# Prof. Jayanth R. Varma’s Financial Markets Blog

A blog on financial markets and their regulation

## Bayesians in finance redux

March 7, 2010

Posted by on In November last year, I wrote a brief post

about Bayesians in finance. The post was brief because I thought that

what I was saying was obvious. A long and inconclusive exchange with

Naveen in the comments section of another

post has convinced me that a much longer post is called for. The

Bayesian approach is perhaps not as obvious as I assumed.

When finance professors walk into a classroom, they want to build

on what the statistics professors have covered in their courses. When

I am teaching portfolio theory, I do not want to spend half an hour

explaining the meaning of covariance; I would like to assume that the

statistics professor has already done that. That is how division of labour is supposed to work in a

pin factory or in a university.

Unfortunately, there is a problem with this division of labour

– most statistics professors teach classical statistics. That is

true even of those statisticians who prefer Bayesian techniques in

their research work! The result is that many finance students wrongly

think that when the finance professors talk of expected returns,

variances and betas, they are referring to the classical concepts

grounded in relative frequencies. Worse still, some students think

that the means and covariances used in finance are sample means and

sample covariances and not the population means and covariances.

In business schools like mine where the case method dominates the

pedagogy, these errors are probably less (or at least do less damage)

because in the case context, the need for judgemental estimates for

almost everything of interest becomes painfully obvious to the

students. The certainties of classical statistics dissolve into utter

confusion when confronted with messy “case facts”, and

this is entirely a good thing.

But if cases are not used or used sparingly, and the statistics

courses are predominantly classical, there is a very serious danger

that finance students end up thinking of the probability concepts in

finance in classical relative frequency terms.

Nothing could be farther from the truth. To see how differently

finance theory looks at these things, it is instructive to go back to

some of the key papers that established and developed modern portfolio

theory over the years.

Here is how Markowitz begins his Nobel prize winning paper

(“Portfolio Selection”, Journal of Finance, 1952) more

than half a century ago:

The process of selecting a portfolio may be divided into two stages.

The first stage starts with observation and experience and ends with

beliefs about the future performances of available securities. The

second stage starts with the relevant beliefs about future performances

and ends with the choice of portfolio.

Many finance students would probably be astonished to read words

like observation, experience, and beliefs instead of terms like

historical data and maximum likelihood estimates. This was the paper

that gave birth to modern portfolio theory and there is no doubt in

Markowitz’ mind that the probability distributions (and the

means, variances and covariances) are subjective beliefs and not

classical relative frequencies.

Markowitz is also crystal clear that what matters is not the

historical data but beliefs about the future – historical data

is of interest only in so far as it helps form those beliefs about

the future. He also seems to take it for granted that different people

will have different beliefs. He is helping each individual solve his

or her portfolio problem and is not bothered about how these choices

affect the equilibrium prices in the market.

When William Sharpe developed the Capital Asset Pricing Model that

won him the Nobel prize, he was trying to determine the market equilibrium

and he had to assume that all investors have the same beliefs but did

so with great reluctance:

… we assume homogeneity of investor expectations: investors are

assumed to agree on the prospects of various investments – the

expected values, standard deviations and correlation coefficients

described in Part II. Needless to say, these are highly restrictive

and undoubtedly unrealistic assumptions. However, … it is far from

clear that this formulation should be rejected – especially in

view of the dearth of alternative models

But finance theory quickly went back to the idea that investors had

different beliefs. Treynor and Black (“How to use security

analysis to improve portfolio selection,” Journal of

Business, 1973) interpreted the CAPM as saying

that:

…in the absence of insight generating expectations different from

the market consensus, the investor should hold a replica of the market

portfolio.

Treynor and Black devised an elegant model of portfolio choice

when investors had out of consensus beliefs.

The viewpoint in this paper is that of an individual investor who

is attempting to trade profitably on the diiference between his

expectations and those of a monolithic market so large in relation to

his own trading that market prices are unaffected by it.

Similar ideas can be seen in the popular Black Litterman model

(“Global Portfolio Optimization,” Financial Analysts

Journal, September-October 1992). Black and Litterman started

with the following postulates:

- We believe there are two distinct sources of information about

future excess returns – investor views and market equilibrium.- We assume that both sources of information are uncertain and are

best expressed as probability distributions.- We choose expected excess returns that are as consistent as

possible with both sources of information.

