Asset Prices Always Fall After Hitting Amaximum

8/12/2019 Asset Prices Always Fall After Hitting Amaximum

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Definition of 'Mean Reversion'

A theory suggesting that prices and returns eventually move back towards the mean or average.

This mean or average can be the historical average of the price or return or another relevant

average such as the growth in the economy or the average return of an industry.

Explaination

Mean reversion is a mathematical concept sometimes used for stock investing, but it can be

applied to other assets. In general terms, the essence of the concept is the assumption that both a

stock's high and low prices are temporary and that a stock's price will tend to move to the

average price over time.

Using mean reversion in stock price analysis involves both identifying the trading range for a

stock and computing the average price using analytical techniques taking into account

considerations such as earnings, etc.

When the current market price is less than the average price, the stock is considered attractive for

purchase, with the expectation that the price will rise. When the current market price is above the

average price, the market price is expected to fall. In other words, deviations from the average

price are expected to revert to the average.

Stock reporting services commonly offermoving averages for periods such as 50 and 100 days.

While reporting services provide the averages, identifying the high and low prices for the study

period is still necessary.

Mean reversion has the appearance of a more scientific method of choosing stock buy and sell

points than charting, because precise numerical values are derived from historical data to identify

the buy/sell values, rather than trying to interpret price movements using charts (charting, also

known astechnical analysis).

Someasset classes, such as exchange rates, are observed to be mean reverting; however, this

process may last for years and thus is not of value to an investor.

Mean reversion should demonstrate a form of symmetry since a stock may be above its historical

average approximately as often as below.

A historical mean reversion model will not fully incorporate the actual behavior of a security's

price. For example, new information may become available that permanently affects the long-

term valuation of an underlying stock. In the case of bankruptcy, it may cease to tradecompletely and never recover to its former historical average.

Mean Reversion

In DeBondt and Thaler the hypothesis of investor over and reaction is related to the concept of

mean reversion. It is a tendency of stochastic process to remain near, or veer towards a long-run
http://en.wikipedia.org/wiki/Moving_averagehttp://en.wikipedia.org/wiki/Technical_analysishttp://en.wikipedia.org/wiki/Asset_classeshttp://en.wikipedia.org/wiki/Asset_classeshttp://en.wikipedia.org/wiki/Technical_analysishttp://en.wikipedia.org/wiki/Moving_average


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average value. According to riskglossary.com, interest rates and implied volatilities tend to

exhibit mean reversion, so do stock market returns. Figure 2 illustrates the difference between

mean reverting and non-mean reverting behaviour. (Source: riskglossary.com downloaded 5th

March, 2008)

Figure 1 Mean reverting and non-mean reverting behaviour

The mean reversion model may be exploited to make extra gains. In general terms, the idea is

that high or low periods in the stock market are only temporary. First, we identify a trading

range for an instrument, and then compute the mean. Sophisticated calculations relate to assets

and earnings of the firm. When the spot price in the stock exchange goes below the average

price, the stock becomes attractive for investors who rush to place purchasing push to raise its

price. When they over-react, the high priced stock is expected to fall. Eventually, prices will

converge to the mean in the long-run. How long is the long-run? That is an issue open forargument.

Foster and Stine (2003) studied the incremental added-value of mean-reverting trading strategies.

They introduce a test to determine whether a particular investment strategy can yield profits

superseding returns of a buy-and-hold investment in the S&P index. They regress excess returns

of the selected strategy against the excess returns from the buy-and-hold investment in the S&P

index. The obtained t-statistic as well as the p-value of the intercept indicates whether adding a

new strategy leads to a significant improvement in the performance of the portfolio. They adjust

the p-values using the Bonferonni1 corrections for multiple comparisons. If the regression

intercept were statistically significant, then that means that the particular strategy did add valueto the original strategy of buy-and-hold the S&P index. The concept behind this test is that a

strategy that gives a positive mean return and is not too highly correlated to the benchmark index


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(S&P in this case) can be linearly combined with the index to obtain a better mean-variance

return profile. Put simply, any strategy that proves to be an adequate supplement to diversify

holdings in the benchmark index can add value.

Chua et al. (2004) examined this strategy on their mean-reverting yield-curve strategies. Their

objective was to test the profitability of their strategies that build on the notion that the yieldcurve mean-reverts to an unconditional yield curve. The results showed that a number of these

yield-curve trading strategies can yield high profits. This was especially true of the trading

strategies that focused on mean-reversion of the yield spreads and curvatures. These strategies

managed to substantially outperform two commonly used benchmarks of investing. When

transaction costs were included in the model, profitability of the trades against the benchmarks

dropped, yet significant results were sustained for some of these strategies. The authors

suggested applying structured derivative trades that mirror the underlying cash flows in order to

reduce the frequency of the trades, thus lower transaction costs considerably.

