What's new

Quant Question Set Round 1


New Member

Question regarding the Quant quiz round 1.

1/Could you explain further why "period returns do not tend to be normal, but cumulative leves to tend to be normal"?

2/ Mean reversion in return. The question is "How does a forecast model compare to true volatiltiy, when the forecast applies the square root rule to forecast VaR, and when there is mean reversion, respectively, (i) in returns and (ii) in return volatiliy?" Can you elaborate further that why forecast overstates in return?


David Harper CFA FRM

David Harper CFA FRM
Staff member
Peach - These are strong questions, you seem to be in control of this section!

1/ the answer actually reads: "Period returns do not tend to be lognormal; rather, cumulative price leves [levels] do tend to be lognormal"

I have a blog article on this here. But the short of it is, if we assume periodic returns are normally distributed then price levels are lognormally distributed. This because, under continuous compounding, the periodic return from today's price S(0) to tomorrow's price S(1) is given by LN[S(T)/S(0)]. Note how that can be negative; e.g., $2 to $1 = LN(1/2) = -70%. If this is 'normal' then S(T)/S(0) is lognormal. That's exactly what lognormal means: if the natural log of a variable is normally distributed, the variable is lognormal. And, note that S(T)/S(0) cannot be negative. Nor can the price level S(T). So again, if periodic returns are normal, future price *LEVELS* are lognormal (or price relatives, which refers to ST/S0)

But since you focused me on the question (where the lognormal is the wrong answer), it occurs to me, having now been thru the entire 2007 curriculum, this is an important LO. Linda Allen says actual asset returns are unlike normal distribution because actual (empirically observed) returns are fat-tailed (leptokurtosis; i.e., kurtosis > 3 or 'excess kurtosis' >0 ), unstable (time-varying) and skewed. The fat-tailed and time-varying "problems" are thematic and resurface more than once. Make sure to understand the fat-tail problem (it could be due to time-varying mean or volatility; Allen says it must be that volatility shifts over time. Jorion says both could shift). Under operational risk, the TAN distribution is exactly a fix to this problem: a gamma distribution is grafted onto the normal distribution expressly to give the normal a fat tail.

2/ My previous article here elaborates on this (see bottom). Briefly, the square root rule only works under i.i.d. (independent & identically distributed returns/variables). The 'independence' part of this is *violated* when there is mean reversion or autocorrelation. That is, to be non mathy about it, "mean reversion" is when gravity pulls the variable toward its long run trend. "Serial correlation" keeps the variable sticky. If you think about volatility as scaled by the square root rule (under i.i.d.) you can imagine it disperses without any 'mean reversion' pull. So, the scaled (forecast by square root rule) volatility will overstate the actual volatility, which is *dampened* by the gravitational pull of mean reversion. I admit the question, which is based on the Allen reading, is confusing because the accuracy of a long horizon volatility *depends* in the case of return volatility mean reversion and simply overstates in the case of return mean reversion. That is just because we are talking about forecast in the context of GARCH(1,1) which contains a term for the long-run volatility (but there is no corresponding term for the long-run average return, so it does not depend on our state versus such a thing). Okay, but frankly this last nuance has a slim/no chance of being tested. The important points here are....

In regard to a long forecast of volatility, TWO different dynamics can render useless the square-root-to-scale rule.

1. The return volatilities probably are not i.i.d.
2. Our starting point (today's volatility) probably is not equal to the long-run variance.

Put another way, we can use square root rule to forecast only if both today's variance happens to equal the long run variance and return volatilities are i.i.d.

To summarize the testable ideas mentioned:

* If the natural log of a variable is normally distributed (e.g., periodic returns), the variable is lognormal (e.g., price relatives or price levels)
* Linda Allen asks "can normality [assumption of normally distributed returns] be salvaged?" and answers "no" to her own question. Returns are not normal because they are fat-tailed (leptokurtosis), unstable (time varying or "not identical") and skewed
* The square root rule (i.e., variance scales directly with time) is very restrictive. It requires i.i.d. which implies no mean reversion and no autocorrelation
* i.i.d. also means "identically" distributed. But note how TIME-VARYING problem violates this! The square root rule gets knocked out quickly: as Allen says, "volatility is stochastic [i.e., time-varying or "not identical"] and, in particular, autoregressive." Volatility is sticky...


New Member

On the issue of mean reversion, I have some issues.

