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EC371 Economic Analysis of Asset Prices
Topic #6: Arbitrage & the APT
R. E. Bailey
Department of Economics
University of Essex
Outline
Contents
1
2
3
The Arbitrage Principle
1
1.1
Arbitrage: an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
State prices and the RNVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Factor Models
5
2.1
5
Approximate factor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Arbitrage Pricing Theory
6
3.1
APT in a single factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.2
Risk premia in the APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.3
APT and CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Reading: Economics of Financial Markets, chapters 7 and 8.
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The Arbitrage Principle
Warning:
‘arbitrage’ is a word often used in different ways, only one of which is precise. To
avoid ambiguity, in EC371 if ‘arbitrage’ appears on its own it is understood to take the precise
meaning used in this note. You are strongly recommended to follow this usage in any EC371 term
paper or exam answer – otherwise you may lose valuable marks.
Other meanings of ‘arbitrage’ are vague, something like “investment strategies that involve buying and selling assets, such that payoffs roughly offset, making the strategies low risk but allowing
a positive net payoff, on average”. In EC371 (and EC372), you may wish to refer to them as ‘approximate arbitrage’ or ‘low risk arbitrage-type’ strategies. But they are not interpreted as arbitrage
strategies in the sense used in EC371. The important lesson is: be precise in your use of language.
Absence of Arbitrage Opportunities
• Hypothesis: you can’t get something for nothing – notice that this is an hypothesis, not an
assertion of fact. Sometimes you may be able to get something for nothing, but it would be
farfetched to claim that you could do so all, or even most, of the time.
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• Arbitrage is an application of this hypothesis
– all “something for nothing” opportunities are taken.
• Having been taken, in equilibrium, no profit remains
– ‘equilibrium’ = absence of arbitrage opportunities (AoAO)
• Assumption: frictionless markets
– zero transactions costs and no institutional restraints on trading
• AoAO: generalisation of the Law of One Price (LOP)
– LOP: the same goods (assets) have the same price
• Prediction: systematic links among asset prices (this note is about exactly how to characterise
these ‘links’).
Arbitrage in an uncertain world
• Arbitrage portfolio:
1. Zero initial outlay
2. Risk-free, i.e. non-negative payoff in every state
• In equilibrium (“you can’t get something for nothing”): AoAO:
– Zero payoff for every arbitrage portfolio in every state, OR
– No arbitrage portfolio exists
• Why is AoAO important? – Because it requires only mild assumptions about investors (hence
it is very general)
• ‘Arbitrage’ is sometimes interpreted as “payoffs zero on average”
Very important: this is NOT the sense that it is used here.
1.1
Arbitrage: an example
Arbitrage: an example
Assets:
B
C
A
State 1
10
8
9
State 2
8
0
12
Price
3
2
pC = ?
Arbitrage portfolio, xA , xB , xC :
3xA + 2xB + pC xC = 0
10xA + 8xB + 9xC ≥ 0
8xA + 0xB + 12xC ≥ 0
2
Assets:
B
C
A
State 1
10
8
9
State 2
8
0
12
Price
3
2
pC = 3
Arbitrage portfolio, xA , xB , xC :
3xA + 2xB + pC xC = 0
10xA + 8xB + 9xC = 0
8xA + 0xB + 12xC = 0
• Two states and three assets A, B, C.
• Arbitrage portfolio: xA , xB , xC .
• AoAO → pC = 3
Example: remarks
1. Arbitrage principle generalizes law of one price to multiple assets and uncertainty
2. Almost all zero initial outlay portfolios are risky i.e. they have postive payoffs in some states
and negative payoffs in others
3. There may exist no arbitrage portfolios. If there is one arbitrage portfolio, there are infinitely
many. (Suppose that an arbitrage portfolio can be found; then scale the portfolio up or down in
any non-zero proportion; now you can check that the new portfolio also satisfies both criteria
for an arbitrage portfolio.)
4. Assume frictionless markets – with market frictions, AoAO makes no definite predictions.
The presence of market frictions does not imply that there is an arbitrage opportunity. Beware
of this common misunderstanding.
5. Arbitrage links asset prices – it is a partial theory of prices
1.2
State prices and the RNVR
The Arbitrage Principle
Three propositions, all equivalent to the Arbitrage Principle
I There exists an investor who prefers more wealth to less and for whom an optimal portfolio can be
constructed
II There exist positive state prices – a price for each state
III Risk Neutral Valuation Relationship
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Proposition I.
The arbitrage principle holds in frictionless asset markets if and only if there exists an investor who
prefers more wealth to less and for whom an optimal portfolio can be constructed.