The Momentum Effect

Fama and French (1996) have also tested two versions of momentum strategies. DeBondt and

Thalers (1985) meanreversion anomaly is contradicted by Fama and French (1996). Fama andFrench tested their 3 factor model (see chapter 4.3) and found that portfolios formed based on

past winner and losers demonstrated continuation or momentum rather than the contrarian effect

of DeBondt and Thaler. A previous study by Jegadeesh and Titman (1993) corroborates the

concept of momentum. The sample periods they use are the 19651989 period. The strategythey tested, that picks stocks according to their last 6-month returns and holds them for 6 months,

realises a compounded on average excess return of 12.01% per year. The authors find further

evidence indicating profitability of their strategies is not due to systematic risk. Moreover, testresults also indicate that abnormal profits are not attributed to lead-lag effects resultant from

delayed stock price adjustments to universal factors. When compared to the CAPM benchmark,

results confirm the momentum effect as being quite large and reliable.

In their answer to the mean reverting theory, the authors suggests that the evidence of initialpositive and later negative relative strength returns suggests that common interpretations of

return reversals as evidence of overreaction and return persistence (i.e., past winners achieving

positive returns in the future) as evidence of underreaction are probably overly simplistic. A

more sophisticated model of investor behaviour is needed to explain the observed pattern of

returns. One interpretation of [their] results is that transactions by investors who buy pastwinners and sell past losers move prices away from their long-run values temporarily and

thereby cause prices to overreact. (Jegadeesh et. al. 1993 p. 90) This interpretation is consistentwith other studies that explored the implications of the so-called "positive feedback traders" on

market price. Jegadeesh and Titman also look at the question why it is possible that the market

underreacts to information about the short-term prospects of firms but overreacts to information

about their long-term prospects. They bring up two reasons for The nature of the information


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available about a firm's short-term prospects, such as earnings forecasts, is different from the

nature of the more ambiguous information that is used by investors to assess a firm's longer-term

prospects. The evidence in this paper does not allow us to distinguish between these two

hypotheses about investor behaviour. (Jegadeesh et. al. 1993 p. 90)

Other Anomalies

Other anomalies include the following.

Standard & Poors Index Effect

Financial literature refers to the Standard & Poors index effect as being observable as soon as acompanys inclusion in the index is announced. The perception was induced by the increaseddemand from index fund partakers for the stocks involved in index composition changes.

Kappou, Brooks and Ward (2007) studied the S&P 500 inclusions and examined the impact of

potential overnight price adjustments after the announcement of an S&P 500 index change. The

authors found evidence of a significant overnight price change that diminish[ed] the profitsavailable to speculators although there [were] still profits available from the first day afterannouncement until a few days after the actual event. (Kappou et. al. 2007 p. 21)

Initial Public Offerings

The IPO effect was studied as it appeared to cause an anomaly. Knopf and Teall studied the IPO

underpricing anomaly in an attempt to explain returns of the initial trading day. They found

evidence in support of the asymmetric information theories of IPO underpricing. When theinvestor bought the listed shares in the last period of the offering day, he would realise below

average returns on the long-run. This effect was present even when companies were segmented

according to industry or according to capitalisation. The underperformance was compared to

several benchmark indexes. The reasons are that on the day of the offering prices jump, and

buying at the closing price will yield worse than other investment opportunities. The companies

that enlist to the stock exchange are usually over-valued, because it is not worth offering under-

valued shares. Moreover, it is not uncommon to window-dress financial statements of

companies prior to being listed for an IPO. For instance, they defer certain costs in order to

improve their net profit and report more favourable results. Further explanation refers to the loss

of efficiency when a companys division loosens control over its achievement.

Closed-end funds

A closed-end fund is a financial instrument that is listed on a certain day, at a certain quantity,

and then initial offering takes place in a secondary market typically from a broker. The value of

the investments in the fund, and the premium (or discount) placed on it by the market are what


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determine the price of a share in a closed-end fund. The total value of all the securities in the

fund is divided by the number of shares in the fund. This is called the net asset value (NAV).

