Limits of Correlation, CAPM, VaR, and Other Finance Models

# What concepts are the most dangerous ones in quantitative finance work?


# What concepts are the most dangerous ones in quantitative finance work?












There are a few things that form the common canon of education in (quantitative) finance, yet everybody knows they are not exactly true, useful, well-behaved, or empirically supported.

So here is the question: which is the single worst idea still actively propagated?

Please make it one suggestion per post.

## Answer by vonjd (score 86)

https://quant.stackexchange.com/a/169

> Correlation

Correlations are notoriously unstable in financial time series - yet one of the most used concepts in quant finance because their is no good theoretical substitute for it. You could say theory is not working with it yet neither without it.

For example the concept is used for diversification of uncorrelated assets or for the modelling of credit default swaps (correlation of defaults). Unfortunately when you need it most (e.g. a crash) it just vanishes. This is one of the reasons that the financial crises started because the quants modeled the cds's with certain assumptions concerning default correlations - but when a regime shift happens this no longer works.

Edit See my follow-up question: What is the most stable, non-trivial dependence structure in finance?

## Answer by Shane (score 58)

https://quant.stackexchange.com/a/157

> CAPM as an allocation strategy.

Market efficiency was predicated on several falicious ideas, including:

- Everyone can borrow (and lend) at the same rate, indefinitely (i.e. no matter their leverage)

- All information is known instantaneously by all market participants.

- There are no transaction costs.

- Rational behavior.

One conclusion is that the higher the beta, the higher the return, but this has clearly been shown to be violated.

While it is useful for segmenting $\alpha$ and $\beta$ (and for portfolio/strategy evaluation), it simply isn't entirely reliable as a portfolio allocation strategy.

As Fama/French concluded in "The Capital Asset Pricing Model: Theory and Evidence" (2004):

> The CAPM, like Markowitz's (1952, 1959) portfolio model on which it is built, is nevertheless a theoretical tour de force. We continue to teach the CAPM as an introduction to the fundamental concepts of portfolio theory and asset pricing, to be built on by more complicated models like Merton's (1973) ICAPM. But we also warn students that despite its seductive simplicity, the CAPM's empirical problems probably invalidate its use in applications.

Note that CAPM adds many assumptions to Markowitz's fundamental model to built itself. Therein lies its fallacy because as said above, those are difficult assumptions. Markowitz' model itself is fairly general in that you can inject 'views' of higher returns or greater volatility etc into the basic framework (or not!) and still be quite rooted in reality for mid-long term horizons.

## Answer by Dirk Eddelbuettel (score 47)

https://quant.stackexchange.com/a/158

Everybody's favourite whipping boy: Identically and independently distributed returns, i.e. draws from $N(\mu, \sigma)$ to describe returns.

We could of course split this is arguing

- identically distributed (and mixture modeling as well as robust methods help)

- independently distributed (and everybody agrees that there is some serial correlation though a formal good model is hard to come by)

- the Normal assumption (and everybody agrees on fatter tails) yet $N(\mu, \sigma)$ makes things so temptingly tractable

## Answer by Andrey Taptunov (score 33)

https://quant.stackexchange.com/a/172

> Value at Risk

The great idea to have systematic indicator for risk exposure but the problems arise when

- it's used as main or single indicator without looking at other risks (e.g Credit risk or Liquidity risk). Emanuel Derman wrote about it recently in his blog:

> But they (GS) did it not with a new formula or a single rule. They did it by being smart rather than doctrinaire. They were eclectic; they had limits on all sorts of exposures -- on VaR, on the fraction of a portfolio that hadn't been modified in a year ... There isn't a formula for avoiding future losses because there isn't one cause of future losses.

- VaR becomes the purpose of risk management - not the situations when when losses exceed VaR “An airbag that works all the time, except when you have a car accident.” (c) D.Einhorm

- It's focused on managable risk in a normal situations with the assumption that tomorrow will be like today and yesterday and without taking rare events into account.

- It becomes another parameter (like profit) that could be gamed (the same profit but with low risk).

