Lecture 8: Capital Asset Pricing Model
Economics of Finance
School of Economics, UNSW
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Systematic vs Idiosyncratic Risk
sABC = sp + si
• sp = βsM : systematic risk – non-diversifiable
• si: idiosyncratic risk – diversifiable
• β ≡ x: share invested in the market portfolio to replicate e
Capital Asset Pricing Model
Capital asset pricing model (CAPM) is a model used to determine an appropriate expected return of any asset
• only systematic risk is valued
• replicate any desired expected asset return ej using the market portfolio (fraction βj) and the risk-free asset (fraction 1–βj)
ej =βjeM +(1−βj)rf =rf +βj(eM −rf)
What if ej < rf
Negative β
Still the same
ej =βjeM +(1−βj)rf =rf +βj(eM −rf)
Alternative interpretation of β
To infer βj, regress the actual (historical) excess asset return,
Rj −rf, on excess market return RM −rf:
Rj −rf =αj +βj(RM −rf)
From econometrics, we remember that regression coefficient βj = cov(Rj,RM)
Therefore, βj indicates how the specific asset co-moves with the
• β > 1 asset is more volatile than the market
• 0 < β < 1 asset is less volatile than the market
• β < 0 asset moves in opposite direction – rare and useful
What about αj? It should be 0 in theory. “Chasing” α.
Security market line
With different β value, the required return for any asset is e=rf +β(eM −rf)
Example: security market line
Country stock market indices averaged over 1988-2017
Example: pricing with CAPM
Gordon’s stock price model:
of dividend growth
Canuseeˆj =rf +βj(eM −rf)toinferthefairstockprice.
Dj current dividend ej expected rate of return gj expected rate
The standard deviation
CAPM gives us guidance about the expected return, what about standard deviation (or risk)?
The standard deviation of Rj : sj = βj sm + si
• reflects an idiosyncratic risk in addition to systematic risk
• recall that market portfolio and CML reach the highest Sharpe ratio
• any other securities are stochastically dominated by M and rf combinations
• this does not necessarily mean other securities will cease from the market, as they will be traded to construct M.
Arbitrage Pricing Theory (APT)
CAPM provides good benchmark, but reality is more complicated: market risk is just one factor, but there are others
Rj =rf +βj,1f1+,...,+βj,KfK +εj,
• Rj is the expected return of the asset (or portfolio) j
• εj idiosyncratic, unexplained part of return E(εj) = 0, E(Rj) = ej
• rf is the risk-free rate
• fk is the factor risk premium
• βj,k is the sensitivity of portfolio j to factor k
• K is the number of factors.
Assumptions (similar to standard OLS):
• exogeneity: εj and factors fk are independent • εj for different assets are independent
This is not pure arbitrage, but statistical arbitrage
Example: applying the model to sector and region
Rj = rf + βj,mfm + βj,sfs + βj,rfr + εj, • fm is the market risk premium (em − rf );
• fs is the risk premium for a particular sector, e.g., ASX health care sector index
• fr is the risk premium for a particular region, e.g., MSCI Asia Pacific index
• βs are sensitivities to the factors
Example: Fama and French Model – 3 Factor Model
Rj = rf + βj,mfm + βj,SMBfSMB + βj,HMLfHML + εj,
• fm is the excess return on the market portfolio
• fSMB is the size factor attributable to the company’s market capitalisation, Small minus Big
• fHML is the value factor driven by the difference between High minus Low book-to-market stocks
Extensions:
• momentum
• operating profitability
• firm investment factor (agressive vs conservative) • time-varying factors – dynamic factor models
Smart Betas: Bridge between Active and Passive Portfolios
Benefits of smart beta products
• Customisable products
• Lower transaction costs relatively to active portfolio
• Sharpe ratio is maximised by market portfolio
• Optimal risky is independent from individual preferences –
the same market portfolio for all
• Rational individual only hold portfolios on CAL
• CAPM is based on replicating portfolio
• takes into account systematic risk
• Factor models offer valuable extension to a simple CAPM
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