BRIEF SURVEY OF STOCHASTICPORTFOLIO THEORY

IOANNIS KARATZAS

Department of Mathematics, Columbia Universityand

INTECH Investment Technologies LLC, Princeton

SQA / MAFN Talk, Columbia University 19 March 2015

For Information Purposes Only

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SYNOPSIS

This lecture offers an overview of Stochastic Portfolio Theory(SPT), a rich and flexible framework for analyzing portfoliobehavior and equity market structure.

SPT was introduced by E. Robert Fernholz in a series ofpapers in the 1990’s, then consolidated in his 2002 monographby the same title.. Considerable progress has occurred since; some of it issurveyed in a review paper that can be downloaded fromhttp://www.math.columbia.edu/∼ik/FernKarSPT.pdf

The theory is descriptive as opposed to normative; is consistentwith observable characteristics of actual markets and portfolios;and provides a theoretical tool which is useful for practicalapplications.

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SPT does not rely on such normative assumptions as

• absence of arbitrage,

• economic equilibrium, or

• existence of an equivalent martingale measure.

At the same time, it is not incompatible with them.

. More specifically, SPT explains under what conditions itbecomes possible to outperform a cap-weighted benchmarkindex – and then, exactly how to do this by means of simpleinvestment rules.

These typically take the form of adjusting the capitalizationweights of an index portfolio to more efficient combinations.

They do it by exploiting the natural volatilities of asset prices, andneed no forecasts of mean rates of return (in “model-free” fashion).

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FRAMEWORK

Asset capitalizations X1(·), · · · , XN(·) . We denote by

dXi (t) = Xi (t + dt)− Xi (t)

the change in the capitalization of asset i = 1, · · · ,Nover the short time interval [t, t + dt] . Then

dXi (t)

Xi (t)=

Xi (t + dt)

Xi (t)− 1

is the arithmetic return of asset i over this interval.Let us stipulate that, given all available informationup to time t , this random quantity has conditional

mean αi (t) · dt and variance Sii (t) · dt .

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For two different assets i 6= j , the arithmetic returns

dXi (t)

Xi (t),

dXj(t)

Xj(t)have conditional covariance Sij(t) · dt

given all available information on asset capitalizations up to time t .

• We have then the conditional Variance / Covariance matrix

S(t) ={

Sij(t)}1≤i , j≤N

and the vector of conditional mean arithmetic rates of return

a(t) ={αi (t)

}1≤i≤N

,

for all the different assets in this market.

• These quantities are the so-called “local characteristics”(the theory allows them to be random in their own right).

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LOGARITHMIC RETURNS

The logarithmic return of asset i over the time interval[t, t + dt] is the quantity

d log Xi (t) = log Xi (t + dt)− log Xi (t) = log

(1 +

dXi (t)

Xi (t)

).

Given all available information up to time t , this randomquantity has conditional variance Sii (t) · dt , and conditionalmean γi (t) · dt given by

γi (t) = αi (t)− 1

2Sii (t) ,

the growth rate of asset i . For different assets i 6= j , thelogarithmic returns have conditional covariance Sij(t) · dt .

(“Elementary” exercise in stochastic calculus.)

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EXAMPLEStock XYZ doubles in good years (+100%) and halves in bad years(-50%). Years good and bad alternate independently and equallylikely (to wit, with probability 0.50), thus the mean arithmeticrate of return is

α =1

2(+100%) +

1

2(−50%) =

1

2− 1

4= 0.25 ,

and the mean logarithmic rate of return is

γ =1

2(log 2) +

1

2

(log

1

2

)= 0 .

On the other hand, log 2 ' 0.7 , so the variance of log-returns is

S =1

2(0.7)2 +

1

2(−0.7)2 ' 0.50 ,

and indeed, as in the previous slide:

(0.25) = 0 + (1/2)(0.50) or α = γ + (S/2) .

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PORTFOLIOS

A collection of weights π1(t), · · · , πN(t) which satisfy

π1(t) + · · ·+ πN(t) = 1 for all t ≥ 0

(i.e., we assume here for concreteness that portfolios are fully in-vested). These are random quantities, determined on the basis ofinformation available only up until time t ; they will denote theproportions of current wealth W (t) invested at time t in thevarious assets.

. We call a portfolio π(·) long-only, if

π1(t) ≥ 0 , · · · , πN(t) ≥ 0 holds for all t ≥ 0 .

We shall deal with long-only portfolios from now on,for the rest of this talk.

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The wealth W π(·) that corresponds to such an investmentstrategy and to initial wealth W π(0) = 1 , satisfies

dW π(t)

W π(t)=

N∑i=1

πi (t)dXi (t)

Xi (t).