To grasp why Proposition I holds, suppose that the arbitrage principle fails in the sense that
there exists a portfolio that (a) requires zero initial outlay, and (b) yields a non-negative payoff in
every state, with a positive payoff in at least one state. Now identify an investor who prefers more
wealth to less in each state. The investor must be willing to hold the portfolio in question: it costs
nothing, is risk free and yields a positive payoff in at least one state. But — here is the crucial point
— the investor would seek to magnify this portfolio (keeping the asset proportions the same) to an
unbounded extent (because more wealth is preferred to less). Formally, the investor has no optimal
portfolio. (Infinite wealth, a fantasy that dreams are made on, is just that: a fantasy.)
State prices and the RNVR
• What is a ‘state price’?
– Consider a state k (any state).
– qk is the price of an asset with payoff of $1 in state k and zero in all other states.
• Proposition II: AoAO is equivalent to existence of qj such that asset prices, pj , satisfy:
pj = q1 v1j + q2 v2j + · · · + q` v`j for all j.
• RNVR: AoAO is equivalent to existence of πk and δ such that: pj = δ(π1 v1j + π2 v2j + · · · +
π` v`j ) for all j.
• More compactly: pj = δE∗ [vj ]
• The πk are ‘martingale equivalent’ (or artificial) probabilities
– they don’t necessarily correspond to anyone’s beliefs
– but their existence is equivalent to AoAO
The Existence of State Prices
A state price, qk , is defined to be the price of an asset that has a payoff of one unit of wealth in state
k and zero in every other state.
Assets with this property are known as Arrow securities (after Kenneth Arrow, who first analysed
them).
Thus there is one Arrow security for each state: qk is the price of the Arrow security that pays ‘1’ in
state k and zero in every other state. Thus an Arrow security is just like an insurance contract: pk is
the insurance premium for a contract with payoff in state k.
Do Arrow securities “really exist”? Some people think that they are a good approximation to what’s
observed; others disagree. But the question is misplaced: what matters is that conceptually they help
to understand the implications of arbitrage. Whether Arrow securities actually exist is irrelevant.
If it helps your understanding, think of state prices as being implicit in the sense that they are not the
prices of any actual assets but, instead, can be inferred from the payoffs of assets that are traded.
Proposition II.
The arbitrage principle is equivalent to the existence of positive state prices, q1 , q2 , . . . , q` such that
pj = q1 v1j + q2 v2j + · · · + q` v`j ,
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j = 1, 2, . . . , n
(1)
This result is often called the linear pricing rule. In words: in the absence of arbitrage opportunities
the price of each asset must be equal to the sum of its payoff in each state multiplied by a state price
corresponding to that state.
The linear pricing rule is an equivalence: if arbitrage opportunities are absent, state prices exist;
if state prices exist, arbitrage opportunities are absent.
Proposition III (RNVR). The linear pricing rule is equivalent to the existence of:
1. a risk-free rate of return, r0 , with associated discount factor, δ ≡ 1/(1 + r0 ), and
2. probabilities, π1 , π2 , . . . , π` , one for each state, such that
pj = δE∗ [vj ]
j = 1, 2, . . . , n
(2)
The symbol vj denotes the list of payoffs, one for each state, for asset j. (In this context, vj is a
‘random variable’: a set of outcomes, each with its associated probability.) The expectation, E∗ [vj ],
is the payoff of asset j in each state weighted by the probability of that state and summed over the
states:
E∗ [vj ] ≡ π1 v1j + π2 v2j + · · · + π` v`j
(3)
The asterisk ∗ superscript appears as a reminder that the probabilities in (3) are purely artificial: they
are an implication of the proposition and need not correspond to any investor’s beliefs.
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Factor Models
Note: factor models on their own are quite separate from the arbitrage principle, and do not rely
on it in any way. It just happens that, together with the arbitrage principle, a factor model yields
interesting predictions – it is the APT.
2.1
Approximate factor models
The factor models are ‘approximate’ inasmuch as an unobserved random error, εj , is assumed (see
below).
Approximate factor models
• Factor models: asset prices depend on a small number of factors
– usually expressed in terms of rates of return
• Single factor model: rj = bj0 + bj1 F1 + εj for all j
bj0 , bj1 are parameters. Assume: E[εj | F1 ] = 0 =⇒ εj is uncorrelated with F1 .
• Multiple factor model: rj = bj0 + bj1 F1 + bj2 F2 + · · · + bjK FK + εj
• What are the factors, Fi ? Factors are the determinants of assets’ rates of return.
• bj1 , bj2 , . . . bjK : ‘factor loadings’.