Even if we stick to the market consensus, the CAPM beta itself has

to be interpreted with care. The derivation of the CAPM makes it clear

that the beta is actually the ratio of a covariance to a variance and

both of these are parameters of the subjective probability

distribution that defines the market consensus. Statisticians

instantly recognize that the ratio of a covariance to a variance is

identical to the formula for a regression coefficient and are tempted

to reinterpret the beta as such.

This may be formally correct, but it is misleading because it

suggests that the beta is defined in terms of a regression on past

data. That is not the conceptual meaning of beta at all. Rosenberg and

Guy explained the true meaning of beta very elegantly in their paper

(“Prediction of beta from investment fundamentals”,

Financial Analysts Journal, 1976) introducing what are

now called fundamental betas:

It is instructive to reach a judgement about beta by carrying out an

imaginary experiment as follows. One can imagine all the various

events in the economy that may occur, and attempt to answer in each

case the two questions: (l) What would be the security return as a

result of that event? and (2) What would be the market return as a

result of that event?

This approach is conceptually revealing but is not always practical

(though if you are willing to spend enough money, you can access the

fundamental betas computed by firms like Barra which Barr Rosenberg

founded and later left). In practice, our subjective belief about the

true beta of a company involves at least the following inputs:

- The beta is equal to unity unless there is enough reason to

believe otherwise. The value of unity (the beta of an average stock)

provides an important anchor which must be taken into account even

when there is other evidence. It is not uncommon to find that simply

equating beta to unity outperforms the beta estimated by naive

regression. - What this means is that betas obtained by other means must be

shrunk towards unity. An estimated beta exceeding one must be reduced

and an estimated beta below one must be increased. One can do this

through a formal Bayesian process (for example, by using a Bayes-Stein

shrinkage estimator), or one can do it purely subjectively based on

the confidence that one has in the original estimate. - The beta depends on the industry to which the firm belongs. Since

portfolio betas can be estimated more accurately than individual

betas, this is often the most important input into arriving at a

judgement about the true beta of a company. - The beta depends on the leverage of the company and if the

leverage of the company is significantly different from that of the

rest of the industry, this needs to be taken into account by

unlevering and relevering the beta. - The beta estimated by regressing the returns of the stock on the

market over different time periods provides useful information about

the beta provided the business mix and the leverage have not changed

too much over the sample period. Since this assumption usually precludes very

long sample periods, the beta estimated through this route typically

has a large confidence band and becomes meaningful only when combined

with the other inputs. - Subjective beliefs about possible future changes in the beta

because of changing business strategy or financial strategy must also

be taken into account.

Much of the above discussion is valid for estimating Fama-French

betas and other multi-factor betas, for estimating the volatility

(used for valuing options and for computing convexity effects), for

estimating default correlations in credit risk models and many other

contexts.

Good classical statisticians are quite smart and in a practical

context would do many of the things discussed above when they have to

actually estimate a financial parameter. In my experience, they

usually agree that (a) there is a lot of randomness in historical

returns; (b) the data generating process does not remain unchanged for

too long; (c) therefore in practice there is not enough data to avoid

sampling error; and (d) hence it is desirable to use a method in which

sampling error is curtailed by fundamental judgement.

On the other side, Bayesians shamelessly use classical tools

because Bayes theorem is an omnivore that can digest any piece of

information whatever its source and put it to use to revise the prior

probabilities. In practical terms, Bayesians and classical

statisticians may end up doing very similar stuff.

The advantage of shifting to Bayesian statistics and subjective

probabilities is primarily conceptual and theoretical. It would

eliminate confusion in the minds of students on the ontological status

of the fundamental constructs of finance theory.

I am now therefore debating in my own mind whether finance

professors must spend some time in the class room discussing

subjective probabilities.

How would it be like to begin the first course in finance with a

case study of subjective probabilities – something like the

delightful paper by Karl Borch (“The monster in Loch

Ness”, Journal of Risk and Insurance, 1976)? Borch

analyzes the probability that the Loch Ness

monster exists (and would be captured within a one year period)

given that a large company had to pay a rather high 0.25% premium to

obtain a million pound insurance cover from Lloyd’s of London

against that risk? This is obviously a question which a finance

student cannot refuse to answer; yet there is no obvious way to

interpret this probability in relative frequency terms.

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This post along with the your conv with Naveen have been very insightful. Having just completed a MSc, I would like to add that my first lecture was about subjective probabilities and risk aversion co-efficients(CARA). This I think would lead to better appreciation of s.d. as a volatility measure( Something is better than nothing 😛 )