To wit: the portfolio’s arithmetic return is the “weighted average”,according to its weights π1(t), · · · , πN(t) , of the individual assets’arithmetic returns.

This wealth W π(t) is a random quantity, with conditional meanand conditional variance rates given, respectively, by

απ(t) :=N∑i=1

πi (t)αi (t) , S ππ(t) :=N∑i=1

N∑j=1

πi (t) Sij(t)πj(t) .

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Likewise, the logarithmic return

d log W π(t) = log W π(t + dt)− log W π(t) = log

(1 +

dW π(t)

W π(t)

)of the wealth corresponding to the portfolio π(·) over theinterval [t, t + dt] , is a random quantity, with the sameconditional variance Sππ(t) · dt as in

S ππ(t) =N∑i=1

N∑j=1

πi (t) Sij(t)πj(t) ,

the quantity we saw before, and conditional mean γπ(t) · dt ,the growth rate of the portfolio.

What is this new growth rate ?

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Here

γπ(t) =N∑i=1

πi (t) γi (t) + γπ∗ (t)

is the growth rate corresponding to the portfolio π(·) , where thequantity γπ∗ (t) is the excess growth rate

γπ∗ (t) :=1

2

N∑i=1

πi (t) Sii (t)−N∑i=1

N∑j=1

πi (t) Sij(t)πj(t)

> 0 .

Difference of the weighted average of the asset variances∑Ni=1 πi (t) Sii (t) , minus the portfolio’s conditional variance

S ππ(t) =N∑i=1

N∑j=1

πi (t) Sij(t)πj(t) .

(“Moderate” exercise in stochastic calculus.)11 / 33

THE PARABLE OF TWO STOCKS

Suppose there are only two, perfectly negatively correlated, stocksA and B. We toss a fair coin, independently from day to day; whenthe toss comes up heads, stock A doubles and stock B halves inprice (and vice-versa, if the toss comes up tails).

Clearly, each stock has a growth rate of zero: holding any one ofthem produces nothing in the long term.

• What happens if we hold both stocks? Suppose we invest $100in each; one of them will rise to $200 and the other fall to $50, fora guaranteed total of $250, representing a net gain of 25%; thewinner has gained more than the loser has lost.

If we rebalance to $125 in each stock (so as to maintain the equalproportions we started with), and keep doing this day after day, welock in a long-term growth rate of 25%.

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Indeed, taking N = 2 and

γ1 = γ2 = 0 , S11 = S22 = −S12 = −S21 = 0.50

andπ1 = π2 = 0.50

in

γπ =N∑i=1

πi γi +1

2

N∑i=1

πi Sii −N∑i=1

N∑j=1

πi Sij πj

=

1

2

(π1(1− π1

)S11 + π2

(1− π2

)S22

)− π1π2S12

we get the same growth rate that we computed a moment ago:

γπ = γπ∗ = 0.25 .

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MORAL OF THIS PARABLE

• In the presence of “sufficient intrinsic volatility”, setting targetweights and rebalancing to them can capture this volatility andturn it into “alpha” (that is, growth).

And this can occur even without precise estimates of modelparameters, even without refined optimization.

We shall encounter several variations on this parableduring the remainder of the talk.

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MARKET PORTFOLIO

The most celebrated long-only portfolio is the market portfolio(say, S&P 500 Index) µ(·) , given by

µi (t) :=Xi (t)

X1(t) + · · ·+ XN(t), i = 1, · · · ,N .

This portfolio buys at time t = 0 the same number of shares in allassets, and just holds on to them (we are assuming here that eachasset has just one share outstanding, so that capitalization andasset price are the same).

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Holding the market portfolio amounts to “owning the entiremarket”, in proportion of course to our initial investment of $1:

W µ(t) =X1(t) + · · ·+ XN(t)

X1(0) + · · ·+ XN(0).

. The excess growth rate of this market portfolio plays a specialrole, when it comes to deciding whether the market can beoutperformed. This excess growth rate

γµ∗ (t) =1

2

N∑i=1

µi (t) Sii (t)−N∑i=1

N∑j=1

µi (t) Sij(t)µj(t)

is also a measure of the market’s intrinsic relative variance.

