Finite State Approach to Modern Portfolio Theory
Introduction
Underlying all of modern finance is the idea of diversification. Holding a diversified collection of assets
permits an investor to better control risk and, for a given level of risk, achieve better investment
performance. But what, in theory and in practice, is diversification? How does one diversify and how
does measure the amount of diversification in a portfolio of assets?
In order to more clearly see the meaning of diversification, we will treat the possible outcomes of the
investment process as being finite in number. The usual Capital Asset Pricing Model (CAPM) formulation
assumes an uncountably infinite number of possible outcomes, any one of which is impossible to
identify or describe. This means it is not possibly to speak meaningfully about good or bad or mediocre
economic environments – only rates of return and their probability distributions survive in the usual
CAPM treatment.
For many purposes, it may be easier to think of the future as containing, at most, a finite number of
possible outcomes. In this way, one can discern different economic environments and assign prices to
securities with known performance characteristics in each of the possible economic environments.
The No-Arbitrage Principle
The most important principle in modern finance theory is the “no-arbitrage principle.” This principle has
supplanted the notion of market equilibrium in finance since the 1970s. The idea of a “market
equilibrium” has always been problematic in finance, because it is difficult to imagine a real world
market equilibrium. Much of the interest in finance centers around change and activity – not around
steady states or equilbria, concepts that dominate much of economic theory.
What is the “no-arbitrage principle?”
The No-Arbitrage Principle: Given a set of prices, there is no investment strategy
that produces a certain return with net investment of zero (or that produces an
infinite return from a finite investment).
It is very important to note that arbitrage might be possible with one set of prices but not possible with
another set of prices. The definition of the “no-arbitrage” principal is contingent upon a given set of
asset prices – one for each asset. A market that is in equilibrium automatically satisfies the noarbitration principle, but a market may be in disequilibrium and nevertheless satisfy the no-arbitrage
principle.
The no-arbitrage principle implies that if there is an investment strategy that does not involve risk then
that strategy will earn the ‘risk-free rate of return.’ There is no requirement that financial markets be in
equilibrium. Indeed, the main significance of the no-arbitrage principle is that markets can be active,
even chaotic, and still be consistent with the no-arbitrage principle. As long as there is no “free lunch”
in the market place, the no-arbitrage principle will be satisfied.
As we will discover in what follows, most important conclusions of finance theory follow from an
application of the no-arbitrage principle. The principle of efficient markets follows directly from this
principle and the Capital Asset Pricing Model and its descendent theories can, with some additional
assumptions, be established by using this simple principle.
In this chapter, we will introduce some of the elementary concepts of modern finance – states,
securities, state prices and market efficiency – in a very primitive context.
States of the World
A state of the world is a description of the environment. For example, if weather is our main interest,
we could define two states of the world: (i) it is raining; (ii) it is not raining. Obviously, there are more
than two states regarding the weather. But, how many states we identify will depend upon the problem
that we are working on. If we are trying to decide whether to put on rain gear when leaving the house,
then the only states that we need consider might just be the two identified in this paragraph.
If you own a bond, there might be only two states that interest you. The first state might be the state in
which the bond pays you the interest payments that it promises. The second state might be the state in
which the bond defaults. If you own shares of a company’s common stock, you might envision a state in
the next period in which the company’s earnings grow versus a different state in which the company’s
earnings don’t grow.
The states of the world that are needed are the states that are most useful to the problem that you are
considering. Whether finance is our interest or the day’s weather is our main concernpa, we are
generally free to pick the states any way we like. The strength of our analysis will often hinge on how
well we pick the states that make up our analysis.
Let’s suppose that we identify three states that might prevail in the future:
State 1: s1 -- the good state – this will be the state where most securities do well
State 2: s2 – the bad state – this will be the state here most securities perform poorly
State 3: s3 – the ‘neutral’ state – this will be the state where mediocrity reigns
Many finance issues can be discussed with this simple description of states. The importance of states is
that the payoff of securities will depend upon which state occurs. We can think of these states as future
states – tomorrow, for example. But, today we don’t know for certain which state is going to occur. We
have to decide today which securities to own without knowing for certain which state we will be in
tomorrow.
