risk measurement chapter 7

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Practical Financial

OptimizationChapter 7

Index FundsStavros A. Zenios

HERMES European Center of Excellence on Computational Financeand Economics

University of Cyprus

The Wharton Financial Institutions Center

University of Pennsylvania

©2002, S.A.Zenios

Overview

Preview

Basics of Index Funds

Indexation Models

International Index Funds



©2002, S.A.Zenios

7.2 Basics of Market Indices

Index:

A statistics that summarizes the relative changes of

a set of variables

Geographical criteria

Sector Growth

IndustryValue

Narrow viewBroad view

©2002, S.A.Zenios


Why use an index?

Comprehensive measure of market trends

Benchmark Passive management of a portfolio

Tracking Indices

Active portfolio management performance

769 all-equity actively managed funds during 1983—1989underperformed the S&P500 by 200 to 500 basis points.

¼ of the best performers continued to perform equally well

¼ of the best performers were worst performers in thesubsequent period



©2002, S.A.Zenios


Rules driven indices

detailed rules determine the choice of securities in the index

predictable composition

Discretionary indices

An index designed by a committee

Considerations:

trading analysis, liquidity, ownership, fundamental analysis, market

capitalization, sector representation

Example: the pronounced bias towards stocks after 1999

©2002, S.A.Zenios

7.2 Basics of Market IndicesCreating an index

Practical considerations

Liquidity of securities

No barriers to entry

Stable composition

Representation of the universe of securities amanager actually invests in

Weights

Equally-weighted

Price-weighted

Cap-weighted



©2002, S.A.Zenios

7.2 Basics of Market IndicesExamples

Example Indices: NYSE (3000 stocks listed), S&P 500, GBI Broad Index, S&P MidCap

400, S&P SmallCap 600, Salomon Brother’s Index of mortgage-backed

securities

Composition of the JP Morgan’s GBI Broad

©2002, S.A.Zenios

7.2 Basics of Indices

Notation

l r wr w R

r w

U i

K j

K

j

l

j j

l

I

K

j j

K

j j

scenariounder indextheof return),(

returnsweights,)(,)(

securitiesallof universitythe

factors)(risk indexin thesecuritiesthe,1,2,

1

11

∑=

==

=

∈

= K



©2002, S.A.Zenios

7.2 Basics of index fundsImplementing an index fund

Creating an index fund Matching an index, i.e. its growth rate mimics that of the index

Enhanced-index or index plus

Challenges

1. Large number of instruments to choose from

2. Transaction costs (net return can be non-positive)

3. Reinvestment of the proceeds

4. Liquidity risk

Models: Structure-based models

Co-movements based models

©2002, S.A.Zenios

7.3 Indexation modelsThe linear programming approach

cw

Step 1

Cells with given

characteristics are

consolidated

Step 2

Stocks are entered

in each cell



©2002, S.A.Zenios

7.3 Indexation modelsIndex tracking model

=

otherwise,0cellthto belongs bondif ,1functionindicator ji

ijδ

( )

0

1

,,,2,1,

tosubject

Maximize

1

≥

=

==

∑

∑

=

∈

x

x

K k jw x

x F

m

i

i

jiij

X x

KKδ

©2002, S.A.Zenios

7.3 Indexation modelsCo-movements

Approach

The return of the index and the returns of the securities in the universe of the

available securities are uncertain

Replication of the index response to the risk factors

Tracking error:

Positive and negative deviations:

∑ ∑= =

−=

−=m

i

K

j

l

j ji

l

i

l

l

I

l

P

l

r w xr r w x R

r w Rr x Rr w x R

1 1

),,(

),(),(),,(

ε

ε

−=

−=

∑

∑

=−

=+

m

i

i

l

i

l

I

l

l

I

m

i

i

l

i

l

xr r w R y

r w R xr y

1

1

),(,0max

),(,0max

l l l l y yr w x R y −+ −== ),,(



©2002, S.A.Zenios

7.3 Indexation modelsCo-movements: Tracking Model

constrainttracking),(

:modeltrackingsided-Two

0

1

)constraint(trackingall,),(

subject to

Maximize

1

1

1

1

ε ε

ε

≤−≤−

≥

=

Ω∈−≥−

∑

∑

∑

∑

=

=

=

=∈

l

I

m

i

i

l

i

m

i

i

l

I

m

i

i

l

i

m

i

ii X x

r w R xr

x

x

l r w R xr

xr

©2002, S.A.Zenios

7.4 International Index FundsMotivation

Composite indices

Strategic asset allocation – long time horizons

Tactical asset allocation – short time horizons

Risk Factors

Local market risk factors (sub-indices)

