people respond to incentives. integrity. transparency. results. renato staub may 2010 portfolio...

People respond to incentives.

Integrity.

Transparency.

Results.

Renato Staub

May 2010

Portfolio Design

Presentation to the Swiss CFA SocietyGeneva, May 11, 2010

Zurich, May 19, 2010


Objectives

• Here, we deal with the expectation based market allocation.

• In classical portfolio management, this is referenced as ‘active’ allocation.

• We want to improve the asset allocation (AA) design process. In particular,

we are:

• Recognizing that history may be a questionable guide to the future

by using historical parameters to simulate the opportunity

• Calibrating the amount of risk taken, given the opportunity, in

order to reasonably expect to achieve the risk budget over time

• Providing a framework for checking the consistency across

capabilities

• Making the investment process more transparent

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Concepts

• Given the value/price signals over time, we are looking for the

• Appropriate expectation based AA strategy

• According amount of risk

• Composition of this strategy

• We combine the following concepts:

• Value/Price as based on discounted cash flow models

• Random walks

• Mean reversion

• Information analysis

• We integrate them in a single framework.

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Nature of Asset Allocation

• There are concepts that do not entail imbalances, e.g. Black-Scholes.

• That is, the market goes up or down with equal probability.

• By contrast, AA assumes that markets

• Deviate from their intrinsic value

• Revert in the long run.

• Without mean reversion, AA cannot add value.

• Hence…4

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PV


… we want to ride this wave!

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Eye-Catcher

• We simulate

• Value/Price of a mean reverting market

• Monthly over 1000 years

• The market’s average return equals 8.5% (f1).

• The upper chart entails all

• Under-valuations >20% and <30%

• The lower chart shows the subsequent

• Ann. 3-year returns (f2).

• We observe: f2 >> f

1.

• Without mean reversion, f2 would equal f

1.

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100 200 300 400 500 600 700 800 9000.2

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

100 200 300 400 500 600 700 800 900-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Terminology

• We deal with two key inputs, i.e.

• A market’s price, P

• Its fundamental value, V

• P can be observed for liquid markets.

• V cannot be observed. It must be estimated by a concept.

• We model in log space, i.e. v = log(V) and p = log(P).

• The value-price relationship is defined as follows:

vp = log(V/P) = log(V) – log(P) = v-p

• Assuming that p reverts to v over the duration d, this implies

xr = (v-p) / d

• xr is the expected return component due to price correction.

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Numerical Example

• The following parameters are given:

• V = 100 (by definition)

• P = 80

• d = 3 years

• V/P = 100/80 = 1.25

• vp = log(V/P) = v-p = log(100/80) = 0.2231

• xr = (v-p) / d = 0.2231 / 3 = 7.44%

• That is, we expect an additional log. return of 7.44% p.a. due to price

correction.

• However, this is a raw expectation.

• Later on, there will be a further modification to xr.

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Plan

• We simulate the vp evolution over a long time.

• We calibrate the simulation such that the resulting

• vp span

• vp volatility

• Reversion time

are in line with practical experience.

• We infer the information embedded in this process.

• We translate this information into a portfolio.

• We investigate the portfolio, in particular

• Its performance

• Other important properties

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0 2000 4000 6000 8000 10000 12000-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


Process - p

• We assume that the price follows a random walk.

• The shocks are based on our forward looking covariance matrix.

• The market price is shocked proportionally to its size, that is

pt+1

= pt +

• However, in order to avoid infinite dispersion, we must ensure mean

reversion.

• To that end, we adjust the equation as follows

pt+1

= pt (1-

pp) +

p

• pp

is p’s gap sensitivity of mean reversion.

• From time series analysis, we know

• < 0 The process is stationary

• >= 0 The process is non-stationary10

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Process – v

• Nobody knows v, and this is why we estimate it.

• We assume it to fluctuate around its (unknown) ‘true’ intrinsic value, i.e.

vt+1

= vt (1-

vv) +

v

• vv

is v’s gap sensitivity of mean reversion.

• But in practice, we often review a model in case of a large vp discrepancy.

• This applies in particular to markets of low model confidence.