Securities and Payoffs
Now that we have states defined, we can proceed to talk about securities. A security will be completely
defined by identifying the payoff that the security provides for each state that may occur. We will use
capitals to identify securities:
Security 1: X1 – pays off in the good state; pays a little in the neutral state, goes bust in the bad
state
Security 2: X2 – pays moderately in the good state and the neutral state and goes bust in the bad
state
Security 3: X3 – pays less than S1 in the good state but loses less in the bad state
We still have not yet defined a security. We need to get more specific about what the payoffs are in
each state to get a complete definition of each security.
We will use the Greek symbol, p, to represent payoffs. Each security has a specific payoff in each state.
This means that p will have two subscripts – one that identifies the security and the second will indicate
in which state the payoff is intended. For example:
p1,2 is the payoff to security S1 in state s2.
If there are three states and three securities, we will have nine payoffs:
p1,1
p1,2
p1,3
--the payoffs to security one
p2,1
p2,2
p2,3
--the payoffs to security two
p3,1
p3,2
p3,3
--the payoffs to security three
The No-Arbitrage Principle
Now that we have defined the states and the securities we can give a more formal definition of the noarbitrage principle.
We need the concept of an investment portfolio. An investment portfolio, in this simple situation, will
be defined by three numbers: φ1, φ2, φ3. Each φ indicates the amount of that security contained in the
portfolio. The prices1 of the three securities are P1, P2, P3. An investment portfolio, then is defined as:
{ φ1, φ2, φ3}
-- a portfolio
With a cost of:
1
All prices used in this text will be assumed to always be non-negative without explicitly saying so.
P 1 φ1 + P 2 φ2 + P 3 φ3
The payoff to the portfolio in state one will be:
φ1 p1,1 + φ2 p2,1 + φ3 p3,1
The payoffs to the portfolio in states two and three are given by:
φ1 p1,2 + φ2 p2,2 + φ3 p3,2
φ1 p1,3 + φ2 p2,3 + φ3 p3,3
Following Stephen Ross’s treatment in his Princeton Lectures2 with slight changes, we give the following
definition of the no-arbitrage principle:
Definition: An arbitrage opportunity is a portfolio with no negative payoffs and a positive payoff in
some state, where the cost of the portfolio is not positive. Formally
P 1 φ1 + P 2 φ2 + P 3 φ3 ≤ 0
and
φ1 p1,1 + φ2 p2,1 + φ3 p3,1 ≥ 0
φ1 p1,2 + φ2 p2,2 + φ3 p3,2 ≥ 0
φ1 p1,3 + φ2 p2,3 + φ3 p3,3 ≥ 0
where at least one of the last three inequalities is strict (meaning > 0), otherwise the budget constraint
equation (the first equation in the definition) is strict (meaning < 0).
In words, an arbitrage opportunity is situation where a costless portfolio can produce a positive return
or a negative cost portfolio fails to provide a negative return. Logically this takes case of the situation
where a finite cost portfolio would yield an infinite return because you could achieve the infinite return
by taking an arbitrarily large position in the portfolio. The no-arbitrage principle is the idea that there is
no arbitrage portfolio, given the payoffs and the security prices. We can state that formally as:
Definition: The No-Arbitrage Principle: Given security prices and the security payoffs for each state of
the world, there does not exist an arbitrage opportunity as defined above.
The relationship between competitive equilibrium and the no-arbitrage principle
2
Stephen A. Ross, NeoClassical Finance, Princeton University Press. See Chapter One.
Clearly, if markets are in equilibrium, the no-arbitrage principle must be satisfied. Otherwise
participants will be attempting to take advantage of the available arbitrage opportunities and the
markets will not remain in equilibrium. But, is it possible for markets to be out of equilibrium, yet satisfy
the no-arbitrage principle? The answer to that question is yes. There is no requirement that markets be
in equilibrium. Markets can be quite chaotic and yet there could be an absence of arbitrage
opportunities. This means that the no-arbitrage principle is a weaker assumption that the usual
markets-in-equilibrium assumption that dominates most of economic theory. In finance, you don’t
really need markets to be in equilibrium to establish most of the main conclusions of the theory.
