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Abstract

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.
Working Paper
Series
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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.
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Luenberger, D.G. Investment Science (1998).