Summary Lectures Advanced Asset Pricing 2024
School of Business and Economics, Vrije Universiteit Amsterdam
Advanced Asset Pricing (E_FIN_AAP)
A. Pitkäjärvi and M. de Vries
, Knowledge Clips 1 – Portfolio Theory, Utility, CAPM, and Fama-MacBeth
Methodology
Portfolio theory
Traditional process of investing
Form expectations (beliefs)
The process of selecting a portfolio: observation and experience à beliefs about the
future performances of available securities à choice of portfolio
Modern Portfolio Theory (MPT) – how to invest money in the financial market
• Given expected returns, risks, and correlations
• Two approaches
o Maximize expected return given a certain level of risk
o Minimize risk given a certain level of expected return
• Focus on the optimal combination of assets
o Diversification
o Irrespective of the investor’s utility function
Utility function – how much to invest of someone’s money in the financial market based
on the risk aversiveness of the client
Portfolio return (n = 2): 𝐸"𝑟! $ = 𝑥" 𝐸(𝑟" ) + 𝑥# 𝐸(𝑟# )
Portfolio var and risk (n = 2): 𝜎 $ "𝑟! $ = 𝑥"$ 𝜎 $ (𝑟" ) + 𝑥"$ 𝜎 $ (𝑟" ) +
2𝑥" 𝑥# 𝜎(𝑟" )𝜎(𝑟# )𝜌(𝑟" , 𝑟# ) = 𝑥"$ 𝜎 $ (𝑟" ) + 𝑥"$ 𝜎 $ (𝑟" ) + 2𝑤" 𝑤# 𝐶𝑜𝑣(𝑟" , 𝑟# )
• With examples correlation = 1, 0, and -1
Minimum Risk Point
• Take 𝑥# as (𝑥" − 1)
• Take derivative of portfolio risk formula and set equal to zero 0 à calculating
minimum of the parabola
What are the beliefs?
• Expected returns
• Variance
• Covariance
Rule of thumb: match the investment horizon with the historical returns that are used to
estimate the expected returns.
,Tangency portfolio: line from risk-free rate at the x-axis (zero risk) to the e]icient frontier.
• This line is the new e]icient frontier, or the capital market line (CML)
Utility function
Step 2 of the traditional process of investing: utility function + budget constraint
Based on portfolio theory, we know that all investors should hold a combination of the
risk-free asset and the market portfolio. But which combination of these two assets
should be held?
Utility function: tells something about the utility/happiness that someone gains from a
certain investment. The curve shows the investments where the person is indi]erent, all
the points give the same utility. It is the indiLerence curve.
Expected Utility Theory (von Neuman Morgenstern)
• Decision maker faced with risky outcomes of di]erent choices will behave as if
he is maximizing the expected value of some function defined over the potential
outcomes.
• Expected utility: weighted average utility of di]erent outcomes with probabilities
vNM axioms of rationality (building blocks of expected utility theory)
, Wealth and utility
Quadratic (mean-variance) utility
𝑏 𝑏
𝑈(𝑊) = 𝑎𝑊 − 𝑉 [𝑊 ] = 𝑎𝑊 − 𝑊 $
2 2
• When wealth increases, the utility increases (a person is happier with more
money)
• When wealth increases, the marginal utility decreases (risk aversion)
• Only unknown is r(m), so the risk only depends on r(m)
• Interpretation 𝐸(𝑟% ): if expected return on market increases, we want to increase
the amount invested in the market portfolio 𝑥% .
• Interpretation 𝑉𝑎𝑟;𝑟% − 𝑟& <: if the variance of excess return increases (market
becomes riskier), we decrease the amount invested in the market portfolio 𝑥% .
School of Business and Economics, Vrije Universiteit Amsterdam
Advanced Asset Pricing (E_FIN_AAP)
A. Pitkäjärvi and M. de Vries
, Knowledge Clips 1 – Portfolio Theory, Utility, CAPM, and Fama-MacBeth
Methodology
Portfolio theory
Traditional process of investing
Form expectations (beliefs)
The process of selecting a portfolio: observation and experience à beliefs about the
future performances of available securities à choice of portfolio
Modern Portfolio Theory (MPT) – how to invest money in the financial market
• Given expected returns, risks, and correlations
• Two approaches
o Maximize expected return given a certain level of risk
o Minimize risk given a certain level of expected return
• Focus on the optimal combination of assets
o Diversification
o Irrespective of the investor’s utility function
Utility function – how much to invest of someone’s money in the financial market based
on the risk aversiveness of the client
Portfolio return (n = 2): 𝐸"𝑟! $ = 𝑥" 𝐸(𝑟" ) + 𝑥# 𝐸(𝑟# )
Portfolio var and risk (n = 2): 𝜎 $ "𝑟! $ = 𝑥"$ 𝜎 $ (𝑟" ) + 𝑥"$ 𝜎 $ (𝑟" ) +
2𝑥" 𝑥# 𝜎(𝑟" )𝜎(𝑟# )𝜌(𝑟" , 𝑟# ) = 𝑥"$ 𝜎 $ (𝑟" ) + 𝑥"$ 𝜎 $ (𝑟" ) + 2𝑤" 𝑤# 𝐶𝑜𝑣(𝑟" , 𝑟# )
• With examples correlation = 1, 0, and -1
Minimum Risk Point
• Take 𝑥# as (𝑥" − 1)
• Take derivative of portfolio risk formula and set equal to zero 0 à calculating
minimum of the parabola
What are the beliefs?
• Expected returns
• Variance
• Covariance
Rule of thumb: match the investment horizon with the historical returns that are used to
estimate the expected returns.
,Tangency portfolio: line from risk-free rate at the x-axis (zero risk) to the e]icient frontier.
• This line is the new e]icient frontier, or the capital market line (CML)
Utility function
Step 2 of the traditional process of investing: utility function + budget constraint
Based on portfolio theory, we know that all investors should hold a combination of the
risk-free asset and the market portfolio. But which combination of these two assets
should be held?
Utility function: tells something about the utility/happiness that someone gains from a
certain investment. The curve shows the investments where the person is indi]erent, all
the points give the same utility. It is the indiLerence curve.
Expected Utility Theory (von Neuman Morgenstern)
• Decision maker faced with risky outcomes of di]erent choices will behave as if
he is maximizing the expected value of some function defined over the potential
outcomes.
• Expected utility: weighted average utility of di]erent outcomes with probabilities
vNM axioms of rationality (building blocks of expected utility theory)
, Wealth and utility
Quadratic (mean-variance) utility
𝑏 𝑏
𝑈(𝑊) = 𝑎𝑊 − 𝑉 [𝑊 ] = 𝑎𝑊 − 𝑊 $
2 2
• When wealth increases, the utility increases (a person is happier with more
money)
• When wealth increases, the marginal utility decreases (risk aversion)
• Only unknown is r(m), so the risk only depends on r(m)
• Interpretation 𝐸(𝑟% ): if expected return on market increases, we want to increase
the amount invested in the market portfolio 𝑥% .
• Interpretation 𝑉𝑎𝑟;𝑟% − 𝑟& <: if the variance of excess return increases (market
becomes riskier), we decrease the amount invested in the market portfolio 𝑥% .
