Market Drawdowns

180 Years of
Market Drawdowns
IHES 2015 — 30 June 2015
Robert J. Frey, Ph.D.
FQS Capital & Stony Brook University

Bio
Worked as Systems Engineer and Program Manager in Defense Industry.
Ph.D. in Applied Mathematics, Stony Brook University.
Recruited into Morgan Stanley’s APT (Pairs Trading) Group in mid-80s.
Started Kepler Financial in 1989; Kepler acquired by Renaissance in 1992.
Managing Director with Renaissance; Developed Nova and Equimetrics Funds which
were subsequently absorbed into Medallion. Retired in 2004.
Former Advisory Board Chair of the University of Chicago’s Program on Financial
Mathematics.
Founder and Co-Director, Program in Quantitative Finance, Applied Mathematics and
Statistics, Stony Brook University.
FQS Capital, Investment Manager and Consulting Services, out-growth of Family Office.
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Myopia
Day-to-day details
overwhelm us narrowing
our focus.
Frequently, relatively
simple models with
longer horizons yield
very useful insights.
You don’t design a house
based on the weather
report.
Source: http://keplerianfinance.com/2013/06/the-relevance-of-history/
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Market Drawdowns
An investment’s drawdown behavior is an important element
of its behavior.
We will focus on a single market: the S&P 500 Total Return
from 1835 to 2015 (Global Financial Data).
Important questions:
How can drawdowns be modeled and analyzed?
How stable is this aspect of performance?
What insights can we develop examining the drawdowns in
an important market over an extended period?
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Drawdowns
Cumulative log returns:
t
⎧
c = ⎨c(t) = ∑ log[W s
/W s−1
]
⎩ s=1
t ∈1,…, T
⎫
⎬
⎭
Drawdown state:
d = { d(t) = max ⎡
⎣
0,max ⎡⎣ c(s) s ∈{1,…, t} ⎤ ⎦ − c(t) ⎤
⎦
t ∈1,…, T }
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Drawdowns
Drawdown partition:
δ = { δ i
= { d(t i−1
+1),…,d(t i
)} i
∀d(t) ∈δ i
,(d(t) = 0) ∨ (d(t) > 0) }
Drawdown process:
D =
⎧
⎪⎛
⎨⎜
⎩⎪ ⎝
Δ 1
= max[δ 1
]
τ 1
= t 1
⎞ ⎛
⎟ ,… Δ i
= max[δ i
]
⎜
⎠ ⎝ τ i
= t i
− t i−1
+1
⎞ ⎫
⎪
⎟ ,… ⎬
⎠ ⎭⎪
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Full Dataset
1835-2015
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One spends about 75% in a
drawdown state.
And more than half the
time in a major drawdown.
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Max Drawdown Distribution
Gamma-Exponential Mixture:
∞
∫0
⎛
⎝
⎜
e −λ/β β −α λ −(1+α ) ⎞
Γ α ⎠
⎟
[ ]
( λe −λx
)dλ
Variant of a Pareto Distribution (Lomax)
f [Δ] = αβ(1+ βΔ) −(1+α )
F[Δ] = 1− (1+ βΔ) −α
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Gaussian
Simulation
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Stable
Simulation
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Pre Great Depression
1835-1928
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27

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Post World War II
1950-2015
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Comparisons
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Comparing Drawdowns
The basic character of the max drawdown
process is reasonably similar across time.
The recent period does show slightly lower
incidence of larger drawdowns.
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Conclusions
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Conclusions
Market drawdowns, as a driver of “regret” may play a greater role than we
realize.
The character of the market over the past 180 years does not seem to have
changed greatly.
A realistic statistical model awaits. We are unable to adequately model
returns to reproduce the observed market.
It would be a mistake to view the Great Depression as an “outlier”.
Taking the long historical view provides insights that are valid today.
Any study such as this suffers from survivor bias and the fact that we are
likely to have underestimated our α exponents.
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Thank You
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