II.MODERN PORTFOLIO THEORY THRU MARKOWITZ AND TOBIN

A. Mean - Variance Analysis Fundamentals

{Note: This entire section presupposes that the reader is broadly familiar with the elementary statistical concepts of mean, variance and covariance. It is not possible to understand modern portfolio theory without these concepts.}

1. What is an asset?

Prior to 1952 (the date of publication of Harry Markowitz's seminal work on Portfolio Selection) assets were divided into financial assets and real assets and were identified and described in great detail. Financial assets were stocks and bonds; real assets were land, housing, machinery, and so forth. Markowitz viewed things differently. Markowitz simplified our notion of assets by focusing upon the financial rewards that assets generate for their owners. Specifically, Markowitz defined an asset by simply specifying information about the financial returns that investors might anticipate from owning the asset. All of modern portfolio theory follows the Markowitz asset definition. Consider an asset, A:

Define asset A as providing the return of $3 when a fair coin flip yields heads and $ 1 when a fair coin flip yields tails:

RETURN MATRIX FOR ASSET AEVENT RETURN PROBABILITY OF RETURNHEADS $ 3 1/2TAILS $ 1 1/2

Using this simple asset, we can talk about some of the concepts that will prove to be crucial to an understanding of modern portfolio theory:

a. The Expected Return of an Asset

The concept of expected return involves the notion of the average return that an asset generates for its owner. In the simple example of Asset A, half the time the asset will return $ 3 and the other half of the time the asset returns $ 1. This means that, on average, Asset A will return $ 2. $ 2, then, is the expected return of Asset A.

Generally, the expected return is written symbolically as:

which can be read as the expected return of Asset A.

For a particular asset, we assume that we know the probability of any particular return occurring. We refer to the probability of a specific return occurring for Asset A as:

We can now show a simple formula that generates the expected return of Asset A:

In the present case:

E [RA ]=(R A1⋅P [RA1 ] )+(R A2

¿ P [R A2 ] )

= ($ 3 times 1/2) plus ($ 1 times 1/2) = $ 2

It is common practice to use the Greek symbol, µ, to represent mean, so that E(RA) º A

b. The Return Variance of an Asset

On average, Asset A, earns $ 2. A natural question arises as to how risky is this asset? Risky means roughly how much, typically, do the asset returns different from the expected return of the asset? Does the return vary much? Note that Asset A never returns $2. Instead half the time it returns $ 1 more than its average and half the time it returns $1 less than its average. It would seem that the return varies from its average by $ 1.

Formally, variance is defined in the following way:

σ2

RA=Var [ RA ]=E { (E [ RA ]−R A )2}

In words, the variance is the expected squared difference between the asset's return and its expected return. In our case this becomes:

σ2

RA=Var [ RA ]=E {( $2−RA )2}

which is:

σ2

RA=Var [ RA ]=∑ {($2−RA )2¿ P (RA ) }=($1⋅1/2 )+ ($1⋅1/2)=$1

The reason that we square the difference between the variance and its mean is to be sure that the difference between the return and the mean is always a positive number. This ensures that large positive variations are not simply cancelled out by large negative variances.

The standard deviation, which is used often instead of variance, is defined as the square root of the variance. (When Wall Streeters use the expression volatility, they measue volatility by calculating the standard deviation).

StddDev=σ=√σ2 = volatility

c. Means and Variances in the Continuous Case

Our examples, thus far, of means and variances assumed that there was some simple probability function over a finite number of alternative returns possibilities. It is often easier to assume that things are continuous. This means that an(uncountably) infinite number of different return values is possible, not just a finite (or countably infinite) number of different alternative values. The standard way of representing probabilities in this case is to assume a probability density function defined over a set of alternative values of return. Suppose the return can be any real number from to . Now, let:

f ( x )

be a function defined for all real numbers. Then assuming that f ( x ) is integrable (which means that the following expression is meaningful):

∫−∞

+∞

f ( x ) dx

Now, we define the mean x, where x is a random variable, as:

μx≡E ( x )=∫−∞

+∞

x⋅f ( x ) dx

Variance of x is defined as:

σ x2=Var ( x )=∫

−∞

+∞

( x−E ( x ))2 f ( x )dx

Note the following general facts about the continuous random variable version of these definitions:

1. f ( x ) is a probability density function. One might expect f ( x ) to be a continuous function. generally,

however, f ( x ) is not continuous -- discontinuities are permitted, but are restrained in certain ways so that the integration performed in our definitions are meaningful. All of the typical ways of performing integration by formulae still work in this world of possibly discontinuous probability density functions.

2. The probability interpretation in the continuous case is to ask about the probability that x lies in some particular interval, a¿ x¿ b. We say that the probability that x lies between a and b, where a is some number less than b is:

P [a≤x≤b ]=∫a

b

f (x ) dx

One implication of this interpretation is that the probability that x is any specific number is zero. For example the probability that x = 6 is:

P [x=6 ]=∫6

6

f (x ) dx=0

3. In order to fit with our intuitive notion of what probabilities are all about, we require that:

f ( x )≥0

for all x, and:

∫−∞

+∞

f ( x )=1

The first condition just avoids the awkward notion of negative probabilities. The second expression just says that x must achieve some number for sure (with probability one).

d. Covariances -- The Most Significant Concept in Modern Portfolio Theory

The most important concept in modern portfolio theory is the statistical notion of covariance. Imagine two random variables x and y. The basic question is how are these two random variables related? Does x tend to exceed its average at the same time y does? Is x larger than average generally when y is less than its average? Is there no apparent relationship between x and y? These questions are addressed by calculating the covariance of x and y:

Cov ( x,y )≡E [ (x−E ( x ) ) ( y−E ( y ) ) ]

If x and y move together, than the covariance will be a positive number. If x and y are inversely related (that is, x is larger than its average typically when y is less than its average), the covariance is a large negative number. If x and y are essentially unrelated, the covariance is approximately zero.

Note that if x = y, then

Cov ( x,x )=Var ( x )=σ x2

A useful normalization of covariance is the correlation coefficient, (x,y):

ρ( x,y )≡Cov (x,y )σ x⋅σ y

Correlation coefficient is confined to the range -1 to +1:

−1≤ ρ≤+1

(The above follows from Schwartz’s inequality). The interpretation of the correlation coefficient is very straightforward. Two assets that follow the same pattern have corr coeff’s equal to 1. Two assets that are polar opposites in their return patterns have corr coeff equal to -1. Unrelated assets have corr coeffs close to zero.

We have said that covariance is the most important concept in modern portfolio theory. We cannot, however, develop the fundamental role played by covariance without more details.

e. So, What Then Is An Asset ?

An asset is a probability distribution of returns -- nothing more, nothing less. We will need to assume that means and variances are known for all assets and that covariances are known for all pairs of assets.

It will be useful to divide all assets into two conceptual groups: assets that are risky versus assets that are riskless. How to define a riskless asset? For our purposes a riskless asset is an asset whose return variance is zero. All other assets, those whose return variances are positive are risky assets. (By definition, a variance cannot be negative).

