TOC
1. APT Foundation
1.1 Factor Model
One Factor Model
The singe factor model
where is the macroeconomic factor, is the error term and is called the factor loading. In this way, we decompose the sources of uncertainty into uncertainty about the economy as a whole and uncertainty about the firm in particular.
is the firmโs specific risk, thus
Risk arises from two uncorrelated sources thus is
Multi Factor Models
Factor categories include external factors, extracted factors, firm characteristics, etc.
Chen, Roll and Ross model
where
- IP: Growth of industrial production
- EI: Changes in expected inflation
- UI: Unexpected inflation
- CG: Unexpected changes in the difference between returns on corporate and government bonds
- GB: Unexpected changes in the difference between returns on long and short-term government bonds
Fama and French (1993)
where
- : Market returns, the is the factor of the CAPM
- SMB: Size factor (small minus big factor). Return of a portfolio of small stocks in excess of the return on a portfolio of large stocks
- HML: Book-to-Market factor (High minus low factor). Return of a portfolio with a high B/M ratio in excess of that with a low B/M ratio
1.2 Arbitrage
Arbitrage is a practice of taking advantage of imbalance between two or more markets and making a risk-free profit. A portfolio is an arbitrage opportunity if
- for at least one
Principle of capital market: no arbitrage.
Law of one price: two items that are the same (having the same cash flow) cannot sell at different prices.
2. APT
2.1 Linear Relationship
APT assumptions
- The universe of assets is large: there are an infinite number of securities; these securities differ from each other in nontrivial ways.
- Returns are generated by multi-factor model
- No arbitrage.
One Factor Case
Suppose return generating process is and there are 2 risky assets and a risk-free asset
Try to construct a risk-free portfolio from 2 risky assets and observe if the arbitrage opportunity open. Put in asset 1 and in asset 2, then the return of the portfolio is
there are no fluctuations iff , thus
the return of this portfolio is then
If , the arbitrage strategy is buy 1 unit ( in and in ) sell 1 unit
Many Factors Case
Assume there are factors and at least portfolios. Asset returns are fully described by factors
Eliminate constant
The return can be broken down into an expected return and an unexpected new. The unexpected news is the systematic risk. The specific risk can be diversified away and is not โpricedโ.
2.2 APT
Construct factor portfolios
Factor Portfolio: a portfolio which has a beta equal to 1 on a given factor and 0 on any other factor. They are the benchmark portfolios for a multi-factor security market line. For the two factor model, 3 factor portfolios are:
- Risk-free one: both are zero
- A portfolio with
- A portfolio with
Suppose we have 3 arbitrary assets in the market, to replicate the factor portfolios, we can construct a portfolio to match the factor loading (betas) of the factor portfolio.
For risk-free asset, we construct portfolio:
solve for and thus
For factor portfolio 1, we construct portfolio:
solve for and thus
For factor portfolio 2, we constrcut portfolio:
solve for and thus
Replication with Factor Portfolios
Take any other asset that has factor loadings , and expected rate of return . Using factor portfolios, we solve
thus
The expected return of this porfolio is
The no-arbitrage condition boils down to
Define
APT Theorem
In the absense of arbitrage, there must exist constants that, together with the factor loadings, describe every assetโs mean return
where
- is the expected return of asset
- is the k-th factor risk
- is the return for bearing 1 unit of risk associated with factor (factor risk premium)
The meaning of
- For risk-free asset:
- For factor portfolio 1:
- For factor portfolio 2:
It is equal to
2.3 Examples
Assume there are two assets , with return generating process (where )
Find the weights to construct risk-free asset
thus
Similarly, for factor portfolio 1
To rule out arbitrage opportunity, the risk-free return will be
The mean return of the factor F1-portfolio will be
thus APT
Direct approach: assume APT holds, then
thus
which yields the same APT equation
2.4 APT and CAPM
Consistency
Difference:
- The APT applies to well-diversified portfolios and not necessarily to all individual stocks
- General equilibrium condition (CAPM) v.s. No-arbitrage (APT)
- The APT can accomodate multiple sources of systematic risk.
2.5 Tesing the APT
There is a two-step procedure as in Fama and MacBeth (1973, JPE)
The first step is to use time series regression to estimate the bโs
The second step is to use cross section regression to estimate the priceโs () of each risk
3. Application
3.1 MM Theory
Application in Corporate Finance.
Modigliani-Miller (MM) Theorem: Under a set of appropriate conditions, the corporate financial policy are irrelevant.
โ The market value of a company is calculated using its earning power and the risk of its underlying assets and is independent of the way it finances investments or distributes dividends.
3.2 Forward and Future
Forward: agreement to buy or sell an asset for a certain price at a certain time. The forward price is the delivery price that would make the contract worth exactly zero.
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