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Working Paper

Series

_______________________________________________________________________________________________________________________

National Centre of Competence in Research

Financial Valuation and Risk Management

Working Paper No. 374

The Joy of Volatility

Michael A.H. Dempster Igor V. Evstigneev

Klaus Reiner Schenk-Hoppé

First version: August 2006

Current version: April 2007

This research has been carried out within the NCCR FINRISK project on

“Behavioural and Evolutionary Finance”

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1

The Joy of Volatility

Michael A.H. Dempster1, Igor V. Evstigneev2, Klaus Reiner Schenk-Hoppé3

1Judge Business School, University of Cambridge, UK,

2Economic Studies, School of Social Sciences, University of Manchester, UK,

3School of Mathematics and Leeds University Business School, University of Leeds, UK.

Modern portfolio theory regards the return of an asset as its upside, while volatility

is seen as its downside. This view is shared by the majority of investors who dislike

volatile markets. Recent results in financial mathematics, however, show that

volatility is actually good, rather than bad, for financial growth. Very simple active

portfolio management opens up this profitable opportunity to generate growth

from volatility.

Can there be any profitable investment when all assets in a market destroy, rather

than create, value? Thanks to volatility, the answer is yes – even if one does not wish to

risk bankruptcy by going short (i.e. by selling what one does not own). This result can

be illustrated with the aid of a simple model of a financial market with only two risky

assets, whose returns in each period are determined by flipping a fair coin (Figure 1).

Placing one’s money in Asset 1 will, on average, reduce one’s investment by one-

tenth within 10 periods. Buying and holding Asset 2 will result in losing as much as

one-third of one’s investment within the same time span. Since growth rates depend on

the logarithm of the (gross) return, investing in the apparently profitable Asset 1, with

net returns of +40% and -30%, actually results in a loss of money. Any buy-and-hold

investment, which purchases both assets, but does not update the positions, can only do

as well, in the long term, as the best-performing asset. Poverty is the inevitable fate of

the passive investor.

2

Consider making an investment according to a simple active management style:

buying or selling assets so as to always maintain an equal investment in both. On

average, wealth will double in 80 periods and grow without limits. This investment style

rebalances wealth according to a constant proportions strategy. It succeeds, where buy-

and-hold fails, because of the volatility of asset returns.

It has recently been proved mathematically1,2,3 that, with stationary asset returns,

every constant proportions rebalancing strategy beats the corresponding market index

(defined as the weighted average of individual asset growth rates). In particular, if one

only invests in the assets growing at maximum rate, or if the market is volatile enough,

any such strategy will beat the best buy-and-hold portfolio. Though examples of this

phenomenon have been reported for quite some time4, and the best rebalancing strategy

is well-known for performing at least as well as any buy-and-hold portfolio5, it is

surprising to learn that no conjecture has ever been made as to the validity of a general

growth-volatility link.

The power of rebalancing strategies is often claimed to be the result of “buying

low and selling high.” However this is the gambler’s fallacy, arguing that the longer the

run of black numbers, the higher the odds of red numbers at the next spin of the roulette

wheel. When returns are determined by the flip of a coin, an asset’s upside and

downside potential does not change over time. Such an asset is not cheap or expensive

at any point in time. Nor does arbitrage (the opportunity to get ‘something for nothing’)

drive this phenomenon – the market in the example is free of arbitrage. Finally, all

investors have equal opportunities, unlike in Parrondo’s paradox6 where some investors

(depending on their wealth) are given favorable odds, this excludes the dynamics of the

wealth distribution as an explanation.

3

The engine that generates growth from volatility is in fact an elementary

mathematical relation, the Jensen inequality. It describes the effect of interchanging

concave (or convex) functions and weighted averages, i.e. expected values. Constant

proportions rebalancing strategies combine random returns in fixed proportions. The

logarithm of this financially engineered return is higher than the combination of the

assets’ individual growth rates, because the logarithm is a strictly concave function. For

fixed, deterministic returns, both quantities are equal, and no excess growth can be

achieved.

This financial market phenomenon closely resembles observations on stochastic

resonance7, a theory in physics that has various applications, e.g. in biology and

neurophysiology. Similar to amplifying a weak signal by adding noise to a nonlinear

system, constant proportions strategies combine two random processes to achieve an

increase in the growth rate. The required nonlinearity is provided by the compound

return on the investment and the logarithmic function that appears in the growth rate of

wealth.

While the implications of the growth-volatility link for asset pricing and portfolio

theory are still being explored, active investors should enjoy the bumpy ride of

volatility. However, as with any investment advice, a word of caution is in order:

Constant proportions strategies do well in the long term but, over short time horizons,

their superior performance cannot be guaranteed!

4

Figure 1. Performance of passive and active investment styles

Asset 1 Asset 2

Heads Tails Heads Tails

Net return on assets

+40% -30% -20% +15%

Growth rate of assets ½ ln(1.4) + ½ ln(0.7) ≈ -0.010 ½ ln(0.8) + ½ ln(1.15) ≈ -0.042

Buy-and-hold strategies cannot do better than Asset 1 in the long-term. All passive styles lose.

Heads Tails

Net return on active

investment (1:1 mix) ½ 40% + ½ (-20%) = +10% ½ (-30%) + ½ 15% = -7.5%

Growth rate of investment ½ ln(1.10) + ½ ln(0.925) ≈ +0.0087

The constant proportions strategy generates positive growth. It beats all passive styles.

Simulation of wealth dynamics

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

0 1000 2000 3000 4000

Active:

Rebalancing

Logarithm of wealth

Passive:

Holding Asset 1

Passive:

Holding Asset 2

Number of coin flips

5

1. Dempster, M.A.H., Evstigneev, I.V. & Schenk-Hoppé, K.R. Volatility-induced

financial growth. Quant. Fin. (forthcoming).

2. Dempster, M.A.H., Evstigneev, I.V. & Schenk-Hoppé, K.R. Exponential growth of

fixed-mix strategies in stationary asset markets. Fin. & Stoch. 7, 263–276 (2003).

3. Evstigneev, I.V. & Schenk-Hoppé, K.R. From rags to riches: on constant proportions

investment strategies. Int. J. of Th. and Appl. Fin. 5, 563–573 (2002).

4. Luenberger, D.G. Investment Science (1998).

5. Algoet, P.H. & Cover, T.M. Asymptotic optimality and asymptotic equipartition

properties of log-optimum investment. Ann. of Prob. 16, 876–898 (1988).

6. Harmer, G.P. & Abbott, D. Losing strategies can win by Parrondo’s paradox. Nature

402, 864 (1999).

7. McClintock, P.V.E. Random fluctuations: unsolved problems of noise. Nature 401,

23–25 (1999).

Correspondence and requests for materials should be addressed to K.R. Schenk-Hoppé

(k.r.schenk-hoppe@leeds.ac.uk).

Financial support by the National Centre of Competence in Research "Financial Valuation and Risk

Management" (NCCR FINRISK) is gratefully acknowledged.