A Geometric Approach toStable Asset Allocation

Paul DoustGlobal head of Quantitative Analysis

2[ Quant Congress USA, New York, July 2008 ] RBS

Objectives of this presentation

1) To prove that the RBS Geometric Mean Variance framework is superior to Black-LittermanIt will be shown that Black-Litterman is like a constrained mean-variance framework, however an actual constrained mean-variance framework is capable of finding better allocations than Black-Litterman. A potential advantage of Black-Litterman might derive from its Bayesian approach, however in practice the Bayesian aspects of Black-Litterman don't work very well.

2) To justify a constrained mean-variance approach from a Capital Asset Pricing Model (“CAPM”) standpoint

3) To present a new way of giving different weights to an investor’s market views

[ This presentation corresponds to the February 2008 Risk Magazine article “Geometric mean-variance”, by Paul Doust. ]

3[ Quant Congress USA, New York, July 2008 ] RBS

Introduction to Asset Allocation

All market participants need to make sure their portfolios contain a sensible mix of potential return versus risk–many investors use a selection of models to achieve this

Markowitz first introduced the mean-variance approach to asset allocation in 1952 (2 slides + 1 CAPM slide)–But traditional mean-variance often produces unstable results

Black and Litterman of Goldman Sachs developed their popular approach in the early 1990’s (3 slides)–Extension of mean-variance in a Bayesian framework

–Hard to implement in practice, with a few other issues too

2007 RBS develops Geometric Mean Variance (20 slides)–A new approach, combining the best ideas from Markowitz mean-

variance and Black-Litterman.

4[ Quant Congress USA, New York, July 2008 ] RBS

Markowitz Mean-Variance Shows how rational investors use diversification to

optimise their portfoliosMean-variance optimisation is part of Modern Portfolio Theory, which also includes the Capital Asset Pricing Model (“CAPM”) where the terminology alpha and beta comes from

Matrix notation:R = column vector of stochastic asset returns

w = column vector of portfolio weights

= column vector of expected excess returns = E(R rf)

= covariance matrix of returns = E((R E(R))(R E(R))′)

I = column vector filled with 1’s [NB: ′ = transpose]

= risk tolerance scalar

Maximise return w′ while minimising variance w′wMaximising utility function U(w) = w′ w′w/2implies w = -1I′w = 1 constrained solution: w = -1 (I (1 I′-1)/(I′-1 I))

5[ Quant Congress USA, New York, July 2008 ] RBS

Markowitz Mean-Variance

Practical problems with mean-variance:–Often sensitive to small changes in the inputs, especially when

some assets are highly correlated

–Easy to produce unrealistic leveraged positions

Markowitz Efficient Frontier

Standard deviation of expected excess return

Efficient frontierwithout cash

Efficient frontierincluding cash

Market portfolio

100% cash

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Capital asset pricing model equilibrium point 0

CAPM: i = i m where i = Cov(Ri,Rm)/m2

Writing wm = column vector of weights of the market portfolio, then Rm is

Rm = wm′R

so denoting the CAPM excess return vector by 0, the formula i = i m is equivalent to

0 = wm/m where m = m2 /m

Writing CAPM as 0 = wm/m shows that it is consistent with unconstrained mean variance (w = -1).

Using = 0 produces a stable result (i.e. w = wm), so Black-Litterman devised a way of tilting this allocation away from w = wm based on how an investor’s views differ from the returns implied by 0

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Black-Litterman asset allocation framework Black-Litterman uses Bayesian inference to blend a

probability distribution for where E()=0, with a distribution for which is derived from investor views(1) Assume ~ N(0, /N) where N >> 1

(2) Assume P~ N(Q, /N) where column vector Q are the investor’s expected returns on portfolios set up in the rows of matrix P, and /N defines the investor’s confidence in the views

Example of P and Q: with 3 views on 5 assets

– View 1: Excess return of asset 2 over cash will be 2.5%

– View 2: Return on asset 4 will be 0.5% above return on asset 3

– View 3: Average excess return on all 5 assets will be 1.75%

P matrix, 3x5 Asset 1 Asset 2 Asset 3 Asset 4 Asset 5 Q vector Expected return

View 1 0 100% 0 0 0 View 1 2.5%

View 2 0 0 -100% 100% 0 View 2 0.5%

View 3 20% 20% 20% 20% 20% View 3 1.75%

8[ Quant Congress USA, New York, July 2008 ] RBS

Black-Litterman asset allocation framework The resulting asset allocation formula is

w = wm + P′(/p + PP′)-1(Q - P0)

where p = N/(1+N). P0 is the expected equilibrium excess returns for the view portfolios, and PP′ is the covariance matrix of the returns of the view portfolios P.

