Summary asset pricing
1
Summary Asset Pricing – Annamarie de Ruijter – Vrije Universiteit – 2019/20
,Introduction and Portfolio Theory
In corporate finance managers decide on how to maximize the value of the firm and in asset pricing
investors decide on how to maximize the value of their portfolios. Managers need markets to raise
capital and investors provide capital for corporations. Financial markets produce information,
monitor investments; facilitate trading, diversification and management of risk, mobilize and pool
savings, ease the exchange of goods and services; and allocate capital to its most productive use.
Capital is scarce because there are more projects than there is financing so banks must select. Asset
pricing looks at the determinants of stock returns (differences) and how we can use this as an
investor to optimize the performance of stock portfolio, in the equity market only. The traditional
process of investing:
1. Form expectation about risk and return (portfolio theory).
2. Utility function and budget constraints (mean-variance utility).
3. Buy and sell orders (what % of wealth in risk-free and market portfolio?).
4. Market returns (CAPM).
The Capital Asset Pricing Model (CAPM) is a set of predictions about the equilibrium (all investments
offer the same reward-to-risk ratio) expected return on risky assets. The model assumes investors
seek mean-variance optimal portfolios. Security analysis to obtain an efficient portfolio therefore
isn’t required, investors can simply hold the market portfolio. This implies that the risk premium on
any individual asset or portfolio is the product of the risk premium on the market portfolio and the
expected return-beta relationship (the contribution of one stock to the variance of the market
portfolio). The beta of a security is here the
right measure of its risk and measures its
contribution to the variance of the market
portfolio and therefore risk premium is
proportional to its beta. Graphically the
individual asset risk premiums as a function
of asset risk result in the security market
line (SML) with a slope of 1 (m = 1). Fairly
prices assets are plotted on this line,
because their expected returns are
proportional to their risk in market
equilibrium ( = 0).
The CAPM assumes that security markets are ideal/efficient, in the sense that:
1. Investors are price-takers (individual trades don’t affect prices), rational and have
homogeneous expectations (one input list in choosing the market portfolio).
2. There are no taxes or transaction costs.
3. All risky assets are publicly traded (anyone can invest in anything and information is costless
and available to all).
4. Investors can borrow and lent any amount at a fixed risk-free rate.
5. The market portfolio is efficient and the risk premium on a risky asset is proportional to its
beta (zero alpha).
6. Returns are normally distributed.
The price of a stock is the expected value of discounted dividends: P = E(Div) / (1 + R)^t.
1. Neo-classical: investors are rational/same expectations and markets are efficient.
2. Behavioral: investors are boundedly rational/feedback interaction loops and markets can
deviate from efficiency.
Risk preferences and the tradeoff between expected return and volatility of a portfolio are expressed
as a utility function, with a penalty if the investor is risk averse. This tradeoff is graphically
2
Summary Asset Pricing – Annamarie de Ruijter – Vrije Universiteit – 2019/20
, represented by indifference curves, where a favorable portfolio has a higher expected return and
lower variance. Higher indifference curves correspond to higher levels of utility. Often used is the
Neuman Morgenstern or workhorse utility, investors will try to maximize the expected value and it
results in a probability-weighted average of utility over the possible outcomes:
W1 = (Rf + Xm x (Rm – Rf)) x W0 where W = wealth and Xm = % invested in the market 0 Xm 1
Where max E(U(W1)) = aE(W1) – b/2 x V(W1) gives Xm = a(E(Rm) – Rf) / b x W0 x V(E(Rm) – Rf) So
the demand for the market portfolio increases with expected return and decreases with risk, risk free
rate, risk aversion and wealth. Mean-variance utility to determine weight in market portfolio.
The modern portfolio theory focusses on the optimal
combination of assets through diversification, that is
reducing overall risk and ensuring stability by including
additional securities in the portfolio. This is mainly reducing
firm specific or idiosyncratic (micro-economic) and
influences but can’t eliminate systematic or market risk
(macro-economic). These two are uncorrelated, the firm risk
is independent to shocks in het global economy.
After the characteristics of all securities (expected returns,
variances, covariances) are specified, the investor chooses
the proportion of his investment budget in to risky and risk-
free asset (capital allocation). All possible portfolios and risk-return combinations are plotted in a
plane as a straight line called the capital allocation line (CAL). The slope of the line equals the
increase in expected return of the complete portfolio per unit of additional standard deviation
(reward-to-volatility ratio). A minimum-variance portfolio has a standard deviation smaller than
either of the individual component asset, this is diversification.
Deciding the efficient frontier of risky assets is known as
the Markowitz portfolio optimization model. The
frontier states different sets of risky portfolios, but we’re
only interested in the highest expected return for any
level of risk. The portfolio with the lowest standard
deviation lies the farthest to the left and the bottom half
is discarded, because it’s inefficient. The risky portfolio is
the same for everyone, only the capital allocation
between the portfolio and risk-free asset will vary due to
risk aversion and personal preference (separation
property).
Axioms of rationality:
1. Completeness: always have an opinion (still holds with prospect theory)
a. L > M, M > L or M L
2. Transitivity: consistency in preferences (not necessarily holds with prospect theory)
a. If L > M and M > N then L > N
3. Continuity: every function is continuous so that you can combine assets (still holds)
a. If L > M > N then PL + (1 – P)N M
4. Independence: however, the choice is presented, you remain consistent in your preferences
(doesn’t necessarily hold with prospect theory)
a. If L > M then PL + (1 – P)N > PM + (1 – P)N
Assets Q and R have a different risk but
same expected return, so some additional
risk for choosing R is not rewarded because
3
Summary Asset Pricing – Annamarie de Ruijter – Vrije Universiteit – 2019/20
1
Summary Asset Pricing – Annamarie de Ruijter – Vrije Universiteit – 2019/20
,Introduction and Portfolio Theory
In corporate finance managers decide on how to maximize the value of the firm and in asset pricing
investors decide on how to maximize the value of their portfolios. Managers need markets to raise
capital and investors provide capital for corporations. Financial markets produce information,
monitor investments; facilitate trading, diversification and management of risk, mobilize and pool
savings, ease the exchange of goods and services; and allocate capital to its most productive use.
