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Risk-neutral Valuation: A Gentle Introduction (1)
Joseph Tham
Abstract
Risk-neutral valuation is simple, elegant and central in option
pricing theory. However, in teaching risk-neutral valuation, it is
not easy to explain the concept of risk-neutral probabilities.
Beginners who are new to risk-neutral valuation always have
lingering doubts about the validity of the probabilities. What do
the probabilities really mean? Are they real or fictional? Where do
they come from? What is the relationship between the risk-neutral
probabilities and the actual probabilities? Does it mean that all
investors are risk-neutral? When is it appropriate to use the
risk-free rate as the discount rate?
From a pedagogical point of view, in the beginning it is best to
avoid the use of probabilities because probabilities can be a
barrier to understanding. Instead, it is far preferable to
introduce the idea of state prices and then show that the approach
with risk-neutral probabilities is equivalent to the use of state
prices.
In this teaching note, we use simple one-period examples to
explain the intuitive ideas behind risk-neutral valuation. It is a
gentle introduction to risk-neutral valuation, with a minimum
requirement of mathematics and prior knowledge. We will provide the
motivation and the rationale for calculating state prices and we
will show that the risk-neutral approach is simply another way of
looking at the issue of state prices. JEL codes D61: Cost-Benefit
Analysis G31: Capital Budgeting H43: Project evaluation Key words
or phrases Risk-neutral valuation Currently, Joseph Tham (in
collaboration with Ignacio Vlez-Pareja) is writing a book on cash
flow valuation. Previously, he taught at the Fulbright Economics
Teaching Program (FETP) in HCMC, Vietnam and worked with the
Program on Investment Appraisal and Management (PIAM) at the
Harvard Institute for International Development (HIID). Email
address: [email protected].
This teaching note is dedicated to the proverbial grandmother
who is diligent, well read and intelligent but has not taken any
course in finance. We have erred on the side of over-explanation
and repetition rather than brevity, conscious of the risk of
boredom and unavoidable loss of the soul in wit. Critical comments
and constructive feedback for clearer explanations and further
clarification on obscurities are welcome.
The author wishes to thank for Ignacio Vlez-Pareja for comments
that substantially improved the teaching note. The author assumes
responsibility for all remaining errors.
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Introduction
Risk-neutral valuation is simple, elegant and central in option
pricing theory. With
risk-neutral probabilities, we can estimate the current value of
an investment project (or
opportunity) with any payoff structure in the future. In the
discrete case, using the risk-
neutral probabilities (that we obtain with a method to be
explained later), we calculate the
expectation of the payoffs for the asset under the states of
nature that can occur and
discount the expectation with the risk-free rate.1 The answer is
the correct no-arbitrage
value, that is, the value that should prevail in the market
under the perfect conditions that
underlie a competitive model.2 The asset value that should exist
under perfect conditions is
a good place to begin the analysis.3
In teaching risk-neutral valuation, it is not easy to explain
the concept of risk-
neutral probabilities. Beginners who are new to risk-neutral
valuation always have
lingering doubts about the validity of the probabilities. What
do the probabilities really
mean? Are they real or fictional? Where do they come from? What
is the relationship
between the risk-neutral probabilities and the actual
probabilities? Does it mean that all
investors are risk-neutral? When is it appropriate to use the
risk-free rate as the discount
rate?
1. For example, suppose two states of nature can occur, and we
know the risk-neutral probabilities for the
two states of nature. In addition, we know the risk-free rate
and the payoffs for the asset under the two states of nature. Then,
to compute the expectation, we simply multiply the payoffs for the
assets under the two states of nature by the respective
probabilities, and discount with the risk-free rate.
2. Later, we will present simple numerical examples of
risk-neutral valuation. 3. We know that the real world is far from
perfect and we do not wish to underestimate (or neglect to our
peril) the huge gap between the features of the real world and
the assumptions of a model in a perfect world. The gap between
theory and practice is much larger in countries without
well-functioning capital markets and it would not be difficult to
ridicule a model that has been constructed for a perfect world.
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From a pedagogical point of view, in the beginning it is best to
avoid the use of
probabilities because probabilities can be a barrier to
understanding. Instead, it is far
preferable to introduce the idea of state prices and then show
that the approach with risk-
neutral probabilities is equivalent to the use of state prices.4
In this teaching note, we use
simple one-period examples to explain the intuitive ideas behind
risk-neutral valuation. It
is a gentle introduction to risk-neutral valuation, with a
minimum requirement of
mathematics and prior knowledge. We will provide the motivation
and the rationale for
calculating state prices and we will show that the risk-neutral
approach is simply another
way of looking at the issue of state prices.5 In addition, we
explain the important
distinction between expectation and expected value. The
methodology with state prices is
also a very appropriate way to think about risk-analysis.
In Section One, we present an overview of the main issues that
are involved in the
valuation of a risky investment project. In Section Two, we
discuss the issue of state prices
in a simple one-period economy with two assets, one risky and
one risk-free. The existence
of the risk-free asset is not necessary; it only simplifies the
exposition. In appendix C, we
present the analysis with two risky assets. Using the portfolio
replication approach, we
construct two new assets, J and G, with special payoff
structures and show the relationship
between these new assets and the state prices. In Section Three,
we show that the risk-
neutral approach and the use of state prices are equivalent.
4. Once the relationship between state prices and risk-neutral
probabilities is clear, then we can stress that
from computational and conceptual points of view, it is easier
to think in terms of risk-neutral probabilities.
5. I will not discuss the valuation of derivatives, such as call
and put options, but the ideas presented here
can be adapted easily for the valuation of such securities.
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Section One: An overview
To motivate and focus the discussion, we state the following
simple objective.
Objective: How do we value an investment project (or
opportunity) A, with two possible
outcomes?
To be specific, assume that there are only two states of nature:
an up state and a
down state. The free cash flow (or payoff) for project A in the
up state is FCFU and the free
cash flow (or payoff) for project A in the down state is FCFD.
At a minimum, we will
assume that all the investors in the economy agree on the values
of the payoffs, that is, they
agree on the values of FCFU and FCFD. However, there is no
consensus on the probabilities
for the two states of nature.6
Assumptions about available information
To a large degree, our success in achieving the stated objective
in valuing project A
will depend on our assumptions about the available information.
Consider an extreme case.
Suppose the only information that is available is the payoff
structure for investment A and
there is no other information.7 What can the investor do? There
are various criteria for
decision making under uncertainty and we will not present them
here. For simplicity, we
will assume that the investor makes her decision based on the
expected return and the
variance (or standard deviation) of the project. In such a
situation, the investor will simply
have to use her subjective assessment of the probabilities for
the payoffs from investment
A under the two states of nature. Based on these probabilities
she can calculate the
6. If there is no agreement on both the values of the payoffs
and the probabilities for the states of nature,
then we cannot make any progress at all!
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expected return and variance. There is no recourse to anything
else. Other investors
looking at the same project A may have different assessments
about the likelihood of the
payoffs under the states of nature and there may be no consensus
among the informed
investors on whether it is a acceptable project or not. To
calculate the expected return
and the variances, the investor uses her personal probabilities
about the payoffs under the
two states of nature. It is extremely difficult if not
impossible to value an investment in
isolation. In other words, if we have to use only the
information on the payoffs for the
project, it will be a totally subjective decision, based on the
past experiences and
preferences of the investor.
Previous experience with similar projects
If the investor has experience in investments that are similar
to project A, she will
try and compare the risk profile of investment A with her
previous experiences and based
on the comparison, make a subjective decision. Suppose
previously, she had invested in a
similar project H and the return on project H was 20%. Now, the
investor must decide to
what extent, the previous project H is similar to the current
investment A. Project H may
have been of a different scale, in a different place or time
period. Moreover, the investor
must decide whether the risk of the current investment A is
higher or lower than the risk of
the previous investment H, and then accordingly, again, make a
subjective adjustment to
the required rate of return. Project H becomes a base line
comparison. Thus, the
comparison will be based on the considered judgment of the
investor using the expected
returns and the variances of the two projects. If the investor
perceives project A to be more
7. The project may be new, with no relevant external
information. The situation may apply to projects in
many countries without well-functioning capital markets and
limited availability of public information.
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risky than project H, she may decide that the hurdle rate should
be 25%, and would only
agree to invest in project A if the expected return is higher
than 25%. On the other hand, if
the investor perceives project A to be less risky than project
H, she may decide that the
hurdle rate should be 18%, and would only agree to invest in
project A if the expected
return is higher than 18%. She will have to make a subjective
assessment of the tradeoff
between the expected return and the variance.
