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: ThamJx@yahoo.com.

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.

3

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.

16

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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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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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.

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

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

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

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

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.

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)

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)

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.

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)

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).

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.

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)

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)

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.

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

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.

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

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)

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

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

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.

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.

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

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.

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)

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)

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.

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,

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.

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%.

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.

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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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

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

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