• Technically, it means a gap sensitivity of v vs. p, that is

vt+1

= vt (1-

vv) + (v

t-p

t) (1-

vp) +

v = v

t+1 = v

t (2-

vv-

vp) - p

t (2-

vp) +

v

• We reference this effect as ‘chasing’: the (perceived) value chases the

price.

• This means a narrowing of vp, i.e. the perceived opportunity becomes

smaller.

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Process – Combining v and p

• Combining v and p, we get

vpt+1

= vt (2-

vv-

vp) - p

t (2-

pp -

vp) +

v +

p

• The next chart is a vp simulation based on monthly data over 1000 years.

• Notably, vp is confined to a certain bandwidth - because of mean

reversion.

• The breadth of the band depends on the input parameters.

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0 2000 4000 6000 8000 10000 12000-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


Process - Information

• As a result of mean reversion, there is a positive correlation between the

• Expected return from correction (i.e. the signal)

• Observed subsequent return from correction

• Hence, we are interested in the correlation between the two, that is

R = corr(Ei[xr(i,i+d)], xr(i,i+1))

where xr(i,i+d) is the return from correction between time i and time i+d.

• Because of the substantial noise components, this correlation is small.

• In other words, our signal, xr, is far from perfect.

• We define

R = IC = Information Coefficient

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Process - Property

• Shape Invariance Theorem:

vp evolutions with identical b‘s entail identical information.

• As an example, the following charts portray two markets. The

• Blue market has shock components of 2s.

• Red market has shock components of s.

• Mean reversion parameters of both markets are identical.

• As expected, both ICs inferred from simulation equal 0.118192.

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0 200 400 600 800 1000 1200-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

VolatilityMean

ReversionChasing I C

15.294% 0.02 0.01 0.118192

7.647% 0.02 0.01 0.118192


Process - Calibration

• The main calibration parameters are the

• Volatilities

• Mean reversion parameters

• The chart to the right shows the percentiles

of our simulation based vp distribution.

• As a reference, our valuation considered U.S. equity

• 60% overvalued in 2000 (techno bubble)

• 50% undervalued in 2009 (credit crisis)

• The according percentiles relative to the market

risks are wider for equity than for bonds.

• This is because we set stronger mean reversion

for bonds than equity.15

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min. 1% 50% 99% max

EQ US -46% -35% 0% 52% 92%

EQ UK -45% -33% 0% 52% 106%

EQ EMU -51% -37% 0% 58% 108%

EQ SWI -41% -32% 0% 50% 80%

EQ J AP -51% -38% 1% 60% 125%

EQ AUS -43% -32% -1% 54% 109%

EQ CAN -45% -34% 0% 56% 120%

EQ EMA -62% -46% 0% 89% 156%

BD 10Y US -21% -12% 0% 13% 23%

BD 10Y UK -19% -13% 0% 14% 24%

BD 10Y EMU -18% -12% 0% 13% 25%

BD 10Y CH -16% -10% 0% 11% 22%

BD 10Y J AP -18% -11% 0% 12% 20%

BD 10Y AUS -19% -13% 0% 16% 28%

BD 10Y CAN -19% -12% 0% 15% 25%

BD 5Y US -14% -7% 0% 8% 14%

BD 2Y US -7% -4% 0% 4% 7%

HY US -25% -16% 0% 19% 32%

BD EM -24% -17% 0% 20% 34%

RB 10Y US -15% -8% 0% 9% 15%

PercentilesMarket


FLAM and PAR

• The Fundamental Law of Active Management (FLAM) infers:

• A portfolio’s information ratio (IR) equals the IC of the comprising

markets (i.e. signal projection) times the square root of the

number of independent market bets available. *)

• The Proportional Allocation Rule (PAR) concludes:

• To allocate efficiently, we should allocate proportional to the signal.

**)

• While

• FLAM describes performance

• PAR tells us how to achieve it

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*) Assuming the various markets have identical ICs **) Assuming the signals are uncorrelated and have identical volatilities


PAR – Schematic Example

• At time 0

• The price equals p and the value equals v

• According to PAR, we buy an amount of (v-p)

• Between time 0 and 1

• The price changes by

• Hence, the value price discrepancy changes to (v-p-)

• At time 1’

• According to PAR, we adjust the quantity by -

• Between time 1 and 2

• The price changes by -

• We have the quantity (v-p-) at price p

• The net gain equals 217

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PAR – Schematic Example

• The same principle works for two up moves combined

with two down moves.