The Existence of State Prices
Definition: A state price is a price of security whose payoff is exactly one unit for that state and zero for
all other states. If there were state prices for each state, these could be used to price the securities in
the following manner:
P1 = q1p1,1 + q2p1,2 + q3p1,3
P2 = q1p2,1 + q2p2,2 + q3p2,3
P3 = q1p3,1 + q2p3,2 + q3p3,3
The price of security one, P1, for example, should be the total payoff in each state(pi,j, where I represents
the security and j represents the state) times the state price for that state. Similarly, P2 and P3 are
calculated in the same manner. (The qi’s are the state prices in the equations above).
Imagine a security that paid exactly one unit in one of the three states, but absolutely nothing in the
other states. The security that paid one unit in s1 would have price q1. Similarly for states 2 and 3. A
portfolio that consists of these three securities would pay exactly one unit regardless of which state
occurs. We call that portfolio a security – the risk-free security in fact, since it’s payoff is always the
same no matter what the future holds.
The Fundamental Theorem of Finance: The No-Arbitrage Principle is Equivalent
to the Existence of State Prices
Restating the ‘Fundamental Theorem,’ if we can calculate the state prices given the security prices, then
the no-arbitrage principle must hold and if we know that the no-arbitrage principle holds, then we can
calculate state prices from knowledge of security prices. Presumably the state prices are not identical
for each state. Different states will generally have different state prices. This has profound implications.
This means the value of a unit of account is worth more in one state than in another when determining
the price of a security.
So far, we have said nothing about the likelihood of any state occurring. Surprisingly, our Fundamental
Theorem is independent of the likelihood of any particular state’s occurrence. But, for the moment
suppose we know the probabilities associated with the occurrence of each of our three states. Then, we
can define the “expected return” to be the sum of the returns weighted by their respective probabilities,
where pij is the return in state j of security i:
Expected Return of security i ≡ prob(1) pi1 + prob(2) pi2 + prob(3) pi3
If you have two different securities that have identical expected returns, but one has all of its return in
the good state and the other has all of its return in the bad state, then the latter will have a higher
security price. This makes portfolio diversification much easier to visualize. Diversifying assets are those
that produce returns in states for which most of your assets aren’t doing very well. In the extreme,
insurance is an example where the good state and bad state might be mirror images of one another.
Let’s suppose we have the three state prices: q1, q2, and q3. If no one security can be created by a linear
combination of the other two, then there will always be some portfolio that, for each state, provides
one unit in that state and zero units in the other states. The price of that portfolio would equal the q for
that state. If there is a (hypothetical) security that provides one unit in each state, the price of that
security would be equal to q1 + q2 + q3. The return for this security is one and the cost is the sum of the
q’s:
r = Return of security that pays one unit in each state =
1
(𝑞1 + 𝑞2 + 𝑞3 )
This security is known as the risk free security and its return will be the risk free rate, r. Notice that r is
uniquely defined by q1, q2, and q3.
If we know that we can solve for : q1, q2, and q3, then the no-arbitrage principle must be true. This is
very simple to show mathematically. The converse proposition is more difficult: that the no-arbitrage
principle implies the existence of state prices. The proof relies on a separation theorem between two
non-overlapping convex sets.