Finding a comparable investment opportunity
To make progress on achieving our stated objective, we need to
specify some
additional information. For example, based on previous
experiences, the investor may
compare the current investment opportunity with similar
investment opportunities. The
other investment opportunity is called a comparable. The idea of
a comparable or a
replicating portfolio is very simple. If we can find a similar
investment Y, that is traded
(or marketed) and has a payoff structure that is identical with
project A, then the value of
project A should be approximately equal to the value of the
comparable project Y. In
practice, it may not be easy or possible to find investment
opportunity Y. For the two
investments to have the same value, we must invoke the law of
one price. Similar
investment projects should have approximately similar values. In
other words, there must
not be any arbitrage opportunities.8 Again, we are making very
strong assumptions. We
need to assume nothing less than perfect capital markets.9
8. In the real world, due to market imperfections there are
always opportunities for arbitrage. It is precisely
the relentless pursuit of arbitrage opportunities by investors
that leads to the short-live span of such opportunities.
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Other assets in the economy
We will assume that there are two assets in the economy, other
than the investment
opportunity A that we are trying to value. Since there are two
states of nature,
mathematically, we need two assets to find a solution, as we
will explain later. What
information do we need to know about these two assets? First,
all the investors in the
economy must agree on the payoff structures for these two assets
under the two states of
nature. Second, we will assume that there are perfect capital
markets and the law of one-
price holds. Third, we must know the current market values for
these two assets.
Replication portfolio
Rather than directly valuing investment A, we will make a
detour. We will use the
information on the two assets in the economy to estimate the
state prices for the two states
of nature. The state prices will facilitate the valuation of
investment A. Several questions
arise. What is the meaning of state prices and how do we
estimate the state prices?
We will present the mathematics for estimating the state prices
later. But assume
that it is possible to estimate them. Basically, using the
information on the two assets in the
economy, we use the replicating portfolio approach to construct
two new basic assets, J
and G, which have very simple payoff structures. Formally, the
state price for the up state
is the value of asset J and the state price for the down state
is the value of asset G. We
value asset J and G by using combinations of the two assets in
the economy to replicate the
appropriate payoff structures for the two basic assets, J and
G.
9 . These assumptions are no more stringent than the assumptions
that we normally make with CAPM or the
M & M world for estimating the cost of capital.
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Asset J and asset G
Suppose we ask the following question. In a competitive market,
at the end of year
0, how much would an investor be willing to pay to obtain $1 in
the up state at the end of
year 1 and $0 dollar in the down state at the end of year 1?
With the assumption of perfect
markets, there will be one correct price. Otherwise there would
be arbitrage opportunities.
This correct price is the state price for the up state. We will
name it (1).
Similarly, in a competitive market, at the end of year 0, how
much would an
investor be willing to pay to obtain $0 in the up state at the
end of year 1 and $1 dollar in
the down state at the end of year 1? Again, with the assumption
of perfect markets, there
will be one correct price. This correct price is the state price
for the down state. We will
name it (2).
Once we have the state prices, we can proceed with the valuation
of project A.
Suppose we wish to value an investment opportunity that pays
$100 in the up state and $50
in the down state. Using the basic assets, J and G, we can value
this very easily. To match
the $100 in the up state (and $0 in the down state), we buy 100
units of asset J, which is
equal to 100*(1). To match the $50 in the down state (and $0 in
the up state), we buy 50
units of asset G, which is equal to 50*(2). Since the portfolio,
which consists of 100 units
of J and 50 units of G, exactly matches the payoff structure for
the investment, in a
competitive market, the value of the portfolio must be the value
of the investment.
Value of investment = 100*(1) + 50*(2) (1a)
Now we go back to the original question. How do we value project
A? Simple. To
match the payoff of FCFU in the up state and $0 in the down
state, we buy FCFU units of
asset J, which is equal to FCFU*(1). To match the payoff of $0
in the upstate and FCFD in
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the down state, we buy FCFD units of asset G, which is equal to
FCFD*(2). Since the
portfolio, which consists of FCFU units of asset J and FCFD
units of asset G, exactly
matches the payoff structure for the investment, the value of
the portfolio must be the value
of project A.
Value of project A = FCFU*(1) + FCFD*(2) (1b)
Section Two: A simple economy
Keeping the above discussion in mind, we move on to the details
of a specific
example. First, we make a detour and calculate the state prices
for the simple economy in
which we have investment project A. Consider a very simple one
period economy without
taxes and two assets: one risky and one risk-free.10 The risky
asset is coffee and the risk-
free asset is a government bond. An investor can invest in bags
of coffee or buy and sell
government bonds. Let the expected return on the government bond
be rf, the risk-free rate,
and let the expected return on the coffee be . Since the coffee
is risky, it reasonable to
assume that the expected return on the coffee is greater than
the risk-free rf. Assume that
there is no expected inflation and the value of rf is 10%.11
Even though this single period
economy is extremely simple, it provides plenty of insights into
the ideas behind risk-
neutral valuation.
10. A single period economy without taxes, two assets and two
states of nature may seem overly simplistic.
Nevertheless, we can obtain important insights from this simple
model and the essential ideas carry over to a more complex model
with multiple assets, multiple periods and continuous probability
distributions for the states of nature.
11. A stochastic or non-deterministic risk-free rate would
unnecessarily complicate the exposition.
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States of nature
Again, for simplicity, we will assume that all the investors in
the economy agree
that there are only two possible states of nature at the end of
year 1. Relative to the price of
coffee at the end of year 0, the price of coffee at the end of
year 1 can either increase or
decrease. We will call the state of nature with the price
increase the up state and the state
of nature with the price decrease the down state.12 Let pU be
the probability of the up
state and let (1-pU) be the probability of the down state. We
will call the set of two
probabilities the probability measure P, where P = {pU, (1-pU)}.
Individual investors may
hold subjective sets of probabilities for the states of nature
at the end of year, and there
may be no consensus among the investors. For the moment, we do
not discuss whether
these probabilities are subjective or objective.
Coffee and bond prices at the end of year 1
We will denote the prices for a bag of coffee at the end of year
1 by VC(1,1) and
VC(1,2). The first parameter in the parenthesis refers to the
time period and the second
parameter refers to the state of nature. Thus, VC(1,1) is the
price for a bag of coffee in year
1 under the up state, and VC(1,2) is the price for a bag of
coffee in year 1 under the down
state.13 The price for a bag of coffee in year 0 is VC(0,1). For
a simple example, the
notation may appear unnecessarily complex. However, the notation
that we introduce now
will prove very useful when we extend the analysis to multiple
periods.
12. Both states of nature have positive probabilities of
occurrence, and since only two states of nature are
possible at the end of year 1, the sum of the probabilities of
the two states must be equal to 1. 13. In general, V(i,j) refers to
the price for a bag of coffee under the jth state of nature (or
node) in the ith
period. For example, V(3,2) would be the price for a bag of
coffee in the second node of year 3.
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To be specific, we assume that the price for a bag of coffee in
the up state is $1,200
and the price for a bag of coffee in the down state is $800.
More importantly, as stated
previously, we assume that all investors in the economy agree on
the prices for a bag a
coffee at the end of year 1. However, there may be no agreement
on the set of probabilities
for the states of nature at the end of year 1. For the moment,
we do not specify the price for
a bag of coffee in year 0.
The process for the coffee price is shown.
Figure 1: Process (or tree) for the price of a bag of coffee
Year 0 1 VC(1,1) = 1,200 VC(0,1) = ? VC(1,2) = 800
We will denote the prices for a government bond at the end of
year 1 by B(1,1) and
B(1,2). Again, the first parameter in the parenthesis refers to
the time period and the
second parameter refers to the state of nature. The process for
the price of a government
bond is shown.
Figure 2: Process (or tree) for the price of a government
bond
Year 0 1 B(1,1) B(0,1) B(1,2)
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Thus, B(1,1) is the price for a government bond in year 1 under
the up state, and
B(1,2) is the price for a bag of coffee in year 1 under the down
state. Since we have
assumed that the bond is risk-free, at the end of year 1, the
price of the government bond
under both states of nature will be the same. That is, the value
of B(1,1) will be equal to
B(1,2). In general, if the bond is risky, B(1,1) need not be
equal to B(1,2).
Probabilities for the states of nature
Next we examine the relationship between VC(0,1), the price for
a bag of coffee in
year 0 and , the expected return from a bag of coffee. We can
ask the question, what is the
value of , the expected rate of return from investing in a bag
of coffee? In other words, if
we were to buy a bag of coffee, hold it for a year and sell it
at the end of year 1, what
would be the rate of return? The answer depends on VC(0,1), the
price for a bag of coffee
in year 0 and P, the set of probabilities for the states of
nature at the end of year 1. It would
seem that the expected return would be indeterminate because
each investor will have
her own assessment of the set of probabilities for the states of
nature at the end of year 1.
Recall that we have assumed that the value of VC(0,1) is not
unknown. However, even if
we knew the value of VC(0,1), we still would not know the
probabilities for the states of
nature. It would seem that the expected return on a bag of
coffee must depend in some way
on the values in the sets of probabilities P = {pU, (1-pU)} that
are held by the investors in
the economy. To narrow the analysis, we will assume that the
price for a bag of coffee in
year 0 is known and the probabilities for the states of nature
in year 1 are unknown.