2. Portfolio Analysisa. The Concept of a Portfolio

A portfolio is simply a collection of different assets. A portfolio is an asset that is created by combining other assets. (It will turn out that a portfolio is an asset and an asset is a portfolio with one asset). As we shall see, we can calculate means and variances for portfolios just as we can for ordinary individual assets.

i. The Simplest Portfolio -- A single asset.

Call the portfolio, Pl, consisting solely of asset X. Then:

The Mean of PlμPλ

≡E (P λ )=E ( X )≡μX is the mean return of X.

The Variance of Pl σ P λ

2 =σ X2 ¿ E (X−E ( X ) )2

is the variance of X.

What this simple portfolio shows is that the concept of a portfolio broadens the class of assets and includes all individual assets. It will turn out that everything said about assets will apply to portfolios and everything said about portfolios will apply to individual assets. Put another way, a collection of assets creates a new asset with its own mean and variance.

Imagine a portfolio that consists of equal amounts of asset X and asset Y:

ii. A More Complicated Portfolio -- Two assets equally weighted. Containing assets X and Y.

The mean of :

E (P1/2 (X ) ,1/2(Y ))=1/2E ( X )+1/2E (Y ) is the mean of

It would be very convenient if the variance of our equally weighted portfolio were as simple to calculate as the mean, but it is not. The variance of a portfolio of more than a single asset is complicated.

The variance of :

Begin with the definition of variance that was given earlier:

σ P2=E [ (P−E ( P ) )2 ]

The return to P is given by the following equation:

The mean of P:

μP≡E ( P )=1/2E ( X )+1/2E (Y )

Substituting these two expressions into the variance definition:

σ P2=E [ (1/2 ( X )+1/2 (Y )−1/2E ( X )−1/2E (Y ) )2 ]=E [ (1/2 ( X+Y )−1/2 {E ( X )−E (Y ) })2 ]

We now square everything inside the squared parenthesis (using the well known expansion of (a-b)2 into a2 + b2 - 2ab):

=E [1/4⋅(X2+Y 2+2XY )+1/4⋅(E2 ( X )+E2 (Y )+2E ( X ) E (Y ) )−1/2⋅( X+Y )⋅(E ( X )+E (Y ) ) ]

We can further simplify the above equation to:

=[1/4 · {E ( X2)+E (Y 2)+2E (XY ) }+1/4 · {E2 ( X )+E2 (Y )+2E ( X ) E (Y ) }−1/2 · {E2 ( X )+E2 (Y )+2E (XY ) }]

=1/4 (E ( X )2−E2 ( X ) )+1/4 (E (Y )2−E2 (Y ) )+1/2 (E ( X ) E (Y )−E ( XY ) )

=1/4 σ X2 +1/4 σY

2+1/2Cov (X,Y )

It is well to stop and note the important difference between how the portfolio variance is calculated compared to the much simpler formulation for the portfolio mean. The mean turned out to be simply the linear combination of the individual asset means. In this specific case of a portfolio with half of its value in one asset and the other half in another asset, the portfolio mean is half of the mean of the first asset added to half of the mean of the second asset. Nothing could be simpler. But, with variance, the story is different and this difference is a persistent theme of modern portfolio theory. Portfolio variance is not simply the linear combination of the individual stock variances.

iii. A Portfolio with Two Assets in Any Combination

We will need some more Greek letters. Let:

represent the fraction of the portfolio made up by asset 1 represent the fraction of the portfolio made up by asset 2

Clearly . We shall for the moment assume that there is no short selling or borrowing so that:

.

With this new notation, we can now calculate the return, (RP), of a portfolio that contains essentially arbitrary, non-negative amounts of two specific assets:

where the return from asset 1 is R1 and the return from asset 2 is R2.

The mean from this two asset portfolio is very simply calculated as the linear combination of the mean returns of the two assets:

μP≡E (RP )=α 1 E (R1 )+α2 E (R2 )

Variance, however, is different and more complicated. Note the following general formula for the variance of a portfolio of two assets:

where is the covariance between X1 and X2.

b. A Geometric View of Mean-Variance Analysis

It is often more useful to draw pictures rather than write out Greek symbols. In order to draw pictures of portfolios, we need to use a slightly different variable than variance. We need to work with the square root of the variance which, as we noted earlier, is called the standard deviation:

For any asset, associate with that asset its mean return and the standard deviation of return:X1→ (μ1 , σ1 )

X2→ (μ2 , σ 2)

X3→ (μ3 ,σ 3)

etc.Our next step is to consider a two dimensional diagram with one axis for the mean return and the other axis for the standard deviation of return.

In the diagram, we plot three assets: X1, X2, and X3, showing their means and standard deviations along the appropriate axes. Any asset can be depicted as a single point in the diagram.

What modern portfolio theory is really all about is how to combine different assets in a useful way to create a desirable portfolio. What we want to know now is how to combine assets X1, X2, and X3 with each other in order to create various new portfolios. Consider only assets X1 and X2 and think of the various ways of combining X1 and X2 in order to create different portfolios:

What portfolios can be created by combinations of assets X1 and X2?

First, let us clear away the question of units. How much of the asset X1 is described by the single point in the above diagram? Implicitly, we have assumed some dollar value of total assets. The point X1 in the diagram above assumes that all of these total assets are placed in asset X1. Similarly, the point X2 describes the mean and standard deviation that applies when all of the total assets are placed in asset X2. Algebraically, if W represents total dollar wealth, and P1 and P2 represent prices of the two assets, then the asset descriptions in the above diagram satisfy:

Second, given the amount of dollar wealth, W, what portfolios are affordable? Recall our Greek symbols, 1 and 2. These symbols will represent the fraction of total dollar wealth invested in asset 1 and asset 2, respectively. This means that the following choices of 's are permissable for affordable portfolios:

We will assume that we are interested in portfolios that exactly exhaust our total wealth so that the weak inequality of the previous equation is replaced by the strict equality in the equation below (this just means that no useful purpose is served by putting money under the mattress or burying it in the backyard):

(We continue to assume that 1+2 = 1 and that neither can be negative in sign.)

Now, back to our main question of interest: Where, in the mean-standard deviation diagram, can we locate the portfolios that can be created by combinations of X1 and X2?

To answer this question, we begin by calculating the return from any portfolio that consists solely of asset 1 and asset 2 in the combination implied by the value of the 's:

Y represents the return from the portfolio created by the combination of the two assets.

The expected return of Y is then:

μY≡E (Y )=E (α1 R1+α 2 R2)=α 1 E ( R1)+α2 E (R2)=α 1 μ1+α 2 μ2

or:

The portfolio mean is the linear combination of the means of the individual assets. This will always be true and this fact greatly simplifies our analysis. Suppose that 1 = 2 = ½.

The mean of the half/half portfolio is designated along the vertical axis as . , as indicated in the diagram. If only the following equation were true:

or, perhaps the following might be true:

Alas and alack, neither of the above equations are generally true. Neither the standard deviation or variance of a portfolio can be constructed as a simple linear combination of the standard deviations or variances of the underlying assets that comprise the portfolio, except under very special circumstances.

