I think DH means that it’s returns that mean revert, but the reversion is observed through the effect on prices.
bchadwick Wrote: ------------------------------------------------------- > I think DH means that it’s returns that mean > revert, but the reversion is observed through the > effect on prices. Generally speaking, returns follow a random walk (i.e they don’t mean revert either).
Yields, perhaps, but not stock returns. Stock prices may be random walks (with drift).
> nope, prices don’t revert I guess it depends on what you’re pricing. We probably agree that the price of an ounce of gold doesn’t mean revert. But we can price a lot of things, no? E.g., what would you pay for a 1-year $1,000 IOU from the US government? What would you pay for $1,000 of 10y term life insurance? For $1 of corporate earnings, rising in expected amount with the average market return? At times people have paid $22 for this. Do we think these prices revert to means?
bchadwick Wrote: ------------------------------------------------------- > Yields, perhaps, but not stock returns. Do you mean yields exhibit mean reversion? > Stock prices may be random walks (with drift). Not sure what you mean here. A RW, even with drift, is not a mean reverting process.
DarienHacker Wrote: ------------------------------------------------------- > > nope, prices don’t revert > > I guess it depends on what you’re pricing. We > probably agree that the price of an ounce of gold > doesn’t mean revert. Actually I think you could make an interesting argument here. 
> But we can price a lot of things, no? E.g., what > would you pay for a 1-year $1,000 IOU from the US > government? What would you pay for $1,000 of 10y > term life insurance? For $1 of corporate earnings, > rising in expected amount with the average market > return? At times people have paid $22 for this. You’re abusing the term “price” here but ok, let’s just say that the 22x PE will “mean revert” to 15x in a year. How do you trade that fact and make money?
justin88 Wrote: ------------------------------------------------------- > You’re abusing the term “price” here but ok, let’s > just say that the 22x PE will “mean revert” to 15x > in a year. How do you trade that fact and make > money? it depends if mean reversion means an earnings increase, price decrease or a combination of both… haha. we may just stay at 22x until we top out of this cycle. all the while stock prices could track earnings up. whats the average P/E multiple for a cycle top? its gotta be around 22x… i think mean reversion when speaking of P/Es is a complex topic. is it possible that there are multiple means, for example, a mean for low cycle, a mean for middle cycle and a mean for end cycle, and that tracking the P/E mean for a whole cycle is of limited use? if you believe we’re mid-cycle, why would you think 22x is necessarily an outsized P/E when you’re targeting 22x two years from now at end cycle?
justin88 Wrote: ------------------------------------------------------- > bchadwick Wrote: > -------------------------------------------------- > ----- > > Yields, perhaps, but not stock returns. > > Do you mean yields exhibit mean reversion? > > > > Stock prices may be random walks (with drift). > > Not sure what you mean here. A RW, even with > drift, is not a mean reverting process. Yields might be random walks, but stock returns aren’t. Stock prices can be random walks, but not stock returns. If today’s yield is 3%, then the best prediction for tomorrow’s yield is also 3%. The best prediction for next years’ yield is 3% (unless you want to add in a drift, or some kind of autocorrellation structure too). But it is possible that yields are random walks (though since yields can’t go below 0, it’s some kind of transformation that is a random walk). If last year’s stock return is 25% (as it was, approximately, in 2009), we don’t conclude that the stock market has started returning 25% per year and will just be perturbed up and down a bit from that, which is what we would conclude if it were a random walk. We think that 25% was an unusually high year, and that the chances of getting a similar year to that are (due to momentum) a bit higher than normal, but it’s much more likely that we will get a return closer to the historical average of 8%. In other words… that stock returns mean revert (we could do the reverse logic for an unusually bad year). Stock RETURNS are almost certainly NOT random walks, because one period’s return does not display persistent predictive power over the long term (persistence). Stock PRICEs might be geometric random walks, because there is price persistence. Today’s price becomes the base on which the entire future sequence is constructed. If today’s price is 10% higher than expected, then prices in the future sequence are going to be 10% higher than they otherwise would be (to the extent that price doesn’t impact future returns). If today’s RETURN is 10% higher than expected, it might (due to momentum) tell us a little about what tomorrow’s price will be, but it tells us nothing about what the return is expected to be a year from now.