In your investment risk module, especially under the hedge funds section (Lo42.2 define pair trading), you say that statistical arbitrage is a kind of strategy that attempts to profit from ""mean reversion"". This means that equities display mean reversion?

In some practice questions under investment risk, the question (I forget which but it is there somewhere) was which of the asset classes display mean reversion, and the choice was interest rates.

Linda Allen says that interest rates and spreads are "predictable" because of mean reversion effects, while equities (as well as currencies) are unpredictable in the short term, while extent of predictability for the long term is still small. This effectively means that only interest rates and spreads display mean reversion.

So what is the final word on equities? mean reverting? If all asset classes tend to their fair values over the long term, then it would imply that all of them are mean reverting to the long term fair value!!! and not just interest rates.

Pl clarify



David Harper CFA FRM

David Harper CFA FRM
Staff member

Fantastic observations because, to ponder mean reversion is to involve at least a dozen learning outcomes in the curriculum (high testability).

First the pair trade: Jaeger does characterize STATISTICAL ARBITRAGE as a sub-class of EQUITY MARKET NEUTRAL by saying the arbitrageur hopes for mean reversion. The pair trader hopes that traded prices will 'mean revert' to their (intrinsic) fair values. The FRM will not ask whether prices mean revert or not; it can be debated.

Rather, the key idea here is: if you assume efficient market hypothesis (EMH), at least in semi-strong form, then you conclude asset prices will mean revert. If you do not (i.e., non EMH framework), you are NOT obliged to find them reverting. PLEASE NOTE, this is key idea in Dowd's model risk reading. That is, if you are an EMH *true believer*, then your problems can be fixed with a BETTER MODEL; but, if not, tinkering the model is not the answer because you do not think such a holy grail exists, instead you are better to game the market (i.e., get one step ahead of other model users).

But more broadly, we should make a difference between empirical observations ("historically, do these things revert?") and the various MODELS discussed in the curriculum. On the empirical side, the FRM will not ask you if equities mean revert. The question does not have a decisive answer. On the interest rate side, the assigned readings are a consensus and do say "YES" interest rates mean revert. Jorion explains this by saying that bond prices converge toward their par (make sure you know why!), although he does not explain the nontrivial leap from price reversion to rate reversion. No matter: the consensus view, reflected in at least four readings, is that EMPIRICALLY (what do we observe) interest rate LEVELS tend to (empirically) mean revert.

But we also care about the MODELS - which are mathematical specifications that may or may not match empirical reality. First, we should be careful about WHAT EXACTLY REVERTS. In the case of stocks, we could have three meanings:

1. Price level
2. Price returns
3. Price return volatility

Regarding the PRICE PATH of stocks, the models do not expect mean reversion. You can get here several ways. The CAPM is all upward drift, no reversion (and intuitive, prices should go up by cost of equity capital). Importantly, the Brownian motion (as a Wiener process) does not mean revert.

Regard PRICE RETURNS, models like CAPM and multi-factor models implicitly assume mean reversion; e.g., in CAPM the return "reverts" to a function of beta, with the error fluctuation around zero. But mean reversion might be a bit of a misnomer here. Maybe better to just stay that the models assume there is a MEAN return level with a distribution around the mean.

Regarding price return VOLATILITY, it depends on the model: EWMA does not assume volatility reversion, GARCH(1,1) is different from EWMA because it adds a term for mean reversion.

On the INTEREST RATE side, all of the models for INTEREST RATE path processes DO ASSUME MEAN REVERSION. The two-factor interest rate model assume both the short and long-rates have their own mean reversion.

(Finally, since we are on mean reversion, keep in mind that for stock prices, the phenomenon of mean reversion invalidates the "square root rule" [that variance scales with time] and inhibits our ability to extrapolate long range volatility. The square root rule required i.i.d. but mean reversion contradicts the idea of 'independent' returns).

In summary:
* Mean reversion could refer to (i) traded prices reverting to their intrinsic values (yes they will, says the EMH true believer), (ii) price level, (iii) price returns, or (iv) price return volatility
* Both empirically and in the models, it is assumed that interest rates mean revert
* In regard to equities, the typical price path process (GBM) does not have mean reversion. The more sophisticated return volatility model [GARCH(1,1)] does assume reversion, but EWMA does not.
* Mean reversion violates i.i.d., and i.i.d. is necessary for the square root rule


New Member

Your reply is a superb exposition on mean reversion. You summarised it very well.

Frankly this kind of learning will never happen from reading assigned readings.

many thanks