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Theory

Covariance: how close two variables move Together

The reward-to-volatility = sharpe ratio

Serial correlation of daily returns is close to zero => very hard to predict from their past

Value-at-Risk (VaR): a measure of downside risk ->Measures the potential loss over a specified

horizon such that there is a (low) probability α that the actual loss will be larger

No clear guidelines as to the choice of sample

length m: small m means that the VaR will be more influenced by recent events; large m is needed for precise estimates

- No way to extrapolate the 1-day VaR to a longer n-day horizon (except if nonoverlapping n-period returns are considered to re-calculate the ...view middle of the document...

e. restrictions on borrowing or investing in the risk-free asset)

- Combinations of portfolios on the efficient frontier are also efficient

- Any portfolio on the efficient frontier has a companion (zero-beta) portfolio with which it is uncorrelated

- Returns on the efficient frontier can be expressed as linear combinations of any two frontier portfolios:

ARBITRAGE PRICING

THEORY

Recall that according to the single factor model, the risky asset’s return can be decomposed into:

- An expected component

- Two unexpected components: exposure to a common macroeconomic factor and firm-specific events

- We have used the return of the market portfolio to summarize the impact of macro factors

- Possible macro-economic factors:

- Interest rates

- Unexpected changes in GDP

Multifactor models

- Allow for more than one factor – thus introduce different sensitivities of assets to the separate sources of systematic risk

- Factors may include unanticipated changes in:

- GDP

- Interest rates

- Inflation

- Estimate the loadings for each factor using multiple regression

Main difference with the single-factor model:

- Factor risk premium can be negative

Arbitrage Pricing Theory

- An arbitrage opportunity: arises when an investor can construct a sure-profit portfolio with

zero net investment

- e.g. different prices of the same security on different exchanges

- The law of one price: if two assets are equivalent, they should have the same market

price

- In efficient markets, profitable arbitrage opportunities will quickly disappear

Equilibrium

- The dominance argument of CAPM:

- Investors hold mean-variance efficient portfolios

- If under/over-priced securities: investors shift their portfolios which creates pressure on prices to restore equilibrium

- The arbitrage argument of APT:

- When there is an arbitrage opportunity, any investor, regardless of risk aversion, will want to take an

infinite position in it -> pressure on equilibrium prices

- NB. No assumption that investors have meanvariance preferences!

Hence, a well-diversified portfolio is one that consists of a large number of securities, so that

its nonsystematic risk is negligible

Comparing APT and CAPM

- APT applies to well diversified portfolios and not necessarily to individual stocks

- With APT it is possible for some individual stocks to be mispriced - not lie on the SML

- APT is more general in that it gets to an expected return and beta relationship without

the assumption of the market portfolio

- APT can be extended to multifactor models

Random walks and the Efficient Market Hypothesis (EMH)

- Are future changes in stock prices

predictable?

- Stock prices follow a random walk: price

changes are random and unpredictable

- Stocks already reflect all available

Information -> The efficient market hypothesis

EMH and competition

- Stock prices fully reflect all publicly available information

- If information is costly to...

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