 - 0.0338 = 8.34% Market Sharpe ratio = 0.0834 0.2036 = 0.41 Example: A Practical CML 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 0.000 0.050 0.100 0.150 0.200 0.250 0.300 P o rt fo lio E xp e ct e d R e tu rn E (r ) Portfolio Standard Deviation σ Capital Market Line Investor A Investor C Market – T-bills Excel: “CML and SML” Investor A Investor C % in market 0.25 0.2036 = 122.8% 0.14 0.2036 = 68.8% % in T-bills 1 − 1.228 = −22.8% Borrow 22.8% at 1 − 0.688 = 31.2% Invest 31.2% in T-Bills Expected Return () 0.0338 + 1.228 × 0.0834 = 13.62% 0.0338 + 0.688 × 0.0834 = 9.11% Risk 25% 14% Sharpe Ratio 0.1362 − 0.0338 0.25 = 0.41 0.0911 − 0.0338 0.14 = 0.41 Risk Aversion () 0.0834 1.228 × 0.20362 = 1.64 0.0834 0.688 × 0.20362 = 2.93 12 ❑ We know from week 1 (1.4) that: ➢ An individual asset’s effect on portfolio return is simply proportional to its weight ➢ However, its effect on portfolio risk depends on its covariance with other portfolio assets ❑ We have already concluded that under the CAPM assumptions, every investor would hold for their risky asset allocation. Therefore: ➢ The attractiveness of an individual asset should be assessed based on its contribution to market return and market risk • Individual asset contribution to market return simply depends on its return (and weight) • However, its contribution to market risk depends on its covariance with all other assets in the market ❑ So, the risk of an individual asset is no longer measured just by its own variance, but rather its covariance with the market ➢ In equilibrium, this is now our key measure of risk ➢ This is a key insight of the CAPM The Market Portfolio 13 ❑ Let’s restate the previous slide mathematically (using risk premium instead of raw return): ❑ The risk premium on the market portfolio (RM ) is: = σ=1
() = σ=1 () ℎ =
σ=1
ℎ ℎ ➢ So, asset ’s contribution to ’s risk premium is () ❑ The risk on the market portfolio ( 2 ) is:
2 = , = σ=1 , = σ=1 , ➢ So, asset ’s contribution to ’s risk (variance) is , Under CAPM Risk is Measured by Covariance 14 Derivation of the CAPM FINS5513 15 ❑ Where have the CAPM assumptions taken us so far?: 1. In equilibrium, all rational investors would hold 2. If investors are holding , the risk measure for an individual asset is its covariance with ❑ From our previous slide, given individual asset risk is measured as covariance with the market, let’s define a reward-to-risk ratio for any asset as: ′ ′ = () ( , ) = () ( , ) ❑ Next, consider this: in a world of homogeneous expectations, there would be no reason for an investor to buy an asset with a lower reward-to-risk ratio than another asset ➢ If one asset had a higher reward-to-risk ratio than another, all investors would buy it, until it’s reward-to-risk ratio was in parity with all other assets • Therefore, in equilibrium, all assets should have the same reward-to-risk ratio: () ( , ) = () ( , ) (, … ) Deriving the CAPM 16 ❑ In equilibrium, would have the highest Sharpe ratio and would therefore also have the highest reward-to-risk ratio given by: () ( , ) = ()
2 ❑ In equilibrium, any individual asset’s reward-to-risk ratio should equal ’s reward-to-risk ratio (the highest possible reward-to-risk ratio). If this were not the case, the price of the asset would adjust until its ratio is in parity with all other assets within and with itself: () ( , ) =
2 Rearranging: = ,
2 () Defining: = ,
2 Then: = Often stated as the CAPM equation: − = [ − ] Deriving the CAPM 3.2 Interpreting the CAPM FINS5513 18 ❑ The market risk premium − or can be regarded as the average risk premium among all assets in the market ❑ The market risk premium depends on the average risk aversion of all market participants ➢ Recall that each individual investor chooses an allocation to the optimal portfolio (in a general equilibrium) such that = −
2 , ℎ ℎ ℎ ➢ In the CAPM economy, all borrowing is offset by lending, so on average = 1 Therefore: = 2 ➢ Therefore, the market risk premium tends to expand when average risk aversion in the market rises (to encourage risk taking) and contract when average market risk aversion falls Market Risk Premium 19 Beta FINS5513 20 ❑ Let’s review the CAPM equation: − = [ − ] Which can be restated as: = ❑ So, asset ’s expected return above the risk-free rate (its “risk premium”) equals: × ➢ Risk is measured by its relative contribution to the variance of , which is it’s : • = ,
2 which tells us how asset moves with ➢ Price of Risk is the market risk premium: − • There is one price of risk in the market ❑ In equilibrium, more risky assets are compensated with higher expected returns to make them equally favourable to less risky ones ➢ One of the reasons the CAPM is widely used is because it is simple and intuitive ➢ It basically states that asset ’s risk premium is linearly related to the market risk premium according to it’s Interpretation of the CAPM 21 ❑ Rather than running covariances between each individual asset in the portfolio, as we do under the Markowitz model, we estimate each asset’s return as a function of its covariance with one common factor - the market return under CAPM ➢ This significantly reduces the calculations we need to make ➢ Importantly, the relates each asset’s return to the market return ❑ can be < 0 and can be > 1: ➢ > 1 means that , > 2 - the asset contributes more risk than the average asset (or is riskier than the market) ➢ < 1 means that , < 2 - the asset contributes less risk than the average asset (or is less risky than the market) ❑ Derived by regression – we will see how in lecture 4 and the iLab ❑ One of the most important concepts in finance and widely used in industry What is ? 22 ❑ Assume in a CAPM equilibrium the market’s expected return () = 11.0%, the risk-free rate = 3.0% and the market’s variance 2 = 0.0256. Applying CAPM, Asset A has expected return of () = 12.6% and Asset B has expected return of () = 8.6%. a) What is the market risk premium? b) What is the of Asset A and B c) What is the covariance of Asset A and B with the market return (respectively)?