ASSET PRICING MODELS
Dalia Sharaf
CAPITAL MARKET THEORY
• It is an extension of portfolio theory (MPT).
• Each investor is assumed to diversify his/ her portfolio according to Markowitz
Model and chose a location on efficient frontier according to preferences.
• CMT adds assumptions due to real world complexity.
CMT ASSUMPTIONS
1. All investors can borrow or lend money at RFR
2. All investors have homogenous expectations (same info, same inputs)
3. All investors have same one period horizon
4. No transaction costs
5. No personal income taxes
6. No inflation
7. Investors are price takers
8. Capital markets are in equilibrium
INTRODUCTION OF RISK-FREE ASSET
• Investors can invest part of their wealth in the risk-free asset and the
remainder in any of the risky portfolios on the Markowitz efficient set.
• Risk-free asset is one that has a certain to be earned return and a variance of
zero.
• Ex: Treasury Bill
RISK-FREE BORROWING AND LENDING
Expected Return
L
M
RF
Y
Risk
B
RISK-FREE BORROWING AND LENDING
• RF 100% invest in risk free assets.
• RF-M all lending portfolios given the Markowitz efficient set.
• M 100% invest in risky portfolio.
• M-L borrowing additional investable funds and investing them to seek higher
expected returns and assume greater risk.
• Conservative investors lend, aggressive investors borrow.
EQUILIBRIUM RISK-RETURN TRADEOFF
• Capital Market Line (CML) specifies equilibrium between expected return and
risk for efficient portfolios.
• Security Market Line (SML) specifies equilibrium between expected return and
systematic risk.
CML
• It shows equilibrium conditions that prevail in the market for efficient
portfolios consisting of the optimal portfolio of risky assets and the risk-free
asset.
• Only efficient portfolios consisting of risk-free asset and portfolio M lie on
CML.
• Portfolio M is the market portfolio of risky securities, it contains all securities
weighted by their respective market values.
• All combinations are on the CML, therefore investors will end up with a
portfolio somewhere on the CML based on their risk tolerance.
UNDERSTANDING CML
UNDERSTANDING CML
• When investors combine the risky assets with the risk free one they must be
compensated by risk premium for this additional risk.
• This is determined by the slope according to the investors level of risk.
• The slope of the CML is the market price of risk for efficient portfolios. It is
known as the equilibrium market price of risk.
EXAMPLE
• What is the slope of CML, if the expected return on
portfolio M is 13%, with a standard deviation of 20% and
RF of 5 %?
•
𝐸 𝑅𝑀 −𝑅𝐹
𝜎𝑀
SOLUTION
• 𝑠𝑙𝑜𝑝𝑒 =
𝐸 𝑅𝑀 −𝑅𝐹 0.13−0.05
=
=0.40
𝜎𝑀
0.20
• The market demands 0.40 for each percentage increase in portfolio
risk.
EQUATION FOR CML
• To calculate return of portfolio it would be the risk free rate + risk premium
(which is the product of market price of risk and amount of risk for the
portfolio under consideration).
• 𝐸 𝑅𝑃 = 𝑅𝐹 +
𝐸 𝑅𝑀 −𝑅𝐹
𝜎𝑃
𝜎𝑀
• Where:
 𝐸 𝑅𝑃 = expected return of any efficient portfolio on the CML
 RF= the rate of return on risk free asset
 𝐸 𝑅𝑀 = the expected return on market portfolio M
 𝜎𝑀 = the standard deviation of the returns on the market portfolio
 𝜎𝑝 = the standard deviation of the efficient portfolio being considered
MORE INFO ON CML
• CML must always be upward sloping as price of risk will always be positive.
• This holds ex ante (expected) when it is formulated for expected return of
risk averse investors.
PORTFOLIO M
• Why do all investors hold identical portfolios?
Based on assumptions, they all use same Markowitz analysis on the same set of
securities and have same expected returns, covariances and time horizon. So they
will arrive at same optimal risky portfolio.
• In equilibrium all risky assets must be in portfolio M because all investors are
assumed to arrive at and hold same risky portfolio. IF an asset is excluded its
price will decline until it becomes an attractive opportunity and investors will
purchase it and it will be included.
• Therefore portfolio M is completely diversified and hence contains only
systematic risk.
• All assets in M are included in proportion to their market value.
SEPARATION THEOREM
• Investor has 4 options (RF, M, lend, borrow)
• Investing decision is deciding which securities to hold but here we already
know that portfolio M is optimal
• Separation theorem: states that investment decision (which risky assets to
select) is separate from financing decision (how to allocate funds between riskfree asset and risky assets).
• Financing decision is a technical one depending on preferences of investor.
CAPM
• CML only applies to efficient portfolios.
• TO calculate expected return or risk for a single security or for an inefficient
portfolio the expected return- beta form of the Capital Asset Pricing Model
(CAPM) is needed.
• Adding different securities to a well-diversified portfolio will affect it differently,
what matters is its contribution to the riskiness of the portfolio which is
measured through covariance.
• Under CMT all investors hold portfolio M so the risk that matters when
considering a security is its covariance with M.
ER OF A SECURITY
• The expected return of any security is related to its covariance with the
market portfolio
• 𝐸 𝑅𝑖 = 𝑅𝐹 +
𝐸(𝑅𝑀 )−𝑅𝐹
𝜎2 𝑀
𝐶𝑜𝑣𝑖,𝑀
 Where,
 𝐸 𝑅𝑖 = The expected return on any individual security I
 RF= the rate of return on the risk-free asset
 𝐸(𝑅𝑀 )= the expected return on market portfolio M
 𝜎 2 𝑀 = the variance of the returns on the marker portfolio
 𝐶𝑜𝑣𝑖,𝑀 = the covariance of the stock with the market

𝐸(𝑅𝑀 )−𝑅𝐹
𝜎2 𝑀
𝐶𝑜𝑣𝑖,𝑀 = risk premium
BETA
• The relevant risk for a security is its covariance with the market portfolio.
• It is more convenient to use a standardized measure of systematic risk that
cannot be avoided through diversification.
• Beta relates the covariance of an asset with the market portfolio, to the
variance of the market portfolio, as such:
• 𝛽𝑖 =
𝐶𝑜𝑣𝑖,𝑀
𝜎2 𝑀
• Beta is relative measure of risk, the risk of an individual stock relative to the
market portfolio of all stocks.
BETA
• Security with beta > 1 means more
volatility .
• Security with beta <1 means less volatility
SML
• SML is how CAPM shows risk and expected return of an asset to be related.
• There is a linear relationship between them, called the SML.
• The slope of this line is rrr (k) on market index - RF
SML
𝑘𝑖 = 𝑅𝑖𝑠𝑘𝑓𝑟𝑒𝑒 𝑟𝑎𝑡𝑒 + 𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
= 𝑅𝐹 + 𝛽𝑖 [𝐸 𝑅𝑀 − 𝑅𝐹]
where,
𝑘𝑖 = the required rate of return on asset i
𝐸(𝑅𝑀 )= the expected return on the market portfolio
𝛽𝑖 = the beta coefficient for asset i
EXAMPLE
• Assume that the beta for IMB is 1.15. Also assume that RF is 0.05 and that the
expected return on the market is 0.12. The required return from IBM can be
calculated as:
• 𝑘𝐼𝐵𝑀 = 0.05 + 1.15 0.12 − 0.05 = 13.05%