• For all four combinations, i.e.

• up-up-down-down

• up-down-up-down

• down-down-up-up

• Down-up-down-up

• PAR results in a net gain of 22 .

• Exposure Theorem:

Following PAR at constant volatility, the extra return grows with time.

• On the other hand, the size of the extreme vp is irrelevant.

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Slide in – Geometrical Representation of Risk

• Risk can be depicted by the length of a vector.

• And correlation equals the cosine of the angle between two risks.

• In the following triangle, the labels mean

• Risk of Portfolio 1 (Pf1): s1

• Risk of Portfolio 2 (Pf2): s2

• Correlation between Pf1 and Pf2: cos(j)

• Relative Risk (Tracking Error)

between Pf1 and Pf2: s1 - s2

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s1

s1 - s

2

s2

j


Signal - Information

• We represent both the signal and the observation by vectors.

• The more the signal points in the direction of the observation, the better it

is.

• Hence, a signal’s prediction quality equals its

• Projection onto the observation axis

• The projection equals the

• Correlation between signal and observation

• Cosine of angle j

• Again, “correlation” and “cosine” are two

different labels for the same thing.

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Sig

nal

Observationj

Projected Signal


Signal - Modification

• xri is the naïve expected extra return for market i

• But only its projection will materialize statistically

• Hence, we IC-correct it

• Further, a return must always be put in relation to its distribution.

• Thus, we divide xri by the volatility of market i.

• Ultimately, the “true” substance of xri equals

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xri

i

ii

i σ

ICxr = s


Signal - Modification

• Assume an equity and a bond market have an si of equal size.

• Identical allocation to equity and bonds implies more portfolio risk from

equity.

• Since PAR assumes identical risks, we must rescale one more time by the

risk:

• Ultimately, the suggested allocation equals

• The portfolio risk is linear in f.

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2

i

ii

iσ

ICxr f= w

2

i

ii

is

σ

ICxr =


Allocation – Vector Approach

• All bets are made vs. cash (only).

• That is, in case of cash, C, and two markets, A and B, the vector approach

• Bets A vs. C

• Bets B vs. C

• Does not bet A vs. B

• Hence, the set of all bets is a one-dimensional structure.

• This is why we call it “Vector Approach”.

• The vector approach is mainly a bet of cash vs. the entire market.

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Allocation – Matrix Approach (First Scenario)

• The matrix approach makes all possible bets, that is, also the bet A vs. B.

• The relative risk between A and B is marked in red.

• The triangle ABC is positioned such that its corners

touch the corresponding iso-vp lines.

• As B is more undervalued than A, we go long B vs. A.

• Based on the vp differential and the risk

geometry, this bet will be of average size.

• That is, the bet A vs. B has mainly implications

in terms of diversification.

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VP=20%

VP=7%

VP=0%

C

B

A


Allocation – Matrix Approach (Second Scenario)

• The vp differential between A and B is unchanged.

• But A and B are correlated much stronger vs. the first scenario.

• Hence, we scale by a smaller risk distance.

• As a result, A vs. B is by far the strongest bet.

• That is, its implication goes beyond diversification only.

• Rather, it is supposed to beef up return.

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VP=20%

VP=7%

VP=0%

C

BA


Signal – Hosting the Matrix Approach

• We build the difference between two IC corrected extra returns.

• And the risk is the relative risk between the two markets.

• That is:

• wij is the position based on the relative bet between market i and market j.

• The above equation also serves the vector approach, in which case j labels

cash.