And there are two exremely great articles about VaR among the best I've ever read:

- Risk Management - What Led to financial meltdown. NY Times

- Derivatives Strategy - Rountable on Limits of VaR

## Answer by gappy (score 25)

https://quant.stackexchange.com/a/202

Easily, the

> Efficient Market Hypothesis

For many reasons. First, many adherents and critics support it for the wrong (often ideological) reasons. This applies even to well-known economists like John Quiggin. Second, because even fewer people know the extent and scope of the anomalies. The literature can get very technical. So even smart people rejecting the EMH, or publishing anomalies, end up being over-optimistic about their ability to beat the market.

## Answer by TheBridge (score 20)

https://quant.stackexchange.com/a/161

> Perfect delta hedging

In my opinion delta hedging is also a dangerous one, but it definitely should teach though. In the BS framework, it is an allegedly perfect way of covering the risk incurred by buying (or selling) a derivative product (such as call and put in simplest cases). Nevertheless due to several real world facts this doesn't work that well in practice :

- discrete time rebalancing of porfolio

- constant volatility so much things have been said on this I won't comment any further

- possibility of market jumps (not little ones) this affects deeply your daily P&L

- transction costs affects the cost of the rebalancing portfolio in a way that is not negligeable

- liquidity, if you are holding big positions in derivatives, your delta hedging will impact the price dynamics

- etc...

The main advantage of the BS delta hedging is that it presents though the big principles of hedging the rest is a matter of sophistication and derivatives trader's vista (or chance).

## Answer by gappy (score 20)

https://quant.stackexchange.com/a/203

My second best is

> Copulas

I won't go as far as declaring gaussian copula The formula that killed Wall Street" (warning: lousy article), but will defer to T. Mikosch in his very good paper on misuses of copulas.

## Answer by Paul Young (score 18)

https://quant.stackexchange.com/a/206

Backtesting - pure and simple. Its the logical and obvious thing to do right? Yet, so many pitfalls lie in wait. Be very careful people. Do it as little as possible and as late as possible.

## Answer by ast4 (score 15)

https://quant.stackexchange.com/a/224

This isn't particularly insightful, but worth pointing out in this thread. Many people get caught up in the elegance and beauty of the mathematics and tend to be disconnected from the real world.

## Answer by Richard Herron (score 9)

https://quant.stackexchange.com/a/165

That value stocks are necessarily riskier than growth; that there has to be a hidden risk factor that we haven't yet found. The Lakonishok, Shleifer, and Vishny abstract says it better than I can:

> For many years, stock market analysts have argued that value strategies outperform the market. These value strategies call for buying stocks that have low prices relative to earnings, dividends, book assets, or other measures of fundamental value. While there is some agreement that value strategies produce higher returns, the interpretation of why they do so is more controversial. This paper provides evidence that value strategies yield higher returns because these strategies exploit the mistakes of the typical investor and not because these strategies are fundamentally riskier.

## Answer by Brian B (score 9)

https://quant.stackexchange.com/a/366

Corporate Actions do not happen.

That is to say, both the models and psychology tend to ignore the possibility of such behavior as takeovers, spinouts, significant changes in leverage (ratio of debt to equity) by issuing or redeeming bonds, and the like.

Now, there are desks (such as merger arb) that specifically play these, and fundamental analysts discuss and sometimes "model" them (if you can call their relatively simple spreadsheets models). But you'll find that the difficulty of including them in options models keeps them unincorporated, and plenty of traders fail to make the necessary mental adjustments.

## Answer by otto (score 8)

https://quant.stackexchange.com/a/163

> To trust yourself.

Concepts must be based on logical ideas and proper premises. It is easy to forget a premise and then misuse a model such as CAPM as asset-allocation method as suggested by Shane so `Y-Recheck-things`. Do not make things personal. Do not abuse models with too complicated schemes (you may abuse some basic assumption) -- and even then don't expect pretend `to know`, rather `to engineer`.

## Answer by user567 (score 8)

https://quant.stackexchange.com/a/688

Persistent autocorrelations in volatility processes are due to long term memory only. I cannot help but sigh at the hundreds of papers which work under this assumption. Haven't people heard about regime shifts?

