index capm and apt
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disadvantage of full markowitz model
- uses too many estimates - hard for analyst to specialize
- the cumulative error may result in a portfolio that is inferior to that derived from a single index model
- uses past returns as estimates for the future is not reliable
1. Individual investors are price takers
2. Single-period investment horizon
3. Investments are limited to traded financial assets
4. No taxes and no transaction costs
5. Information is costless and available to all investors
6. Investors are rational mean-variance optimizers
7. There are homogeneous expectations
8. investors can borrow or lend any amount at a fixed, risk-free rate
the slope of CML is the...
sharpe ratio of the market portfolio
1. All investors will hold the same optimal portfolio of risky assets – market portfolio M.
2. The borrowing and lending cancel out, so the market portfolio equals the entire wealth of economy.
3. Market Risk Premium depends on the market variance adjusted by the average degree of risk aversion.
4. An individual security’s risk premium is proportional to the risk premium of the market portfolio.
5. CAPM is an equilibrium model,therefore all the assets included in the market portfolio should have same risk reward ratio.
CAPM-Implication - 1. All investors will hold the same optimal portfolio of risky assets - market portfolio, ie..; why is the market portfolio the optimal portfolio?
- The M portfolio lies in the efficient frontier and the highest CAL;
- As they are rational mean variance optimizers, they will come up with the same portfolio to hold when it eventually comes to an equilibrium. As it is a close economy, all the individual investors add up to form the whole market. As they hold the same portfolio, then the portfolio itself will become the market portfolio.
CAPM-Implication -Market Risk Premium depends on the market variance adjusted by the average degree of risk aversion. - why?
Because lending cancels out with borrowing, the average position is y=100%
CAPM-Implication -4. An individual security’s risk premium is proportional to the risk premium of the market portfolio.- why?
β measures the contribution of the asset to the market portfolio risk over the total market portfolio risk.
β captures the... because β=cov(ri, rm) / var(rm)
the contribution of the asset risk to the portfolio variance
CAPM-Implication - 5. CAPM is an equilibrium model,therefore all the assets included in the market portfolio should have same risk reward ratio. - implications
1. all risky assets are found along SML, because all of them has the same risk reward ratio
2. zero alpha
what is the slope of Security market line?
market risk premium
- same beta if market index = market portfolio
- no alpha; alpha is the excess
- the M portfolio VS more general as it provides the expected return and beta relationship without the assumption of the (unobservable) M; any well-diversified portfolio that is highly correlated with macro factors can serve as the benchmark portfolio
- CAPM applies to individual stocks VS APT applies to well-diversified portfolios too
- APT calculated the expected return by using the macro factors directly and not the index portfolio proxy as it the case Index Model or the market portfolio in CAPM.
- APT allows for mispricing in individual securities
- the market portfolio is unobservable, therefore it is difficult to test its mean-variance efficiency
- Impossible to get the expected return for all risky assets in the portfolio according to CAPM.
compute α and rules to buy a stock
- α = analyst expectation - CAPM required return;
- positive α
sell the stock when it is..
- -Security returns can be described by a factor model
- -There are sufficient securities to diversify away unsystematic risk
- -Well-functioning security markets do not allow for the persistence of arbitrage opportunities
why does APT justifies the use of index models in est. parameters?
index portfolio is well-diversified port. highly correlated with macro factors?
referring to the SML, when are there no arbitrage opportunities?
all portfolios that are a combination of port of β = 1 and risk free, must lie at SML
R square(for well diversified portfolio)
capm individual stock risk premium
E(ri) - rf = betai * (market risk premium),
apt individual securities realized return
ri=E(ri) + sum of (beta*change of macro factors)
APT expected return of individual securities
E(ri) = rf + sum of (beta*macro factors)
betai = (2 ways)
2 points along SCL, cal
) * corr(rB
CML/CAL and SCL and SML x-ax
- 1. sigma
- 2. excess return of market
- 3. beta
unsystematic variance of a portfolio in apt
continuously compound rate
rcc= ln (1+EAR)
Skew: positive number
Kurtosis: positive number
what are their meaninngs?
- Skew: right tail
- Kurtosis: fat tail, higher chance of large gain and losses
5% Var meaning;
the corresponding SD values for 1% Var and 10% Var
- 5% prob. (95% CI) of getting return lower than 1.65 SD below the mean;
- 2.33 and 1.28
1. current equity
price)call 3. (the formula4. equity5. amount
- 1. current price*Q - current loan
- 2. end price*Q
- 3. current equity/end position <= maintenance margin
- 4. min P*Q - current loan
- 5. initial equity - current equity
buy on margin - ROE
with borrowing: (current equity-initial equity) /initial equity
without borrowing: (current price-initial price)/initial price
when will you leverage? given the expectation of a stock price
if the expected price is greater than the initial price, leveraging will yield a higher return than no leverage
where the correlation of firm specific risks in a portfolio are assumed to be zero, the systematic and unsystematic risk of the portfolio is
given a target return, find the optimal weights between risky and risk-free asset, if you find that the weight for the risk-free is negative,
drop the risk free and only considers risky assets.
given the corr(A, B) = -1: implication
- a risk-free portfolio can be created
- return for this portfolio is risk-free rate
- set variance = 0, simplify to wasda-wbsdb=0
covariance of a stock with the market in CAPM is the measure of
risk of the stock
a zero-beta stock: implication
its return is the risk-free rate
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