The market price of a fund share is frequently higher or lower than the NAV. When the fund's

share price is higher than NAV it is selling at a premium; conversely when it is lower, it is

selling at a discount to the NAV.Closed-end fund shares, in contrast to open-end funds, are

priced by the market, often substantially diverging from the NAV of the fund assets. Open-end

funds are only available for buying and selling at the close of business each day, at the calculated

NAV, and for which orders must be placed in advance, before the NAV is known.

The market prices of closed-end funds are often ten to twenty percent different than the NAV

while the value of an exchange traded fund would only very rarely differ from the NAV by more

than one-fifth of a percent. Malkiel (1977) says that this is in startling contrast to securitiesmarket efficiency. He develops some theoretical principles concerning the valuation of shares of

closed-end investment companies. His conclusion is that while the structure of discounts can be

partially explained on the basis of theoretical principles, the sizes of the discounts are far largerthan warranted. The years he tested from 1970 to 1990 demonstrated an average discount

ranging from 5% to 20%. Malkiel made several conclusions as to what justifies these premiums

and discounts. Discounts were related to unrealized appreciation (during the period when fundshad unrealized appreciation) and to distribution policy with respect to capital gains, as well as to

portfolio policies concerning investing in letter stock and foreign securities. (Malkiel 1977p.857) However, the general explanation he gives is a psychological one: generally, closed-end

companies sell at discounts because they must be bought through brokers- brokers who are not

enthusiastic to sell them. The problem stems from investor habits whereas investment funds are

not all that popular. It is unlikely to occur to the mind of the average investor to buy such a fund.

Rather, the public is sold fund shares by the brokers or salesmen. Brokers prefer to sell thosetypes of securities that earn them the largest amount of commission. Closed-end funds include

way less commission than other financial products. Moreover, it is very likely for such a sale to

be a one-time deal as the investors are even less likely to trade closed-end as they do with

ordinary shares. Malkiels on the other hand explains, that this is still not justifiable reason for

the closed-end funds to be discounted to this extent, and thereby the current mispricing should

provide individual investors great opportunities. It seems likely, that the pricing of closed-end

investment-company shares does illustrate an example of a market imperfection in the valuation

of capital assets.

Weather

The weather anomaly is not related to weather derivatives, contrarily, it is concerned with the

direct relationship between investment ambiance and meteorological factors. The empirical

investigation of Saunders in 1993 revealed the effect of weather on human (investor) behaviour.

He tested the null hypothesis that stock prices from exchanges in New York City have not been

systematically affected by local weather. Saunders showed that there is a negative connection


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between the cloudy appearance and the yield of stock markets. His conclusion was that thediscovery that the weather in New York City has a long history of significant correlation with

major stock indexes supports the view that investor psychology influences stock prices. The

causal linkage in these correlations is strongly supported both by the extensive experimental and

survey literature indicating that weather influences mood. (Saunders 1993 p.1344) Theestablishment of causal direction between a temporal, economically insignificant, local, mood

influence and asset prices explains some of the surprisingly large economic impact this anomaly

instigates. The empirical findings of Saunders support arguments for the inclusion ofeconomically neutral behavioural variables in models of asset-pricing and cast doubt on the

hypothesis that security markets are entirely rational. (Saunders 1993 p.1345)

ASSET PRICES ALWAYS FALL AFTER HITTING

AMAXIMUM

As discussed in the introduction, there are many possible definitions of mean reversion, whichseems to mean different things to different people. The broadest definition is probably as

follows:

Definition 1: An asset model is mean reverting if asset prices tend to fall

(rise) after hitting a maximum (minimum).

The definition is popular because it is linked to a straightforward test. Look at historically

extreme stock market highs, and establish whether the market subsequently fell.

Using this definition, many analysts can convince themselves that stock marketsobviously mean revert. For example, (so the thinking goes), the stock market was clearly

overvalued in the summer of 1987 and also in the later 1990s. This overvaluation

explains the subsequent falls. Or, a common perception is that equity markets mean

revert because it so happened that equities rose after the 1974 market low was reached.

The trouble with this definition is its breadth. It is a truism that markets fall after hitting amaximum because a local maximum necessarily hits a higher value than those on nearby

dates. Any process at all is mean reverting in this sense. So although the 1987 crash

might satisfy this definition of mean reversion, this proof is of no use in refining the class

of appropriate models. It is helpful therefore to move to a narrower definition, which will

be harder to validate empirically, but also distinguishes better between classes ofstochastic processes.

Before we move on, however, it is worth making the point that it can be perfectlyconsistent with an efficient and arbitrage free market to have some processes in an asset

model where all market participants do indeed know that a high or low point has been

reached.