bchadwick Wrote: ------------------------------------------------------- > Yields might be random walks, but stock returns > aren’t. > > Stock prices can be random walks, but not stock > returns. I don’t know too much about yields tbh, but I would guess that yields are probably similar to prices, which are generally modeled as lognormally-distributed random variables. Returns are modeled by random walks/geometric brownian motion (or some more-sophisticated derivative thereof), and prices follow from returns. > If today’s yield is 3%, then the best prediction > for tomorrow’s yield is also 3%. The best > prediction for next years’ yield is 3% (unless you > want to add in a drift, or some kind of > autocorrellation structure too). But it is > possible that yields are random walks (though > since yields can’t go below 0, it’s some kind of > transformation that is a random walk). I’m not sure what you’re saying here. Whether the predicted value is the same as the current value is not relevant. It is neither indicative nor non-indicative of something being a random walk. > If last year’s stock return is 25% (as it was, > approximately, in 2009), we don’t conclude that > the stock market has started returning 25% per > year and will just be perturbed up and down a bit > from that, which is what we would conclude if it > were a random walk. We think that 25% was an > unusually high year, and that the chances of > getting a similar year to that are (due to > momentum) a bit higher than normal, but it’s much > more likely that we will get a return closer to > the historical average of 8%. In other words… > that stock returns mean revert (we could do the > reverse logic for an unusually bad year). That is not mean reversion. What you are referring to is generally called “regression toward the mean”, which in particular is not a causal phenomenon. Mean reversion on the other hand is causal. If returns did mean revert, then a 25% up year would change the forward distribution of following year’s returns, skewing it lower than a historical average. > Stock RETURNS are almost certainly NOT random > walks, because one period’s return does not > display persistent predictive power over the long > term (persistence). Stock PRICEs might be > geometric random walks, because there is price > persistence. > > Today’s price becomes the base on which the entire > future sequence is constructed. If today’s price > is 10% higher than expected, then prices in the > future sequence are going to be 10% higher than > they otherwise would be (to the extent that price > doesn’t impact future returns). If today’s RETURN > is 10% higher than expected, it might (due to > momentum) tell us a little about what tomorrow’s > price will be, but it tells us nothing about what > the return is expected to be a year from now. OK, you’ve kind of got it but you’re also a bit confused here. Stock prices and returns are BOTH random walks, with prices being geometric random walks (when there’s drift). As you’ve pointed out, prices exhibit exponential growth, so one can take the log of the price series to linearize it. This is why prices are lognormal; this linearization is transforming the price lognormal distribution to a returns normal distribution (with vanilla mu and sigma that we all know and love). These mu, sigma parameters can then be used in a random walk/geometric brownian motion model. Pretty much all of modern derivative pricing (e.g. Black Scholes, and its child theories) is based on these ideas. Hope that helps.
OK, justin88, I see what you mean about mean-reversion vs reversion-to-the-mean. The difference is a bit subtle, but significant, and you’re right, I would say that stock returns are reverting to the mean rather than mean-reverting. However, I maintain that if stock returns are a random walk, they should not even revert to the mean. The key problem with random walks is that the long-term expected mean constantly changes with time and therefore the sequence isn’t stationary. We do agree that stock prices are a random walk, though, so that’s good.
justin88 - Can you help clarify/add? 1) Is it correct to say the Ornstein–Uhlenbeck process is considered a ‘mean-reverting’ process? This is used in modeling rates (i.e. Hull-White). 2) Is it correct to say the Wiener process is not a ‘mean-reverting’ process but it shows a regression to the mean? GBM is used to model stock prices/returns. 3) Is it fair to say the key distinction between OU and the Wiener process is the drift term? This is what differentiates between ‘mean-reverting’ or not. I’m a bit confused as I was under the impression a stationary process by definition mean-reverts - is that not the case? To apply this I was further under the impression that stock return time series are stationary and mean-reverting - though the correct way to look at this is they are a stationary random walk, right? A white-noise process is an example of a stationary random walk, right? Thanks for the input, your posts have been helpful.
