# arbitrage pricing theory apt by abe24

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```									                         Lecture 9
The Arbitrage Pricing Theory, APT

AIM OF LECTURE 9
Introduce the main arguments behind the APT.
Learn how to form risk-free arbitrage portfolios.
Understand how non-arbitrage arguments can pin down the
structure of expected returns.
Show how the APT may reduce to the CAPM in a special case.

Main reference: Copeland and Weston, chapter 7 (part J).

9.1 INTRODUCTION
The APT offers an alternative to the CAPM. APT does not require assumptions about utility
or the distribution of security returns. The APT relies on the following assumptions:
1. Returns are generated according to a linear factor model.
2. The number of assets are close to infinite.
3. Investors have homogenous expectations (same as CAPM)
4. Capital markets are perfect (i.e. perfect competition, no transactions costs [same as
CAPM])

The assumptions 1 and 2 are subject to criticism.

9.2 THE LINEAR FACTOR MODEL
Asset returns are assumed to be generated by a linear factor model:

Ri = ai + bi1F1 + bi2F2 +...+ bikFk + εi
˜            ˜       ˜           ˜    ˜                    (LFM)

˜ ˜     ˜
The return on asset i depends linearly on k factors (F1,F2,...,Fk) and unsystematic
(idiosyncratic) risk εk. The extent to which the return on the asset depends on the factors is
˜

Furthermore it is assumed that
ε
E[˜ i] = 0                                      (i)
ε˜
E[˜ iεj] = 0                                    (ii)
˜
E[Fk] = 0                                       (iii)
ε˜
E[˜ iFk] = 0                                    (iv)
˜ ˜
E[FkFm] = 0                                       (v)
Financial Markets: Theories & Evidence 2

(i) says that the idiosyncratic risk is zero on average, (ii) that idiosyncratic risk is uncorrelated
across different assets, (iii) that the factor is zero on average [this is just a normalisation], (iv)
that the idiosyncratic risk is uncorrelated with the factors, (v) that the different factors are
uncorrelated.

Taking expectations of the linear factor model (LFM) we obtain [given (i) and (iii)]

˜
E[Ri] = ai

Using this in (LFM) we obtain

Ri = E[Ri] + bi1F1 + bi1F2 +...+ bikFk + εi
˜      ˜        ˜       ˜           ˜    ˜                         (7.39)

which is equation (7.39) in Copeland and Weston.

Examples of factors could be macroeconomic variables like GDP growth, unemployment rate,
taxes, fiscal policy, etc.

9.3 ARBITRAGE PRICING WITH ONE FACTOR AND
NO IDIOSYNCRATIC RISK
Before establishing the APT we will look at the concept of arbitrage in a more simple
framework. Let us assume that there is no idiosyncratic risk, and that there is only one factor
(Single Factor Model).

˜           ˜
Ri = ai + biF                                     (SFM)

Form an arbitrage portfolio in two assets, i and j.
˜     ˜         ˜
Rp = wRi + (1-w)Rj
˜                 ˜
= w(ai + biF) + (1-w)(aj + bjF)
˜          ˜
= w(ai - aj) + w(bi - bj)F + aj + bjF
Chose w = -bj/(bi - bj) then
˜
Rp = w(ai - aj) + aj
but this portfolio is risk free and therefore must earn the risk-free rate of return
˜
Rp = Rf = w(ai - aj) + aj
⇔
aj - Rf = -w(ai - aj)
or substituting for -w gives
bj
aj - Rf =     (ai - aj)
bi - bj

⇔
Financial Markets: Theories & Evidence 3

(bi - bj)(aj - Rf) = bj(ai - aj)
⇔
bi(aj - Rf) - bjaj + bjRf = bjai - bjaj
⇔
bi(aj - Rf) = bj(ai - Rf)
⇔
aj - R f         ai - Rf
=          ≡λ                                    (FRP)
bj              bi

λ is the factor risk premium, and is the same for all assets (independent of i,j).

Rewrite (FRP) for asset i and we have

ai - Rf = biλ

˜
Remember that E[Ri] = ai, therefore

E[Ri] - Rf = biλ
˜                                                (SIM)

The excess return on asset i is equal to the factor loading bi (i.e. the sensitivity to factor risk)
times the factor risk premium. Since we have only one factor here we refer to the equation
above as the single-index model. All assets in this economy must lie on the arbitrage pricing
line in the figure below

Figure 9.1

[Available in a separate file (Figure91.pdf) on
www.econ.rochester.edu/Wallis/Renstrom/Eco217.html]

Pick an asset or portfolio with b=1, and let the expected return on such an asset be δ, then
(FRP) gives
λ = δ - Rf

Substitute into (SIM) and we have

˜
E[Ri] - Rf = bi(δ - Rf)

This looks very similar to the CAPM! In fact we may view CAPM as a special case of APT
when there is only one risky factor. (It is not quite the same though, because we have ignored
the idiosyncratic risk).
Financial Markets: Theories & Evidence 4