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To illustrate this, consider a trivial deterministic model. In a deterministic world stocksand cash must have identical returns and cash rates must deterministically follow the

current path of forward interest rates. Here everyone can know that interest rates (bond

prices) or stock returns have reached a high (low) or low (high) point without violating

the trivial market efficient and arbitrage free conditions. If the current forward interest

rate curve is smoothly upward or downward sloping then this could in turn be viewed as a

form (possibly the simplest conceivable form) of mean reversion.

MEASUREMENT OF MEAN REVERSION

Statistical Model Comparisons

As observed in the Introduction to this paper, unlike means, standard deviations andcorrelations, there is no single accepted numerical measure of mean reversion. To say

that a model has mean reversion of 5%, or of 500, is meaningless. Several comparisons

exist of model statistics. See for example Smith (1996), Lee and Wilkie (2001) orPriceWaterhouseCoopers (2003). None of these comparisons extend to measures of mean

reversion.

It does not help that mean reversion means different things to different people. In theparlance of econometrics a mean reverting process should be stationary, but as we have

seen, this is not a useful definition given that the main focus of attention is likely to be

the mean reversion or otherwise of stock prices. A broader usage of the term hastherefore become common in the asset modelling community, but this usage conflicts

with classical time series terminology. This confusion over terminology has hitherto

complicated the creation of standard parameter definitions.

However, the development in the previous section provides us with a useful way forwardin defining a standard diagnostic tool to establish the extent to which a particular model

displays mean reversion. Such a diagnostic tool would clearly have value to regulators,

end users and others seeking to understand key differences between various models.

Short and long term volatility

What then would be the generic parameters such a benchmark model should show? Letus suppose that the mean reversion acts like a time-varying drift as discussed above. Over

very short time periods, the drift is difficult to observe, because it is a drift per unit timeand you do not have many units of time. Or put another way, as you reduce the time

interval the volatility of the process (typically of order dt) increasingly dominates thedrift (or order dt) and nuances of the drift process will be dwarfed by the crude volatility

term.

So, sampled at frequent intervals over short time spans, a mean reverting process couldlook a lot like a random walk, with some volatility S. This is our first generic parameter.


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As short-term volatility is easy to compute and we have few constraints on data the data

may become rather granular if we look on a second by second basis but within reasonable

limits we can get quite a good handle on short term volatility by using small time steps.

We might expect different models to show some consensus on this parameter, at least for

two alternative models calibrated to the same data set.

Let us now look at very long holding periods. A mean reverting model shows dependenceof returns from one period to the next, but that influence has to decay if the two periods

are separated by a large time interval. Over periodsseparated by a wide interval, we might

expect something close to independence, because in a market efficient environment

information that impacts prices on one day is not likely to be hugely correlated with

information that happens many years later. Therefore, over very long time scales, most

models should look something like a random walk. The volatility of this long term

random walk will not necessarily be equal to the short term volatility S but insteadwould take some different value L, which forms our second parameter.

As this second parameter describes volatilities over long holding periods, it isintrinsically difficult to measure and will depend heavily on calibration. Whereas we

have (almost) unlimited access to non overlapping short time periods (e.g. hourly data)

there are not many independent ten, twenty or thirty year time periods to work with. We

can expect L to vary from one model to the next, according to the selected model-building methodology.

The difference between long and short-term volatilities already gives one way to describemean reversion. Strong mean reversion is characterised by longterm volatility well below

the short-term volatility. For a random walk, the long and short term volatilities are equal.

Measuring Volatilities: History and Simulations

6.9 There are two situations requiring volatility measurement. If we are calibrating aparticular model we need to measure volatility in actual data. If we are comparing two or

more different models (lets assume that they are black boxes) by looking at simulated

data generated by the models.

6.10 The first situation relates to either historic volatility, based on one observed timeseries or implied volatility derived from derivative prices. Essentially the major problem

in the case of taking measurements from historic data is sampling error. The observed

history is only one possible outcome of thousands that could have occurred, and there is

no guarantee the observed history is representative in any way. Lack of data is a major

problem, resulting in subtle biases we will later consider. The problems may be different, and generally less acute in the case of calibrating against

derivative price data, but they still exist. If a model is calibrated against derivative prices

then the choice of derivatives may be important the model may accurately price thechosen calibration instrument but be hopeless at pricing other instruments. The issue of

time homogeneity is also important a time decaying volatility structure may be

indistinguishable from the effects of mean reversion in a time inhomogeneous model.


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This distinction is unimportant for derivative pricing on day one, but it matters greatly if

a model is marched forward in time.