The variance of a two asset portfolio is:

Recall the definition of the correlation coefficient (which equals the covariance divided by the product of the standard deviations):

If we now substitute for the covariance in our equation for variance, we will get:

c. The Special Case When the Correlation Coefficient Equals One

If , a remarkable fact emerges:

σY2=α

12 σ12+α

22 σ22+2 α1 α 2 σ1 σ2=(α1 σ1+α 2 σ2)

2

Taking the square roots of both sides of the above equation gives us:

We obtain the remarkable result that if the correlation coefficient equals one, then the standard deviation is, indeed, the linear combination of the standard deviations of the two assets that comprise the portfolio. This means that all the portfolios that can be constructed from two perfectly correlated assets lie on the straight line between the two assets in our mean-standard deviation diagram:

What is the interpretation of ? It is helpful to think about the value of the correlation coefficient when both assets are identical. In that case:

So that the correlation coefficient turns out to equal one in the special case where the two assets are identical. Indeed, if the correlation coefficient is one, we are going to be hard put to tell the difference between the two assets (although they can, in principle, have different return distributions). Assets that give us correlation coefficients equal to one can be thought of as assets that are essentially identical.

Thus when , it must be the case that:

and:

d. The Special Case When One of The Two Assets in a Portfolio is a Riskless Asset

Suppose that asset 1 is riskless, meaning that 1 = 0 = 12. Revisit the formula for the variance of a portfolio consisting of only two assets (after substituting in the formula for the correlation coefficient):

If 1 = 0 = 12, then:

taking square roots:

This means that any portfolio that consists of assets 1 and 2 must have a standard deviation that is the standard deviation of the risky asset multiplied by the fraction of the portfolio invested in the risky asset. In terms of the mean-standard deviation diagram, this means that all the portfolios in question must lie on the straight line connecting asset X1 and asset X2.

e. A Portfolio With Many Assets

Suppose we have n different assets, where n is some positive integer. Suppose that assets X1, X2, X3,....Xn have the random variable returns R1, R2, R3,.......Rn. We assume again some total dollar amount of wealth, W, so that each of our random variables are standardized to represent the return that would obtain if the entire amount of wealth, W, was invested in that specific asset. In order to create a portfolio, we need to specify the amount of each asset that will be in the portfolio -- we need some 's: 1,2,3,....,n.

The return, Y, of the portfolio is given by:

The mean, Y, of the portfolio is given by:

The variance, Y2, of the portfolio is given by:

The variance, Y2, can be simplified to:

σY2=∑i=1

n∑ j=1

n(α i α j σ i , j )

Note that in the above formulation, it should be clear that i,i º i2. This means that variance can be split into two summations, one part consisting solely of variances and the other consisting of covariances (that are not variances):

which, given i,i º i2, means that:

This final equation for the variance of any arbitrary portfolio is a very important equation. It shows that the overall portfolio variance is determined by a weighted sum of the individual variances and a weighted sum of the various covariances. The interesting question is which of these summations is more important -- the weighted sum of variances or the weighted sum of covariances? The answer is remarkable and surprising.

f. Large Portfolios with Equal Asset Weightings

What really determines portfolio variance? Ultimately the answer to that question lies at the heart of modern portfolio theory. A partial answer is given here in the form of a limit exercise. Suppose we consider a portfolio that has equal weights for each asset. This means that we assume that 1 = 2 = 3 = ...... = n. The common value of all the 's will be 1/n. Thus we are assuming that:

for all i.

We now substitute into our previous equation for variance the common value of 1/n for the 's:

σY2=∑i=1

n ( 1n )

2σ

i2+ ∑i=1

n ∑j=1, i≠ j

n (1n )( 1n )σ i , j

This equation can be rewritten (by extracting constants from inside the summations):

σY2=

1n

∑i=1

nσ

i2

n+ ∑i=1

n ∑ j=1, i≠ j

n ( 1n )( 1

n )σ i , j

Examine closely the first term in the above equation:

This is essentially 1/n multiplied times the average variance. As n gets larger and larger what happens to this expression? The answer is it gets smaller and smaller. If n is large enough this entire expression gets as close as anyone might like to zero. In the limit, individual asset variances do not matter to the overall computation of portfolio variance under the assumption that assets are equally weighted in the portfolio and we make the portfolio have enough stocks.

Mathematically:

limn→∞

( 1n

∑i=1

nσ

i2

n )=0

What, then determines portfolio variance, if individual asset variances become irrelevant in the limit? Interestingly the answer is approximately that overall portfolio variance, in the limit, is determined by the average covariances of the assets comprising the portfolio.

Consider the second term in the prior equation for variance:

∑i=1

n ∑j=1, i≠ j

n ( 1n )( 1

n )σ i , j

This can be rewritten:

( 1n )∑i=1

n ∑ j=1 , i≠ j

n (1n )σ i , j

Multiply this expression by (n-1)/(n-1), which should not change its value:

( n−1n )∑i=1

n∑ j=1, i≠ j

n ( 1(n−1 )n )σ i , j

It may not be completely obvious, but the bracketed expression is the average of all the covariances, so that:

∑i=1

n∑j=1, i≠ j

n ( 1(n−1 )n )σ i , j=σ i , j

where is the average covariance. (Notice that there are n timesn-1   i,j's in the summations).

If we now recapitulate the results of this section, we find that as the portfolio gets larger and larger (in the sense of adding additional stocks to the portfolio), the total portfolio variance will depend only upon the average covariance between the various stocks.

Taking limits of the variance sums and the covariances sums separately:

limn→∞

σY2=lim

n→∞( 1n

∑i=1

nσ

i2

n )+ limn→∞

(∑i=1

n∑ j=1 ,i≠ j

nσ i , j)

We have already shown that the limit of the expression summing variances is zero and that the limit of the expression summing covariances is equal to the average covariance:

The mathematics, then, gives us the conclusion that the overall portfolio variance, in the limit, is equal to the average of the covariances. The variance of individual stocks become irrelevant as more stocks are added to the portfolio.

We said earlier that covariance is the most important concept in modern portfolio theory. The analysis in this section gives a preliminary indication why this is true.

3. Which Portfolios Are Better Than Others?

It is natural to assume that investors like higher earnings but do not like higher risk. Ultimately, that creates an important trade off for every investor: how to generate higher return without incurring higher risk? We need to be very precise about what we mean when we say that investors prefer higher returns, even though the meaning of that statement may appear obvious.

Before we wander off into utility theory, it might be instructive to consider a few simple examples involving games of chance.

a. Flipping Coins

Return to our example from Section IV-B, asset A:

Define asset A as providing the return of $3 when a fair coin flip yields heads and $ 1 when a fair coin flip yields tails:

RETURN MATRIX FOR ASSET AEVENT RETURN PROBABILITY OF RETURNHEADS $ 3TAILS $ 1

A natural question to ask is what is asset A worth? One of the best ways to tackle this question is think of asset A as a game of chance that you have an opportunity to play. The game has a price that you must pay each time you play the game. What is the right price for the game represented by asset A? On average, we expect the player of the game to earn $ 2. Imagine that you could play this game at a fixed price per game as often as you might like.

