Unconstrained Multi-Asset InvestingSeeking diversification through efficient portfolio construction using

cash-based and derivative instruments

22.01.13

“Uncertainty is the only certainty there is, and knowing how 

to live with insecurity is the only security”

John Allen PaulosProfessor of Mathematics at Temple University, Philadelphia, USA

Background

• Working party paper prepared in 2012– Presented at Risk and Investment Management conference, June 2012

• This paper formed core part of sessional paper to be presented to Institute and Faculty audiences in London and Edinburgh in 2013

• Themes– Measurement of portfolio diversification– Diversification benefits of unconstrained multi‐asset investment– Use of derivatives to implement investment strategies– Risk measurement of an unconstrained multi‐asset portfolio

• Question– Where does the actuarial profession believe it has most to add in the realm 

of portfolio construction?

Traditional approach fails to deliver consistent returns

Annualised rolling 3 year returns

-15.0%

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

1989 1992 1995 1998 2001 2004 2007 2010

Ann

ualis

ed %

retu

rn

Typical return expectation

Annualised rolling 3 year returns

-15.0%

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

1989 1992 1995 1998 2001 2004 2007 2010

Ann

ualis

ed %

retu

rn

Typical return expectation

Annualised rolling 3 year returns – Standard Life Balanced Fund

Uncertainty is the only certainty there isSource: Standard Life Investments

Source: IPD UK Monthly Property Index, All Property; Federal Reserve Trade-Weighted Exchange Value of US Dollar vs 6 Countries; Dow Jones UBS – Commodity Index; Barclays Capital Global Corporate Index, Excess Returns; Barclays Capital Emerging Markets Index, Excess Returns; Barclays Capital US High Yield Index, Excess Returns; Standard Life Investments, 31 December 2011

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Property Currency Basket Commodities

IG Corporate

Bonds

EmergingMarket Bonds

High Yield Bonds

2000-2007 2008-2011

Correlation of Global Equities with other market sectors

Finding diversity is increasingly difficult

Portfolio diversification measures

• No single measure of diversification

• Growing academic literature on measures of systemic risk

• Need for some definite measures of diversification:1. Assess current state of the world2. Observe how it is changing through time

• We want to consider measures for:1. Diversification in a given portfolio2. Diversification potential in an investment universe

Introduction to principal components analysis

•Analysis is based on the weighted covariance matrix:1.Covariance matrix of asset returns: 2. Portfolio weights

Introduction: Eigenvalues

• For every linear combination of assets, represented by a vector v, we can compute the size of the projected variance

• The first eigenvalue is the projected variance associated to the vector for which this projection is maximised. This is typically interpreted as the market

• Later eigenvalues are associated with vectors that maximise this expression under the constraint that they are orthogonal to vectors computed earlier

• This process creates an ensemble of uncorrelated strategies called principal portfolios. This process is called principal component analysis

Effective Number of Assets

• We define the effective number of assetsbased on the entropy of the eigenvalues of the weighted covariance matrix

• Entropy H(p) is a measure of the information in the probability distribution p. It has been used widely in physics, information theory, signal processing, etc

• Hence the effective number of assets is a measure of the concentration of eigenvalues

• It takes values between 1 and N

Effective number of assets:

Diversification within a given portfolio

Diversification Potential

• The diversification potential is the maximum over the effective number of assets as a function of the portfolio weights

• Well defined optimisation problem since the measure varies smoothly with the portfolio weights w

• Because of symmetries, we are looking for the local minimum near the weights that define an equal weighting in terms of risk

• Can be solved with a local optimiser (Newton–Raphson  type)

Diversification potential:

The maximum amount of effective assets that any given portfolio universe could have

Effective number of stocks in FTSE 100Market cap weighted

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

Dec 00 Apr 02 Sep 03 Jan 05 Jun 06 Oct 07 Mar 09 Jul 10 Nov 11

Effe

ctiv

e nu

mbe

r of s

tock

s

3000

3500

4000

4500

5000

5500

6000

6500

7000

FTSE

100

inde

x

FTSE100 index Effective number of stocks; cap weighted (lhs)

Effective number of stocks is low compared to universe

Effective number of stocks in FTSE 100Equally weighted and maximum potential

0

10

20

30

40

50

60

Dec 00

Apr 02

Sep 03

Jan 05

Jun 06

Oct 07

Mar 09

Jul 10

Nov 11

Effe

ctiv

e nu

mbe

r of s

tock

s

3000

3500

4000

4500

5000

5500

6000

6500

7000

FTS

E 1

00 in

dex

FTSE100 index Effective number of stocks; cap weighted (lhs)