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2

ij

jjii

ijσ

ICxrICxr f= w


Simulations – Valuation

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EQ US EQ UK EQ EMU

EQ SWI

EQ JAP

EQ AUS

EQ CAN

EQ EMA

BD 10Y US

BD 10Y UK

BD 10Y EMU

BD 10Y CH

BD 10Y JAP

BD 10Y AUS

BD 10Y CAN

BD 5Y US

BD 2Y US

HY US BD EM

RB 10Y US

-20%

-10%

0%

10%

20%

30%

40%

50%

60%


Simulations – Matrix Approach

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EQ US EQ UKEQ

EMU

EQ

SWI

EQ

J AP

EQ

AUS

EQ

CAN

EQ

EMA

BD

10Y

US

BD

10Y

UK

BD

10Y

EMU

BD

10Y

CH

BD

10Y

J AP

BD

10Y

AUS

BD

10Y

CAN

BD 5Y

US

BD 2Y

USHY US BD EM

RB

10Y

US

SB US

EQ US 28.8% 0.4% 0.1% 0.4% 0.5% -0.1% 0.0% -0.4% 0.8% 0.8% 0.6% 0.7% 0.9% 0.7% 0.8% 0.7% 0.6% 0.8% 0.6% 0.7% 0.5% 10.1%

EQ UK 16.2% -0.4% -0.2% 0.0% 0.2% -0.4% -0.3% -0.5% 0.4% 0.5% 0.3% 0.4% 0.6% 0.3% 0.4% 0.3% 0.3% 0.3% 0.2% 0.4% 0.2% 2.8%

EQ EMU 26.0% -0.1% 0.2% 0.2% 0.3% -0.2% -0.1% -0.4% 0.5% 0.5% 0.4% 0.4% 0.6% 0.4% 0.4% 0.4% 0.3% 0.4% 0.3% 0.4% 0.3% 5.4%

EQ SWI 15.6% -0.4% 0.0% -0.2% 0.1% -0.4% -0.3% -0.5% 0.3% 0.3% 0.2% 0.3% 0.4% 0.2% 0.3% 0.2% 0.2% 0.2% 0.1% 0.3% 0.2% 1.7%

EQ J AP 6.9% -0.5% -0.2% -0.3% -0.1% -0.4% -0.4% -0.5% 0.1% 0.1% 0.0% 0.1% 0.2% 0.0% 0.1% 0.1% 0.0% 0.0% 0.0% 0.1% 0.0% -1.5%

EQ AUS 25.1% 0.1% 0.4% 0.2% 0.4% 0.4% 0.1% -0.3% 0.7% 0.7% 0.6% 0.6% 0.8% 0.6% 0.7% 0.6% 0.5% 0.6% 0.5% 0.6% 0.4% 9.2%

EQ CAN 25.8% 0.0% 0.3% 0.1% 0.3% 0.4% -0.1% -0.3% 0.5% 0.6% 0.4% 0.5% 0.6% 0.5% 0.6% 0.5% 0.4% 0.5% 0.4% 0.5% 0.3% 6.8%

EQ EMA 52.1% 0.4% 0.5% 0.4% 0.5% 0.5% 0.3% 0.3% 0.6% 0.6% 0.5% 0.5% 0.7% 0.6% 0.6% 0.5% 0.5% 0.6% 0.5% 0.5% 0.4% 10.2%

BD 10Y US -8.2% -0.8% -0.4% -0.5% -0.3% -0.1% -0.7% -0.5% -0.6% 0.7% -2.1% -0.5% 1.8% -1.9% -0.8% -1.9% -1.1% -1.0% -1.5% -0.1% -0.9% -13.2%

BD 10Y UK -9.0% -0.8% -0.5% -0.5% -0.3% -0.1% -0.7% -0.6% -0.6% -0.7% -4.3% -1.1% 1.2% -1.8% -0.9% -1.3% -1.1% -0.9% -1.4% -0.4% -1.0% -18.1%

BD 10Y EMU -6.2% -0.6% -0.3% -0.4% -0.2% 0.0% -0.6% -0.4% -0.5% 2.1% 4.3% 3.1% 3.4% 0.5% 1.5% 1.5% 0.5% 0.4% -0.3% 1.4% 0.1% 15.4%

BD 10Y CH -6.4% -0.7% -0.4% -0.4% -0.3% -0.1% -0.6% -0.5% -0.5% 0.5% 1.1% -3.1% 2.3% -0.8% 0.1% -0.2% -0.7% -0.4% -0.9% 0.4% -0.9% -6.2%