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Price known, probabilities unknown
Suppose we assume that the market price for a bag of coffee in
year 0 is $1,000 and
an investor believes that P = {pU, (1-pU)} = {50%, 50%}. In
other words, the investor
believes that it is equally likely that the price for a bag of
coffee at the end of year 1 can
increase to $1,200 or decrease to $800. Based on the assessment
of this investor, we can
calculate the expected return. Let (1,1) be the return in the up
state, let (1,2) be the
return in the down state and let (0,1) be the expected return in
year 0.
(1,1) = VC(1,1) VC(0,1) VC(0,1)
= 1,200 1,000 = 20.00% (2a) 1,000 (1,2) = VC(1,2) VC(0,1)
VC(0,1)
= 800 1,000 = -20.00% (2b) 1,000
The return in the up state is positive 20%, the return in the
down state is negative
20%, and with equal probabilities for the states of nature at
the end of year 1, the expected
return on a bag of coffee is 0%. To obtain the expected return,
we multiply the returns in
the states of nature by their respective probabilities.14 Let
EP{(1,1:2)} denote the
14. In EXCEL, we can use the SUMPRODUCT function to find the
expected return.
Rather than taking the expectation of the returns under the two
states of nature at the end of year 1, we can calculate the
expected return in an equivalent way. Let EP{VC(1,1:2)} denote the
expectation of the coffee prices at the first and second states of
nature at the end of year 1, with the probability measure P. In the
inner parenthesis of EP{VC(1,1:2)}, the first parameter refers to
the time period, namely year 1, and the second parameter, with the
colon, refers to the coffee prices in the set of nodes in that
period, namely the first and second nodes. We calculate the
expectation by multiplying the coffee prices under the two states
of nature by the corresponding probabilities. Then P = {pU, (1-pU)}
= {50%, 50%} VC(0,1) = $1,000
EP{VC(1,1:2)}= p*VC(1,1) + (1 p)*VC(1,2)
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expectation of the return to a bag of coffee under the first and
second states of nature at the
end of year 1, with the probability measure P.15
P = {pU, (1-pU)} = {50%, 50%} (3a)
Expected return (0,1) = EP{(1,1:2)} = p*(1,1) + (1 p)*(1,2)
= 50%*20% + 50%*-20%
= 0.00% (3b)
Clearly, if the price for the bag of coffee is correct, then the
probabilities must be
mistaken because no investor would buy a bag of coffee with an
expected return of 0%
when the risk-free return is 10%. Alternatively, if the
probabilities are correct, then the
price for a bag of coffee must be mistaken.16 The price for a
bag of coffee in year 0 must
be low enough to enable the investor to obtain a return higher
than 10%.
Interpretation of expected return
It is extremely important to understand clearly the meaning of
the expected return.
At the end of year 1, the return of 0% does not occur. The
return at the end of year 1 will
be either 20%, if the up state of nature occurs, or 20% if the
down state of nature occurs.
If the expected return of 0% does not occur, what is the meaning
or interpretation of the
= 50%*1,200 + 50%*800 = 1,000.00 = EP{VC(1,1:2)} - VC(0,1)
VC(0,1) = 1,000.0 1,000.0 = 0.00% 1,000.0 15. In the inner
parenthesis of EP{(1,1:2)}, the first parameter refers to the time
period, namely year 1, and
the second parameter, with the colon, refers to the returns in
the set of nodes in that period, namely the first and second
nodes.
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0% expected return? Should we even use the expected return as a
criterion for decision
making in investment projects? The 0% return is a return in a
probabilistic sense. If we
were to invest in bags of coffee on several occasions and the
prices for bags of coffee
under the two states of nature do not change, then on average,
the return that we would
obtain, based on all the investments, would be 0%. There is no
guarantee that the return
would be 0%. If the number of times that we invested in bags of
coffee increased, we
would expect that the return would tend to 0%.
Analogy with coin toss
An analogy with the toss of a coin would be useful. Suppose we
toss a coin and
receive a return of 20% if the outcome is heads and lose 20% if
the outcome is tails. If we
toss a coin five times in a row, there is no guarantee
whatsoever that the expected return is
0%. The expected return would depend on the number of heads and
tails. Alternatively,
suppose we were to toss the coin 200 times in a row. Then it
would be reasonable to
assume that the expected return would be close to zero percent.
And if we were to increase
the number of tosses to 2000, then the result would be even
closer to 0%. It is extremely
important to bear in mind this probabilistic interpretation of
the expected return.
Set of probabilities for another investor
There is no presumption that all investors hold the same set of
probabilities for the
two states of nature at the end of year 1. Suppose another
investor has a different
assessment of the probabilities for the states of nature at the
end of year 1 and believes that
16. Earlier, we had observed that since the investment in coffee
is riskier than the investment in government
bonds, the return on the coffee must be higher than 10%, the
risk-free rate.
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P = {pU, (1-pU)} = {60%, 40%}. That is, this investor believes
that the probability of the up
state of nature is higher than the down state of nature. Using
the new set of probabilities,
the expected return is 4%.
P = {pU, (1-pU)} = {60%, 40%} VC(0,1) = $1,000 (4a)
EP{VC(1,1:2)} = p*VC(1,1) + (1 p)*VC(1,2)
= 60%*1,200 + 40%*800 = 1,040.00 (4b)
= EP{VC(1,1:2)} - VC(0,1) VC(0,1) = 1,040.0 1,000.0 = 4.00% (5)
1,000.0
Again, the expected return of 4% is lower than the risk-free
rate of 10% and thus the new
set of probabilities is also questionable.
Relationship between p and
The following table shows the relationship between p, the
probability of the up
state of nature and , the expected return on a bag of coffee,
holding the price of coffee for
a bag of coffee at the end of year 0 constant at $1,000. Based
on the table, we can
determine that the probability of the up state of nature would
have to be higher than 75% to
obtain an expected return higher than 10%. However, we still
have not find one correct set
of probabilities. Investors may have sets of assessments where
the probability of the up
state of nature ranges between 75% and 95%, with the
corresponding expected returns
ranging from 10% to 18%.
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Table 1: Relationship between p, the probability of the up state
of nature and the expected return on a bag of coffee
Probability of 50% 0.0% Up state of 55% 2.0% nature, p 60%
4.0%
65% 6.0% 70% 8.0% 75% 10.0% 80% 12.0% 85% 14.0% 90% 16.0% 95%
18.0%
Probabilities known, price unknown
In the previous analysis, we had assumed the price for a bag of
coffee at the end of
year 0 was known and the probabilities were unknown. Now we can
conduct a two-way
analysis, and examine the relationship between the expected
return and VC(0,1), the price
for a bag of coffee at the end of year 0, for different values
of p, the probability of the up
state of nature.
On the vertical axis of the table, we have the different values
for VC(0,1) and on the
horizontal axis, we have different values for p. We already know
that with p = 75% and
VC(0,1) = 1,000, the expected return is 10%. Holding the value
of VC(0,1) constant at
$1,000 and moving along the row in the table, we see that with
an increase of 5 percentage
points in the value of p, the expected return increases by 2
percentage points. For
example, if the value of p increases from 80% to 85%, the value
of increases from 12%
to 14%. Holding the value of p constant at 80%, and moving down
the column in the table,
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18
we see that expected return increases from 12% to 21.1% if the
price for a bag of coffee in
year 0 decreases from $1,000 to $925.
Table 2: Relationship between the expected return and VC(0,1),
for different values of p p 75.0% 80.0% 85.0% 90.0% 95.0%
V(0,1) 1,000.0 10.0% 12.0% 14.0% 16.0% 18.0% 975.0 12.8% 14.9%
16.9% 19.0% 21.0% 950.0 15.8% 17.9% 20.0% 22.1% 24.2% 925.0 18.9%
21.1% 23.2% 25.4% 27.6% 900.0 22.2% 24.4% 26.7% 28.9% 31.1%
Using the table, for given values of VC(0,1) and p, we can
determine the expected
return . Suppose the value of VC(0,1) is $950 and the value of p
is 80%. We can verify
that the expected return (0,1) is 17.9%.
P = {pU, (1-pU)} = {80%, 20%} VC(0,1) = $950 (6a)
EP{VC(1,1:2)} = p*VC(1,1) + (1 p)*VC(1,2)
= 80%*1,200 + 20%*800
= 1,120.00 (6b)
= EP{VC(1,1:2)} - VC(0,1) VC(0,1) = 1,120.0 950.0 = 17.89%
(7)
950.0
Based on the analysis up to this point, the results are not
encouraging. The expected
return on an investment in a bag of coffee depends both on the
price at the end of year 0
and the probabilities for the state of nature. And there is no
guarantee of any kind that
investors would agree on the probabilities for the states of
nature at the end of year 1 or on
the price for a bag of coffee at the end of year 0.
