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
Rit = Rite + uit
where Rit = actual rate of return on the ith stock,
1 Rite= the expected return on the ith stock and
uit = the unexpected, surprise or news element.
We can further subdivide the surprise or news element uit into
systematic or market risk mt, 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 εit , which affects a single firm:
uit = mt + εit
(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 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 bij or ‘betas’ given below:
mt = ∑ j bit (Fj − EFj)t = b11(F1t − EF1t) + b12(F2t − EF2t) + ·· ·
(27)
where the expectations operator E applies to information at time t −
1 or earlier.
3 A crucial assumption of the APT is that the idiosyncratic or specific
risk εi is uncorrelated across different securities, cov(εi, εj) = 0.
Return on the Portfolio
For simplicity, suppose there is only one systematic risk factor, Ft
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
∑
xR
∑
x R
xb
F
b F
∑
EF
4 EF
ε
x ε
(28)
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 εi .
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 (Ft − EFt ).
A More Formal Approach
We assume that investors have homogeneous expectations and
that the return Rit on any stock is linearly related to a set of kfactors Fit :
5 R
ER
∑
b F
EF
ε
(30)
The impact of any particular Fj depends on the value of bij and this
is different for each security.
This is the source of the covariance between the returns Rit on
different securities.
We assume that we can continue adding factors to (30) until the
unexplained part of the return εi is such that
E(εiεj ) =0 for all i ≠ j and all time periods
(31a)
E[εi(Fj − EFj)] = 0 for all stocks and factors (and all t)
(31b)
6 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
∑
0 for all j = 1, 2, . . . , k
x b
(32)
and the assumption of zero investment implies that
∑
x
0
(33)
It follows from (33) that some xi are less than zero, that is, some
stocks are held short and the proceeds invested in other securities.
7 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
…
∑
xR
∑
x R
∑ xb
∑
∑
F
x ER
∑
xb
F
EF
EF
8 b F
∑
EF
x ε
ε
(34)
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 rt.
This is because the riskless asset is included in the set of assets
available.
9 On the other side, the expected return on any security-i may be
written as a linear combination of the factor weightings bij. For
example, for a two-factor model,
ERi = λ0 + λ1bi1 + λ2bi2
(37)
We noted that bi1 and bi2 in (30) are specific to security-i.
The expected return on security-i weights these security-specific
betas by a weight λj that is the same for all securities.
Hence, λj 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 ERi = ERz + bi1E(R1 − Rz) + bi2E(R2 − Rz)
(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
bij).
In addition, each factor loading (e.g. bi1) is ‘weighted’ by the
expected excess return E(R1 − Rz), 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 F1.
11 Implementation of the APT
The APT is a factor model and may be summed up in two
equations:
R
ER
∑
a
λ
b F
∑
ε
(42a)
b λ
(42b)
where λ0 = rt or ERz.
Note that in (42b), the λ0 and λj are constant across all assets (in
the cross-section).
12 The time-series equation (42a) is not a prediction equation for Rit
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 λ0 must be estimated and as it is
the expected return on a portfolio when all betas are zero, λ0 is the
zero-beta rate.
The bij are interpreted as the amount of exposure of asset-i to the
factor j and λj is interpreted as the ‘price’ of this risk exposure.
So the expected return premium on asset-i equals the unit
exposure βij to factor Fj times λj, the price of this beta risk.
The APT may be implemented in the following (stylised) way.
13 A ‘first-pass’ time series regression of Rit on a set of factors Fit (e.g.
inflation, GDP growth, interest rates) will yield estimates of ai and
the bi1, bi2, 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 bi vary over the m securities
and are therefore the right-hand-side variables in (42b).
Hence, in equation (42b), the bij are the variables that are different
across the m securities.
The λj are the same for all securities and, hence, these can be
estimated from the cross-section regression (42b) of Ri on the bij
14 (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