Exchange rate risk

Applications:

Global indices

Corporate bond indices



©2002, S.A.Zenios

7.4 International Index FundsIntegrated-Disintegrated model

Structuring the

indexed portfolio

Determine weights

for each market j in

the global index and

holdings for each

bond i

Bond picking

decisions based

on the weights

xij

Determine weights

for each market i in

the global index

Determine weights for

each bond i in the

market j

Structuring the

indices

Global indexreturn

Index return Index return Index return

Market

return

Market

return

Market

return

Ω1

Ω j

Ω K

U 1 U 1 U K

I 1 I j I K

γi

I G

wi1 wij wiK γi

©2002, S.A.Zenios

7.4 International Index FundsNotation

Notation:

Markets

Sample from the universeΩ j of securities:

Weights of the security i in the jth sub-index.

Overall (global) index weights

Weight of the global index in the jth sub-index

K j ,,1K=

K jU j ,,1,K

=ijw

∑=

= K

j

j j

1

1, γ γ

jwm

i

jij all,1

∑=

= γ



©2002, S.A.Zenios


Problem of index fund manager Determine the z j of the portfolio value invested in the jth

market, and pick the bonds from each marketΩ j to add to

the portfolio with proportion xij

Disintegrated or decentralized model:

Pick the z j and then choose the xij so that

Integrated approach

Jointly determine the z j and xij

∑=

= K

i

jij z x1

©2002, S.A.Zenios


Local return:

Local portfolio return:

Exchange rate:

Exchange rate appreciation:

Portfolio return in base currency:

Return of the composite index:

l

ijr

∑=

=m

i

ij

l

ij

l

pj xr r x R

1

),(

∑=

= K

j

l

pj

l

j

l

p r x Rer x R1

),(),(

0/ j

l

j

l

j E E e =

( ) ( )∑=

= K

j

l

j j

l

j

l

I r w Rer R1

,, γ γ

l

j j E E ,0



©2002, S.A.Zenios

7.4 International Index FundsIntegrated Indexation model

( )

( ) ( )

∑

∑∑

∑

∑ ∑

=

= =

=

Ω∈ =∈

=

≥

=

Ω∈≤−≤−

m

i

ij j

K

j

m

i

ij

K

j

l

I

l

pj

l

j

l

K

j

l

pj

l

j

l

X x

x z

x

x

l r Rr x Re

r x Re p

1

**

1 1

1

1

0

1

all,,,

subject to

,max

ε γ ε

©2002, S.A.Zenios

7.4 International Index FundsBreaking up the model

Strategic allocation:

Model to determine the optimal z j :

∑ ∑∑= ==

==

K

j

j

m

i

l

ijij

l

j

K

j

j

l

pj

l

j

l

pj p z r we z r w Rer w R z R1 11

),()),(,(

0

1

all,),(),(

subject to

),(max

1

1

1

≥

=

Ω∈≤−≤−

∑

∑

∑ ∑

=

=

Ω∈ =∈

z

l r R z r w Re

z r w Re p

K

j

j

K

j

l

I j

l

j

l

j

l

K

j

j

l

pj

l

j

l

X x

ε γ ε



©2002, S.A.Zenios

7.4 International Index FundsBreaking up the model

Tactical allocation for international index funds

( ) ( )

0

all,,,

subject to

),(Maximize

solve,,2,1eachfor

1

*

≥

=

Ω∈≤−≤−

=

∑

∑

=

Ω∈∈

x

z x

l r w Rr x R

r x R p

K j

m

i

jij

l

j

l

pj

l

l

pj

l

X x

ε ε

K

©2002, S.A.Zenios

7.4 International Index FundsThe value of integration



©2002, S.A.Zenios

7.4 International Index FundsOperational model for international index funds

Motivation:

Accounts for transaction costs in portfolio revision

Allows cash infusion or withdrawal

Liquidity and diversification constraints

Decision variables are defined in terms of face

values than as proportions of total assets

©2002, S.A.Zenios


Notation

face value invested in security i in the jth currency

face value sold of security i in the jth currencyface value of security i in the jth currency that remains asinventory in the indexed portfolio

risk free investment in the base currency

initial holdings in the risk free asset in the base currency

face value of initial inventory of security i in the jth currency

current bid price of security i in the jth currency

current ask price of security i in the jth currency

ij x

ij yij z

+v

ijb0

0v

b

ij P 0a

ij P 0



©2002, S.A.Zenios


Initial value of the portfolio

Inventory balance

Cashflow balance

Value at the end of the holding period

Rate of return

∑ ∑= =

+= K

j

m

i

ij

b

ij j b P E vV 1 1

00

0

00

ijijijij y xb z −+= 0

∑ ∑ ∑= = =

+

−+=

K

j

m

i

m

i

ij

a

ijij

b

ij j x P y P E vv1 1 1

00

0

0

( )∑ ∑= =

+ ++= K

i

m

i

ij

b

ij

l

ij

l

j

l

T z P r E vV 1 1

01

0

0),,,,(V

V V r v z y x R

l l l

p

−=+

©2002, S.A.Zenios


( )

( ) ( )0,,,

allfor ,,,,,,

,...,1,,...,1all,

subject to

,,,,max

1 1

0

1

0

0

1

0

0

0

,,,

≥

Ω∈≤−≤−

=

−+

===+−

+

+

= = =

+

=

Ω∈

+

∈

∑ ∑ ∑∑

∑+

v z y x

l r Rr v z y x R

v y P E v x P E

K jmib y x z

r v z y x R p

l

I

l l

p

K

j

K

j

m

i

ij

b

ij j

m

i

ij

a

ij j

ijijijij

l

l l

p

l

X v z y x

ε γ ε



©2002, S.A.Zenios

7.5 Models for Corporate Bond

Index Funds

Challenges

Diverse sources of risk inherent in the bond market

The number of securities in the index is large

Risks

Changes in the term structure of risk free rates

Changes in the term structure of credit spreads

Changes in the credit rating of the bond The likelihood that the bond will go into default

The amount recovered if a bond goes into default

©2002, S.A.Zenios

7.5 Models for Corporate Bond

Index Funds

International indexation can track corporate

bond indices

Subindices are not country-specific but by

Credit rating

Sector

Maturity date

Currency risk is eliminated

that is, 1=l

j E



©2002, S.A.Zenios

7.5 Models for Corporate Bond Index

Funds – Enhanced index funds

©2002, S.A.Zenios


Funds – Enhanced index funds



©2002, S.A.Zenios


Funds – Tracking the Merrill Lynch Index

Value of a 100USD investment in the Merrill Lynch US Treasury index and in

indexed portfolios constructed using treasury securities only

©2002, S.A.Zenios


Funds – Tracking the Merrill Lynch Index

Tracking Performance of index-plus portfolios that track the Merrill Lynch

US Treasury index by investing in both treasury securities and corporatebonds



©2002, S.A.Zenios


Funds – Salomon Brothers Index

Value of 100USD investment in the Salomon Brothers Index of

mortgage-backed securities, and in indexed portfolios using a single

period and a stochastic model

©2002, S.A.Zenios


Funds – Salomon Brothers Index

Tracking errors of the two indexed portfolios on Salomon Brothers Index

of mortgage-backed securities of the two portfolios constructed using a

single-period and a stochastic programming model vs. the index



©2002, S.A.Zenios


Funds – Callable bond index

Value of a 100USD investment in the callable bond index and

in indexed portfolio developed using a single-period and multi

period model

©2002, S.A.Zenios


Funds – Callable bond index

Tracking errors of the indexed portfolios developed using a single

period and a stochastic programming model tracking the callable

bond index.

risk measurement chapter 7

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