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19
Agreement on price by all investors
At this point, we invoke the law of one price and assume that
all investors agree
that the correct price for a bag of coffee at the end of year 0
is $950.17 However, we still
do not assume any agreement on the probabilities for the states
of nature at the end of year
1. That is, individual investors may still have their own sets
of probabilities for the states
of nature at the end of year 1.
With VC(0,1) = $950, we can calculate the returns under the two
states of nature at
the end of year 1. Let (1,1) be the return in the up state and
let (1,2) be the return in the
down state.
(1,1) = VC(1,1) VC(0,1) VC(0,1)
= 1,200 950 = 26.32% (8a) 950 (1,2) = VC(1,2) VC(0,1)
VC(0,1)
= 800 950 = -15.79% (8b) 950
Based on the agreed price, the return in the up state of nature
is 26.32% and the
return in the down state of nature is 15.79%. However, these
returns are still insufficient
for us to make a decision on whether we should or should not
invest in a bag of coffee. We
would like to calculate some kind of expected return and for
that calculation, we need to
specify relevant probabilities for the states of nature at the
end of year 1. But there is no
agreement on the probabilities and consequently no agreement on
the expected return.
Nevertheless, we have made some progress in the analysis.
17. Recall that there is no disagreement among the investors
about the prices for a bag of coffee under the
two states of nature at the end of year 1.
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20
The agreement on the correct price is a huge assumption, but if
it is true, then it is
all the information that we need for trading bags of coffee, and
as we will show later, with
this information, we can value project A. However, there is a
whiff of circularity in the
discussion. Suppose we were interested in investing in coffee.
The correct price for
coffee is the unknown variable and the whole point of the
analysis is to determine the
correct price for a bag of coffee at the end of year 0! If we
assume that all investors
agree on the price for a bag of coffee at the end of year 0,
then we have simply assumed
away the task that we were supposed to undertake in the first
place.
Our task here is to value project A. Using only the information
on project A, we
cannot make progress in the analysis. We have to use some other
information about assets
in the economy in order to value project A. Here we are using
the information on the
coffee price and the price for a government bond to estimate the
probabilities for the
states of nature at the end of year 1. Armed with this
information, we will return to the
valuation of project A.
We have assumed that we know the correct price in year 0 for a
bag of coffee is
$950. If the market for coffee is highly, but not perfectly,
competitive, the existence of a
single price for a bag of coffee may not be an unreasonable
assumption. For trading
purposes, we really do not need the expected return. If we were
a coffee trader, we would
buy coffee if someone was selling at a price less than $950 and
we would sell coffee if the
buyer were willing to pay more than $950. In reality, there are
always imperfections in the
market and one never knows the correct price. However, in
reasonably competitive
markets, such deviations from the correct price do not last for
long. Better-informed
traders, relative to others in the market, would take advantage
of arbitrage opportunities
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21
offered by discrepancies in prices, and in competitive markets,
there would be strong
continual tendencies towards a single price for a homogenous
product, such as a well-
defined bag of coffee.
Agreement on probabilities
Even if we could agree on the price for a bag of coffee at the
end of year 0, how
would we agree on the probabilities for the states of nature at
the end of year 1? Each
investor may have different (subjective) values for the
probabilities in the set P = {pU, (1-
pU)}. For example, suppose at the end of year 0, all investors
agree that the price for a bag
of coffee should be $950. The expected return will depend on the
probabilities for the
states of nature. Based on the numbers in the table above, the
expected return would
range from 15.8% for p = 75% to 24.2% for p = 95%. How can we
narrow the range of
disagreement among the investors, if all the investors hold
subjective probabilities for the
states of nature at the end of year 1?
The interesting fact is that we only need agreement on the value
of VC(0,1), the
price for a bag of coffee in year 0. We do not need agreement on
the probabilities for
the states of nature. Nevertheless, in appendix B, we briefly
discuss how we might reach
agreement on the probabilities by invoking an external model,
such as CAPM. Again, we
must stress that the agreement on the probabilities for the
states of nature is not required.
Law of one price (or absence of arbitrage)
However, we have to begin the analysis at some point. If we
assume perfect
markets (a big IF), then there will be one price in the market.
Arbitrage opportunities from
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22
discrepancies in prices may exist but these discrepancies will
be short-lived in a
competitive market.
The surprising fact is that in real-neutral valuation, we do not
need agreement on
the probabilities for the states of nature if there is agreement
on the price for a bag of
coffee at the end of year 0. Of course, in practice, we do not
know the correct price.
However, if the markets are competitive and there are pressures
toward one price, then we
can derive some useful results in valuation.
Estimation of state prices
We will begin the analysis at the end of year 0 and assume that
the price for a bag
of coffee is known. Assume that the market for coffee is
competitive and coffee dealers are
willing to sell or buy a bag of coffee at $950. What is the
meaning of the state prices? First,
we will give the formal definitions of the state prices and then
we will derive the
expressions for the state prices and explain how the state
prices can be useful in valuation.
To be specific, with the state prices we can price an asset at
the end of year 1 with any
payoff structure. To motivate the definitions of state prices,
we will introduce two new
assets, J and G. The usefulness of the two assets will be stated
now but a true appreciation
will come later. With linear combinations of the two new assets
J and G, we will be able to
price an asset with any payoff structure at the end of year
1.
Asset J
Consider a new asset J that has a very special payoff structure
at the end of year 1.
At the end of year 1, the payoff for asset J is $1 if the up
state occurs and the payoff is $0 if
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23
the down state occurs. Suppose an investor were to buy asset J
at the end of year 0. At the
end of year 0, what would be the appropriate price for an
investor to pay to buy asset J? In
other words, what is the value of J(0,1)?
In symbols,
J(1,1) = 1 and J(1,2) = 0 (9)
The process (or tree) for asset J is shown.
Figure 3: Process (or tree) for asset J
Year 0 1 J(1,1) = 1.0 J(0,1) = ? J(1,2) = 0.0
Asset G
Consider a new asset G that has a very special payoff structure
at the end of year 1.
The process (or tree) for asset G is shown.
Figure 4: Process (or tree) for asset G
Year 0 1 G(1,1) = 0.0 G(0,1) = ? G(1,2) = 1.0
At the end of year 1, the payoff for asset G is $0 if the up
state occurs and the
payoff is $1 if the down state occurs. Suppose an investor were
to buy asset G at the end of
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24
year 0. At the end of year 0, what would be the appropriate
price for an investor to pay to
buy asset G? In other words, what is the value of G(0,1)?
In symbols,
G(1,1) = 0 and G(1,2) = 1 (10)
Pricing asset J
How do we price asset J? We will price asset J at the end of
year 0 by constructing
a replicating portfolio Z. At the end of year 0, we construct a
portfolio that consists of C
bags of coffee and B amount of government bonds that replicates
the payoff structure of asset J under both states of nature at the
end of year 1. That is,
ZJ(1,1;C,B) = J(1,1) = 1 (11a)
ZJ(1,2;C,B) = J(1,2) = 0 (11b)
In ZJ(1,1;C,B), the notation for the portfolio Z, there are
three parameters. The first two parameters are the same as before.
The third parameter after the semi-colon refers
to the proportions of the two assets that make up the portfolio.
For notational simplicity,
we may drop the third parameters in some of the following
expressions. But it is important
to keep in mind that ZJ is really a portfolio of bags of coffee
and units of government
bonds.
Value of ZJ in year 0
Recall that we assume that an investor can borrow and lend as
much government
bonds as they would like. Let ZJ(0,1) be the value of the
portfolio at the end of year 0,
which is equal to the sum of the value of the coffee and the
value of the government bonds.
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25
At the end of year 0, the value of the coffee is C times V(0,1)
and the value of the
government bond is B times B(0,1). Furthermore, we have assumed
that V(0,1), the price for a bag of coffee at the end of year 0 is
$950. Suppose B(0,1), the price for a government
bond at the end of year 0 is $600. Then,
B(1,1) = B(1,2)
= B(0,1)*(1 + rf) = 600*(1 + 10%) = 660.00 (12)
Then the value of the portfolio Z at the end of year 0 is given
below.
ZJ(1,1;C,B) = Z(0,1) = C*V(0,1) + C*B(0,1) (13) Next, we will
invoke the law of one price. If the markets are competitive and
there
are no arbitrage opportunities, then the value of the portfolio
ZJ(0,1) at the end of year 0
must be equal to the value of the asset J at the end of year 0.
In symbols,
ZJ(0,1) = J(0,1) (14)
The expression for the value of ZJ(0,1) has two unknown
parameters, C and C.
Thus, we will be able to value asset J if we know the values of
the two parameters, C and
C.
System of two equations and two unknowns
Recall that we have two assets and there are two states of
nature. By setting up a
system of equations with two equations and two unknowns, we can
determine the values
for C and B. We will assume that a suitable solution exists.
