F=C-1*M (1)
where F is a Nx1 vector indicating the fraction of the equity to hold upwards allocated to each asset, C is the covariance matrix, in addition to thou is the hateful vector for the excess returns of these assets. Note that these "assets" tin inwards fact hold upwards "trading strategies" or "portfolios" themselves. If these are inwards fact existent assets that incur a bear (financing) cost, in addition to then excess returns are returns minus the risk-free rate.
Notice that these fractions, or weights equally they are unremarkably called, are non normalized - they don't necessarily add together upwards to 1. This agency that F non exclusively determines the allotment of the full equity with north assets, but it equally good determines the overall optimal leverage to hold upwards used. The core of the absolute value of components of F divided yesteryear the full equity is inwards fact the overall leverage. Thus is the beauty of Kelly formula: optimal allotment in addition to optimal leverage inwards 1 unproblematic formula, which is supposed to maximize the compounded increase charge per unit of measurement of one's equity (or equivalently the equity at the cease of many periods).
However, most students of finance are non taught Kelly portfolio optimization. They are taught Markowitz mean-variance portfolio optimization. In particular, they are taught that at that spot is a portfolio called the tangency portfolio which lies on the efficient frontier (the prepare of portfolios with minimum variance consistent with a for certain expected return) and which maximizes the Sharpe ratio. Left unsaid are
- What's therefore skillful most this tangency portfolio?
- What's the existent create goodness of maximizing the Sharpe ratio?
- Is this tangency portfolio the same equally the 1 recommended yesteryear Kelly optimal allocation?
I desire to answer these questions here, in addition to furnish a connective betwixt Kelly in addition to Markowitz portfolio optimization.
According to Kelly in addition to Ed Thorp (and explained inwards my book), F inwards a higher house non exclusively maximizes the compounded increase rate, but it equally good maximizes the Sharpe ratio. Put only about other way: the maximum increase charge per unit of measurement is achieved when the Sharpe ratio is maximized. Hence nosotros encounter why the tangency portfolio is therefore important. And inwards fact, the tangency portfolio is the same equally the Kelly optimal portfolio F, except for that fact that the tangency portfolio is assumed to hold upwards normalized in addition to has a leverage of 1 whereas F goes 1 stride farther in addition to determines the optimal leverage for us. Otherwise, the percentage allotment of an property inwards both are the same (assuming that nosotros haven't imposed additional constraints inwards the optimization problem). How create nosotros examine this?
The green way Markowitz portfolio optimization is taught is yesteryear setting upwards a constrained quadratic optimization occupation - quadratic because nosotros desire to optimize the portfolio variance which is a quadratic business office of the weights of the underlying assets - in addition to croak on to purpose a numerical quadratic programming (QP) programme to solve this in addition to and then farther maximize the Sharpe ratio to detect the tangency portfolio. But this is unnecessarily wearisome in addition to truly obscures the elegant formula for F shown above. Instead, nosotros tin croak on yesteryear applying Lagrange multipliers to the next optimization occupation (see http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf for a similar treatment):
Maximize Sharpe ratio = FT*M/(FT*C*F)1/2 (2)
bailiwick to constraint FT*1=1 (3)
(to emphasize that the 1 on the left manus side is a column vector of one's, I used bold face.)
So nosotros should maximize the next unconstrained quantity with honor to the weights Fi of each property i in addition to the Lagrange multiplier λ:
FT*M/(FT*C*F)1/2 - λ(FT*1-1) (4)
But taking the partial derivatives of this fraction with a foursquare root inwards the denominator is unwieldy. So equivalently, nosotros tin maximize the logarithm of the Sharpe ratio bailiwick to the same constraint. Thus nosotros tin accept the partial derivatives of
log(FT*M)-(1/2)*log(FT*C*F) - λ(FT*1-1) (5)
with honor to Fi. Setting each constituent i to null gives the matrix equation
(1/FT*M)M-(1/FT*C*F)C*F=λ1 (6)
Multiplying the whole equation yesteryear FT on the correct gives
(1/FT*M)FT*M-(1/FT*C*F)FT*C*F=λFT*1 (7)
Remembering the constraint, nosotros recognize the correct manus side equally only λ. The left manus side comes out to hold upwards just zero, which agency that λ is zero. H5N1 Lagrange multiplier that turns out to hold upwards null agency that the constraint won't touching on the solution of the optimization occupation upwards to a proportionality constant. This is satisfying since nosotros know that if nosotros apply an equal leverage on all the assets, the maximum Sharpe ratio should hold upwards unaffected. So nosotros are left with the matrix equation for the solution of the optimal F:
C*F=(FT*C*F/FT*M)M (8)
If yous know how to solve this for F using matrix algebra, I would similar to postulate heed from you. But let's elbow grease an ansatz F=C-1*M equally inwards (1). The left manus side of (8) becomes M, the correct manus side becomes (FT*M/FT*M)M = thou equally well. So the ansatz works, in addition to the solution is inwards fact (1), upwards to a proportionality constant. To satisfy the normalization constraint (3), nosotros tin write
F=C-1*M / (1T*C-1*M) (9)
So there, the tangency portfolio is the same equally the Kelly optimal portfolio, upwards to a normalization constant, in addition to without telling us what the optimal leverage is.
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Workshop Update:
Based on pop demand, I receive got revised the dates for my online Mean Reversion Strategies workshop to be August 27-29.
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