risk measurement chapter 7
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8/8/2019 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
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©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
7.2 Basics of Market Indices
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
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©2002, S.A.Zenios
7.2 Basics of Market Indices
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
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©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
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©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
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©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 −+ −== ),,(
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©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
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©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
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©2002, S.A.Zenios
7.4 International Index FundsNotation
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
7.4 International Index FundsNotation
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
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©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
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©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
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©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
7.4 International Index FundsOperational model for international index funds
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
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©2002, S.A.Zenios
7.4 International Index FundsOperational model for international index funds
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
7.4 International Index FundsOperational model for international index funds
( )
( ) ( )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
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©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
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©2002, S.A.Zenios
7.5 Models for Corporate Bond Index
Funds – Enhanced index funds
©2002, S.A.Zenios
7.5 Models for Corporate Bond Index
Funds – Enhanced index funds
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©2002, S.A.Zenios
7.5 Models for Corporate Bond Index
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
7.5 Models for Corporate Bond Index
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
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©2002, S.A.Zenios
7.5 Models for Corporate Bond Index
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
7.5 Models for Corporate Bond Index
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
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©2002, S.A.Zenios
7.5 Models for Corporate Bond Index
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
7.5 Models for Corporate Bond Index
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.
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