9.4 ARBITRAGE PRICING WITH k FACTORS AND
NO IDIOSYNCRATIC RISK
The above results generalise into the case with many risky factors.
˜            ˜       ˜           ˜
Ri = ai + bi1F1 + bi2F2 +...+ bikFk
We may form an arbitrage portfolio w, which has the property that w′1=0, i.e. all portfolio
n
weights sum to zero: i=1 wi = 0. This means that no net wealth is invested. The return on the
portfolio is
˜
Rp =          n
i=1
˜
wiRi =    n
i=1   wi ai +     n
i=1
˜
wi bi1F1 +    n
i=1
˜
wi bi2F2 +...+    n
i=1
˜
wi bikFk
n
The portfolio can be made risk free if                   i=1   wibik = 0 for all factor loadings k
A portfolio that requires no investment and that is risk-free must give zero return (otherwise
one could scale up the investment activity and earn infinite profit, and this cannot be an
n
equilibrium). The non-arbitrage condition is i=1 wi ai = 0.

We then have the following conditions:
n
i=1 wi = 0 (arbitrage portfolio)

n
i=1   wi ai = 0 (non-arbitrage condition [i.e. zero return])
n
i=1   wi bi1 = 0
n
i=1   wi bi2 = 0
(risk-free portfolio)

n
i=1   wi bik = 0

We may write these conditions in matrix form:

The matrix is of order (2+k)×n, and the vector of weights of order n×1, and the vector of
zeros (2+k)×n. Since the product is a vector of zeros, it implies that any row of the matrix
can be written as a linear combination of the other rows (i.e. the rows are linearly
dependent).1

1
The easiest way to see this is when the number of assets is 2+k. With n=2+k the matrix is a square matrix. By
premultiplying the product above with the vector of weights w′, we obtain a quadratic form, but this quadratic
form sums is zero: w′0=0. If a quadratic form is zero it means that the matrix is singular (its determinant is
zero), and the rows of the matrix are linearly dependent.
Financial Markets: Theories & Evidence 5

For example, the second row may be written as a linear combination of the first row and the
last k rows:

[a1, a2,..., an] = λ0[1, 1,...,1] + λ1[b11, b21,..., bn1] + λ2[b12, b22,..., bn2] +...

+ λk[b1k, b2k,..., bnk].

So for any ai:
ai = λ0 + λ1 bi1 + λ2 bi2 + λk bik
First, an asset (say f) with zero sensitivity to all factors (i.e. bf1 = bf2 =...= bfk = 0) will have
af = λ0 equal to the risk-free rate of return, so λ0 = Rf.
˜
Second, E[Ri] = ai.
Therefore
E[Ri] - Rf = λ1 bi1 + λ2 bi2 +...+ λk bik
˜                                                            (7.46)
This is equation (7.46) in Copeland and Weston.
Thus, the excess return of asset i (risk premium of asset i) is a linear combination of factor
risk premia, λ1, λ2,...,λk. If we can find assets or portfolios which has got unit sensitivity to
one factor but zero sensitivity to all the others we may rewrite relationship (7.46). For
example, denote δk the expected return on a portfolio which has got bk = 1 and has got zero
correlation with all other factors (i.e. all other b’s zero), then we have

λk = δ k - Rf
We may substitute for all λk above

˜
E[Ri] - Rf = (δ1-Rf)bi1 + (δ2-Rf)bi2 +...+ (δk-Rf)bik                 (7.47)

This is equation (7.47) in Copeland and Weston.

9.5 THE ARBITRAGE PRICING THEORY
In the above two sections we ignored the idiosyncratic risk for illustrative purposes. One of
the main arguments of APT is that with a "large" number of assets the idiosyncratic risk of
a well diversified portfolio is close to zero. To make the argument go through we would need
an infinite number of assets. If we have a finite number of assets the APT as given by the
equations (7.46) and (7.47) above is only an approximate pricing rule.

Finite number of assets also raises some issues concerning the concept of arbitrage. In the
above two examples we were able to construct arbitrage portfolios without risk. This is not
possible if idiosyncratic risk is present why the APT relies on the concept of asymptotic
arbitrage. Investors form arbitrage portfolios which contain idiosyncratic risk, then when we
take limits of these portfolios (as the number of assets go to infinity) they become free of
residual risk.
Financial Markets: Theories & Evidence 6

Copeland and Weston do not discuss this (because of the mathematical complications)
and they do not actually prove the APT. Instead they try to give you the intuition in equation
(7.42a) which says that the portfolio weight in each asset is of order 1/n.

9.6 SUMMARY
The APT relies on less restrictive assumptions than the CAPM. However, there are some
drawbacks.
1. The linear factor model. (It’s not as restrictive as it may seem, we could allow for
functions of factors).
2. Infinite number of securities.
It is important to understand why APT is a theory of arbitrage pricing. We obtained the
pricing relationships by forming arbitrage portfolios. It’s also important to understand that we
rely on a large number of securities, to be able to form arbitrage portfolios with close to zero
residual risk.

Copeland and Weston give a nice summary on page 222.

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