• Without further assumptions, there is no way of knowing what the factors should be. This
is the main weakness of factor models. In a crucial sense any factor model is an incomplete
specification of assets’ returns, unless the factor model is accompanied by a theory that defines
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the F1 , F2 , . . . , FK , or states how each Fi is to be identified with its measurable empirical
counterpart (i.e. a set of ‘bridge principles’ that ties purely conceptual Fi with something that
can be found in samples of data.
A Single Factor Model
rj
rj = bj0 + bj1 F1
bj0
F1
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Figure 1: A single factor model
• In a one factor model, F1 measures the factor. See Figure 1.
• There is a separate line for each asset.
• That is, bj0 and bj1 differ across assets. (Strictly, some of the factor loadings might be the
same among some assets, but not all would be equal – otherwise it would be as if there is just
a single asset.)
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3.1
Arbitrage Pricing Theory
APT in a single factor model
Arbitrage Pricing Theory, APT
• APT = Factor model + Arbitrage Principle.
• APT prediction: cross section of returns depends on factor loadings.
• From Factor model to APT – the conditions:
1. Identify a portfolio with zero initial outlay.
2. Eliminate systematic (factor) risk.
3. Eliminate unsystematic risk (approximately).
4. AoAO ⇒ portfolio return ≈ zero.
• APT prediction: µj = λ0 + λ1 bj1 for all j.
λ0 = r0 if a risk-free asset exists
λ1 : ‘risk premium’ of factor 1.
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µj
6
µj = r0 + λ1 bj1
r 0 + λ1
r
0
-
1
bj1
Figure 2: APT prediction in a single factor model
3.2
Risk premia in the APT
Risk premia in the APT
• APT prediction: µj = r0 + λ1 bj1
See Figure 2.
• Factor risk premium = λ1
• Interpretation: µj − r0 = λ1 when bj1 = 1.
• Multidimensional figure needed for multi-factor model. In words: a factor risk premium equals
the risk premium for any asset (could be a portfolio) such that its rate of return varies one-forone with values of the factor.
3.3
APT and CAPM
APT and CAPM
• Relationship between CAPM and APT predictions?
• CAPM prediction: µj − r0 = (µM − r0 )βj
• APT (single factor model): µj − r0 = λ1 bj1
• Suppose that F1 = rM − r0 – i.e. the single factor is excess return on the market
• Then λ1 = µM − r0 and bj1 = βj .
• Hence, CAPM can be given an APT interpretation.
• CAPM and APT may also be consistent in a multifactor model.
• But they are founded on different assumptions.
Watch out: The CAPM is certainly compatible with the arbitrage principle – arbitrage opportunities
are not present anywhere in the CAPM. The crucial difference between the APT and CAPM is the
‘Factor Model’ used in the APT.
It is possible for the CAPM and APT to be compatible with one another even if the return on the
market portfolio is not one of the factors, indeed even if the factors are not portfolio returns at all. To
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see this, assume that both CAPM and APT are true. Suppose (just to keep the algebra simple) that
there are just two factors.
Assuming that the APT holds, consider two assets (portfolios, perhaps) labelled P and Q such
that: P has risk premium equal to λ1 ; and Q has risk premium equal to λ2 . That is,
µP − r0 = λ1
µQ − r0 = λ2
Assuming also that the CAPM holds, its prediction for portfolios P and Q is
µP − r0 = βP (µM − r0 ) = λ1
(4)
µQ − r0 = βQ (µM − r0 ) = λ2
(5)
As always, the APT implies
µj − r0 = λ1 bj1 + λ2 bj2 ,
j = 1, 2, . . . , n
Substitute for λ1 and λ2 from (4) and (5) to obtain
µj − r0 = [βP (µM − r0 )]bj1 + [βQ (µM − r0 )]bj2
= [βP bj1 + βQ bj2 ](µM − r0 )
= βj (µM − r0 )
(6)
where βj = βP bj1 + βQ bj2 , for j = 1, 2, . . . , n.
Equation (6) is the CAPM prediction for asset j. This shows that both the APT and CAPM
could hold in the same capital market — not that they must, of course. Both could be consistent
with the same set of observations, though they might not be. In a sense, the APT is more general
than the CAPM because, if the CAPM is true, the predictions of the two are indistinguishable –
“observationally equivalent” in an empirical context. However, there are circumstances for which
the APT predictions could hold when those of the CAPM do not.
Summary
Summary
1. The arbitrage principle (AoAO) asserts that zero initial outlay and zero risk implies zero return.
2. The arbitrage principle holds very generally and, in frictionless markets, links asset prices in
predictable ways.
3. Factor models assume that rates of return are linear functions of a small number of factors.
4. The APT marries the arbitrage principle to factor models to make predictions about assets’
risk premiums.
5. CAPM and APT are compatible but CAPM is more restrictive.
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