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. Indeed, it can be shown that the excess growth rate of themarket portfolio has an interpretation as “cap-weighted averagerelative variance”, in that

γµ∗ (t) =1

2

N∑i=1

µi (t)Sµii (t) > 0 ,

with

Sµij (t) := Sij(t)− Sµi (t)− Sµj (t) + Sµµ(t) ≥ 0 , 1 ≤ i , j ≤ N

the variances/covariances of the different assets – not in absoluteterms, but relative to the market. Here as before,

Sµi (t) :=N∑

k=1

µk(t) Sik(t) , S µµ(t) =N∑i=1

N∑j=1

µi (t) Sij(t)µj(t) .

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• This excess growth rate turns out to be the “fuel” requiredfor alpha generation: If this quantity dominates a positive lowerbound, even on the average, as in

1

T

∫ T

0γµ∗ (t) dt ≥ g > 0 ,

over a given, finite time-horizon [0,T ], then it is possible tooutperform the market over this horizon: W π(T ) > W µ(T ) .

You can find a long-only portfolio π(·) that allows you toturn this intrinsic volatility into “alpha” (growth).

This can be constructed at any given time only on thebasis of the prevailing market weights, and even withoutestimation or optimization – in a “model-free” manner.

(We’ll come back to this in a moment.)

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Here is a plot of the cumulative excess growth∫ T0 γ∗µ(t) dt for

the U.S. equities market over most of the twentieth century.

Slope of the advantage increases during periods of higher relativevolatility.

0.0

0.5

1.0

1.5

2.0

2.5

1927 1932 1937 1942 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002

YEAR

CUM

ULAT

IVE

EXCE

SS G

ROW

TH

Figure 1 : Cumulative Excess Growth for the U.S., 1926-1999.

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EXAMPLE: ENTROPY-WEIGHTING

Consider the Gibbs entropy (or diversity) function

H(m) :=N∑i=1

mi log(1/mi

)over the positive unit simplex. With c > 0, consider also thelong-only “entropy-weighted portfolio”

ηi (t) :=µi (t)

(c + log(1/µi (t))

)c + H(µ(t))

, i = 1, · · · ,N

whose relative performance with respect to the market is given by

log

(W η(T )

W µ(T )

)= log

(c + H(µ(T ))

c + H(µ(0))

)+

∫ T

0

γµ∗ (t) dt

c + H(µ(t)),

a remarkable formula. (“Serious” exercise in stochastic calculus.)

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The first term (blue; change in diversity) on the RHS of thisdecomposition

log

(W η(T )

W µ(T )

)= log

(c + H(µ(T ))

c + H(µ(0))

)+

∫ T

0

γµ∗ (t) dt

c + H(µ(t)).

is bounded, since

0 < c < c + H(·) ≤ c + log n.

Thus, under the condition

1

T

∫ T

0γµ∗ (t) dt ≥ g > 0

the second term (brown; “source of alpha”) on the RHS will soon-er or later overtake the first (with sufficiently large real numbersc > 0 and T > 0), hence the outperformance (red; relative return)

W η(T ) > W µ(T ).

(Pictorial illustration coming... .)21 / 33

OPEN QUESTION

Suppose you know

γµ∗ (t) ≥ g > 0 , 0 ≤ t <∞

holds for some real number g .

Is it then possible to outperform the market portfolioover arbitrary time-horizons ?

We do not know. In a couple of (very interesting) special cases,the answer is affirmative; but I am afraid these are two separate,additional talks... .

If you find out and let us know, we’ll be grateful.

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REMARK: We have not imposed any conditions on the randomcovariance matrix S(t) =

{Sij(t)

}1≤i , j≤N

of the different assets.

. If we assume that all its eigenvalues are bounded away from zeroand away from infinity on [0,T ], then the condition

γµ∗ (t) ≥ g , ∀ 0 ≤ t ≤ T

for some g > 0 , is equivalent to the condition

max1≤i≤N

µi (t) ≤ 1− δ , ∀ 0 ≤ t ≤ T

for some δ ∈ (0, 1) .

. And in this case the market portfolio CAN be outperformedover ARBITRARY time horizons [0,T ] with T ∈ (0,∞) .

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EQUAL-WEIGHTED PORTFOLIO

The choice of portfolio weights

pi = 1/N , i = 1, · · · ,N ,

leads to the equal-weighted portfolio (“Value Line” Index)

ϕi (t) = 1/N , i = 1, · · · ,N .

This stands at the opposite extreme of market weighting: keepsequal proportions of wealth W ϕ(·) in all assets at all times.

Holding the equal-weighted portfolio involves considerableamounts of trading, as one keeps trying to “buy low and sell high”:shedding assets that have risen in value, to buy assets that havelagged behind.

(Gist of “Statistical arbitrage” strategies.)

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Historically, equal-weighted portfolios have tended to outperformtheir capitalization-weighted counterparts (market portfolios).