If it costs $ 1 to play the game, would you play? Of course, because you could never lose. Either you win $ 1 or $ 3 depending upon the outcome of the flip of the coin.

Suppose it costs $ 1.50 to play the game, would you still play? The answer is yes, again, for most people. Even though, it is now possible that you will lose by flipping tails, you will still expect to win $ 3.00 for a net gain of $ 1.50 each time your coin flip turns out to be heads. The half of the time you expect to lose by flipping tails will cost you a net of $ .50 (you pay $ 1.50 to play when you win only $ 1.00). So, most people will willingly play this game, since on average they make $ .50 times however many times they play the game.

Suppose it costs $ 3.00 to play the game, would you play? The answer is no. Since, now you can never win. The $ 3 you earn from a heads flip generates no net gain above the $ 3 cost to play; the $ 1 earnings from a tails flip implies a $ 2 net loss.

This kind of reasoning leads to the conclusion that the price of this game, call it PG will satisfy the following condition:

Indeed the price of the game can not be anything different from $ 2.00. Below $ 2.00 there will be excess demand to play the game; above $ 2.00 new business will be set up offering the game to any customers willing to play for more than $ 2.00 -- hence, excess supply for the game.

We get the interesting result that the price of the game equals its expected value. This suggests that expected value is what investors ought to be interested in. When choosing among portfolios, then, perhaps investors should choose the portfolio that has the maximum expected value among all the portfolios that they can afford. How good is this as a guide to investment behavior?

II. The Bernoulli Paradox (or St. Petersburg Paradox)

Imagine the following game. You are permitted to flip a (fair) coin until you sucessfully flip heads for the first time. Upon the first heads flip the game is over and you are awarded:

( $ 2 )N

where N is the first heads flip.

If heads occurs on the first flip, then you win $2. If the first heads occurs on the second flip, you win $4. And so on.

First Coin Flip That Is A Heads Earnings

1 $ 22 $ 43 $ 8- -- -N $ 2N

To go any further, we must ask what is the likelihood of flipping a heads on the first, second, third, and so forth flips of the coin.

The odds of flipping a heads on the first flip is clearly 1/2.

P[H] = 1/2

What about the odds of winning on the second flip? That would require that the first flip be tails followed by a heads on the second flip. This should happen about 1/4 of the time, since 1/2 the time tails will occur on the first flip and half of the time the second flip will yield a heads. Hence, the result of 1/4.

P[TH] = 1/4

Success on the third flip will have a probability of 1/8.

In general the probability of success on the Nth flip (defined as the first time a heads result occurs) will be:

P [N ]=( 12 )

N= 1

2N

We now have enough information to define the expected value of this game. If RN is the return if heads first occurs on the Nth flip and P[RN] is the probability of the first heads flip occurring on the Nth flip, then the expected value of this game is given by:

E [Bernoulli Game ]=∑N=1

∞(RN⋅P [ N ] )=∑N=1

∞ (2N⋅12N )=∑N=1

∞1=∞

The value of the game is infinite!

We suggested in the last section, that most people are willing to play any game that has an expected value higher than the price that they are required to pay to play the game. If that were true for the Bernoulli game then people would be willing to pay any arbitrarily large amount of money to play the Bernoulli game. This result is referred to as the Bernoulli Paradox, sometimes called the St. Petersburg's Paradox.

Why is the Bernoulli Paradox important? It would be convenient if we could rank assets according to their expected value. For example, we might wish to say that an investor prefers asset i to asset j if and only if:

This is what is meant by ranking assets according to expected value. The Bernoulli Paradox shows that something is awry with this kind of ranking. The Bernoulli game has an infinite value so that playing that game should be preferred to accepting any finite dollar amount of money. Not many people would turn away arbitrarily large amounts of money to play the Bernoulli game -- hence the paradox. What the paradox means is that ranking assets according to their expected value is not an acceptable solution to the problem of investor preferences.

III. The sure win gambling strategy

Imagine that you are given the opportunity to place a bet on the flip of a fair coin. A fair coin is defined as a coin that has exactly 50 percent chance of flipping a heads and a 50 percent chance of flipping a tails.

It has long been known that a particular betting strategy seems to be a sure thing. Suppose you bet $ 100 on the first coin flip that the coin will flip heads.

If the first flip is heads you will win $ 100.

Imagine that the first flip is tails. You lose $ 100. Then double your bet to $ 200 and bet that the second flip will turn up heads. If the second flip is heads, you will win $ 200. Since you lost $ 100 on the first flip, you will win a net of $ 100.

Imagine that the first two flips are tails. Then you lose $ 100 on the first flip and $ 200 on the second flip. Therefore, bet $ 400 on the third flip, again betting that the coin will flip heads on the third flip. If the flip turns up heads, you win $ 400 minus the losses of $ 100 on the first flip and $ 200 on the second flip, means, once again, you end up winning a net of $ 100.

Continue this process until you eventually flip heads.

With probability one, you will eventually flip a heads, so that you must with certainty win $ 100 eventually. This result can only fail to materialize if you never, ever flip a heads no matter how many coin flips you engage in. This latter pattern has a probability of zero, so with probability one, you will make $ 100 by eventually flipping heads.

Notice the similarity of this argument with the Bernoulli Paradox discussed earlier.

Economists have wrestled with the above gambling conundrum because it seems to suggest that there is always a free lunch available in gambling. We will note later that this gambling strategy violates the condition known as the “no arbitrage” assumption that underlies virtually all of modern finance theory.

The way economists get around this gambling sure thing is to postulate that there is a limit to how much an individual can borrow (or lose, since we assume every individual begins with a finite amount of wealth before they begin to gamble). If there is a limit to how much you can lose, then after enough flips of tails you can’t play anymore. This artificial termination of the gambling experiment, which is probably realistic, is not aesthetically very pleasing but is necessary to avoid this awkward counterexample to the usual notion that you can’t get something for nothing.

IV. The Concept of Preference Orderings

In order to develop a theory of investor activity, we will need to think seriously about the basis upon which investors choose different portfolios. In other words, what do people want? This is a vast subject. In ordinary economics this is the subject matter of utility theory or preference theory.

In elementary economics, we usually assume that each individual has a preference ordering over all possible consumption bundles. Let us pause for a moment and tell this story in two dimensions, meaning two

consumption goods, x1 and x2. It should prove useful to think of this problem in the form a diagram with x1 on one axis and x2 on the other axis. With this interpretation, any consumption bundles, y or y', are just points in the positive regions of our diagram as indicated below:

Now consider all points like y or y'. What is this set? It is the set of all pairs {y1,y2} where y1 and y2 are not permitted to be negative numbers. Notationally, this is often stated in the following way:

{ y=( y1 ,y2) : y1≥0, y2≥0 }The above notation can be read literally as "the set of all y, where y is a pair of numbers both of which are not permitted to be negative numbers." (Implicitly, we will always assume that we are talking about real numbers in the mathematical sense.)