Suppose the up state occurs at the end of year 1. Then at the end
of year 1, the value of the portfolio ZJ(1,1) is equal to
C times V(1,1) and the value of the government bond is B times
B(1,1).
ZJ(1,1) = C*V(1,1) + B*B(1,1) (15a)
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26
Suppose the down state occurs at the end of year 1. Then at the
end of year 1, the
value of the portfolio ZJ(1,2) is equal to C times V(1,2) and
the value of the government
bond is B times B(1,2).
ZJ(1,2) = C*V(1,2) + B*B(1,2) (15b) We obtain two equations by
setting the values of the portfolio Z under the two
states of nature at the end year 1 equal to the corresponding
values of asset J under the two
states of nature.
C*VC(1,1) + B*B(1,1) = J(1,1) = 1 (16a)
C*VC (1,2) + B*B(1,2) = J(1,2) = 0 (16b) By using basic
algebraic manipulations, we can solve the two equations and
obtain
the values for the two unknown parameters, C and C. Subtracting
one equation from the other, we obtain
C = 1 (17a) VC(1,1) VC(1,2)
Substituting the numerical values for V(1,1) and V(1,2), we
obtain that
C = 1 = 1 = 0.00250 (17b) 1,200 800 400
We obtain an expression for B by substituting the expression for
C in one of the
above equations, and solving for B.
B = -VC(1,2)/B(1,2) (18a) VC(1,1) VC(1,2)
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27
Substituting the appropriate numerical values, we obtain
that
B = -800/660 1,200 - 800
= -0.0030303 (18b)
With the values of C and B, we calculate the value of
Z(0,1).
ZJ(0,1) = C*V(0,1) + B*B(0,1) = 0.0025*950 + -0.0030303*600
= 2.37500 + -1.81818
= 0.55682 (19)
Thus, using the two equations and two unknowns, we find the
values of parameters
C and B, calculated the value of ZJ(0,1), and thereby find the
value of the state price for
the up state of nature, namely (1).
Next, we will discuss the interpretation of the parameters C and
B and verify that we have been successfully in replicating the
payoff structure for asset J.
Interpretation of the parameters and
What is the interpretation of the parameters and ?18 The
interpretation of and
is as follows. Positive values of mean that we are buying bags
of coffee, and negative
values of mean that we are selling bags of coffee. Positive
values of mean that we are
lending cash (by buying government bonds), and negative values
of mean that we are borrowing cash (by selling government
bonds).
18. For notational simplicity, we have dropped the subscripts
for and . There should be no loss in clarity.
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28
We construct the portfolio Z by buying bags of coffee, which is
valued at $2.375,
and borrowing amount of government bonds, which is valued at
$1.81818. Next we show that the payoffs for portfolio Z match (or
replicate) the payoffs for the asset J.
Suppose the up state of nature occurs. Then the value of the Z
portfolio is $1.
*VC(1,1) + *B(1,1) = 0.0025*1,200 0.0030303*660
= 3.0000 + -2.0000 = 1.0000 (20a)
The value of the coffee is $3, and we pay back $2 for the cash
that we borrowed by
selling government bonds. The net result is $1.
Suppose the down state of nature occurs. Then the value of the Z
portfolio is $0.
*VC(1,2) + *B(1,2) = 0.0025*800 0.0030303*660
= 2.0000 + -2.0000 = 0.0000 (20b)
The value of the coffee is $2, and we pay back $2 for the cash
that we borrowed by
selling government bonds. The net result is $0.
We have assumed competitive markets. In a competitive market, to
avoid arbitrage
opportunities, assets with identical payoff structures must have
the same price. Since in
year 1, under both states of nature, the payoff structure for
the portfolio Z matches the
payoff structure for the asset J, the value of the portfolio in
year 0 must be the no-arbitrage
price for the asset J in year 0. That is, ZJ(0,1) = J(0,1)
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29
Pricing asset G
How do we price asset G? We will price asset G at the end of
year 0 by
constructing a similar replicating portfolio ZG. We will use the
same approach that we had
used for finding the value of asset J.
At the end of year 0, we construct a portfolio that consists of
bags of coffee and amount of government bonds that replicates the
payoff structure of asset G under both
states of nature at the end of year 1.19 That is,
ZG(1,1) = G(1,1) = 0 (21a)
ZG(1,2) = G(1,2) = 1 (21b)
At the end of year 0, the value of the coffee is times V(0,1)
and the value of the
government bond is times B(0,1). Then the value of the portfolio
at the end of year 0 is given below.
ZG(0,1) = *VC(0,1) + *B(0,1) (22a) Next, we will invoke the law
of one price. If the markets are competitive and there
are no arbitrage opportunities, then the value of the portfolio
ZG(0,1) at the end of year 0
must be equal to the value of the asset G at the end of year 0.
In symbols,
ZG(0,1) = G(0,1) (22b)
The expression for the value of ZG(0,1) has two unknown
parameters, and . Again, we can set up a system of equations with
two equations and two unknowns, we can
determine the values for and . Suppose the up state occurs at
the end of year 1. Then at
the end of year 1, the value of the portfolio ZG(1,1) is equal
to times V(1,1) and the value
of the government bond is times B(1,1).
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30
ZG(1,1) = *VC(1,1) + *B(1,1) (23) Suppose the down state occurs
at the end of year 1. Then at the end of year 1, the
value of the portfolio ZG(1,2) is equal to times V(1,2) and the
value of the government
bond is times B(1,2).
ZG(1,2) = *VC(1,2) + *B(1,2) (24) We obtain two equations by
setting the values of the portfolio Z under the two
states of nature at the end year 1 equal to the corresponding
values of asset G under the two
states of nature.
*VC(1,1) + *B(1,1) = G(1,1) = 0 (25a)
*VC(1,2) + *B(1,2) = G(1,2) = 1 (25b) Subtracting one equation
from the other, we obtain
= -1 (26a) VC(1,1) VC(1,2)
Substituting the numerical values for VC(1,1) and VC(1,2), we
obtain that
= -1 = -1 = -0.00250 (26b) 1,200 800 400
We obtain an expression for by substituting the expression for
in one of the
above equations, and solve for .
= VC(1,1)/B(1,1) (27a) VC(1,1) VC(1,2)
19. Note that the values of and for asset G will necessarily be
different from the and for asset J that
we find because the payoff structures are different for the two
new assets.
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31
Substituting the appropriate numerical values, we obtain
that
= 1,200/660 1,200 - 800
= 0.00454545 (27b)
With the values of and , we calculate the value of Z(0,1).
ZG(0,1) = *VC(0,1) + *B(0,1) = -0.0025*950 + 0.00454545*600
= -2.3750 + 2.72727
= 0.35227 (28)
Again, using the two equations and two unknowns, we find the
values of
parameters C and B, calculated the value of ZG(0,1), and thereby
find the value of the
state price for the down state of nature, namely (2).
Interpretation of the parameters and
What is the interpretation of the parameters and ? The
interpretation of and
is as follows. We construct the portfolio ZG by selling bags of
coffee, which is valued at
$2.375, and lending amount of government bonds, which is valued
at $2.72727. Next we show that the payoffs for portfolio ZG match
(or replicate) the payoffs for the asset G.
Suppose the up state of nature occurs. Then the value of the ZG
portfolio is $1.
*VC(1,1) + *B(1,1) = -0.0025*1,200 0.00454545*660
= -3.0000 + 3.0000 = 0.0000 (29a)
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32
The value of the coffee is $3. We promised to sell coffee worth
$3. However, we
receive $3 in cash that we had lent by buying government bonds.
The net result is $0.
Suppose the down state of nature occurs. Then the value of the
ZG portfolio is $0.
*VC(1,2) + *B(1,2) = -0.0025*800 + 0.00454545*660
= -2.0000 + 3.0000 = 0.0000 (29b)
The value of the coffee is $2. We promised to sell coffee worth
$2. However, we
receive $3 in cash that we had lent by buying government bonds.
The net result is $1.
We have assumed competitive markets. In a competitive market, to
avoid arbitrage
opportunities, assets with identical payoff structures must have
the same price. Since in
year 1, under both states of nature, the payoff structure for
the portfolio ZG matches the
payoff structure for the asset G, the value of the portfolio in
year 0 must be the no-arbitrage
price for the asset G in year 0. That is, ZG(0,1) = G(0,1).
State prices for the states of nature
Let (1) be the state price for the up state and let (2) be the
state price for the
down state. Then the value of (1) is equal to the value of the
asset J at the end of year 0,
namely J(0,1).
(1) = J(0,1) = 0.55682 (30a)
And the value of (2) is equal to the value of the asset G at the
end of year 0,
namely G(0,1).
(2) = G(0,1) = 0.35227 (30b)
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33
What is the meaning of the state price (1)? At the end of year
0, the state price for
the up state is the correct value for an asset that pays $1 if
the up state occurs at the end of
year 1 and $0 if the down state occurs at the end of year 1,
namely $0.55682.