Risk-Adjusted Probabilities
Are the relevant probabilities of each state known? We assume they are not. But, it turns out, we can
calculate risk adjusted probabilities and use them to price assets in a manner similar to using state prices
to price assets. These risk-adjusted probabilities are easily calculated, if you know the values of the
state prices. Set Q equal to the sum of the q’s. Then the risk-adjusted probabilities are simply:
π1 = q1/Q
π2 = q2/Q π3 = q3/Q
Because of the way that we defined Q, these risk-adjusted probabilities must sum to 1 and therefore
behave just as normal probabilities – they are non-negative and sum to one. These risk-adjusted
probabilities are so named because if
{z1, z2, z3}
Represents the returns to a security in each of the three states, then its price, P, is given by:
P = π1 z1 + π2 z2 + π3 z3
which is nothing more than the expected value of the returns from holding the security where the
expected value is calculated using the risk-adjusted probabilities, as opposed to the true (unknown)
probabilities. The risk-adjusted probabilities are often described as martingale probabilities, or simply,
the martingale measure. For many problems in finance, it is much simpler to work with the martingale
measure than with the state prices from which the martingale measure is derived.3
The Martingale Property
A random variable satisfies the martingale property if the expected future value of the variable is
identical to its current value, regardless of the future date that is considered:
E[Vt+s] + V for all t and for s > 0
Suppose your wealth at time t is $ 100, i.e., Wt = $ 100. Imagine the following process determines your
future wealth: each period you flip a fair coin and gain $ 1 if heads and lose $ 1 if tails. What is the
expected value of your future wealth at some date t+s in the future (where s> 0)? The answer is $ 100
since you expect to gain $ 1 half the time and lose $ 1 the other half of the time. Your wealth, using the
process, is a martingale.
Stock prices clearly do not satisfy the martingale property. If they did, no one would ever invest in
stocks. But, if you use an adjusted stock price series where the adjusted stock price at any time is the
discounted future value of the stock price using the risk free rates as discounts, the adjusted stock price
3
The martingale measure is obviously related to the martingale property that the best estimate of a future value
of something is its current value. Adjusting prices with the martingale measure gets one to the martingale
property of security prices. This is the correct formalization of the random walk concept in finance.
measure does satisfy the martingale property:
Stock Price
E[P1
P1
P0
to
t1
time
In the above diagram, one period elapses between t0 and t1. The slope of the line from P0 to E[P1] is (1
+ r). It is clear that the expected future price is not equal to the current price. So prices, by themselves,
do not satisfy the martingale property. But, if we slight modify our variables, we can get to the
martingale property. Define a new variable R, by de-trending the price series, in the following way:
R0 = P0
Rt+1 = Pt+1 - rPt
If we extend our single period model, the variable Rt, as defined above, will satisfy the martingale
property, since Rt = E[Rt+s] where s is any positive integer. The martingale property, in this setting,
applies to the de-trended price series. (De-trending the stock price series is equivalent to creating a new
random variable that in time t is the expected current value of the price at time t discounted back to the
present). Actual future stock prices will, it may be supposed, bounce around the trend line as described
in the diagram below:
Stock Price
E[P1]
time
The dots, describing possible actual prices, will not necessarily fall along the trend line, but should track
it with some variation. This is the meaning of ‘random walk’ in stock price series. It does not mean that
the stock price distribution is unknown. Quite the contrary. It means that next period’s price, or any
future price, is unpredictable, even though we may be fully confident about its expected future price.
Think of a coin flip, once again. You have no idea exactly how many of the next ten coin flips will turn
out to be heads. But, your expectation is five. What actually happens is random and will constitute, in
repeated trials a random walk around a mean of five.
If Stock Prices (Adjusted) is a Martingale Process, So What?
There are some strong information assumptions embedded in our treatment thus far. The two most
important embedded assumptions are:
1. The states of the world and their payoffs are known with certainty
2. All market participants agree on what the states are and what the payoffs are
If you combine these assumptions with the idea that martingale probabilities exist (in another words,
state prices exist or, equivalently, the no-arbitrage condition is satisfied), then you can almost taste the
idea of market efficiency. It is not possible to define market efficiency without directly addressing the
preferences of investors, something we don’t intend to do here. But once we are able to define investor
preferences, these assumptions will be enough to guarantee the kind of market efficiency that says that
no investor’s welfare (utility) can be improved without a decrease in the welfare of some other investor.
This is the classic allocative efficiency that is known in general equilibrium theory as a Pareto optimum.
It means that resources aren’t wasted. It doesn’t meant that allocations are ‘fair’ or ‘just’ or anything
like that. It just means there is no ‘free lunch’ left out there could be captured and given to someone.
The existence of state prices and martingale probabilities throw considerable light on other important
issues in finance – diversification, risk management, for example.