Expression (/p + PP′)-1(Q - P0) is a vector, giving the weight of each view portfolio to add into the market portfolio wm, and is of the form

(Covariance matrix)-1(Excess return)

which is the same as the basic mean-variance formula, applied to the returns which the investor expects in excess of the market equilibrium returns which are already incorporated into wm.

9[ Quant Congress USA, New York, July 2008 ] RBS

Problems with Black-Litterman Potentially as unstable as traditional mean-variance

Arbitrary parameter p, and parameters which are hard to determine

A neutral view produces a different result compared to using the framework with a view portfolio deleted from P–Deleting a view portfolio from the P matrix corresponds to limit

where the investor’s view on the return on that portfolio has infinite variance. So switching a portfolio in and out of the P matrix corresponds to switching between infinite and plausible variance, which causes a discontinuity. The conclusion is that a complete set of view portfolios should always be used

Bayesian mechanism doesn’t work properly

Solution: Geometric Mean Variance framework (GMV)–GMV is constructed to have all the benefits of Black-Litterman

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Markowitz allocation formula is w = -1, so use matrix /m to define mappings between w-space and -space

Given a weight vector in w-space let w = w/mbe the corresponding point in -space. Then the utility function becomes

U(w) = -½(w - )′-1(w - ) + constant

–Maximisation of U(w) is a distance minimisation problem in -space since (w - )′-1(w - ) is the square Mahalanobis distance from w to induced by the metric

–Unconstrained solution is w =

Geometric Mean Variance

w-space -space

m-1

/m

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Black-Litterman stable point in GMV

Expand the stable point 0 into a stable region by mapping w where w is the set of the points in w-space which are acceptable allocation results

For example, one choice is w = { w : I′w = 1, w}

The stable point 0 will also be used to define the expected returns on portfolios where the investor has no view

w-space -space

/m

w

12[ Quant Congress USA, New York, July 2008 ] RBS

Black-Litterman blending mechanism in GMV If PP′ it is straightforward to show that varying p

results in a simple linear interpolation in -space between 0 and the investor’s views

Hence Geometric Mean Variance can find better allocations, more closely aligned to the investor’s views

-space

0

Investor’s views

Black-Litterman allocation resulting from varying p.

Geometric mean-variance allocation

13[ Quant Congress USA, New York, July 2008 ] RBS

Black-Litterman view portfolios in GMV

Use view portfolios by defining the expected return Q using

Q = 0 + P′(PP′)-1 (Q - P0)

This satisfies PQ = Q, with the expected return on all portfolios orthogonal to P corresponding to 0. Hence the investor has the same flexibility to specify views with GMV as with Black-Litterman.

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Bayesian aspects of Black-Litterman If an investor chooses (e.g. = PP′) and then

adjusts p to achieve an acceptable result, Black-Litterman is no more than a highly constrained mean-variance framework with the Bayesian aspects ignored.

If an investor tries to use set up to genuinely reflect confidence in the views, the Bayesian aspects don’t work well because–The Bayesian inference works on the distribution of , not R. But

in practice, investors only have intuition about R, not . Only historical returns can be observed, so the separation of the covariance matrix of the mean from the return distribution itself is a theoretical rather than a practical concept.

–One of the usual sanity checks that can be used in connection with Bayesian Inference cannot be used with Black-Litterman because the result only depends on p and not N and N individually.

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Bayesian aspects of Black-Litterman Bayesian inference is a technique for blending two

distributions relating to the same quantity. When used in experimental science, there is a useful sanity check, because Bayesian inference does not make sense if the two distributions do not overlap

However with Black-Litterman, this sanity check cannot be used because for a given p, N and N can always be chosen so that the distributions overlap.

Situation where Bayesian inference makes sense

Situation where Bayesian inference does not make

sense

16[ Quant Congress USA, New York, July 2008 ] RBS

Geometric mean variance Given that the Bayesian aspects of Black-Litterman

don’t work very well, Black-Litterman is effectively a highly constrained mean-variance framework, but a genuine constrained mean variance framework offers more flexibility.–With GMV, investors still only need to specify views on portfolios

of their choice via Q = 0 + P′(PP′)-1 (Q - P0)

–Philosophy of GMV is that asset allocation corresponds to the choice of a point in -space, which implies the allocation result via the mapping back into w-space.

–Effort goes into defining w (and hence ) rather than and p

The identification of the point in -space is valuable information for the investor, which is hidden in conventional constrained mean variance which effectively operates in w-space.

17[ Quant Congress USA, New York, July 2008 ] RBS

CAPM justification of constrained mean-variance

No one knows for certain what should be.

Models like CAPM are useful, but since the markets are not in equilibrium, 0.

