How Delegated Fund Management
Creates Comovements and Priced
Factors
Prof. Michael Stutzer
Burridge Center for Securities Analysis
and Valuation
Leeds School of Business
University of Colorado
Boulder, Colorado
Technical References
•
Proof of the Key Theoretical Result is Found in
:
–
Stutzer,
Fund Managers May Cause Their
Benchmarks to Be Priced Risks
,
Journal of
Investment Management
, 2003, pp. 64

76. Reprinted
in Gifford Fong (ed.),
The World of Risk Management,
World Scientific Publishers, 2006.
[Uses textbook mean

variance analysis]
–
Stutzer,
Benchmark Investing in the ICAPM
, current
CU Working Paper.
[Uses Merton

style continuous time
analysis
]
Empirical References
•
Barberis, Schleifer, and Wurgler,
Comovement,
J. Fin.
Econ
., 2005.
•
Boyer,
Comovement Among Stocks with Similar Book

To

Market Ratios
, Current Working Paper, Brigham
Young Univ. Dept. of Finance.
•
Cremers, Petajisto, and Zitzewitz,
Should Benchmark
Indices Have Alpha? Revisiting Performance Evaluation
,
Current Working Paper, Yale Univ. Dept. of Finance.
Performance Measurement:
CAPM Return vs. Risk Formula
[ ] ( [ ] )
fund Tbill fund market Tbill
E R R E R R

A Positive
is a Performance Measure
–
Underlying Theory
:

Each “Rational” Investor Selects an Expected
Utility Maximizing Portfolio: E[R
p
]

Var[R
p
]

Depends on a Fixed Investment Opportunity Set
Figure 1:
A Rational Investor, as Described in the Previous Slide
An Unfortunate Recent Development
•
Multi

factor (Fama

French) Formula Alpha
[ ] ( [ ] )
( [ ] ) ( [ ] )
m
fund Tbill fund m Tbill
value size
fund HML Tbill fund SMB Tbill
E R R E R R
E R R E R R
HML is a benchmark portfolio that is long “value” and short growth
SMB is a benchmark portfolio that is long small cap and short large cap
“m” denotes the market portfolio.

Depends on a Time

Varying Set of Investment Opportunities, Moving
Around As Two Unobservable “Risk Factors” Change (see Merton’s
ICAPM). HML and SMB Portfolios’ Returns Are Assumed to Proxy for
the Two Risk Factors
A Puzzle
•
It is difficult to establish that the value and
size effects are proxies for the two
supposed risk factors.
•
As a result, there is no foundation for
using the multi

factor
as a performance
measure.
–
This calls into question hordes of academic
studies that base their conclusions on multi

factor
•
Why Do Stocks Added (Deleted) from S&P 500
Start Correlating More (Less) With that Index
When Added (Deleted)? [see Vijh (1994)
Barberis, et.al. (2005).]
•
Why Do Stocks That Switch from S&P/Barra
Growth Index to S&P/Barra Value Index Start
Correlating Less With the Former and More with
the Latter? [Boyer (2004)].
Another Puzzle
•
I will show that the rise of both index and
professionally delegated investing (as opposed
to strictly individual investing) helps explain both
(seemingly unrelated) puzzles.
–
In finance theory, the objectives that motivate
managers of both index funds and active funds are
different than the objectives that are assumed to
motivate individual investors.
–
This leads to differences in asset demands that
change the assets’ prices in ways that solve the two
puzzles.
An Answer to These Puzzles
Delegated Investor Equity Holdings
Gompers and Metrick (2001)
The March of Delegated Investment
Continues
•
Worldwide By 2000: Perhaps $30 Trillion
Total Institutional Investment [Walters
(1999)].
•
Worldwide 2002

2007: Equity Mutual
Funds Grew From $4 Trillion to $12 Trillion
(ICI)
•
Delegated Investing Now Represents a
Strong Majority of Invested Funds. When
Merton Wrote around 1970, It was
Probably Only 20%, and Less When
Markowitz Wrote Around 1960.
The Latest in Delegated Investing: The Founding of
Soros Alpha Hedge Fund for Huns
•
Quantitative Theory of Individual Investing
–
Investor Cares About the Probability Distribution of Return
Resulting From Investing
–
In “Modern Portfolio Theory”, Investor Chooses Portfolio “p” to
Maximize E[R
p
]

Variance[R
p
]
•
Quantitative Theory of Delegated Investing
–
Fund Manager Cares About Probability Distribution of Return
In
Excess of a Fixed Benchmark Return R
b
–
In “Tracking Error Variance (TEV) Theory”, Manager Chooses
Portfolio “p” to Maximize
E[R
p

R
b
]

Variance[R
p

R
b
]; e.g. “b” = S&P 500
•
Index Fund Acts
as

if
it Uses Extremely Large
In Theory, Objectives Differ
Different Objectives
Different Asset Demands
•
Quantitative Theory of Individual Investing
–
Investor Chooses
Tangency Portfolio
with
Portfolio Weights q
p
CovMatrix

1
E[R

R
Tbill
]
•
Maximizes Sharpe Ratio E[R
p

R
Tbill
]/Std.Dev.[R
p
]
•
All Investors Choose Same Portfolio
!
This is Counterfactual!!
•
Quantitative Theory of Delegated Investing
–
Fund Chooses
Tilted Portfolio
With
Portfolio Weights q
pb
q
b
+ cq
p
•
Maximizes Information Ratio E[R
p