Effective number of stocks; equally weighted (lhs) Diversification Potential (lhs)

Maximum potential is three times that of a market cap weighted approach

Effective number of assets in UK pension portfolio

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Mar 04 Aug 05 Dec 06 May 08 Sep 09 Jan 11 Jun 12

Effe

ctiv

e st

rate

gies

100

110

120

130

140

150

160

170

180

190

Bala

nced

Fun

d to

tal r

etur

n in

dex

Total return index Effective number of strategies (lhs) Diversification Potential (lhs)

Low number of effective assets reflects dominance of equity and interest rate risk

Defining the investment challenge

• “How can I build a better diversified investment portfolio but without impacting long term return expectations?”

• Allow a broader global investment universe currencies asset classes and sectors  yield curves across different geographies volatility

• A number of these strategies (such as specific currency pairs) will necessitate the use of derivative contracts

• Allow unconstrained dynamic asset allocation  offers access to the full diversification potential of the chosen investment universe 

Money making strategies and diversification

Strategy correlation with global equity

-0.60

-0.40

-0.20

-

0.20

0.40

0.60

0.80

1.00

Cor

rela

tion

coef

ficie

nt

Least diversification

Most diversification

UK EquityHigh yield creditRussian Equity

Mexican 10 year bonds

Global inflation-linked bondsUK corporate bonds

Europe bond yield steepenerUS Dollar v EuroUS equity large v small cap

Strategy correlation with global equity

-0.60

-0.40

-0.20

-

0.20

0.40

0.60

0.80

1.00

Cor

rela

tion

coef

ficie

nt

Least diversification

Most diversification

UK EquityHigh yield creditRussian Equity

Mexican 10 year bonds

Global inflation-linked bondsUK corporate bonds

Europe bond yield steepenerUS Dollar v EuroUS equity large v small cap

Derivative-based strategies can offer the most diversification potentialSource: Standard Life Investments, July 2012

Portfolio Themes – choice of implementations

• Profit opportunities in areas of growth Implementation choices• US equity Physical or equity futures/options• US technology equity vs US small cap equity Physical and / or equity futures/options • Korean equity Physical or equity futures/options

• Corporate balance sheet strength • US large cap equity vs US small cap equity Physical and / or equity futures/options• European high dividend‐yield equity Physical or equity futures/options• Investment grade and high yield credit Physical or index CDS contracts

• Fiscal advantages of resource‐based economies • Mexican government bonds Physical bonds• Brazilian Real vs Australian Dollar Currency forward• Russian equity Physical• US Dollar vs Canadian Dollar Currency forward

• Challenging Eurozone resolution• US Dollar vs Euro  Currency forward

Wide range of strategies that can make money

Effective factors for unconstrained multi‐strategy portfolio

Portfolio’s greater number of ‘moving parts’ allows for more return consistency

0

2

4

6

8

10

12

14

Mar 04 Aug 05 Dec 06 May 08 Sep 09 Jan 11 Jun 12

Effe

ctiv

e st

rate

gies

110

120

130

140

150

160

170

180

190

200

210

220

Glo

bal e

quity

inde

x

Global Equity index Effective number of strategies (lhs) UK balanced

Portfolio risk managementCombining physical and derivative holdings

• Risk‐based portfolio construction Size of each position Volatility of each position

• Ensures consistency Some assets physically invested others by derivatives Different asset classes have different volatility

• Stand‐alone riskmeasures the risk any of any one investment strategy US Equity has volatility: 23.2% Portfolio allocates 10% investment to US equities  Stand‐alone risk of US equities to portfolio is 2.3%