However, since markets are approximately driven by something like CAPM, investors are not free to choose any because an arbitrary implies an implausible w, so constraints are justified.–Constraints around 0 determine the degree of deviation that is

allowed from CAPM

Another interpretation is that w = -1 can be regarded as imposing a strong constraint on the allowed values of , so that the resulting w looks like a plausible wm

–This turns the “problem” that traditional mean-variance is hard to use into a benefit.

18[ Quant Congress USA, New York, July 2008 ] RBS

Beyond traditional mean-variance Before the true mean has been chosen, one is free to

wander around -space using any metric deemed appropriate–Metrics different from can be used to help decide where the true

mean lies, however once the true mean has been chosen then metric must be used, otherwise one isn’t doing mean-variance

–Give different weights to different views via choice of metric

Regard the investor’s Q as an initial guess at what the mean of the return distribution should be, and assert that the true mean of the return distribution has to lie within some subspace .

– Whatever method is used to choose the true mean , the investor should be happy with the result, i.e. the investor should actively choose the point in which best reflects his views.

19[ Quant Congress USA, New York, July 2008 ] RBS

Beyond traditional mean-variance: other metrics A sensible choice for where the true mean must lie

is = { : m-1w } where w = { w : I′w = 1, w}–For all ww to make sense in a CAPM context a much tighter

constraint would be needed, because in practice this often produces zero-weightings for some assets. But since the assumption is only that the markets are approximately governed by CAPM, a less restrictive constraint makes sense.

Given Q, one way of choosing the true mean is to minimise ( - Q)′-1( - Q) for some metric which the investor is free to choose. For example, if all directions in -space are equally important one would choose = 1.

Different choices of allow the investor to give more or less weight to different views. Hence this matrix plays a similar role to the covariance matrix in Black-Litterman.

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Geometric Mean Variance: example data As an example, look at 10 of the world’s major equity

markets, taking daily data from Jan-01 to Jan 07:

Ccy Index Size (1012$) wm

0

i

USD Dow Jones Wilshire 5000 16.12 48.3% 4.9% 16.7%EUR Eurozone STOXX total market 5.14 15.4% 5.3% 20.3%JPY Japan Datastream market 4.71 14.1% 2.6% 19.1%GBP FTSE All Share 3.64 10.9% 4.1% 16.6%CHF Swiss Market Index 1.00 3.0% 4.6% 19.6%SEK Dow Jones Total Market 0.39 1.2% 5.8% 24.7%NOK Dow Jones Total Market 0.09 0.3% 3.4% 22.2%NZD NZX All Share 0.03 0.1% 0.4% 9.7%AUD S&P/ASX 200 0.88 2.6% 1.1% 11.0%CAD S&P/TSX Composite 1.39 4.2% 3.2% 13.5%

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Geometric Mean Variance: example data

rij USD EUR JPY GBP CHF SEK NOK NZD AUD CAD

USD 1 0.51 0.14 0.47 0.45 0.5 0.29 0.03 0.1 0.67EUR 0.51 1 0.24 0.79 0.78 0.72 0.46 0.15 0.25 0.49JPY 0.14 0.24 1 0.23 0.23 0.25 0.24 0.21 0.47 0.19GBP 0.47 0.79 0.23 1 0.81 0.74 0.54 0.1 0.24 0.52CHF 0.45 0.78 0.23 0.81 1 0.7 0.51 0.09 0.21 0.45SEK 0.5 0.72 0.25 0.74 0.7 1 0.52 0.11 0.26 0.51NOK 0.29 0.46 0.24 0.54 0.51 0.52 1 0.15 0.32 0.37NZD 0.03 0.15 0.21 0.1 0.09 0.11 0.15 1 0.38 0.06AUD 0.1 0.25 0.47 0.24 0.21 0.26 0.32 0.38 1 0.18CAD 0.67 0.49 0.19 0.52 0.45 0.51 0.37 0.06 0.18 1

Covariance matrix given by ij = ijrij, where i shown on previous slide, and where the correlation matrix rij is shown below

22[ Quant Congress USA, New York, July 2008 ] RBS

Geometric Mean Variance: example views Suppose that an investor is bullish on the CAD market

against the USD market, the EUR market against the GBP market, and the EUR market against the CHF market. To express these views, use P as follows:

For these views, the expected returns P0 and the standard deviation of the returns for these views are

P matrix USD EUR JPY GBP CHF SEK NOK NZD AUD CAD

View 1 -1 0 0 0 0 0 0 0 0 1View 2 0 1 0 -1 0 0 0 0 0 0View 3 0 1 0 0 -1 0 0 0 0 0

P0 Std Dev

View 1 (CAD - USD) -1.71% 12.6%View 2 (EUR - GBP) 1.21% 12.5%View 3 (EUR - CHF) 0.72% 13.1%

23[ Quant Congress USA, New York, July 2008 ] RBS

Geometric Mean Variance: example views

To tilt the default allocation wm in favour of the investor’s views, the relevant quantity to define is Q P0, so choose

Suppose also that the result must be constrained so that the weights are all positive and sum to 1, i.e. = { : m-1w } where w = { w : I′w = 1, w}

With this data and these views, now consider–Black-Litterman result with = PP′ compared to GMV.