The second situation we need to consider involves the estimation of volatility fromsimulation output (generated from a black box model, say). Here, lack of data is not a

problem. If sampling error is significant one can simply run moresimulations from the

black box. Instead, the difficulty that arises is the proliferation of different ways of

calculating volatilities.

For example, consider a 1-year volatility. We could measure this as the volatility of thereturn in year 1, or alternatively as the forward start volatility of return between t=20 and

t=21. For most models, the forward start volatility will be higher. This is because for the

forthcoming year, we already know current levels of interest rates and the returns in

immediately preceding years. To some extent this may help us predict the next yearsreturns. Some of the volatility in the first year returns will be reduced by the fact that all

simulations start from common initial conditions. The forward start volatility may be

higher because each simulation starts from a different point at t=20. In exceptional circumstances, however, the forward start volatility may be lower than the

one-year volatility. This can happen in a model with stochastic volatility whose starting

conditions specify a starting volatility well above the long term mean.

From a time series analysis point of view, we are most interested in unconditionalvolatilities that have minimal dependence on initial conditions. In other words, our ideal

definition of a one-year volatility should be forward start volatility, starting some point in

the long future. In practice, simulation data may well be supplied with a finite horizon,

and so some extrapolation may be required to develop the limiting forward start

volatility.

Maximum predictability We have now defined and clarified notions of long and short-term volatility. The next

important property is required to describe where the transition occurs between short-term

behaviour and long-term behaviour. We capture this using the notion of maximum

predictability.

We concluded section 5 with a useful and quite broad definition of mean revertingprocesses as one where returns in one period are negatively correlated with those in other

periods. We have argued that this correlation is likely to be close to zero for very small

periods, and also for very long periods. It follows that there is some finite period length,

which has a largest absolute correlation with the return in some subsequent period. Wecall this the most predictive period.

It seems reasonable to assume that if one interval is most predictive of another, then thesetwo intervals will be adjacent, because every intervening interval results in a loss of

information. For similar reasons, a symmetry argument suggests that the optimally

predictive interval pair will be the same length. This optimally predictive period term is

an important attribute of the time series, as is the value of the optimum correlation


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achieved. This gives two more attributes of a mean reverting series, which we can define

in generic terms.

These attributes are shown below for the Wilkie mode The optimal predictive term may be close to 1 year, or even less. For annually projected

series it is therefore helpful to extrapolate the observed volatilities to shorter time frames,

so that a non-integer optimal predictive length can be estimated.

Thus far, we have defined four numerical attributes of a mean reverting process: Short-term volatility Long-term volatility Optimal predictive term Optimal predictive correlationThese are mutually constrained to some extent. For example, if the long term volatility is close to

the short term volatility, then the process is much like a random walk, and so we would be

surprised to find a large optimal predictive correlation. We are not here suggesting that optimal

predictive term and correlation are perfect measures. Correlation and its structure varies overtime. One needs to be pragmatic and analyse a data set to see whether correlation is so different

from 0 that the concept of optimal predictive correlation adds to the analysis.

Prediction based on Return History

We have examined the distribution of return in one period given the return over aprevious period. The observed correlation gives us a possible predictor of future returns.

Given a predictor of a future price, we can decompose the price variance as:

The variance of the predictor (XX, in our earlier model)Plus residual variance (WW)This decomposition will vary according to the information basis of the

predictor. The predictor could be based on:

A ratio such as the dividend or earnings yield

The historic return over a particular past period

The entire history of historic returns

We have described the first of these using the optimal predictive correlation. The secondform of predictor was discussed, in terms of past data, in section 2. Economically, it is

also relevant to establish the extent to which future movements can be predicted given all

past movements. This reconciles to the economic notion of weak-form market efficiency.

Therefore, we have another potential statistic:

Predictive correlation based on total historyAs this is the result of a multivariate conditional expectation, the calculation is delicate, bothfrom the point of view of inverting large matrices and also the tendency for sampling error to

generate spurious evidence of predictability. The regression actually gives us an r2 statistic from

which we extract a square root to express as a correlation.


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We could choose the time horizon over which to measure the predictive correlation; themost natural horizon for comparison purposes is the previously defined optimal

predictive term.

Information structures

Time series properties are usually defined in terms of forward start volatilities. However,many actuarial investigations start from known initial conditions and therefore the initial

start volatilities are more relevant. An information structure is a way of relating initial

start volatilities to the forward start volatilities.

Asset Prices Always Fall After Hitting Amaximum

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