Arbitrage Pricing Theory (APT) (Text reference: Chapter 11) Topics arbitrage factor models pure factor portfolios expected returns on individual securities comparison with CAPM a different approach 1 Arbitrage in general, arbitrage refers to earning a riskless profit at zero cost the no-arbitrage principle is one of the most important ideas in modern finance although it is most commonly applied in areas such as option pricing, we have implicitly used the no-arbitrage principle many times in this course examples: PV of a perpetuity: 2
foreign exchange: negative forward interest rates: APT exploits the no-arbitrage concept in the context of diversified portfolios 3 Factor Models we can write the actual return observed on asset as: risk can be systematic, or unsystematic: examples of systematic risks: inflation, interest rates, exchange rates, etc. note that any news can be thought of as consisting of an expected part plus a surprise in general, there can be systematic factors. Define:!"# 4
then we can write a factor model for the return on asset as: " where!! # this implies!!!! the market model is a commonly used version with a single factor (the market, e.g. TSE 300) 5 consider what happens in a factor model when we form a diversified portfolio with " securities: $# & % ' ( & % ' ( & % ' ( & % ' ( ) & % ' ( or *# $#,+,+ " 6
Pure Factor Portfolios for simplicity, consider a factor model with two factors: it is possible to combine securities into pure factor portfolios, e.g.,+ and +. For example: 7 consider a pure factor 1 portfolio. Since by definition: + + we can write +, where is the risk premium on a pure factor 1 portfolio. in general, it is possible for many security combinations to be pure factor 1 portfolios, but this doesn t matter. Why? therefore, independent of portfolio composition, any pure factor portfolio has + 8
Expected Returns on Individual Securities we have already seen that the expected return on a security in a factor model can be written as!" #%$&$'$( ) ) now we want to relate this to risk premia on pure factor portfolios by constructing portfolios consisting of the risk free asset and pure factor portfolios, investors can attain any desired systematic risk exposure for example, suppose we have a two factor model and a security * with $+,-/.,$01!2435 an investor with $1,000 could put everything into security * and have an expected return of: 9 suppose instead the investor borrows $1,100 at and invests $800 in a pure factor 1 portfolio and $1,300 in a pure factor 2 portfolio. The expected return from this strategy would be: since these two investments have the same systematic risk, they must offer the same expected return: 10
* problem: consider a two factor APT model with three securities and " A 10% 0.4 1.1 B 15% 1.2 1.8 C 13% 1.6 0.5 How can you construct a portfolio using all three stocks such that the portfolio for each factor is zero? To prevent arbitrage, what is? 11 Comparison With CAPM CAPM can be viewed as a one factor APT model where the factor is the market portfolio APT does not even say how many factors there are, let alone what they are have to use statistical methods to 1. identify set of factors 2. measure expected risk premia on factors 3. measure sensitivities of securities to factors some factors which have been used in practice include inflation, stock indexes, exchange rates, industrial production, spread between short term and long term interest rates, spread between low risk and high risk corporate bonds, etc. CAPM is still more widely used today, but APT is (slowly) gaining broader acceptance 12
A Different Approach both CAPM and APT are risk-based, i.e. they depend on measures of risk and risk premia another approach is to simply look for regular empirical patterns in stock return data for example! + aside: note the relationship to style investing: portfolios with low M/B are called value portfolios, portfolios with high P/E are called growth portfolios problem: why does this happen? Will it last? 13