With this definition of the set of consumption bundles, let us now introduce the notion of a preference ordering and the associated notation. Consider:

The above statement can be read y' is at least as good as y". If we replace this weak inequality with a strong inequality, we could consider:

The above statement can be read y' is better than y". This is sometimes called strict preference, in the sense that y' is strictly preferred to y".

With this notation we can now introduce the concept of a preference ordering over a set of alternatives. Let Y stand for the set of all consumption bundles as introduced above. Then a preference ordering, ³, on Y will always satisfy the following two conditions:

1. For any two elements, y', y" in Y either y'³ y" or y"³ y'

2. For any three elements y, y', y" that satisfy:

y ³ y' and y' ³ y"

it must be the case that:

y ³ y"

The first condition is called completeness. It is called completeness because the condition assures that you can always order any two members of Y. You are never in a situation that you simply cannot compare two different consumption bundles. The second condition is called transitivity. Transitivity is an appealing condition intuitively. Any time we speak of a preference ordering, both completeness and transitivity will be assumed.

When we do elementary economics we make two other assumptions normally. One assumption is a continuity assumption, which says roughly that making small changes in a consumption bundle keeps that bundle in approximately the same place in the preference ordering. It is this assumption that will permit us later to talk of utility functions. But, more of that later. For now, the most interesting assumption that elementary economics makes about consumption bundles is the idea that preference orderings are convex.

i. Convex Preferences Convex preferences are best explained by returning to our diagram with commodities x1 and x2. Imagine that you begin with a bundle, y, and you ask the question: where are all the bundles that are equivalent to y? A bundle y' would be equivalent to y if y ³ y' and y' ³ y simultaneously. This set is represented by an indifference curve in elementary economics:

The curve in the above diagram, pointed to and identified, has a convex shape, which is why such a preference ordering is called convex. If you pick any two points on the curve and connect the points with a straight line, then any point on the straight line (other than the endpoints) will represent a consumption bundle that will be strictly preferred to anything on the indifference curve. We demonstrate this in the diagram below:

The point z in the above diagram lies on the straight line connecting y and y'. Therefore, if preferences are convex, it must be the case that z > y and that z > y'. Mathematically, we say that a preference order is convex if and only if:

Whenever y » y' (meaning that y is equivalent to y')

if z = 1y + 2y' where 1 + 2 = 1 with 1, 2 ³ 0.

then z > y and z > y' (meaning z is strictly preferred to both y and y')

The rest of the elementary theory of consumer behavior consists of showing what a consumer will buy and how that varies as prices vary, assuming the consumer has a preference ordering that is convex and continuous. Here's a bit of nostalgia for refugees from elementary courses in microeconomics:

This bit of nostalgia shows what happens when a utility maximizing consumer finds that the price of x2 has fallen. Before the price falls, the investor would purchase consumption bundle A. After the price drop, the consumer purchases consumption bundle B. Note that the amount of x2 purchased increased after the price drop. As most refugees from elementary microeconomics will recall, this need not always be the case in the world of convex indifference curves.

ii. Why Convex Preferences? -- An Aside

It is not clear why convexity of preferences occupies so much interest in introductory courses in Economics, since nothing is really gained by that assumption and some silly results are permitted. For example, it is possible that demand curves might have the wrong slope if preferences are convex. This tantalizing possibility is explored at length in elementary economics courses. Why this is of any interest is something of a mystery. Who cares if some unusual circumstances might cause the demand for a good to increase after a price increase because consumers feel poorer and now prefer the inferior good. This situation, if it ever occurs in the real world, would clearly be pathological and should scarcely occupy anyone's interest. A better approach to elementary consumer theory would be to assume that preference orderings always have the property that more consumption is preferred to less and then move quickly to demand functions that represent such orderings. Just assume that demand functions are downward sloping, with regard to consumption goods and proceed. Nothing much is gained by making a big deal about "diminishing marginal rate of substitution" in elementary consumer theory, since that assumption (which is equivalent to convexity) does not produce useful implications. (The elaborate discussions of income and substitution effects have no applications in elementary consumer theory other than to demonstrate the curiosum of an upward sloping demand function).

V. Utility Functions

We will find it necessary to use utility functions in order to develop the arguments of modern portfolio theory. What is a utility function? It is nothing more that a numerical representation of a preference ordering. Suppose we have some set of things, Y. Y can be consumption bundles; Y can be assets; Y can be a set of anything(s) at all. Now let ³ be a preference ordering on Y.

A utility function is a function that associates with every member of Y a real number and that has the following property:

If y,y' are members of Y and y ³ y'

then U(y) ³ U(y')

All a utility function does is assign a numerical ordering to the set of alternatives that agrees with some particular preference ordering. In that sense, a utility function always represents a preference ordering.

Note that the definition of an indifference curve, mathematically, is the set of all members of Y such that U(y) equals some specific number. In other words, the utility is constant along an indifference curve.

VI. Utility Functions For Orderings of Assets

If the underlying set, Y, is a set of assets, then we can immediately broaden the set to include all portfolios that can be created out of the underlying set of assets. We now pose the question of what kind of preference orderings to consider for investors. Let us move directly to the utility function U(y) that represents the preference ordering for any particular investor.

What characteristics of the asset, y, should interest the typical investor.

To simplify our presentation, we will restrict our attention to a special set of investors. We shall suppose that all investors have utility functions that depend upon only two variables: mean and variance. Thus for any y in our set Y, we assume that each investor has a utility function: U(y) that depends only upon y and

2y. We assume that U(y, 2y) likes increases in y and dislikes increases in 2y.

Mathematically:

Assume that there is a utility function, U (μ y , σ

y2) that depends solely upon the mean and variance of portfolios. Assume that this function is sufficiently differentiable for whatever purposes it needs to be. Then further assume the following two conditions:

1.

∂U (μ y , σy2 )

∂ μy>0

2.

∂U (μ y , σy2 )

∂σy2

<0

The first condition says simply that the investor, other things equal, prefers to have assets with higher mean returns as opposed to lower mean returns -- an eminently sensible condition. This condition is similar to the assumption in consumer theory that more consumption is preferred to less.

The second condition lies at the heart of finance theory. This assumption is known as risk aversion. An investor that is risk averse always prefers, other things equal, less return variance to more return variance. This assumption is clearly false for some investors. There are some investors (call them gamblers, speculators, whatever) who clearly prefer return variance -- indeed there are some investors who seem to thrive on it. We will not discuss such investors any further. We will only be interested in the activities of risk averse investors. The reason is that investors that prefer risk constitute a minor and insignificant part of the investing community and are pathological (in the sense that upward sloping demand curves in elementary consumer theory are pathological). In everything that follows, we shall assume that all investors have a utility function that satisfies the two conditions given above and in particular satisfies the requirement of risk aversion.

B. The Road To The Sharp-Lintner-Mossin Model

1. Prior to 1952

The earliest views of portfolio management had two distinct strands of thought. First was the idea that with a little forethought and effort it was possible to pick out securities that would perform better than other securities. Graham-Dodd analysis began the notion of fundamental research aimed at identifying the underlying real worth of a company and its securities. Graham and Dodd's work first appeared in the 1930's and is considered the pioneering work that lies behind the very large research efforts of modern investment banking firms who provide views to their customers as to which stocks and bonds (and other securities) to own and which securities to avoid owning. Technical research, begun even earlier by the man who created the Dow Jones Industrial Average and published the Wall Street Journal -- Charles Dow, continues to dominate the conversation of most professional investors. Technical research focuses on the ability of an individual, with effort, to accurately forecast the future prices of securities, based upon the past history of security prices (and other things).