BD 10Y J AP -12.2% -0.9% -0.6% -0.6% -0.4% -0.2% -0.8% -0.6% -0.7% -1.8% -1.2% -3.4% -2.3% -2.7% -1.7% -3.0% -2.8% -1.6% -1.9% -1.7% -2.4% -31.3%

BD 10Y AUS -9.5% -0.7% -0.3% -0.4% -0.2% 0.0% -0.6% -0.5% -0.6% 1.9% 1.8% -0.5% 0.8% 2.7% 1.0% 0.8% 0.1% 0.1% -0.5% 0.9% -0.1% 5.7%

BD 10Y CAN -8.2% -0.8% -0.4% -0.4% -0.3% -0.1% -0.7% -0.6% -0.6% 0.8% 0.9% -1.5% -0.1% 1.7% -1.0% -0.3% -0.5% -0.5% -1.0% 0.2% -0.6% -5.7%

BD 5Y US -5.1% -0.7% -0.3% -0.4% -0.2% -0.1% -0.6% -0.5% -0.5% 1.9% 1.3% -1.5% 0.2% 3.0% -0.8% 0.3% -2.4% -0.5% -1.2% 1.5% -1.7% -3.1%

BD 2Y US -2.6% -0.6% -0.3% -0.3% -0.2% 0.0% -0.5% -0.4% -0.5% 1.1% 1.1% -0.5% 0.7% 2.8% -0.1% 0.5% 2.4% 0.0% -0.7% 2.1% -4.8% 1.8%

HY US -1.2% -0.8% -0.3% -0.4% -0.2% 0.0% -0.6% -0.5% -0.6% 1.0% 0.9% -0.4% 0.4% 1.6% -0.1% 0.5% 0.5% 0.0% -0.6% 0.7% -0.3% 0.6%

BD EM -0.3% -0.6% -0.2% -0.3% -0.1% 0.0% -0.5% -0.4% -0.5% 1.5% 1.4% 0.3% 0.9% 1.9% 0.5% 1.0% 1.2% 0.7% 0.6% 1.2% 0.3% 9.0%

RB 10Y US -7.6% -0.7% -0.4% -0.4% -0.3% -0.1% -0.6% -0.5% -0.5% 0.1% 0.4% -1.4% -0.4% 1.7% -0.9% -0.2% -1.5% -2.1% -0.7% -1.2% -1.9% -11.5%

SB US -0.4% -0.5% -0.2% -0.3% -0.2% 0.0% -0.4% -0.3% -0.4% 0.9% 1.0% -0.1% 0.9% 2.4% 0.1% 0.6% 1.7% 4.8% 0.3% -0.3% 1.9% 12.1%

BUCKET

ALLOCATION vs.

SVALUA-

TION


Simulations – Vector Approach

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EQ US EQ UKEQ

EMU

EQ

SWI

EQ

J AP

EQ

AUS

EQ

CAN

EQ

EMA

BD

10Y

US

BD

10Y

UK

BD

10Y

EMU

BD

10Y

CH

BD

10Y

J AP

BD

10Y

AUS

BD

10Y

CAN

BD 5Y

US

BD 2Y

USHY US BD EM

RB

10Y

US

SB US

EQ US 28.8% 2.7% 2.7%

EQ UK 16.2% 1.2% 1.2%

EQ EMU 26.0% 1.6% 1.6%

EQ SWI 15.6% 0.9% 0.9%

EQ J AP 6.9% 0.1% 0.1%

EQ AUS 25.1% 2.5% 2.5%

EQ CAN 25.8% 1.9% 1.9%

EQ EMA 52.1% 2.4% 2.4%

BD 10Y US -8.2% -5.2% -5.2%

BD 10Y UK -9.0% -5.8% -5.8%

BD 10Y EMU -6.2% 0.3% 0.3%

BD 10Y CH -6.4% -5.2% -5.2%

BD 10Y J AP -12.2% -13.5% -13.5%

BD 10Y AUS -9.5% -0.8% -0.8%

BD 10Y CAN -8.2% -3.4% -3.4%

BD 5Y US -5.1% -9.7% -9.7%

BD 2Y US -2.6% -26.7% -26.7%

HY US -1.2% -1.4% -1.4%

BD EM -0.3% 1.7% 1.7%

RB 10Y US -7.6% -10.8% -10.8%

SB US -0.4% -2.7% -1.2% -1.6% -0.9% -0.1% -2.5% -1.9% -2.4% 5.2% 5.8% -0.3% 5.2% 13.5% 0.8% 3.4% 9.7% 26.7% 1.4% -1.7% 10.8% 67.1%

BUCKETVALUA-

TION

ALLOCATION vs.