Capital is scarce because there are more projects than there is financing so banks must select. Asset
pricing looks at the determinants of stock returns (differences) and how we can use this as an
investor to optimize the performance of stock portfolio, in the equity market only. The traditional
process of investing:
1. Form expectation about risk and return (portfolio theory).
2. Utility function and budget constraints (mean-variance utility).
3. Buy and sell orders (what % of wealth in risk-free and market portfolio?).
4. Market returns (CAPM).
The Capital Asset Pricing Model (CAPM) is a set of predictions about the equilibrium (all investments
offer the same reward-to-risk ratio) expected return on risky assets. The model assumes investors
seek mean-variance optimal portfolios. Security analysis to obtain an efficient portfolio therefore
isn’t required, investors can simply hold the market portfolio. This implies that the risk premium on
any individual asset or portfolio is the product of the risk premium on the market portfolio and the
expected return-beta relationship (the contribution of one stock to the variance of the market
portfolio). The beta of a security is here the
right measure of its risk and measures its
contribution to the variance of the market
portfolio and therefore risk premium is
proportional to its beta. Graphically the
individual asset risk premiums as a function
of asset risk result in the security market
line (SML) with a slope of 1 (m = 1). Fairly
prices assets are plotted on this line,
because their expected returns are
proportional to their risk in market
equilibrium ( = 0).
The CAPM assumes that security markets are ideal/efficient, in the sense that:
1. Investors are price-takers (individual trades don’t affect prices), rational and have
homogeneous expectations (one input list in choosing the market portfolio).
2. There are no taxes or transaction costs.
3. All risky assets are publicly traded (anyone can invest in anything and information is costless
and available to all).
4. Investors can borrow and lent any amount at a fixed risk-free rate.
5. The market portfolio is efficient and the risk premium on a risky asset is proportional to its
beta (zero alpha).
6. Returns are normally distributed.
The price of a stock is the expected value of discounted dividends: P = E(Div) / (1 + R)^t.
1. Neo-classical: investors are rational/same expectations and markets are efficient.
2. Behavioral: investors are boundedly rational/feedback interaction loops and markets can
deviate from efficiency.
Risk preferences and the tradeoff between expected return and volatility of a portfolio are expressed
as a utility function, with a penalty if the investor is risk averse. This tradeoff is graphically
2
Summary Asset Pricing – Annamarie de Ruijter – Vrije Universiteit – 2019/20
, represented by indifference curves, where a favorable portfolio has a higher expected return and
lower variance. Higher indifference curves correspond to higher levels of utility. Often used is the
Neuman Morgenstern or workhorse utility, investors will try to maximize the expected value and it
results in a probability-weighted average of utility over the possible outcomes:
W1 = (Rf + Xm x (Rm – Rf)) x W0 where W = wealth and Xm = % invested in the market 0 Xm 1
Where max E(U(W1)) = aE(W1) – b/2 x V(W1) gives Xm = a(E(Rm) – Rf) / b x W0 x V(E(Rm) – Rf) So
the demand for the market portfolio increases with expected return and decreases with risk, risk free
rate, risk aversion and wealth. Mean-variance utility to determine weight in market portfolio.
The modern portfolio theory focusses on the optimal
combination of assets through diversification, that is
reducing overall risk and ensuring stability by including
additional securities in the portfolio. This is mainly reducing
firm specific or idiosyncratic (micro-economic) and
influences but can’t eliminate systematic or market risk
(macro-economic). These two are uncorrelated, the firm risk
is independent to shocks in het global economy.
After the characteristics of all securities (expected returns,
variances, covariances) are specified, the investor chooses
the proportion of his investment budget in to risky and risk-
free asset (capital allocation). All possible portfolios and risk-return combinations are plotted in a
plane as a straight line called the capital allocation line (CAL). The slope of the line equals the
increase in expected return of the complete portfolio per unit of additional standard deviation
(reward-to-volatility ratio). A minimum-variance portfolio has a standard deviation smaller than
either of the individual component asset, this is diversification.
Deciding the efficient frontier of risky assets is known as
the Markowitz portfolio optimization model. The
frontier states different sets of risky portfolios, but we’re
only interested in the highest expected return for any
level of risk. The portfolio with the lowest standard
deviation lies the farthest to the left and the bottom half
is discarded, because it’s inefficient. The risky portfolio is
the same for everyone, only the capital allocation
between the portfolio and risk-free asset will vary due to
risk aversion and personal preference (separation
property).
Axioms of rationality:
1. Completeness: always have an opinion (still holds with prospect theory)
a. L > M, M > L or M L
2. Transitivity: consistency in preferences (not necessarily holds with prospect theory)
a. If L > M and M > N then L > N
3. Continuity: every function is continuous so that you can combine assets (still holds)
a. If L > M > N then PL + (1 – P)N M
4. Independence: however, the choice is presented, you remain consistent in your preferences
(doesn’t necessarily hold with prospect theory)
a. If L > M then PL + (1 – P)N > PM + (1 – P)N
Assets Q and R have a different risk but
same expected return, so some additional
risk for choosing R is not rewarded because
3
Summary Asset Pricing – Annamarie de Ruijter – Vrije Universiteit – 2019/20