What is the meaning of the state price (2)? At the end of year
0, the state price for
the down state is the correct value for an asset that pays $0 if
the up state occurs at the end
of year 1 and $1 if the down state occurs at the end of year 1,
namely $0.35227.
Suppose we wish to receive $1 with full certainty at the end of
year 1. How would
we accomplish this goal? With the existence of the risk-free
government bond, which
provides the same amount under both states of nature at the end
of year 1, we would
simply buy the appropriate amount of bonds to give me $1 with
full certainty at the end of
year 1.
Alternatively, how could we use combinations of asset J and
asset G to receive $1
with full certainty at the end of year 1? With a moments
reflection, we see that we can
achieve this objective if we invest in one unit of asset J and
one unit of asset G. If the up
state of nature occurs we will receive $1 from asset J (and
nothing from asset G), and if the
down state of nature occurs we receive $1 from asset G (and
nothing from asset J). Thus,
no matter which state of nature occurs, we are assured of
receiving $1. The cost for
investing in one unit of asset J and one unit of asset G is
simply the sum of the two state
prices.
Investment cost = (1) + (2)
= 0.55682 + 0.35227 = 0.90909 (31)
Since the investment is risk-free, we would expect that the
return would be equal
to the risk-free rate rf.
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34
1 + = 1 = 1.1000 (32) 0.90909
And indeed it is the case that the risk-free return is 10%.
What does it mean to invest in one unit of asset J and one unit
of asset G? After all,
the fundamental assets are bags of coffee and units of
government bonds. Asset J and asset
G are based on combinations of bags of coffee and units of
government bonds. Investing in
one unit of asset J means that we buy 0.0025 bags of coffee and
borrow cash by selling
government bonds equal in value to 0.0030303 units of government
bonds. Investing in
one unit of asset G means that we sell 0.0025 bags of coffee and
lend cash by buying
government bonds equal in value to 0.00454545 units of
government bonds. The purchase
and sale of 0.0025 bags of coffee offset each other. The net
effect of investing in one unit
of asset J and one unit of asset G is the purchase of 0.00151515
units of government bonds.
0.00454545 0.0030303
= 0.00151515 units of government bonds (33)
The cash value of the purchase of government bond is
0.00151515*600 = 0.9091 (34)
and the return on the purchase of the government bond is
10%.
Replicating the payoff structure for the investment in
coffee
How do we use the state prices for valuation? Reconsider the
payoff structure for
the investment in a bag of coffee. If the up state of nature
occurs, the payoff is $1,200 and
if the down state of nature occurs, the payoff is $800. We can
replicate the payoff structure
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35
for the investment in a bag of coffee by judiciously selecting a
portfolio ZC that consists of
investments in asset J and asset G.20
Specifically, at the end of year 0, if we invest in 1,200 units
of asset J and 800 units
of asset G, we would be able to replicate the payoff structure
for the investment in a bag of
coffee. The payoffs for portfolio ZC under the two states of
nature are shown below.
Up state of nature: 1,200*{J(1,1) + G(1,1)}
= 1,200*J(1,1) + 1,200*G(1,1) (35a)
Down state of nature: 800*{J(1,2) + G(1,2)}
= 1,200*J(1,2) + 1,200*G(1,2) (35b)
Recall that J(1,1) = 1, J(1,2) = 0, G(1,1) = 0 and G(1,2) = 1.
Suppose the up state of
nature occurs. The payoff from the investment in asset J is
$1,200 and the payoff from the
investment in asset G is zero. Suppose the down state of nature
occurs. The payoff from
the investment in asset J is zero and the payoff from the
investment in asset G is $800.
Thus, we see that the portfolio ZC, which consists of 1,200
units of asset J and 800 units of
asset G, successfully replicates the payoff structure for the
investment in a bag of coffee.
What is the value of the portfolio ZC at the end of year 0? We
find the value of portfolio ZC
by multiplying the units of asset G and asset J by their
respective prices (or equivalently,
state prices) at the end of year 0.
20. Note that previously we constructed the replicating
portfolios by using bags of coffee and units of
government bonds. Now we are using the new basic assets J and G
to construct the replicating portfolios.
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36
Value of portfolio ZC = 1,200*J(0,1) + 800*G(0,1)
= 1,200*(1) + 800*(2)
= 1,200*0.55682 + 800*0.35227
= 668.18 + 281.82
= 950.00 (36)
As expected, the value of portfolio ZC is equal to V(0,1), the
price for a bag of
coffee at the end of year 0.
Due to the special payoff structures for asset J and G, with
positive payoffs under
only one state of nature and nothing in the other state, the
strategy for constructing and
valuing the replicating portfolio is extremely easy. The number
of units in asset J and asset
G are equal to the payoffs under the respective states of
nature, and the value of the
portfolio is the sumproduct of the payoffs and the state prices
for the two states of nature.
Replicating the payoff structure for the investment in
government bonds
We can use the state prices to value the investment in a
government bond as well.
The investment in a government bond provides a payoff of $660
under both states of
nature at the end of year 1. Specifically, if we invest in 660
units of both asset J and asset
G, we will replicate the payoff structure for the investment in
a government bond. Let ZB =
(J,G) represent the replicating portfolio, where the first
parameter J refers to the number
of units invested in asset J and the second parameter G refers
to the number of units
invested in asset G. Then the portfolio ZB = (J, G) = (660, 660)
will replicate the payoff
structure for the investment in a government bond. Again, we
find the value of portfolio ZB
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37
at the end of year 0 by multiplying the units of asset J and
asset G in the replicating
portfolio by their respective prices (or equivalently, state
prices).
Value of portfolio ZB = 660*J(0,1) + 660*G(0,1)
= 660*(1) + 660*(2)
= 660*0.55682 + 660*0.35227
= 367.50 + 232.50
= 600.00 (37)
As expected, the value of portfolio ZB is equal to B(0,1), the
price for a government bond
at the end of year 0.
Replicating the payoff structure for any asset
Now we are ready to show that using the state prices, we can
replicate the payoff
structure for any asset. To be specific, we will return to the
valuation of project A that was
mentioned in Section One.
Let A = (U, D) = (FCFU, FCFD) represent the payoff structure for
the project ,
where the first parameter is the payoff in the up state and the
second parameter is the
payoff in the down state. To value the asset, we would construct
a replicating portfolio ZA
= (J,G), where J = U and G = D. Again, we find the value of
portfolio ZA by
multiplying the units of asset J and asset G by the respective
state prices.
Value of portfolio ZA = J*(1) + G*(2)
= U*(1) + D*(2)
= FCFU*(1) + FCFD*(2) (38)
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38
Stochastic Discount Factors
Another interpretation of the state prices is as follows. Let A
= (U, D) represent
the payoff structure for an investment opportunity. Then we know
that at the end of year 0,
the no-arbitrage price for the investment opportunity is equal
to the value of the replicating
portfolio ZA, which is given as follows.
Value of portfolio ZA = J*(1) + G*(2)
= U*(1) + D*(2) (39)
Let (1) be the discount rate for the up state of nature and let
(2) be the discount
rate for the down state of nature. Then, to find the value of
the portfolio, we could
discount the payoffs in the up and down states by the respective
discount rates (1) and
(2). We call (1) and (2) stochastic discount rates because they
depend on the uncertain
occurrence of the states of nature at the end of year 1.
Value of portfolio ZA = U + D (40) 1 + (1) 1 + (2)
Equating the coefficients in line 40 with the corresponding
coefficients in line 39,
we obtain that
1 + (1) = 1 (41a) (1)
1 + (2) = 1 (41b) (2)
Substituting the relevant numerical values, we find that
(1) = 1 = 1 = 1.79591 (42a) (1) 0.55682
(2) = 1 = 1 = 2.83873 (42b) (2) 0.35227
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39
Section Three: Equivalence between state prices and risk-neutral
probabilities
There is yet another way to think of the state prices and that
is in terms of risk-
neutral probabilities. We can show that the state prices can be
rewritten as follows. See the
algebraic details in appendix A. The state price for the up
state of nature is equal to q
divided by one plus the risk-free rate, and the state price for
the down state of nature is
equal to 1 - q divided by one plus the risk-free rate.
(1) = q (43a) (1 + rf)
(2) = (1 - q) (43b) (1 + rf)
where the expression for q is as follows.
q = (1 + rf)*VC(0,1) VC(1,2) (44) [VC(1,1) VC(1,2)]
The numerator of q is the difference between the price for a bag
of coffee at the end
of year 0, compounded forwarded one-period by one plus the
risk-free rate, and VC(1,2),
the low price in the down state of nature at the end of year 1.
The denominator of q is the
difference between VC(1,1), the high price of coffee in the up
state of nature at the end of
year 1 and VC(1,2), the low price in the down state of nature at
the end of year 1.
