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optimum portfolio

ind curve tangent to mean variance frontier

risk averse - steep ind curves
less risk-averse - flatter

preferences towards risk and return determine the portfolio of risky assets if there is no risk free asset


efficient set

group of portfolios that both minimise risk for a given expected return and maximise expected return for a given level of risk

it is CML if risk-free asset is present


two-fund separation

any risk-averse investor (regardless of their degree of risk-aversion) can form their optimal portfolio by combining two mutual funds. first is the tangency portfolio of risky assets. second is the risk-free asset. all that the degree of risk-aversion dictates is the portfolio weights placed on each of these funds.

the result underlies CAPM


capm assumptions

1) investors maximise utility defined over expected return and return variance
2) unlimited amounts of borrowing and loaning at risk-free rate
3) investors have homogeneous expectations regarding future asset returns
4) asset markets are perfect and frictionless (no taxes on sales or purchases, no transaction costs and no short sales restrictions)


capital market equilibrium

demand for risky assets is identical to their supply
supply is summarised in the market portfolio


market portfolio

portfolio comprising of all assets, where the weights used in construction of the portfolio are calculated as the market capitalisation of each asset divided by the sum of market capitalisations across all assets


equilibrium and capm

two fund separation: all investors hold efficient portfolios and all investors hold risky securities in the same proportions dictated by the tangency portfolio

capital market equilibrium: demand=supply for risky securities

then, the market portfolio is constructed with the same weights
then, market portfolio and tangency portfolio are identical


betas diversifiable undiversifiable

capm: two assets with identical turns should have the same betas, although their variances can differ.
reason: proportion of returns' variances can be eliminated through diversification.
undiversifiable risk is the one that is driven by the variation in the return on the market as a whole. asset's exposure to such risk is summarised by beta

thus, asset's beta measures its relevant risk

sigma^2=beta^2 sigmamarket^2+sigmaE^2


roll critique

statement of the capm states that market portfolio is mean-variance efficient

roll noted that the market portfolio is not observable to the econometricians who wish to conduct a test of the above statement. researchera use proxies like sp500 or nikkei 250. but the true will contain bonds and stocks not included and also non-financial assets such as real estate, human capital, durable goods.
hence, the validity of tests of capm depend critically on the quality of the proxy.
so, critique is that capm is not testable, as market portfolio is unobservale.


first stage of the capm testing

for each asset we run a time series regression of that asset's returns (minus risk free rate) on the market excess return.
check if Ai is zero


second stage of capm testing

single cross-sectional regression for all assets
regressing the average historical return for each asset (minus rf) on the beta for each asset found in the first stage.
G0 zero G1 market premium


empirical evidence: capm

data are generally not supportive of the capm
the relationship between beta and returns is usually positive but typically flatter than it should be.
high beta stocks have returns smaller than predicted by capm

there are certain assets that appear to consistently have non-zero Ai in time-series regressions.

too flat explanation:
measurement error. we see beta plus some measurement error. assets with high beta tend to have very positive measurement error. so, their true beta may be below. same about the low.

it is possible that one factor is not enough to explain al of the variation.

loadings on other factors like p/e b/m firm size have been shown to explain ex post realized returns.


market efficiency

a market is efficient with respect to a given information set O if no agent can make economic profits through the use of a trade rule based on the information set O. economic profits are defined as the returns after costs are adjusted for risk.


definitions of market efficiency in Fama 1991

weak form - prices fully reflect all historical information (past prices and past financial characteristics, macroeconomic conditions)
semi-strong form - prices fully, accurately and speedily reflect all new public information releases. in addition to historical data.
strong form - prices reflect all information, both public and private.


joint hypothesis problem

empirical researchers do not know the true expected returns generation model. so, abnormal returns may be incorrectly measured.
we are in position where we are not sure whether the markets are inefficient or our model of expected returns is wrong.

the null of any test of efficiency is comprised of two components:
informational efficiency
the accuracy of one's model for expected returns


weak form tests

current and past asset returns should have no predictive power for future returns.
so, the expectation of next period's return conditional on past returns is zero.
returns are uncorrelated with their own past values.

can be viewed as test of rwm

calendar effects - pattern in returns related to the day of the week or week or month.
dummy regression


weak form emp results

academic time is staggering

autocorrelation tests:
daily,weekly: positive autocorrelation
fama (1965) and Lo and Mackinlay (1988)
strength of correlation is dependent on the size of stock.
infrequent and non-synchronous trading of small stocks will generate positive autocorrelation even when individual stock returns are uncorrelated over time.

three-five years: negative autocorrelation
fama and French (1988) and Poterba and Summers (1988)
it might be that such long swings in prices reflect mean reversion in expected returns over time, which is not picked by the expected return generating model. (joint hypothesis)

calendar: incredible January effect. statistically positive and greater in jan. again pronounced for small stocks.
taxation impacts
year end effects
effects from remuneration packages of fund managers

day of the week effects French (1988)
strong indications of market inefficiencies
joint hypothesis is not an exit

other variables: lakonishok, shleifer and vishny (1994)
choose shares that are low to earnings dividents book value or cash flows.
value and glamour portfolio, 10 in total. 5 years tracking.
positive relationship to return
value outperforms the glamour by 10-11% per annum
the excess returns over glamour have persisted over 1968-1990 period.

debondt thaler (1985)
allocate stocks on the basis of past performance in excess returns. losers tend to outperform winners. 25% more over 3 years.

difference in value and glamour is not high enough to justify the discrepancy between returns.

trading rules
MA rule can allow for positive excess returns on average.
brock lakonishok and lebaron (1992) apply to us stock index data with some success.



theough holding portfolios of assets we can reduce the risk associated with our position.
the correlations between returns are less than perfect. it implies that when returns on the first are above average, those on the second need not be above average. hence, the returns would cancel each other out to some extent, so the total variance. would be smaller than the weighted average of the individual asset variances.