He said “Weiner process”, he he he.
> Ornstein–Uhlenbeck process is considered a ‘mean-reverting’ process Yes. In fact it’s the most widely used MR process in finance. > Wiener process is not a ‘mean-reverting’ process correct. Its independent increments however are stationary. See http://en.wikipedia.org/wiki/Wiener_process > but it shows a regression to the mean? Not sure what you mean by that, and the phrase was incorrectly intepreted by justin earlier. I’ll address below. > Is it fair to say the key distinction between OU and the Wiener process is the drift term? Depends on your definition of “key”. I’d say the standout feature of O-U is mean reversion. YMMV. > a stationary process by definition mean-reverts yes > stock return time series are stationary as typically modeled, yes (see http://en.wikipedia.org/wiki/Share_price for some caveats however) > and mean-reverting That’s implied by being stationary > they are a stationary random walk contradictory. Random walks are a classic example of nonstationary processes. Back to Justin, before he misleads too many readers… > Whether the predicted value is the same as the current value is not relevant. It is neither indicative nor non-indicative of something being a random walk Well in fact a random walk implies that predicted value is the current value (plus drift), so if the predicted value is not the drifted current value we know we’re not looking at a random walk. > > If last year’s stock return is 25%, We think that 25% was an > > unusually high year, and that the chances of > > getting a similar year to that are (due to > > momentum) a bit higher than normal, but it’s much > > more likely that we will get a return closer to > > the historical average of 8%. In other words… > > that stock returns mean revert (we could do the > > reverse logic for an unusually bad year). > > That is not mean reversion. Well, in fact what Bchad described is exactly mean reversion. Here the relationship stands because sustained equity price outperformance is usually followed by underperformance, as exhibited by long-term stability of P/E. > What you are referring to is generally called “regression toward the mean”, Regression to the mean is an artifact of statistic sample size; it has nothing to do with what Bchad correctly described. You can read http://en.wikipedia.org/wiki/Regression_toward_the_mean if you want to learn the difference. > OK, you’ve kind of got it but you’re also a bit confused here. Stock prices and returns are BOTH random walks, That’s mathematically impossible. > with prices being geometric random walks (when there’s drift). If you mean by this “arithmetic RW with drift is a geometric RW”, that’s wrong. Perhaps by “(when there’s drift)” you mean simply “(with drift)”.
LPoulin133 Wrote: ------------------------------------------------------- > justin88 - Can you help clarify/add? > > 1) Is it correct to say the Ornstein–Uhlenbeck > process is considered a ‘mean-reverting’ process? Yes > 2) Is it correct to say the Wiener process is not > a ‘mean-reverting’ process but it shows a > regression to the mean? GBM is used to model > stock prices/returns. Correct, and yes wrt regression toward the mean. RTTM is not a real phenomenon. It’s just what statistically non-savvy people tend to observe, after observing outliers. For instance if the Knicks normally put up 25 points a quarter, but score 40 points in the first quarter. One could say, in between the first and second quarters, that the average points per quarter for that game will “regress toward the mean”. That is, it’s more likely the Knicks will continue to score 25 points a quarter than 40; it is unlikely they will average 40 points per quarter for the whole game. Therefore the points per quarter number will likely be closer to the mean than to the outlier as more samples come in. > 3) Is it fair to say the key distinction between > OU and the Wiener process is the drift term? This > is what differentiates between ‘mean-reverting’ or > not. No, they’re quite different. Namely OUs mean revert and Wieners (sorry couldn’t resist) don’t. Both can have drift. > I’m a bit confused as I was under the impression a > stationary process by definition mean-reverts - is > that not the case? No, it’s not. A truly mean reverting process has “memory” and cannot be a stationary process. > To apply this I was further under the impression > that stock return time series are stationary and > mean-reverting - though the correct way to look at > this is they are a stationary random walk, right? Returns are modeled as stationary (aka random walk aka wiener aka gbm), but there is some empirical evidence that they are not completely stationary (i.e. they do show some mean reverting tendencies). This is a theory vs practice distinction. > A white-noise process is an example of a > stationary random walk, right? Stationary just means the mean and variance are constant. I’m not familiar with a white-noise process but I’d guess that if it’s a random signal it’s stationary. > Thanks for the input, your posts have been > helpful. Glad to (try to) help.
