Arbitrage Pricing Theory

Transcription

1 Arbitrage Pricing Theory The APT implies that the return on a security can be broken down into an expected return and an unexpected, surprise or news component. For any individual stock, this news component can be further broken down into general news that affects all stocks and specific news that affects only this particular stock. The APT predicts that general news will affect the rate of return on all stocks but by different amounts. The APT may be represented as R it = R e it + u it where R it = actual rate of return on the ith stock, 1

2 R it e = the expected return on the ith stock and u it = the unexpected, surprise or news element. We can further subdivide the surprise or news element u it into systematic or market risk m t, that is, the risk that affects a large number of stocks each to a greater or lesser degree. The remaining uncertainty is unsystematic (idiosyncratic or specific) risk ε it, which affects a single firm: u it = m t + ε it (26) As in the case of the CAPM, we shall find that systematic risk cannot be diversified away because this element of news or new information affects all companies. 2

3 But specific risk can be diversified away and therefore is not priced. The systematic risk may be news about economy-wide variables are, for example, a government announcement that the GDP is higher than expected or a sudden increase in interest rates by the Central Bank. These economy-wide factors F (indexed by j) may have different effects on different securities, and this is reflected in the different values for the coefficients b ij or betas given below: m t = j b it (F j EF j ) t = b 11 (F 1t EF 1t ) + b 12 (F 2t EF 2t ) + (27) where the expectations operator E applies to information at time t 1 or earlier. 3

4 A crucial assumption of the APT is that the idiosyncratic or specific risk ε i is uncorrelated across different securities, cov(ε i, ε j ) = 0. Return on the Portfolio For simplicity, suppose there is only one systematic risk factor, F t and n securities in the portfolio. The return on a portfolio R of n-securities held in proportions xi is, by definition, R x R x R b F EF ε x x b F EF x ε (28) R 4

5 The return on the portfolio is a weighted average of the expected return plus the portfolio-beta (multiplied by the news about factor F) plus a weighted average of the specific risk terms ε i. As the number of securities increases, the last term on the righthand side of (28) will approach zero the specific risk has been diversified away. Hence, the return on the portfolio is made up of the expected returns on the individual securities and the systematic risk, as represented by the single economy-wide news term (F t EF t ). A More Formal Approach We assume that investors have homogeneous expectations and that the return R it on any stock is linearly related to a set of k- factors F it : 5

6 R ER b F EF ε (30) The impact of any particular F j depends on the value of b ij and this is different for each security. This is the source of the covariance between the returns R it on different securities. We assume that we can continue adding factors to (30) until the unexplained part of the return ε i is such that E(ε i ε j ) =0 for all i j and all time periods (31a) E[ε i (F j EF j )] = 0 for all stocks and factors (and all t) (31b) 6

7 Equation (31a) implies that unsystematic (or specific) risk is uncorrelated across securities, while (31b) implies that specific risk is independent of the factors F. Form a zero-beta portfolio of n-assets with zero net investment. The zero-beta portfolio must satisfy x b 0 for all j = 1, 2,..., k (32) and the assumption of zero investment implies that x 0 (33) It follows from (33) that some x i are less than zero, that is, some stocks are held short and the proceeds invested in other securities. 7

8 The next part of the argument introduces the arbitrage element. If investors put up no funds and the zero-beta portfolio earns a non-zero expected return, then a risk-free arbitrage profit can be made. This arbitrage condition places a restriction on the expected return of the portfolio, so using (28), we have R x R x ER b F EF ε x R x b F EF x b F EF x ε (34) 8

9 Using (34) and (32) plus the assumption that for a large welldiversified portfolio the last term on the right-hand-side approaches zero, we have R x R ER (35) Arbitrage arguments then suggest that this riskless return must be zero: ER 0 (36) The riskless portfolio does not earn the riskless rate of interest r t. This is because the riskless asset is included in the set of assets available. 9

10 On the other side, the expected return on any security-i may be written as a linear combination of the factor weightings b ij. For example, for a two-factor model, ER i = λ 0 + λ 1 b i1 + λ 2 b i2 (37) We noted that b i1 and b i2 in (30) are specific to security-i. The expected return on security-i weights these security-specific betas by a weight λ j that is the same for all securities. Hence, λ j may be interpreted as the extra expected return required because of a securities sensitivity to the jth factor (e.g. GNP or interest rates). An alternative to (37) is 10

11 ER i = ER z + b i1 E(R 1 R z ) + b i2 E(R 2 R z ) (41) One interpretation of the APT is that the expected return on a security-i depends on its sensitivity to the factor loadings (i.e. the b ij ). In addition, each factor loading (e.g. b i1 ) is weighted by the expected excess return E(R 1 R z ), that is, the (excess) return on a portfolio whose beta with respect to the first factor is one and with respect to all other factors is zero. This portfolio with a beta of 1 therefore mimics the unexpected movements in the factor F 1. 11

12 Implementation of the APT The APT is a factor model and may be summed up in two equations: R a b F ε (42a) ER λ b λ (42b) where λ 0 = r t or ER z. Note that in (42b), the λ 0 and λ j are constant across all assets (in the cross-section). 12

13 The time-series equation (42a) is not a prediction equation for R it since the factors are also measured at time t, the equation measures the contemporaneous risk exposure to the factors. If there is no risk-free rate, then λ 0 must be estimated and as it is the expected return on a portfolio when all betas are zero, λ 0 is the zero-beta rate. The b ij are interpreted as the amount of exposure of asset-i to the factor j and λ j is interpreted as the price of this risk exposure. So the expected return premium on asset-i equals the unit exposure β ij to factor F j times λ j, the price of this beta risk. The APT may be implemented in the following (stylised) way. 13

14 A first-pass time series regression of R it on a set of factors F it (e.g. inflation, GDP growth, interest rates) will yield estimates of a i and the b i1, b i2, and so on. This can be repeated for i = 1, 2,...,m securities so that we have m values for each of the betas, one for each of the different securities. In the second-pass regression, the b i vary over the m securities and are therefore the right-hand-side variables in (42b). Hence, in equation (42b), the b ij are the variables that are different across the m securities. The λ j are the same for all securities and, hence, these can be estimated from the cross-section regression (42b) of R i on the b ij 14

15 (for i = 1, 2,..., m). The risk-free rate is constant across securities and hence is the constant term in the cross-section regression. 15