S

Zero


Simulations – Valuation and Suggested Allocation

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EQ US

EQ UK

EQ EMU

EQ SWI

EQ JAP

EQ AUS

EQ CAN

EQ EMA

BD 10Y US

BD 10Y UK

BD 10Y EMU

BD 10Y CH

BD 10Y JAP

BD 10Y AUS

BD 10Y CAN

BD 5Y US

BD 2Y US

HY US

BD EM

RB 10Y US

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

Valuation

Allocation (Vector)

Allocation (Matrix)


Allocation - Comparison

• Vector approach:

• xri and w

i have always identical signs.

• Cash is a special bucket, as all bets are made vs. cash.

• Hence, the cash dispersion is massively larger.

• Matrix approach:

• xri and w

i do not necessarily have identical signs.

• Market i may be undervalued but shorted vs. most other markets.

• The reason is: it may be less undervalued than other markets.

• Cash is a bucket like any other.

• Hence, the cash dispersion is much smaller.

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Performance - Comparison

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0.0

2.2

4.5

6.7

9.0

11.3

13.5

15.8

18.0

20.2

22.5

24.7

27.0

29.2

31.5

33.7

36.0

38.3

40.5

42.8

45.0

47.3

49.5

51.8

54.0

56.3

58.5

60.8

63.0

65.3

67.5

69.8

72.0

74.3

76.5

78.8

81.0

83.2

85.5

87.7

90.0

92.2

94.5

96.7

99.0

1

10

100

1000

10000

Performance - Vector vs. Matrix Approach

Vector

Matrix

Time (Years)

Perf

orm

ance

(Log)


Performance - Comparison

• Resulting Information Ratios (IR), based on 20 market bets:

• Vector Approach: 0.93

• Matrix Approach: 1.48

• Overall, the level of the resulting IR seems (too) high.

• This is partially due to the stationarity underlying our framework.

• Realistically, that’s the best possible assumption.

• However, the high IR is also due to efficient use of information.

• Notably, the matrix approach performs much better.

• The reason is its better diversification.

• The portfolio can be scaled through f to any risk/return level at the given

IR.

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Summary and Conclusions

• We develop a formal allocation process that is

• Transparent

• Consistent across markets

• To that end, we

• Calibrate and simulate a mean reverting value/price process

• Extract and translate the embedded signals

• We present two translation approaches

• Vector approach: all market bets are made vs. cash

• Matrix approach: there are also bets between markets

• The matrix approach performs better, as it is better diversified.

• We may deviate from the suggested allocation; the according performance

difference is attributed to non-fundamental factors.34

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Thank you very much for your Attention!

35

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References

[1] Grinold, Richard C. and Ronald Kahn. “Active Portfolio Management.” Probus Publications, Chicago, 1995.

[2] Hull, Jon C., 1993, Options, Futures, and other Derivatives, Second Edition. Prentice Hall, Englewood Cliffs, NJ.

[3] Staub, Renato. “Signal Transformation and Portfolio Construction for Asset Allocation.” Working Paper, UBS Global Asset Management, Sept. 2008.

[4] Staub, Renato. “Deploying Alpha: A Strategy to Capture and Leverage the Best Investment Ideas”. A Guide to 130/30 Strategies, Institutional Investor, Summer 2008.

[5] Staub, Renato. “Are you about to Handcuff your Information Ratio?” Journal of Asset Management, Vol. 7, No. 5, 2007.

[6] Staub, Renato. “Unlocking the Cage”. Journal of Wealth Management, Vol. 8, No. 5, 2006.

[7] Staub, Renato. “Deploying Alpha Potential”. UBS Global Asset Management, White Paper, 2006.

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