Showing posts with label Volatility. Show all posts
Showing posts with label Volatility. Show all posts

Sep 26, 2010

Equity Returns and Mean Reversion

One of the most triggering questions - given the current crisis - is:

Will equity returns recover?

Mean Reversion
In 2009 the S&P-500 index - as most stock market indices - reached the lowest level since the turn of the century. In less than two years time world stock indices had dropped around fifty percent of their value. Since then, stock indices increased about forty percent.

It's tempting to think that this recovery could have been predicted in advance. This suspected predictable effect of recovering stock prices returning to their long-term average, is called: 'Mean Reversion'.

More explicitly: 'Mean Reversion of stock prices' is the effect that abnormal stock prices gradually return to their long-term historical average or equilibrium price.

Reversion Speed
In a 2010 working paper, the Dutch regulator DNB provides an answer to this question of recoverability.  In this paper, authors Spierdijk, Bikker and Van den Hoek analyze 'mean reversion in international stock markets' in seventeen developed countries during the period 1900-2008.

One of the outcomes of this study is not only an interesting country spread between 'mean returns' and volatility (risk, standard deviation), but also a mind boggling country difference in 'reversion speed' (rs).  Reversion speed can be defined as the 'yearly interest speed to return to the long-term average. RS differs strongly per country, as the next slide shows:

Ranked by average return (all %):

Reversion conclusions
The DNB study concludes that in the period 1900-2008:
  • Average Return
    The average World Stock Return is estimated at 8.0% with a volatility of 16.7% (S.D.).

  • Half-Life Reversion Period (HLRP)
    It takes 'World Stock Prices' on average about 14 years  to absorb half (!) of a shock (HLRP), with a confidence interval of [10 years -21 years]

  • High Half-Life Uncertainty
    The uncertainty of the half-lives estimates is very high. This is due to the fact that the lower bounds for the corresponding median unbiased estimators are close to zero. The upper bounds of the confidence intervals for the half-lives are therefore very high.

  • Mean Reversion, a Trading Strategy?
    The relative low value of the mean reversion rate, as well as its huge uncertainty, severely limits the possibilities to exploit mean reversion in a trading strategy

Concluding Remarks
We should keep in mind that - no matter how well investigated - historical data - as always - only have a limited predictive power.

Looking with a 'actuarial eye' at the volatile annual development of the S&P-500 returns and their moving averages, it's hard to deny some kind of visual proof of an increasing volatile yearly return and a structural declining 10- or 15-years average return.....

This 'visual proof', combined with the results of the 'DNB Mean Reversion paper',  is perhaps the best indicator that the future average long term World Stock return of 8% is probably way too optimistic and still includes too much the optimist mood and hope of the last decades of the 20th century...

S&P-500, averages annual returns and inflation 1950-2010

Distribution Rate
Inflation Real
Price Change
Total Return
1950's 13.2% 5.4% 19.3% 2.2% 10.7% 16.7%
1960's 4.4% 3.3% 7.8% 2.5% 1.8% 5.2%
1970's 1.6% 4.3% 5.8% 7.4% -5.4% -1.4%
1980's 12.6% 4.6% 17.3% 5.1% 7.1% 11.6%
1990's 15.3% 2.7% 18.1% 2.9% 12.0% 14.7%
2000's -2.7% 1.8% -1.0% 2.5% -5.1% -3.4%
1950-2009 7.2% 3.6% 11.0% 3.8% 3.3% 7.0%

Key question is : What would be a save 'long-term total return of stocks' as a base for an investment strategy, without the 'Hope Bubbles' of the last two decades of the last century?

Probably a long term stock return of about 6% would turn out to be a save basis for a kind of investment reversion strategy......
However, now we know where we are going, it's absolutely necessary to know where we are now? Unfortunately.... we don't know.... ;-) 

Sources, related links:
- DNB 2010: Mean Reversion in International Stock Markets
- (Dutch) DNB-2010: Herstel aandelenmarkten is niet vanzelfsprekend
- Wikipedia: S&P-500 Annual Returns 
Simple Stock Investing: S&P-500 historical data

Aug 7, 2008

The initial price of volatility

We'll start this blog with some theory about volatility.
A short excerpt form the original
interesting article on Estopedia, called "The Uses And Limits Of Volatility" by David Harper,CFA, FRM

One of the theoretical properties of volatility may or may not surprise you: it erodes returns.

This is due to the key assumption of the random walk idea: that returns are expressed in percentages. Imagine you start with $100 and then gain 10% to get $110. Then you lose 10%, which nets you $99 ($110 x 90% = $99). Then you gain 10% again, to net $108.90 ($99 x 110% = $108.9). Finally, you lose 10% to net $98.01. It may be counter-intuitive, but your principal is slowly eroding even though your average gain is 0%!

If, for example, you expect an average annual gain of 10% per year (i.e. arithmetic average), it turns out that your long-run expected gain is something less than 10% per year. In fact, it will be reduced by about half the variance (where variance is the standard deviation squared). In the pure hypothetical below, we start with $100 and then imagine five years of volatility to end with $157:

The average annual returns over the five years was 10% (15% + 0% + 20% - 5% + 20% = 50% ÷ 5 = 10%), but the compound annual growth rate (CAGR, or geometric return) is a more accurate measure of the realized gain, and it was only 9.49%. Volatility eroded the result, and the difference is about half the variance of 1.1%. These results aren't from a historical example, but in terms of expectations, given a standard deviation of σ (variance is the square of standard deviation, σ^2) and an expected average gain of μ, the expected annualized return is approximately μ - ( σ^2 ÷ 2).

The 'hidden secret' in this article is in the last line.

We may define the Volatility Rate Loss (VRL) here as the difference between the Average Annual growth Rate (AAGR) and the (real) Compound Annual Growth rate (CAGR).


Let's calculate the VRL for several volatilities:

Volatility VRL
σ ( σ^2) / 2
3% 0,03%
5% 0,13%
10% 0,50%
15% 1,13%
20% 2,00%
25% 3,13%
30% 4,50%
35% 6,13%
40% 8,00%

From this simple table, and given the fact that actual volatility of several main indexes (shares) are varying around 20-30% and that volatilities of 40 (or more) of investments in individual companies are no exception, it's clear that we can not neglect the influence of VRL.

For example if we take an investment with an average return of 10% and a volatility ( σ ) of 20%, the average return (of 10%) will show 2% too high.

Another way of putting it (in this last case) is that when you decide to step over from a non-volatile investment to a volatility risk type of investment, your investment should return on average 2% higher to give the same return as your non-volatile investment. In a way VRL is the initial price you pay for volatility sec.

So, when judging returns, don't look at average returns, but always look at the Compound Annual Growth Rate.