Source: Standard Life Investments, 31 March 2012

Absolute Return investing Risk‐based portfolio construction

0%5%

10%15%20%25%

30%35%40%

45%50%

Nominal holding Strategy volatility

0%5%

10%15%20%25%

30%35%40%

45%50%

Nominal holding Strategy volatility

3%

Rus

sian

E

quity

0%

1%

1%

2%

2%

Stan

d-al

one

risk

US

equ

ity

Glo

bal h

igh

yiel

d cr

edit

Kor

ea E

quity

vs

Eur

opea

n E

quity

US

Dol

lar v

s E

uro

Glo

bal i

ndex

-lin

ked

bond

s

UK

cor

pora

te

bond

s

Aus

tralia

n sh

ort-t

erm

in

tere

st ra

tes

Sw

edis

h vs

G

erm

an s

hort-

term

rate

s

3%

Rus

sian

E

quity

0%

1%

1%

2%

2%

Stan

d-al

one

risk

US

equ

ity

Glo

bal h

igh

yiel

d cr

edit

Kor

ea E

quity

vs

Eur

opea

n E

quity

US

Dol

lar v

s E

uro

Glo

bal i

ndex

-lin

ked

bond

s

UK

cor

pora

te

bond

s

Aus

tralia

n sh

ort-t

erm

in

tere

st ra

tes

Sw

edis

h vs

G

erm

an s

hort-

term

rate

s

0%

1%

1%

2%

2%

Stan

d-al

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risk

US

equ

ity

Glo

bal h

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Kor

ea E

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vs

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opea

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quity

US

Dol

lar v

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Glo

bal i

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-lin

ked

bond

s

UK

cor

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Aus

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in

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Sw

edis

h vs

G

erm

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hort-

term

rate

s

Stand-alone risk aligns size of position with its volatilitySource: Standard Life Investments, sample absolute return portfolio, March 2012

Risk‐based portfolio constructionSeek a number of different risks

6.5%

15.9%

0%

5%

10%

15%

20%

25%

US

Equ

ity

Rus

sian

Equ

ity

US

Lon

g B

ond

Yie

lds

(For

war

d S

tart)

Rel

ativ

e V

aria

nce

Inco

me

Hig

h Y

ield

Cre

dit

Fina

ncia

l Sec

tor v

s B

road

Cre

dit

Kor

ean

Equ

ity v

s E

urop

ean

Equ

ity

Long

US

Dol

lar v

s E

uro

Mex

ican

Gov

ernm

ent B

onds

vs

Eur

o

Bro

ad v

s Fi

nanc

ial S

ecto

r Equ

ity

US

Equ

ity L

arge

vs

US

Equ

ity S

mal

l Cap

Glo

bal E

quity

Eur

opea

n E

quity

Glo

bal I

nfla

tion-

Link

ed B

onds

Long

US

Dol

lar v

s C

anad

ian

Dol

lar

US

Equ

ity T

echn

olog

y S

ecto

r vs

US

Equ

ity S

mal

l Cap

Long

US

Dol

lar v

s Ja

pane

se Y

en

UK

Inve

stm

ent G

rade

Cre

dit

UK

Infla

tion

Sec

urity

Sel

ectio

n - a

lpha

Long

Nor

weg

ian

Kro

ne v

s E

uro

FX H

edgi

ng

Eur

opea

n In

vest

men

t Gra

de C

redi

t

Chi

na E

quity

vs

UK

Equ

ity V

olat

ility

Aus

tralia

n S

hort-

term

Inte

rest

Rat

es

Sw

edis

h vs

Ger

man

Sho

rt-te

rm In

tere

st R

ates

UK

Equ

ity

Eur

opea

n B

ond

Yie

ld S

teep

ener

Div

ersi

ficat

ion

Exp

ecte

d V

olat

ility

Vol

atili

ty %

Diversification

Expected volatility

Source: Standard Life Investments, APT system, March 2012

Total aggregated risk 22.4% (total nominal exposure 220%)

Leverage and portfolio risk

Portfolio Leverage factor Strategy volatility Portfolio risk

US equity 1.0 23.2% 23.2%

UK corporate bonds 3.0 7.7% 23.1%

Australian short-term interest rates 10.0 0.9% 9.0%

Portfolio Leverage factor Strategy volatility Portfolio risk

US equity 1.0 23.2% 23.2%

UK corporate bonds 3.0 7.7% 23.1%

Australian short-term interest rates 10.0 0.9% 9.0%

• In theory lowest risk portfolio is the most leveraged

• However, actual outcome is driven by the behaviour of only ONE factor

• Leveraged credit is a strategy that failed in the GFC

Historical scenario analysis

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18

Bank Meltdown 2008 ( September 12 - October 15, 2008)

Subprime Debacle 2007 (July 15 - August 15, 2007)

Emerging Market Sell-Off 2006 (May 1 - June 8,2006)

Bond Sell-Off (June 14 - July 31, 2003)