The second strand of thought was the idea that conservative investors should purchase conservative securities. In practice, retired persons were encouraged to own high grade fixed income investments and utility stocks, whereas younger (presumably, less conservative investors), were urged to purchase common

stocks, considered much riskier than high grade bonds or utility stocks. The corollary was that aggressive investors should purchase more risky securities. Both of these ideas were to come under attack by academic economists during the decade that followed 1952. The beginning of modern portfolio theory dates to the publication in 1952 in the Journal of Finance of an article by Harry Markowitz entitled "Portfolio Selection." Markowitz began an analytic approach to financial markets that has revolutionized the way that academics and financial market participants view investments.

2. The Markowitz Theory of Portfolio Selection

Harry Markowitz addressed the question: What should a rational individual investor do? Specifically, what portfolio should a rational investor purchase given his preferences and wealth and the portfolios that he can afford. The key insight, as we suggested earlier, was that assets were defined by their means and variances (and their covariances with other assets). Markowitz was concerned with the solution to the following problem:

The Markowitz Problem:

Suppose their are a finite number of assets: X1, X2, X3,......Xn with corresponding R1, R2, R3,....Rn random variable returns and that the mean returns and return variances of these assets and their covariances with each other are known, finite numbers.

Imagine an investor who is risk averse and prefers higher mean return to lower mean return.

What portfolio should that investor select?

This problem is very similar to the problem in consumer theory that was reviewed above when a rational consumer asks: what should I buy given my wealth and the available commodity bundles that I can afford?

a. The Construction of the Feasible Set

The first question Markowitz asked and answered was the question as to which portfolios are affordable. We begin by assuming some total dollar value of wealth, W, and then ask what portfolios can I afford? We need prices. Let P1, P2, P3...Pn be the n prices of the n assets, X1, X2, X3, ......Xn. We will need our 's again. Let 1, 2, 3, ....,n be non-negative fractions that sum to 1. They are the fractions of our wealth, W, that we invest in each asset. The affordable portfolios become the set of all {1, 2, 3, ....n} that satisfy:

1. For each i,

2.

3. ∑i=1

n(αi Pi X i )≡α 1 P1 X1+α2 P2 X 2+. .. .. .+α n Pn X n≤W

We will restrict our attention to strict equality for equation 3 above to avoid the money under the mattress phenomenon. We rewrite 3 to be 3':

3'. ∑i=1

n(αi Pi X i )≡α 1 P1 X1+α2 P2 X 2+. .. .. .+α n Pn X n=W

So much for our definitions. Any set of non-negative 's that satisfy 1, 2, and 3' will define a portfolio that is affordable. We will say feasible to mean affordable. Therefore, we now have defined the feasible set -- at least mathematically.

i. Mean - Standard Deviation Diagrams

This story is best told with our two-dimensional diagrams. Interestingly, all of the assets and portfolios that we are interested in can be depicted in two dimensions. The reason for this remarkable fact is that the two dimensions are mean and standard deviation and therefore each asset becomes a single point in our two dimensional diagram that we employed so often in the previous section.

All of the points in the above diagram represent individual assets. Since assets include portfolios, it is possible that some of those points are portfolios consisting of more than one asset. For the moment let us focus on one lone asset. We select and consider asset y in the diagram below:

We can generate other assets in , space simply by considering y all by itself. Why? Because of an assumption that economists often call free disposal. Suppose our asset y in the above diagram has an expected return of 8 percent with a standard deviation of .6 (as indicated in the diagram). Aren't there some other assets that are achievable? Yes. Suppose we consider owning asset y and just throwing away some of what we earn each period. The new return will be lower than 8 percent because we have thrown away some of the generated return each period. Similarly, we could arbitrarily increase the standard deviation of return by postponing receipt of income in certain periods and exaggerating the deviations of returns.

What this means is that by assuming free disposal, we generate all of the assets to the right and below the point y:

All the shaded area to the right and below y are achievable assets merely by selecting y and then throwing away or selectively postponing the returns generated from owning y. This little demonstration obviously applies to each asset. The feasible set, then, will contain all the assets below and to the right of any asset:

To construct the feasible set, we need only consider the assets and portfolios in the upper left corner.

The above demonstration shows that you can focus your attention upon the assets that are in the northwest corner of the diagram:

Points A, B, C, and D above represent assets (portfolios) that are M-efficient (the M stands for Markowitz).

M-Efficient Portfolio -- a portfolio that has the lowest variance (standard deviation) among all feasible portfolios with the same mean return.

The M-Efficient portfolios all lie along the northwest corner of the diagram. What we now intend to demonstrate is that this boundary will be a curve consisting of points with the curvature shown in the preceding diagram.

ii. Efficient Portfolios -- The Boundary of the Efficient Set

We are interested in what the northwest boundary of the efficient set looks like. It was asserted in the previous section that the boundary of the feasible set looked something like that diagram above. We now set out to show that the feasible set is closed and convex. Closed and convex means two things: closed means the boundary is a curve such as we have drawn instead of isolated points and that the curve consists of feasible portfolios. Convex means that the boundary has no segments that look like the curve between B and C in the diagram below:

Can the feasible set have a boundary that looks like the convex curve connecting portfolios B and C in the above diagram? The answer is no and here is why:

Consider two possible situations between B and C.

First, suppose that ρB,C=1 . We know from earlier, that when the correlation coefficient between two assets equals one, then all portfolios that can be created solely out of positive amounts of the two assets will lie on the straight line that connects the two assets. This means that, if we draw a straight line connecting points B and C, any portfolio that consists of some of each of B and C (we are assuming a given wealth here) must lie on the (dotted) straight line indicated in the previous diagram.

We conclude that if ρB,C=1 , then a curved segment like that between B and C will not occur since there will always be portfolios that are feasible that lie on the straight line joining B and C.

This leaves only one other situation. Suppose that

Recall the equation for the variance of a portfolio with two assets with substituted in:

Suppose everything about this portfolio is the same except for the value of . If now is less than it was before, then the variance and, hence, the standard deviation will be less than it was before. Lowering has effect of lowering the standard deviation of the portfolio without altering anything else. This means that the portfolios generated by assets B and C will be located in the same place as those for which , except that the standard deviation will be less than is indicated by the points along that line. They must lie to the left of the dotted line between B and C! This fact means that segments connecting B and C are either

straight lines if ρB,C=1 . Otherwise, such segments cannot have the curvature pictured above. So long as -1<ρB,C<1 , the curvature will be as pictured below between any two assets (or portfolios):

-

VII. The Assumption of Risk Aversion

Markowitz was interested in the investment behavior of a risk averse individual. His assumption was that higher mean returns were preferred to lower mean returns and that lower variance of return was preferred to higher variance of return. This means that we assume that a typical investor has a utility function, U(P) over all possible portfolios that can be formed from existing assets and that the utility function concerns itself only with the mean and variance. It is easier to work with standard deviation (the square root of variance) and we will do so. Thus the utility function of a typical investor:

U ( P )=U (μP , σ P)

satisfies:

1.