## Answer by JohnAndrews (score 6)

https://quant.stackexchange.com/a/3251

In my opinion you should question EVERYTHING.

Recently I read this article Ten Things We Should Know About Time Series by Michael McAleer which is to my opinion a good summary of some common issues in time series analysis.

These ten things are:

- Knowledge of Econometrics and Statistics is Essential

- Be Aware of Measurement Errors

- Test for Zero Frequency, Seasonal and Periodic Unit Roots

- Analyse Fractionally Integrated and Long Memory Processes

- Estimate VARFIMA Models

- Use and Interpret Cointegrating Models Carefully

- Choose Sensibly Among Univariate Conditional, Stochastic and Realized Volatility Models

- Do Not Confuse Thresholds, Asymmetry and Leverage in Volatility

- Do Not Underestimate the Complexity of Multivariate Volatility Models

- Think Carefully About Forecasting Models and Expertise

See the article for a further description of each point.

## Answer by markbruns (score 5)

https://quant.stackexchange.com/a/204

It is dangerous when large proportions of the trading population "religiously" believe [as a matter of unexamined faith] that certain necessary assumptions which govern the accuracy of the models they use will always hold. It is best to really understand how the models have been derived and to have a skeptics understanding of these assumptions and their impact. All models have flaws -- yet it is possible to use flawed models if you can get consistent indicators from several different or contrarian approaches based upon radically different assumptions. When the valuations from different approaches diverge, it is necessary to understand why -- when this happens, it is necessary to investigate the underlying assumptions ... this sort of environment often provides trading opportunities, but the environment can also quite easily be an opportunity for disaster.

Of course, implicit and explicit assumptions are absolutely necessary to sufficiently simplify any mathematical analysis and to make it possible to derive models that can give lots of traders the generally useful trading "yardsticks" that they rely upon. As an example, consider the Black-Scholes model. The Black-Scholes model is ubiquitous; a commonly used "yardstick" for option valuations. The Black-Scholes model of the market for a particular equity explicitly assumes that the stock price follows a geometric Brownian motion with constant drift and volatility.

This assumption of "geometric Brownian motion with constant drift and volatility" is never exactly true in the very strictest sense but, most of the time, it is a very useful, simplifying assumption because stock prices are often "like" this. It might not be reality, but the assumption is a close enough approximation of reality. This assumption is highly useful because of how makes it possible to apply stochastic partial differential equations methodology to the problem of determining appropriate option valuations. However, the assumption of "constant drift and volatility" is a very dangerous assumption in times when judgement, wisdom and intuition would tell an experienced investor "Something is "odd." It's as if we're in the calm before the storm." OR "Crowd psychology and momentum seem to be more palpable factor in the prices right now."

## Answer by Mats Lind (score 4)

https://quant.stackexchange.com/a/29669

Ignoring to account for possibly omitted variables

Ignoring to account for possibly omitted variables has arguably lead to both of the severe problems below:

The fall of the US mortgage market in 2008 as risk on mortgage bond portfolios were grossly underestimated as the strong dependence of their bonds on common variables like the state of the business and credit cycles were ignored and covariations and portfolio variance understated.

The rise and fall of the US junk-bond market fuelled by Milken using statistics covering fallen angels from decades ago to predict default rates on new bonds for which most variables but solvency ratios were other omitted and dissimilar to the variables of the bonds in the statistics.

## Answer by Dave Harris (score 2)

https://quant.stackexchange.com/a/70066

The most dangerous concept is hidden in the underlying rules of mathematics, not economics or finance. Many of the other posts only exist because this concept exists. I know because it took me forever to figure out why it was the real problem.

In the underlying rules of calculus, there is a non-hidden, but not well thought through, assumption that all parameters are known with certainty. Friedman argued that this assumption was acceptable because the markets behaved as if the parameters were known. The difficulty is that the rules of math are very different if that assumption is dropped. Once they change, the economic models work out to be different. Markowitz’s models cannot survive the transition.