Substituting the numerical values, we find that the value of q
is
q = (1 + rf)*VC(0,1) VC(1,2) [VC(1,1) VC(1,2)]
= (1 + 10%)*950 800 = 61.250% (45) 1,200 - 800
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40
We can also rewrite the expression for q in the following way.
Let
u = 1 + gU = VC(1,1) = 1,200 = 1.2632 (46a) VC(0,1) 950 d = 1 +
gD = VC(1,2) = 800 = 0.8421 (46b) VC(0,1) 950 Let u be the ratio of
the high price in year 1 to the current price in year 0 and let
d
be the ratio of the low price in year 1 to the current price in
year 0. Then roughly speaking,
we can say that the price of coffee is expected to increase by
26% in the up state at the end
of year 1 or decrease by 17% in the down state at the end of
year 1.
We can write q, in terms of u and d, as follows.
q = (1 + rf) - d (47) u - d Let A = (U, D) represent the payoff
structure for an investment opportunity, and
let A(0,1) be the no-arbitrage price for the investment
opportunity at the end of year 0. We
know that A(0,1) can be written in terms of the state
prices.
A(0,1) = U*(1) + D*(2) (48)
We can rewrite the expression for A(0,1) in terms of q.
A(0,1) = q*U + (1 q)*D (49) 1 + rf
We observe that the expression for q satisfies all of the
properties for a probability
by showing that the value of q must be strictly greater than
zero and strictly less than one.21
21. First, consider the denominator in the expression for q. We
have assumed that the prices for a bag of
coffee under the two states of nature at the end of year 1 are
positive. Moreover, VC(1,1) the price in the up state is higher
than VC(1,2) the price in the down state. Therefore the difference
between the two prices will be positive.
Next, we will examine the higher and lower limits for the value
of the numerator. The expression for the numerator is given
below.
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41
Moreover, with the interpretation of q as a probability, the
expression for A(0,1) is elegant
and simple. We call Q = {qU, (1 qU)}= {61.25%, 38.75%} the set
of risk-neutral
probabilities. To find the correct no-arbitrage price for a bag
of coffee at the end of year 0,
simply take the expectation of the prices for a bag of coffee at
the end of year 1 with
respect to the set of risk-neutral probabilities in Q, and
discount by the risk-free rate. The
definition of q is a mathematical consequence of the state
prices for the two states of nature
at the end of year 1. It is as if we could assume that all
investors were risk-neutral and we
could discount the expectation of the prices at the end of year
1, taken with respect with the
risk-neutral probabilities, with the risk-free rate. There is no
presumption that investors are
actually risk-neutral. The risk-neutral probabilities are useful
and elegant mathematical
constructs for valuation.
(1 + rf)*VC(0,1) - VC(1,2)
We have assumed that the risk-free rate is positive. Now, the
value of the numerator must be greater strictly greater than zero.
Suppose the numerator is less than or equal to zero.
(1 + rf)*VC(0,1) - VC(1,2) 0 (1 + rf) VC(1,2)
VC(0,1)
It would suggest that the value of VC(0,1) is less than the
value of VC(1,2) and under both states of nature at the end of year
1, with full certainty, the return on a bag of coffee is higher or
equal to the risk-free rate. This cannot be the case. Therefore,
the value of the numerator must be greater than zero. What is the
upper limit on the value of the numerator? Suppose the value of the
numerator is greater than the value of the denominator.
(1 + rf)*VC(0,1) VC(1,2) VC(1,1) - VC(1,2) (1 + rf)
VC(1,1)/VC(0,1)
This means that the risk-free return is higher than the return
from investing in a bag of coffee. However, it cannot be the case
that the risky return from the investment in coffee is less than
the risk-free return. Therefore, the value of the numerator must be
strictly less than the value of the denominator.
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42
Expectation price (or no-arbitrage price) versus expected
value
Note that here we are not calculating the expected value with
respect to the risk-
neutral probabilities and we need to clearly distinguish between
the concept of expectation
and expected value. Previously, when we were discussing the
subjective probability
assessment of the investors, we were calculating the expected
value and there was an
explicit understanding that the expected value would not be
realized. If the up state of
nature occurred, we would realize (1,1), the return in the up
state of nature and if the
down state of nature occurred, we would realize (1,2), the
return in the down state of
nature. The expected value was a result that we would achieve if
we were to repeatedly
invest in a bag of coffee. When we use the risk-neutral
probabilities to calculate the price
for a bag of coffee at the end of year 0, the value of the
discounted expectation is the
correct no-arbitrage price; it is not an expected price.
Conclusion
In a simple competitive economy, with two basic assets and two
states of nature,
we can calculate the state prices for the two states of nature
IF we assume that the law of
one price holds. The state prices represent the values of new
assets at the end of year 0,
with specific payoff structures at the end of year 1. These new
assets are based on
combinations of the two basic assets in the economy, namely the
risky bags of coffee and
the risk-free government bonds. Specifically, with the law of
one price, in equilibrium
there will be no opportunity for arbitrage, and assets (or
investment opportunities) with the
same payoff structure must have the same price. We construct a
portfolio Z that consists
of combinations of the two new assets in such a way that the
payoff structure for portfolio
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43
Z exactly matches the payoff structure for the asset that we are
trying to value.
Mathematically, the state prices can be viewed from another
perspective. With some
rearrangement, the state prices can be reinterpreted as
risk-neutral probabilities.
We use CAPM or a similar asset-pricing model to determine the
expected return
that would be consistent with the no-arbitrage price. Based on
the expected return, we can
determine the objective probabilities (for the state of nature
at the end of year 1) that are
consistent with the expected return. The expectation of the
payoffs at the end of year 1
with respect to the objective probabilities, discounted by ,
equals the no-arbitrage value at
the end of year 0. Equivalently, we can calculate the
no-arbitrage value at the end of year 0
by taking the expectation of the payoffs at the end of year 1
with respect to the risk-
neutral probabilities, discounted by the risk-free rate rf.
There is no assumption that
investors are necessarily risk-neutral. We can conduct valuation
and obtain consistent
results as if all investors were risk-neutral.
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44
APPENDIX A
Algebraic expression for (1)
In this appendix, we derive algebraic expressions for the state
prices by substituting
the appropriate expressions for and into the expression for
(1).
(1) = J(0,1) = *VC(0,1) + *B(0,1) = VC(0,1) +
-VC(1,2)*B(0,1)/B(1,1) (A1) VC(1,1) VC(1,2) VC(1,1) VC(1,2)
We know that B(0,1) = 1/(1 + rf) (A2) B(1,1)
Rearranging line A1, we obtain
(1) = (1 + rf)*VC(0,1) VC(1,2) (A3) [VC(1,1) VC(1,2)]*(1 +
rf)
Define q as follows.
q = (1 + rf)*VC(0,1) VC(1,1) (A4) [VC(1,1) VC(1,2)]
Then (1) = q (A5) (1 + rf)
Algebraic expression for (2)
(2) = G(0,1) = *V(0,1) + *B(0,1) = -VC(0,1) +
VC(1,2)*B(0,1)/B(1,2) (A6) VC(1,1) VC(1,2) VC(1,1) VC(1,2)
We know that B(0,1) = 1/(1 + rf) (A7) B(1,2)
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45
Rearranging line A6, we obtain
(2) = -(1 + rf)*VC(0,1) + VC(1,2) (A8) [VC(1,1) VC(1,2)]*(1 +
rf)
Observe that 1 q is given by the following expression.
1 - q = -(1 + rf)*VC(0,1) + VC(1,2) (A9) [VC(1,1) VC(1,2)]
Therefore, we can write the state price for the down state as
follows.
(2) = 1 - q (A10) (1 + rf)
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46
APPENDIX B
The relevance (or irrelevance) of Capital Asset Pricing Model
(CAPM)
To narrow the range of disagreement among the investors about
the expected return
for investing in a bag of coffee, we need an external model to
provide an answer. The
prices for a bag of coffee at the end of year 1 under the two
states of nature give some
indication of the risk of investing in a bag of coffee. One
possibility for narrowing the
range of disagreement among the investors is to apply an
external model, for example, the
Capital Asset Pricing Model (CAPM). However, any external model
for asset pricing will
be appropriate. Based on the CAPM or any other model that is
agreed upon by the
investors, we can derive an appropriate expected return for
investing in a bag of coffee.22
Suppose all the investors agree that the correct price for a bag
of coffee at the end
of year 0 is $950. Furthermore, they believe that the CAPM is
suitable for estimating the
return on a bag of coffee and based on CAPM, the value for the
expected return is 20%.
If all the investors agree on the price for a bag of coffee at
the end of year 0 and the
expected return based on CAPM, then all the investors will have
to agree on the
probabilities for the states of nature at the end of year 1.