∂U (μP , σP )∂ μP

>0

2.

∂U (μP , σP )∂σ p

<0

We can now draw indifference curves in mean - standard deviation diagrams. The assumptions above assure us that such indifference curves must have a positive slope, meaning that if one has to take on more standard deviation, one must be compensated with higher returns to be equally well off.

This is about all we can conclude from the assumption of risk aversion. Indifference curves slope upward (from left to right) in - diagrams.

VIII. Indifference Curves in Mean - Standard Deviation Diagrams

It is a long standing tradition in economics to use the notion of indifference curves as a short hand and picturesque way of representing how an individual might feel about various possibilities. In elementary consumer theory we use indifference curves to exhibit the behavior of utility maximizing consumers and households. We will find this same approach useful in analyzing propositions in modern portfolio theory.

What is an indifference curve? An indifference curve is a set of different possible things that all are viewed as equally valuable. In consumer theory, the things are consumption bundles. In finance theory the things will turn out to be portfolios. A single indifference curve, as we will use it, will connect, along the curve, all portfolios that are equally desirable to the investor (these are not, in any sense, identical portfolios). Let's consider a diagram that contains several indifference curves:

Consider the indifference curve labeled U1 in the above diagram. If you pick any arbitrary point along that curve, the point will be some particular portfolio and some specific values for and . The concept of the curve is that if you consider another portfolio with a higher value of , then more must be forthcoming to make the investor equally well off. The portfolios along the curve U1 are obviously not identical, but they are equivalent to the investor who willingly trades off increases in standard deviation in order to obtain higher expected returns. The curve simply exhibits the implied trade off from the point of view of the investor. Note that U1 > U2 > U3 > U4. This means the investor gets happier as he moves towards indifference curves in the northwest direction.

IX. Investor Equilibrium -- The Optimum Investor Choice

What is the best portfolio for an investor? The answer involves arranging for the investor to reach the highest possible indifference curve that can be reached by choosing among all feasible portfolios. It is easy to combine the two previous diagrams and show what happens. The investor's best choice is portfolio E. Any other portfolio involves a lower level of utility (intersects an indifference curve that is southeast of the indifference curve drawn through E). The best indifference curve that the investor reaches is labeled Uopt.

We have drawn the boundary of the set of feasible portfolios in a very convenient way. So convenient that only one portfolio stands out as optimal for the investor, portfolio E. Risk aversion alone will not guarantee the uniqueness of the optimizing portfolio. We need to assume that the indifference curves are convex in order to get this unique optimal solution to the investor's choice problem. (Note that the feasible set is strictly concave as long as no two assets have a correlation coefficient equal to one). It is possible, then, under merely the assumption of risk aversion that there is more than one optimal portfolio. All such optimal portfolios will yield the same identical level of utility. For example, view the diagram below, which shows three optimal portfolios E1, E2, and E3 all yielding the level of utility corresponding to indifference curve Uopt.

X. So? What is Markowitz's Contribution?

The results of the above diagram seem pretty empty -- another famous tangency. Elementary economics seems to have an unending supply of tangencies for students to digest and here, apparently, is another one. The main contribution of Markowitz is to pose the problem of investor choice when the assets are essentially probability distributions of returns. If Markowitz's main result, captured in the two preceding diagrams, seems pretty barren, it is. It remained for Tobin to sharpen Markowitz's result to achieve a startling conclusion.

Markowitz was the first to formalize the idea that investment involves a trade off between risk and reward with risk represented by variance (standard deviation) of returns and reward represented by expected (mean) returns. Markowitz defined (somewhat unfortunately) the notion of an efficient portfolio (we call this M-efficient) as a portfolio that has the least return variance among all portfolios with the same mean return. An investor will find his best choice among these efficient portfolios.

That's about it.

3. Tobin's Theorem

James Tobin took Markowitz's analysis one step further and reached a remarkable conclusion. Tobin's Theorem is the following: If at least one asset is riskless (has a zero variance), then all risk averse investors will find that their best portfolio consists of at most two assets: the risk free asset and a specific efficient portfolio. This is a staggering conclusion and it takes some considerable thought to ponder its implications and to understand its meaning. Remember that prior to Tobin and Markowitz, the common notion was that investors that like risk should own risky assets and investors that don't like risk should not own risky assets. That is not what Tobin's Theorem says. Consider Tobin's Theorem mathematically:

Tobin's Theorem:

Assume: (1.) a set of assets with known means, variances and pairwise covariances.

(2.) that investors are risk averse(3.) that at least one asset is riskless (has a zero variance)(4.) All investor can borrow or lend at the riskless rate of interest

Conclusion: Every investor will hold a portfolio that is defined by the following equation:

where 1, 2 are non-negative numbers that sum to one and where X1 is the riskless asset (with the highest expected return among all such riskless assets) and X2 is a unique portfolio. Only the 's vary from investor to investor. The X1 and X2 portfolios are the same for all!

a. A Diagrammatic Theorem of Tobin’s Theorem

The introduction of a riskless asset into Markowitz's framework changes the feasible set of portfolios. This is the central analytic change introduced by Tobin. Obviously all portfolios that were feasible before remain so. New portfolios, however, are now possible that combine assets with the new riskless asset. All such newly created assets will lie on a straight line between Rf and the asset to be combined with it. In particular, a portfolio E* can be identified by finding the tangent to the old boundary of the feasible set. Any portfolio on the line from Rf thru E* is now feasible. Indeed this line becomes the new boundary of the feasible set:

The above diagram establishes Tobin's Theorem (so long as investors are risk averse). Even without drawing indifference curves, it is obvious that an optimal investment portfolio must lie on the boundary of the feasible set, which is now the straight line that passes from Rf thru E*. Portfolios that lie on that line must satisfy, for some pair of 's that sum to one:

where X1 is the risk free asset with expected return equal to Rf and X2 is the portfolio E*. The usual picture of the solution to the investor choice problem is the following:

The optimal portfolio, Popt, is the point of tangency between the indifference curve Uopt and the line from Rf through E*. Uopt is the farthest northwest attainable indifference curve that can be reached by any portfolio bounded by the straight line through E*. It is worth noting that nothing we have said guarantees that there is a unique point like Popt. There may be several portfolios that are optimal, but they are all worth the same to the investor in the sense that they all will achieve the identical indifference curve, like Uopt in our diagram.

[It is worth noting that, even if there is no riskless asset, the two portfolio result is still true. That is, if only riskless assets are available, there will always be two portfolios that can be used to construct any portfolio on the mean-variance efficiency frontier. Tobin’s result is a special case where one of these portfolios (assets) is the risk free rate, but the more general result is true – all efficient portfolios can be constructed by a combination of two, suitably chosen, portfolios. (See Section 6.7 in Luenberger’s, Investment Science, 1998)].

i. What is the Meaning of Tobin's Theorem?