I will be proposing a new stochastic calculus with that assumption dropped at the WEAI conference in June/July 2022. When published, the most recent version of that paper should be in their proceedings. You can find the current version at

> Harris, David E., A Generalization of Stochastic Calculus--A Conjecture (November 29, 2018). Available at SSRN: https://ssrn.com/abstract=3197451 or http://dx.doi.org/10.2139/ssrn.31974.

Consider what happens if you drop that assumption that the parameter is known in the equation where future wealth equals prior wealth times a factor plus a random shock centered on zero with a finite variance. If $$x_{t+1}=Rx_t+\epsilon_{t+1}, R>1,$$ then the sampling distribution for the maximum likelihood estimator of $\hat{\beta}$ is the Cauchy distribution. So the expectation of future wealth is $$E(Rx_t+\epsilon_{t+1})$$ which cannot exist as the integral diverges. You can see papers by Mann and Wald and John White on this. There is also a paper adjacent to this topic by Sen as well.

> Mann, H. and Wald, A. (1943). On the statistical treatment of linear stochastic difference equations. Econometrica, 11:173-200.

> White, J. S. (1958). The limiting distribution of the serial correlation coefficient in the explosive case. The Annals of Mathematical Statistics, 29(4):1188-1197.

> Sen, P. K. (1968). Estimates of the regression coefficient based on kendall's tau. Journal of the American Statistical Association, 63(324):1379-1389.

Now we cannot form expectations, and this is in just one place, so we cannot have all kinds of other things like covariances.

The next problem is that Frequentist statistics are not coherent; they violate the converse of the Dutch Book Theorem. Anyone who understands enough math can Dutch Book any market maker if their client uses a Frequentist formula or estimation technique. I have five training modules on how to do this to get the point across to people on how dangerous this is.

So now we have two problems. We have no covariances or even a mean; we lack coherent methods of setting prices, using the word coherence in the de Finetti sense of the word.

That puts us in Bayesian estimation, but only in the Bayesian version of probability and statistics with subjective, informative, proper prior distributions. It turns out that all of the other priors can be Dutch Booked if you know enough. Now we lack a theory.

Bayesian methods are generative, not sampling-based. The difference doesn’t matter for many mundane tasks, but as it happens, when you leave the exponential family of distributions, you also leave the world of mundane tasks. An AR(1) problem, given a large enough sample size, will map to an AR(1) problem in Bayesian estimation but may very well be an ARIMA(1,2,2) problem on the sampling side of the fence. That implies a very different world than an equilibrium-only world because, in addition to being generative, it also requires a discussion of mutually exclusive and exhaustive states.

So, let us go back to $$ x_{t+1}=Rx_t+\epsilon_{t+1}, R>1.$$ In finance, $x=p\times{q}$. An allocation is the product of a price and a quantity. What if $q_{t+1}$ becomes a different firm in a merger or zero from a bankruptcy? That allocation is now a product distribution, and it is not being treated like one.

Each possible future outcome for a firm would be a “small world” in de Finetti’s axiomatization of probability. We have to be in de Finetti’s world because the Dutch Book Theorem posits that we can have finite additivity. Its converse precludes countable additivity. We cannot use a $\sigma$ algebra if money is at risk.

We cannot have things like continuous hedging because people like me know how to unravel the implied math and make a Dutch Book. In our model, $x_{t+1}$ has multiple possible future states, so now we not only have to predict the value of a future allocation but also that allocation in all possible ending states of the firm.

So to recap, dropping the assumption that parameters are known in an applied finance setting eliminates dynamic hedging, expectations, covariances, Frequentist statistics, quadratic utility in certain types of artificial neural networks, and our models need to discuss how an equilibrium price is created as these are generative models. And, I am just scratching the surface here. The second-order effects are as astonishing.

Without the assumption that the parameters are known, most of financial econometrics and its modeling are lost.

## Answer by guest240502 (score 0)

https://quant.stackexchange.com/a/79232

The idea that probability distributions of asset prices can actually be estimated (with any degree of reliability).