R
b
]/Std.Dev.[R
p

R
b
]
–
Quantitative Practitioners Often Say They Try to Do This
–
Equivalent to Minimizing the Probability of Underperforming the
Time

Averaged Benchmark Return
•
Fund
–
Benchmark Portfolio is Proportional To Tangency
Portfolio
Different Asset Demands
Different Probability Distributions for Assets
•
Total Asset Demands Must Equal Supply
•
Asset Demands of Individuals
+ Asset Demands of Funds = Total Demand
Implies the Following Return vs. Risk Formulae:
(see Stutzer,
J.Inv.Mgmt
., 2003)
( ) [ ( ) ( )]
m b
m Tbill m m b
b B
A
E R R W Var R W Cov R R
W
( ) [ ( ) ( )]
m b
fund Tbill fund m fund b
b B
A
E R R W Cov R R W Cov R R
W
( ) [ ( ) ( )]
m b
Tbill m b
b B
A
E R R W Cov R R W Cov R R
W
General Multi

Factor Model
•
As a Consequence of the Risk vs. Return
Formulae On the Previous Slide:
(see Stutzer,
J.Inv.Mgmt
., 2003).
( ) [ ( ) ] [ ( ) ]
m b
fund f fund m f fund b f
b B
E R R E R R E R R

With
Only
Individual Investors, There is No Summed Term, i.e. the
Formula is the CAPM (with
= 0 in theory).

With Delegated Fund Management, Each Popular Benchmark
Portfolio “b” is an Additional Factor (with
= 0 in theory).

While This Has the Same Mathematical Form as Fama

French,
the Presence of the Extra Factors Are Not Proxies for Risk Factors
That Cause Changes in Investment Opportunities
!
Some Empirical Evidence
•
S&P 500 is a popular benchmark “b”.
•
In order to meet (index funds) or beat (large
cap growth funds) the S&P 500, managers
must hold some of it:
q
pb
q
b
+ cq
p
•
So when stocks are added (deleted) from S&P
500 index, demand for them becomes more
(less) closely connected to demand for the
other 499.
–
As a result, they become more (less) correlated with
the S&P 500 index, i.e.
their S&P 500
coefficient
goes up (down).
Change in a Stock’s
w.r.t.
S&P 500 When It is
Added or Dropped From the S&P
[Barberis, et.al.(2005)]
Changes in
Compared to Changes in
Industry and Size

Matched Firms:
Same Thing Happens
•
S&P/Barra Growth and Value Indices
–
Equally
Capitalized by Low Book

to

Market
Ratio Stocks (“Growth”) and High Book

to

Market Stocks (“Value”).
–
Indices are Reconstructed Semi

Annually in
Order to Maintain The Equal Capitalization
•
Hence Some Stocks Are Reassigned From Growth
to Value (or Vice Versa) Just to Keep the Equal
Capitalization. For example, a growth stock with
positive returns (and hence lower B/M Ratio) could
nonetheless be reclassified as a value stock, to
keep 50% of market cap in the value stock index.
These are called
Index Balancers
.
Change in Stock’s
When Switched From Value to Growth Index
[Boyer(2007)]
w.r.t. Growth Index Increases, While
w.r.t. Value Index Decreases
Change in Stock’s
When Switched From Growth to Value Index
w.r.t. Value Index Increases, While
w.r.t. Growth Index Decreases
What About Expected Returns?
•
The previous evidence demonstrates that the
correlations among stocks’ returns are affected
by delegated investing.
•
Q: If the correlations are changed, is it rational to
presume
that stocks’ expected returns would
not
be affected by delegated investing?
•
A: NO. This Presumption is Not Rational!
Here is some evidence that they are affected:
Recall the Return vs. Risk Formula:
( ) [ ( ) ] [ ( ) ]
m b
fund f fund m f fund b f
b B
E R R E R R E R R
•
Many Trillions are Benchmarked to the S&P 500
•
Gomez and Zapatero (2003) Tested a 2

Factor
Formula using the MSCI US Index for “m” and the S&P
500 as the single benchmark “b”.
•
They Concluded that this 2

Factor Formula
Outperformed the 1

Factor (CAPM) Model
Cremers, Petajisto, and Zitzewitz
(2008)
•
Fama

French/Carhart 4

Factor Model:
R
2
= 29%
•
CPZ 4

Factor Model: S&P 500, Russell Midcap,
Russell 2000 + value

growth as factors:
R
2
= 48%
•
Fama

French/Carhart 4

Factor Model gives the
S&P 500 itself a Positive Alpha!!
–
Same thing happens with some other passive indices
–
This alone casts doubt on the usual interpretation of
alpha as the return to active management
.
Cremers, et.al. (continued)
Recap: The Two Puzzles’ Status
•
Puzzling changes in correlations of stocks going
into and out of benchmark indices is explained
by the rise of delegated investing, via both index
and managed funds.
•
Puzzling non

market causes of expected returns
that do not seem to arise as proxies for risk
variables are also (at least partly) explained by
the rise of delegated investing.
The Fama

French
Has No Clothes!
•
The Multi

Factor
is a relevant performance
measure for individual investors only when the
factors are the portfolios most closely correlated
with risk variables causing changes in the
investment opportunity set.
•
But we just argued that good direct evidence of
that is lacking, and that the rise of delegated
investing provides a plausible alternative reason
for the factors.
–
But if so,
is not necessarily relevant to
individual investors’ welfare.
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