Let us consider various extremes. First, where are the most risk averse investors? They end up with all of their portfolio at the point Rf in the above diagram. They should hold cash, in other words. Their total portfolio has a zero variance and an expected return of Rf. Now, suppose we ask what happens when very risk averse investors become slightly less risk averse. The answer is that these investors move slightly to the right along the line coming from Rf. What do they own? As they move to the right along the line Rf-E*, investors buy some E* giving up some of the riskless asset. Finally as investors are willing to take on more and more risk to achieve higher returns, investors reach the point E*. Achieving the point E* int the diagram means that no cash (riskless asset) is held and the entire portfolio is simply E*.

What happens to the right of the point E*? The interpretation of points along the line Rf-E* that lie to the right of the point E* is that the investor begins to borrow and uses the proceeds to invest in more of the portfolio E*. To the right of E* the investor uses leverage to take on more risk to have the chance of even higher returns.

ii. Borrowing and Lending and Tobin's Theorem

We have assumed explicitly that investors can borrow and lend at the same rate. This assumption often calls for the response that borrowing and lending rates are very different for investors. Generally speaking such a response is factually incorrect. The very largest investors can borrow and lend at approximately the same rate. Consider the repo market where a huge volume of daily borrowing and lending occurs. Further, most derivative products, as we shall observe later, are priced as if borrowing and lending rates were identical. This means, again as a practical matter, that investors are often able to borrow and lend at identical rates. Thus it is not invariably true that borrowing and lending rates differ.

Nevertheless, the question arises. Of what significance is the assumption that borrowing and lending rates are different? We show diagrammatically below an exception to Tobin's Theorem that can be concocted by assuming that borrowing the riskless asset involves higher rates then lending the riskless asset.

In the diagram above, it is asumed that borrowing takes place at higher rates than lending. The line Rf-E*1 reflects the zone where investors (who are relatively more risk averse) own (lend) some of the riskless asset. Between E*1 and E*2, you are back in the pure Markowitz world. All of the portfolios on the boundary between E*1 and E*2 are possible optimal solutions to the investor choice problem depending upon the shape of the indifference curves in this region. Finally, borrowing occurs on the line segment emanating from the point E*2. Thus, there are two Tobin Theorem areas -- to the left of E*1 and to the right of E*2. In between E*1 and E*2 is the old Markowitz region where Tobin's Theorem no longer applies. (It is worth noting that even if the borrowing rate exceeds the lending rate it is possible, depending upon the curvature of the feasible set in the neighborhood of E*, that Tobin's Theorem is still true as stated. We have drawn a case where Tobin's Theorem is not exactly true as stated).

How significant is this result? Not very. In principle, this result implies that leveraged investors should hold a different risky portfolio E*, than unleveraged investors. But, the general conclusion remains basically Tobin's Theorem. As long as investor either (i.) chooses to own some of the risk free asset or (ii.) chooses to borrow, the rest of that invesor's asset will go into a specific portfolio (that varies only between whether (i.) or (ii.) applies).

We conclude that the assumption of equivalent borrowing and lending is not only a good approximation to much of reality, but it is also a fairly harmless assumption. The essential theme of Tobin's Theorem remains even without such an assumption.

XI. The Separation Theorem; The Mutual Fund Theorem

The Tobin Theorem is sometimes called a separation theorem. The term separation means that all assets are separated into two groups: one group contains the risk free asset and the second group contains all other assets – portfolio E*. Tobin's theorem says that a risk averse investor will limit his consideration to linear combinations of these two groups of assets – all other asset combinations are irrelevant to optimal choice. These two groups of assets are separated out from all assets and these two groups alone constitute the ultimate components of the optimal choice.

Sometimes the Tobin Theorem has been called the mutual fund theorem. The mutual fund is E*. It is a specific collection of assets combined evidentally in a very efficient way. Tobin's Theorem is essentially about how all risky assets can be combined in a very efficient way to produce a single efficient portfolio E*. Think of E* as a mutual fund. Every risk averse investor will hold either cash or shares in the mutual fund E* or some combination of both (or possibly lever up and buy more shares in the mutual fund E* than present wealth would allow). The only two assets that end up mattering are cash (our riskless asset) and a mutual fund (E*).

Both of these interpretations are useful. What is happening, largely behind the scenes, in Tobin's Theorem is diversification. As we shall see later when we take up the Capital Asset Pricing Model, the portfolio E* contains a very large number of different assets.

XII. Proving Tobin's Theorem

Let us restate Tobin's Theorem::

Tobin's Theorem:

Assume: (1.) a set of assets with known means, variances and pairwise covariances.

(2.) that investors are risk averse(3.) that at least one asset is riskless (has a zero return variance)(4.) Investor can borrow or lend at the riskless rate of interest

Conclusion: Every investor will hold a portfolio that is defined by the following equation:

where 1, 2 are non-negative numbers that sum to one and where X1 is the riskless asset (with the highest expected return among all such riskless assets) and X2 is a unique portfolio of risky assets. Only the 's vary from investor to investor. The X1 and X2 portfolios are the same for all!

Proof: We are not actually going to prove this theorem, but instead will indicate how the proof generally proceeds. We will break the proof into two parts:

1. We must establish that the choice problem in Tobin's Theorem has a solution. There is a famous theorem in mathematics (called Weierstrass's Theorem) that says that any time you have a continuous function defined for all members of a closed and bounded set, that function will attain a maximum for some element in the set. The choice problem in Tobin's Theorem satisfies the conditions of this famous theorem of mathematics and therefore we can conclude that there is some portfolio that maximizes each risk averse investor's utility function. Indeed, there may well be more than one such portfolio, but, if there is more than one, it will only provide the same utility as any other portfolio that maximizes the utility function. In other words, the utility level reached is unique, although more than one portfolio might achieve that same maximum attainable utility level.

How do we know the choice problem in Tobin's Theorem satisfies the conditions of the Weierstrass Theorem? Is the utility function continuous? Continuous means roughly that if you change the portfolio slightly that will only cause slight changes in the amount of utility that the portfolio generates. Put differently, if the utility function is continuous small changes in the portfolio do not lead to big changes in the utility generated by the portfolio. The answer is yes -- the utility function is certainly continuous. Is the set of feasible portfolios closed and bounded? We have shown that it is bounded from the northwest direction (which is the only direction that matters in this problem). What does closed mean? If there is some portfolio that is very very similar although slightly different from many of the portfolios that are in the feasible set, then that portfolio is also in the feasible set. Is the feasible set closed? Yes. Thus Weirstrass's Theorem applies and our choice problem definitely has some solution. That is, there is at least one feasible portfolio that will maximize the utility function over all possible feasible portfolios.

2. Assuming that we have a portfolio that maximizes the utility function, what do we know about the portfolio? Does it have to lie on the line from Rf through E* and beyond? Yes. If not, then the portfolio would be inefficient -- it would lie to the right of line. This means we could alway achieve another portfolio that had the identical standard deviation but a higher mean. Such a portfolio could not be utility maximizing if it is not on the line from Rf through E*.