1 + = EP{VC(1,1:2)} (B1.1) VC(0,1) 1 + = pU*VC(1,1) + (1
pU)*VC(1,2) (B1.2) VC(0,1)
22. Here we do not wish to discuss the relative merits and
demerits of CAPM and other related asset pricing
models.
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47
In line B1.2, we solve for pU, the probability of the up state
of nature.
pU = [1 + (0,1)]*VC(0,1) VC(1,2) VC(1,1) VC(1,2)
= (1 + 20%)*950 800 1,200 800 = 85.00% (B2)
And we can verify that p = 85% is consistent with = 20% and
VC(0,1) = $950.
With the specification of the expected return at 20%, we have
calculated the corresponding
probabilities for the states of nature at the end of year 1. To
be consistent with CAPM, the
probability of the up state of nature must be 85% and the
probability of the down state of
nature must be 15%. Thus, to resolve the disagreement on the
probabilities for the states of
nature at the end of year 1, we invoke an external pricing
model.
Expected return = pU*(1,1) + (1 pU)*(1,2)
= 85%*26.32% + 15%*-15.79%
= 20.00% (B3)
Using these probabilities and the returns of 26.32% and 15.79%,
we can calculate
the expected return and verify that it is 20%. See line 8a and
8b in the main text. We can
call P = {pU, (1-pU)} = {85%, 15%} the set of objective (or
actual) probabilities for the
states of nature because under our assumptions, all of the
investors agree on the
probabilities. With these assumptions, we have been able to
derive a set of objective
probabilities for the states of nature from the subjective
probabilities held by the
investors.
These two assumptions, namely the agreement on the correct price
at the end of
year 0 and the knowledge of the required expected return, are
very stringent. First,
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48
investors may not trust the results of CAPM, and more
importantly, there may be no
agreement on the price for a bag of coffee at the end of year 0.
The CAPM has come under
severe attack in countries with developed capital markets and it
may not approximate
reality. In countries with developing capital markets, the
relevance of CAPM or related
models is even more questionable. Nevertheless, we need to begin
the analysis with some
assumptions and as a first step, we assume that the CAPM (or
another similar model) is a
stylized model that approximates reality.
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49
APPENDIX C
Absence of a risk-free asset
In the previous sections, we assumed that there were two assets:
a risky asset and a
risk-free asset. However, it is not necessary to assume a
risk-free asset, and in this section,
we will redo the analysis by assuming that there are two risky
assets.
Again, consider a very simple one period economy with two risky
assets and two
states of nature. The first risky asset is coffee and the second
risky asset is rice. Again, we
will assume that there are two states of nature. Let VC(1,1) and
VR(1,1) be the prices for a
bag of coffee and a bag of rice, respectively, in the up state
at the end of year 1 and let
VC(1,2) and VR(1,2) be the prices for a bag of coffee and a bag
of rice, respectively, in the
down state at the end of year 1. Let VC(0,1) and VR(0,1) be the
prices for a bag of coffee
and a bag of rice, respectively, at the end of year 0.
The numerical values for the prices of the two risky assets are
presented below.
Currently a bag of coffee is trading at $950 and a bag of rice
is trading at $600 and all
investors are in agreement about the stated prices.
VC(1,1) = 1,200 VC(1,2) = 800 (C1.1)
VR(1,1) = 660 VR(1,2) = 600 (C1.2)
VC(0,1) = 950 VR(1,2) = 600 (C1.3)
If the up state of nature occurs, the price for a bag of coffee
is $1,200 and if the
down state of nature occurs, the price for a bag of coffee is
$800. Recall that in the up state
the return on a bag of coffee is 26.32%, in the down state the
return on a bag of coffee is
negative 15.79% and the expected return on a bag of coffee is
20%.
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50
If the up state of nature occurs the price for a bag of rice is
$660, and if the down
state of nature occurs, the price for a bag of rice is $600.
Let R(1,1) be the return on a bag of rice in the up state and
let R(1,2) be the return
on a bag of rice in the down state.
R(1,1) = VR(1,1) VR(0,1) VR(0,1)
= 660 600 = 10.00% (C2.1) 600 R(1,2) = VR(1,2) VR(0,1)
VR(0,1)
= 600 600 = 0.00% (C2.2) 600
P = {pU, (1-pU)} = {85%, 15%} VR(0,1) = $600
Expected return = p*(1,1) + (1 p)*(1,2)
= 85%*10% + 15%*0%
= 8.50% (C3)
In the up state the return on a bag of rice is 10%, in the down
state the return on a
bag of rice is 0% and the expected return on a bag of rice is
8.50%.
We will construct a portfolio Z consisting of bags of coffee and
bags of rice that replicate the payoff structure for a risk-free
investment F, where F(1,1) is the payoff if
the up state of nature occurs at the end of year 1 and F(1,2) is
the payoff if the down state
of nature occurs at the end of year 1. Using the portfolio Z, we
can calculate the state
prices for the two states of nature.
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51
State price for the up state
To simplify the number of decimal places, we have increased the
payoff structure
under the two states of nature by 1,000. This simply increases
the values of and by a factor of 1,000.
*VC(1,1) + *VR(1,1) = J(1,1) = 1,000 (C4.1)
*VC (1,2) + *VR(1,2) = J(1,2) = 0 (C4.2) Based on the earlier
discussion, we use simple algebraic manipulations to write
down the appropriate expressions for and and determine the
values for and . We
can verify that is 3.125 and is 4.16667. 3.125*1,200 +
-4.16667*660 = 1,000.00 (C4.3)
3.125*800 + -4.16667*600 = 0.00 (C4.4)
With the values of and , we calculate the value of Z(0,1) which
is equal to the
state price (1).
(1) = Z(0,1) = *V(0,1) + *B(0,1) = 3.125*950 + -4.16667*600
= 468.75 (C5)
State price for the down state *VC(1,1) + *VR(1,1) = G(1,1) = 0
(C6.1)
*VC (1,2) + *VR(1,2) = G(1,2) = 1,000 (C6.2)
We can verify that is -3.4375 and is 6.25. -3.4375*1,200 +
6.25*660 = 0.00 (C6.3)
-3.4375*800 + 6.25*600 = 1,000.00 (C6.4)
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52
With the values of and , we calculate the value of Z(0,1) which
is equal to the
state price (2).
(2) = Z(0,1) = *V(0,1) + *B(0,1) = -3.4375*950 + 6.25*600
= 484.375 (C7)
Risk-free return To obtain a risk-free return, we will invest in
one unit of asset J and one unit of
asset G. The cost for investing in one unit of asset J and one
unit of asset G is simply the
sum of the two state prices.
Investment cost = (1) + (2)
= 468.75 + 484.375 = 953.13 (C8)
Since the investment is risk-free, we would expect that the
return would be equal to
the risk-free rate.
1 + = 1,000 = 1.0492 (C9) 953.13
Thus in this case, the risk-free return is 4.92%. Even though in
this economy there is no
risk-free asset, we can make risk-free investments by investing
in equal units of asset J and
asset G. However, in terms of the fundamental assets, namely
bags of coffee and bags of
rice, what does it mean to invest in one unit of asset J and one
unit of asset G?
Investing in one unit of asset J means that we buy 3.125 bags of
coffee and borrow
cash by selling government bonds equal in value to 4.16667 units
of government bonds.
Investing in one unit of asset G means that we sell 3.4375 bags
of coffee and lend cash by
buying government bonds equal in value to 6.25 units of
government bonds. The net effect
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53
of investing in one unit of asset J and one unit of asset G is
the sale of 3.125 bags of coffee
and the purchase of 2.0833 bags of rice.
3.125 3.4375 = -0.3125 bags of coffee (C10.1)
-4.16667 + 6.25 = 2.0833 bags of rice (C10.2)
The cash value of the coffee is $2,968.75 and the cash value of
the rice is
0.3125*950 = 296.88 (C11.1)
2.0833*600 = 1,249.98 (C11.2)
The net investment is 953.10.
1,249.98 296.88 = 953.10 (C12)
Since the investment is risk-free, we would expect that the
return on the portfolio of
one unit of asset J and one unit of asset G is equal to the
risk-free rate of 4.92%.
1 + = 1,000 = 1.0492 (C13) 953.10
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54
LIST OF SYMBOLS V(i,j) Price for a unit of the asset in the jth
state of nature (or node) of the ith
period pU Probability for the up state of nature at the end of
year 1 pD Probability for the down state of nature at the end of
year 1 where pD = 1 -
pU P The set of probabilities for the two states of nature at
the end of year 1
where P = {pU, (1 - pU)} (i,j) Return on the asset in the jth
state of nature (or node) in the ith period rf Risk-free discount
rate EP{V(i,j:k)} Expectation of the values of the asset in the jth
and kth nodes of the ith
period with respect to the set of probabilities P. The
expectation is equal to sum of the values at the nodes multiplied
by the respective probabilities.
ZJ(i,j;C,B) Replicating portfolio Z to match the payoff
structure for asset J. The first
parameter refers to period i, the seco