–Mean-variance as a strong constraint on

Q P0 =  

CAD – USD = 0.50%EUR – GBP = 0.37%EUR – CHF = 0.32%  

24[ Quant Congress USA, New York, July 2008 ] RBS

Geometric Mean Variance versus Black-Litterman

With = PP′, Black-Litterman result depends on p

The largest p which satisfies all the constraints w is p = 0.17. The point p is the allocation corresponding to the investor’s views, and the GMV result is quite close to this.

p=

p=.17

p GMV

USD 48.3% 46.2% 34.8% 34.9%EUR 15.4% 17.2% 27.7% 27.3%JPY 14.1% 14.1% 13.9% 13.8%GBP 10.9% 9.2% -1.1% 0.0%CHF 3.0% 2.6% 0.2% 0.0%SEK 1.2% 1.2% 1.4% 1.2%NOK 0.3% 0.3% 0.5% 0.4%NZD 0.1% 0.0% -1.2% 0.0%AUD 2.6% 2.5% 1.1% 1.1%CAD 4.2% 6.5% 21.2% 21.3%

25[ Quant Congress USA, New York, July 2008 ] RBS

Geometric Mean Variance versus Black-Litterman

Define distance in -space with ( - Q)′-1( - Q) –This is equivalent to standard constrained mean variance

In this case, it is clear that Geometric Mean Variance has found a better allocation because it is more closely aligned with the investor’s views

-space

0

Investor’s views Q (p )

Black-Litterman allocation with p = 0.17

Geometric mean-variance allocation

26[ Quant Congress USA, New York, July 2008 ] RBS

Working in -space with geometric mean variance The GMV result is a good match to the investor’s views

with most elements of | Q|<0.01% and less than 0.03% in all cases.

Suppose now that the investor wants to become more bullish on view 2 (EUR-GBP) and view 3 (EUR-CHF)

Recalculating the closest , the corresponding elements in P( 0) only change by 0.01%. Stronger views are being inhibited by the constraint w.

Q P0 =  

0.50%0.37%0.32%  

P( 0) =  

0.500%0.354%0.318%  

Q P0 =  

0.50%0.40%0.35%  

27[ Quant Congress USA, New York, July 2008 ] RBS

Working in -space with geometric mean variance

Defining distance in -space by ( - Q)′-1( - Q), more aggressive views on are not reflected in the result.

Inability of constrained system to become more bullish on the EUR equity market against the GBP or CHF markets

0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

0.30%

0.20% 0.30% 0.40% 0.50% 0.60% 0.70%View on (EUR-GBP) or (EUR-CHF)

Fit

tin

g e

rro

r (

- Q

) fo

r G

BP

or

CH

F

EUR vs GBP

EUR vs CHF

28[ Quant Congress USA, New York, July 2008 ] RBS

Working in -space with geometric mean variance

Regard Q as a guess for the true mean , and define distance by ( - Q)′-1( - Q) where is given by

= P′(PP′)-1(PP′)-1P + (1P′(PP′)-1P) ,

where =1 and where .

This definition of is the natural functional form so that controls the weight on the different views, and controls the weights in the rest of -space. Above gives more weight to EUR-GBP and EUR-CHF views, producing

versus

although | Q| grows with biggest entry ~ 0.035%.

=  

100

00.10

00

0.1

Q P0 =  

0.50%0.40%0.35%  

P( 0) =  

0.493%0.390%0.348%  

29[ Quant Congress USA, New York, July 2008 ] RBS

Geometric Mean Variance: conclusion RBS’s new Geometric Mean Variance approach is

preferable to Black-Litterman because–Geometric Mean Variance produces allocations which reflect an

investor’s views more closely

–Geometric Mean Variance is a simpler framework than Black-Litterman, and hence is preferable because "All things being equal, the simplest solution tends to be the best one“ [ Occam's Razor ]

–Geometric Mean Variance retains the attractive features of Black-Litterman, namely the ability to specify different investor views, together with the benefits of focussing on a benchmark portfolio

Interpreting traditional mean-variance as imposing a strong constraint on the allowed values of extends these ideas and allows investors give different weights to their views when the initial Q

30[ Quant Congress USA, New York, July 2008 ] RBS

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