This is done at considerable risk, though, as both terms on theright-hand side of the “remarkable formula”

log

(W ϕ(T )

W µ(T )

)= log

(µ1(T ) · · ·µN(T )

µ1(0) · · ·µN(0)

)1/N

+

∫ T

0γϕ∗ (t) dt

can fluctuate quite a lot; here

γϕ∗ (t) =1

2 N

N∑i=1

Sii (t)− 1

N

N∑i=1

N∑j=1

Sij(t)

> 0

is the excess growth rate of this equally-weighted portfolio.

• Choppy relative performance.

• Long periods of underperformance.

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MODULATED EQUAL-WEIGHTING

It is possible to mitigate these undesirable effects, by finding cleverways to “interpolate” between the market portfolio weights

µi (t) :=Xi (t)

X1(t) + · · ·+ XN(t), i = 1, · · · ,N

and the equal weights

ϕi (t) =1

N, i = 1, · · · ,N ;

say, by fixing a number 0 < λ < 1 and forming

πi (·) ∼= λµi (·) + (1− λ)1

N, i = 1, · · · ,N .

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There are many ways to do this interpolation in a sytematic,“functional” way.

Done well, such pairing with the market portfolio can. retain the “good” characteristics of equal-weighting

(outperformance of the market), as well as. mitigate the bad

(choppiness, long periods of underperformance).

• Please note also that such interpolation or “modulation”, as in

πi (·) ∼= λµi (·) + (1− λ)1

N, i = 1, · · · ,N ,

has also the effect of over-weighing the small-cap stocks andunder-weighing that large-cap stocks, relative to the marketweights(the valleys are exalted – somewhat – and the mountains andhills made somewhat low (er)... .)

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−20

−10

010

2030

40

%

1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004

1

2

3

Figure 2 : Simulation of modulated equal-weightingduring a half-century, 1956–2005.

1: “Change in Diversity”2: “Alpha Source”3: Relative Return

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By analogy with the decomposition

log

(W η(T )

W µ(T )

)= log

(c + H(µ(T ))

c + H(µ(0))

)+

∫ T

0

γµ∗ (t)dt

c + H(µ(t))

we saw before, for the performance relative to the market of the“entropy-weighted portfolio”

ηi (t) :=µi (t)

(c + log(1/µi (t))

)∑Nj=1 µj(t)

(c + log(1/µj(t))

) , i = 1, · · · ,N

with

H(m) =N∑j=1

mj log(1/mj) .

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TO RECAPITULATE:

VOLATILITY CAPTURE

• Results from rebalancing to Target Weights.• Setting these (target weights) does not require forecasts ofreturns, volatilities/covariances, or other factors.• Is mathematically consistent with Stochastic Portfolio Theory(meaning: there are precise statements, and proofs, for all theclaims we have made).

MARKET DIVERSITY

• Why do Relative Returns deviate from the “Alpha Source”?• Because of the change in Market Diversity. This a measureof concentration or dispersion of capital, a function only of theprevailing market-weight configuration at any given moment.• Market Diversity has been mean-reverting for more than 75years (Figure 3).

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−30

−20

−10

010

2030

YEAR

%

1927 1932 1937 1942 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002

Figure 3 : Cumulative Change in Market Diversity during 1927-2004.

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SOME REFERENCES

I Fernholz, E.R. (2002). Stochastic Portfolio Theory.Springer-Verlag, New York.

I Fernholz, E.R., Karatzas, I. & Kardaras, C.(2005). Diversity and arbitrage in equity markets.Finance & Stochastics 9, 1-27.

I Fernholz, E.R. & Karatzas, I. (2005). Relativearbitrage in volatility-stabilized markets.Annals of Finance 1, 149-177.

I Karatzas, I. & Kardaras, C. (2007). The numeraireportfolio and arbitrage in semimartingale markets.Finance & Stochastics 11, 447-493.

I Fernholz, E.R. & Karatzas, I. (2009) StochasticPortfolio Theory: An Overview. Handbook of NumericalAnalysis, volume “Mathematical Modeling and NumericalMethods in Finance” (A. Bensoussan, ed.) 89-168.http://www.math.columbia.edu/∼ik/FernKarSPT.pdf

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In Conclusion

• There is an alpha opportunity, that is always positive in avolatility-capture strategy.

• Volatility-capture strategies, which are Risk-Managed andOptimized, seek to capture volatility more efficiently than anequally-weighted portfolio possibly can.

• These produce more consistent and stable results, and withhigher information ratios.

THANK YOU VERY MUCH

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