Bond Rally (May 1 - June 13, 2003)

Gulf War 2 (March 1-23,2003)

Equity Rally (October 10 - November 27,2002)

Equity Sell-Off (August 23 - October 9, 2002)

Sept 11th

Tech Wreck (April 7-14, 2000)

Russian/LTCM

Asian Crisis 1997

Mexican Crisis 1995

Rate Rise 94

Gulf War 1990

Black Monday 1987

% Move

Multi-asset Portfolio MSCI World (GBP) move over same period

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18

Bank Meltdown 2008 ( September 12 - October 15, 2008)

Subprime Debacle 2007 (July 15 - August 15, 2007)

Emerging Market Sell-Off 2006 (May 1 - June 8,2006)

Bond Sell-Off (June 14 - July 31, 2003)

Bond Rally (May 1 - June 13, 2003)

Gulf War 2 (March 1-23,2003)

Equity Rally (October 10 - November 27,2002)

Equity Sell-Off (August 23 - October 9, 2002)

Sept 11th

Tech Wreck (April 7-14, 2000)

Russian/LTCM

Asian Crisis 1997

Mexican Crisis 1995

Rate Rise 94

Gulf War 1990

Black Monday 1987

% Move

Multi-asset Portfolio MSCI World (GBP) move over same period

Source: UK £ GARS Regulated Unit Trust, RiskMetrics, 30/09/2012

Analyse actual return experience 

0

20

40

60

80

100

120

140

160

Weekly % return

No

of o

bser

vatio

ns

Multi-asset portfolioMSCI World

WORSTthree equity weeks

Equity mulit-asset-16.6% -5.5%-9.3% -3.1%-8.3% -3.0%

BESTthree equity weeks

Equity multi-asset+9.9% +4.3%+8.3% +3.1%+7.6% +3.1%

-7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 810 9 10 1-8-9-100

20

40

60

80

100

120

140

160

Weekly % return

No

of o

bser

vatio

ns

Multi-asset portfolioMSCI World

WORSTthree equity weeks

Equity mulit-asset-16.6% -5.5%-9.3% -3.1%-8.3% -3.0%

BESTthree equity weeks

Equity multi-asset+9.9% +4.3%+8.3% +3.1%+7.6% +3.1%

-7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 810 9 10 1-8-9-10

Distribution of weekly returns of unconstrained multi-asset portfolio v MSCI Global Equities

Source: Standard Life Investments, net performance from 12/06/2006 to 30/11/2012Portfolio performance is based on the £, institutional pooled pension portfolio* Source: Thomson Datastream, MSCI World (£) (net of tracker fund fee)

Summary

• Traditional multi‐asset funds offer insufficient diversification opportunities

• Derivatives significantly broaden the range of investment strategies that a portfolio can utilise

• Combining traditional assets with derivative‐implemented strategies produces more risk efficient portfolios increase the likelihood of ‘good’ outcomes for your portfolio

• Actuaries well placed to add value in understanding and communicating investment diversification

Appendices

Absorption Ratio: related measures

• Several related indicators based on correlations and principal components exist: Billio et. al. (2010): Measuring Systemic Risk in the Finance and Insurance Sectors Golts et. al. (2010): Diversification, Volatilities, and the Supercurrency Fenn et. al. (2010): Temporal Evolution of Financial Market Correlations Meucci (2009): Managing diversification 

• There are several ways of measuring the concentration of eigenvalues: Absorption Ratio Herfindahl‐Hirschman index (HHI) Gini coefficient Entropy

• But which one is mathematically justified? What properties do they satisfy?

Properties of ‘effective factors’

• Invariantmeasure• Depends continuously on the portfolio weights• High level of stability• Varies intuitively under addition of extra assets• Interpretable as the number of independent risk factors required to capture the risks 

of the underlying asset positions without losing information• Rigorous mathematical interpretation:

Known in the signal processing literature as the effective rank Satisfies a property that is analogous to Shannon’s noisy coding theorem

Risk‐based portfolio guidelines

• Limit the amount of risk allocated to any one broad asset group Eg equities, interest rates, credit etc.

• Limit the total aggregate risk that can be taken Risk models underestimate volatility and overestimate diversification benefits in 

stressed markets

• Continuous monitoring of portfolio’s actual risk behaviour versus what is expected

The information shown relates to the past. Past performance is not a guide to the future.  The value of investment can go down as well as up. 

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