DarienHacker Wrote: ------------------------------------------------------- > > a stationary process by definition mean-reverts > > yes This is wrong. I’m not going to address all of your points, but I will point out one additional thing below. > > > If last year’s stock return is 25%, We think > that 25% was an > > > unusually high year, and that the chances of > > > getting a similar year to that are (due to > > > momentum) a bit higher than normal, but it’s > much > > > more likely that we will get a return closer > to > > > the historical average of 8%. In other > words… > > > that stock returns mean revert (we could do > the > > > reverse logic for an unusually bad year). > > > > That is not mean reversion. > > Well, in fact what Bchad described is exactly mean > reversion. Here the relationship stands because > sustained equity price outperformance is usually > followed by underperformance, as exhibited by > long-term stability of P/E. Bchadwick was talking about momentum, which is the “opposite” of mean reversion. Momentum has been a popular trade lately. (Check out at AQR’s momentum fund “AMOMX”.) Empirically, there is evidence that outperformance can lead to future outperformance (momentum) or future underperformance (reversion) in certain limited circumstances. Nonetheless, random walk models (possibly complex ones) are generally used when modeling prices. Again, a theory vs practice distinction.
You can have short term momentum with long term mean reversion. Throw a ball in the air and momentum will keep it going up for a while, but gravity will eventually take over and the ball will “revert” to ground level. That doesn’t mean that momentum doesn’t help predict the ball’s position. Depending on your time horizon, you can predict short term trends with momentum and long term trends with mean reversion, and - assuming that the ball’s “height” is the analogue for its price - both mechanisms can make money. Now imagine that the ball can pass through the ground and that, once on the other side of the ground, the ball feels the same gravity pulling it back “up” to ground level. You can still have momentum and reversion operating simultaneously. Now imagine that the ground “trends” higher (like a rising water level), so that the baseline changes over time (like economic growth). Now you have a trend, mean reversion, and momentum operating simultaneously. This is actually pretty close to my mental model to how the equity market works, although the trending behavior is itself kinda noisy. Now imagine that there’s a gusty wind blowing in somewhat random directions, sometimes pushing the ball higher, sometimes lower, sometimes not doing anything. That’s the analogue to random noise. It’s really not that hard to see how these things can be believed and traded simultaneously.
bchadwick Wrote: ------------------------------------------------------- > It’s really not that hard to see how these things > can be believed and traded simultaneously. Yes but the magnitude of the force from the wind >> magnitude of gravity
to clear up some definitions: 1) processes that are not mean-reverting: random walk = (arithmetic) Brownian motion = Wiener process, used to model stock returns (i.e. log-returns), can be with or without drift, can be positive or negative geometric Brownian motion = used to model stock prices = non-negative = is NOT random walk = the log of the stock price (i.e. the stock return) follows arithmetic Brownian motion (random walk) 2) processes that are mean-reverting: (arithmetic) Ornstein-Uhlenbeck = NOT random walk = can be positive or negative = has been used to model interest rates (Vasicek model?) geometric Ornstein-Uhlenbeck = NOT random walk = positive = often used to model commodites = the log of the asset price follows arithmetic O-U process