数量-组合-cfa二级基础班portfoio management

Topic

Weightings

in

CFA

Level

IISession

NO.ContentWeightingsStudy

Session

1-2Ethics

&

Professional

Standards10Study

Session

3tative

Methods5-10Study

Session

4Economic

ysis5-10Study

Session

5-7Financial

Statement

ysis15-25Study

Session

8-9Corporate

Finance5-15Study

Session

10-13Equity

ysis20-30Study

Session

14-15Fixed

e

ysis5-15Study

Session

16-17Derivative

Investments5-15Study

Session

18Portfolio

Management5-15Study

Session

13Alternative

Investments5152-80Portfolio

ManagementSS18R53

An

introduction

to

multifactor

modelsR54 ysis

of

active

portfolio

managementR55

Economics

and

investment

marketsR56

The

portfolio

Management

Process

and

the

InvestmentPolicy

Statement3-80【梦轩考资

】Portfolio

ManagementR53

An

introduction

to

multifactor

modelsArbitrage

Pricing

Theory

(APT)Multifactor

modelMacroeconomic

FactorFundamental

factor

modelsStatistical

factor

modelsActive

RiskInformation

RiskActive

risk

squared专业提供CFA

FRM全清+讲义4-80Arbitrage

Pricing

Theory

(APT)APTasset

pricing

model

developed

by

the

arbitrage

pricing

theoryAssumptionsA

factor

model

describes

asset

returnsThere

are

many

assets,

so

investors

can

form

well-diversified

portfoliostha iminate

asset-specific

riskNo

arbitrage

opportunities

exist

among

well-diversified

portfoliosExactly

formulaE

(

RP

)

RF

P

,1

(1

)

P

,2

(2

)

...

P

,

k

(k

)5-80Arbitrage

Pricing

Theory

(APT)The

factor

risk

premium

(or

factor

price,

λj)

represents

the

expected

return

inexcess

of

the

risk

free

rate

for

a

portfolio

witha

sensitivity

of

1

to

factor

j

and

asensitivity

of

0

to

all

other

factors.

Sucha

portfolio

is

called

a

pure

factorportfolio

for

factor

j.The

parameters

of

the

APT

equation

are

the

risk-free

rate

and

the

factor

risk-premiums

(the

factor

sensitivities

are

specific

to

individual

investments).6-80Arbitrage

Pricing

Theory

(APT)Arbitrage

OpportunitiesThe

APT

assumes

there

are

no

market

imperfections

preventing

investorsfrom

exploiting

arbitrage

opportunities→

extreme

long

and

short

positions

are

permitted

and

mispricing

willdisappear

immedia

y→

all

arbitrage

opportunities

would

be

exploited

and

eliminated

immedia

y7-80Arbitrage

Pricing

Theory

(APT) -

ExampleExample:

suppose

that

two

factors,

surprise

in

inflation

(factor

1)

and

surprise

inGDPgrowth

(factor

2),

explain

returns.

According

to

the

APT,

an

arbitrageopportunity

existsunlessE(RP

)

RF

βp,1

(λ1

)+βp,2

(λ2

)Well-diversified

portfolios,

J,

K,

andL,

given

in

table.PortfolioExpected

returnSensitivity

toinflation

factorSensitivity

to

GDPfactorJ0.141.01.5KLE(RJ)

0.14

RF

1.0λ1

+1.5λ2E(RK

)

0.12

RF

0.5λ1

+1.0λ2E(RL

)

0.11

RF

1.3λ1

+1.1λ2p,1p,2+0.06βPE(R

)

0.07

0.02β8-80Arbitrage

Pricing

Theory

(APT)The

Carhart

four-factor

model

(four

factor

model)According

to

the

model,

there

are

three

groups

of

stocks

that

tend

to

havehigher

returns

than

those

predicted

solely

by

their

sensitivity

to

the

marketreturn:Small-capitalization

stocksLow

price-to

book-ratio

stocks,

commonly

referred

to

as

“momentum”stocksStocks

whose

prices

have

been

rising,

commonly

referred

to

as

“momentum”stocks9-80【梦轩考资

】Multifactor

ModelMultifactor

models

have

gained

importance

for

the

practical

business

ofportfolio

management

for

two

main

reasons.multifactor

models

explain

asset

returns

better

than

the

market

modeldoes.multifactor

models

provide

a

more

detailed ysis

of

risk

than

does

asingle

factor

model.Passive

management.

ysts

can

use

multifactor

models

to

match

an

indexfund's

factor

exposures

to

the

factor

exposures

of

the

index

tracked.Active

management.

Many tative

investment

managers

rely

onmultifactor

models

in

predicting

alpha

(excess

risk-adjusted

returns)

or

relativereturn

(the

return

on

one

asset

or

asset

class

relative

to

that

of

another)

as

partof

a

variety

of

active

investment

strategies.In

evaluating

portfolios,

ysts

use

multi-factor

modelsto

understand

thesources

of

active

managers'

returns

and

assess

the

risks

assumed

relative

tothe

manager's

ben

ark

(comparison

portfolio).专业提供CFA

FRM全清+讲义10-80Types

of

Multifactor

ModelsMacroeconomic

FactorFundamental

factor

modelsStatistical

factor

modelsMixed

factor

modelsSome

practical

factormodelshave

the

characteristics

ofmore

thanone

of

the

above

categories.

We

can

callsu odels

mixed

factormodels.11-80Macroeconomic

Factor

ModelMacroeconomic

Factorassumption:

the

factors

are

surprises

in

macroeconomic

variables

thatsignificantly

explain

equity

returnsexactly

formula

for

return

of

asset

iWhere:Ri

=

return

for

asset

iE(Ri

)

=

expected

return

for

asset

iFGDPFQSbi1=

surprise

in

the

GDP

rate=

surprise

in

the

credit

quality

spread=

GDP

surprise

sensitivity

of

asset

ibi2

=

credit

quality

spread

surprise

sensitivity

of

asset

iεi

=

firm-specific

surprise

which

not

be

explained

by

the

model.Ri

E(Ri

)

bi1FGDPi

2

QSi

b

F

Regression

(timeseries)ReturnFGDPFQS………………………………………bi1,

bi2Surprise

=

actual

value

–

predicted

(expected)value12-80Macroecon【o梦m轩考i资cwwwF.mxakaoczit.coomr】

1o064d548e42l

-专W业提供hCFAaFtRM全d程o高e清视s频s+讲u义rprise

mean?

Suppose

our

forecast

at

the

beginning

of

the

month

is

that

inflation

will

be

0.4percent

during

the

month.

At of

the

month,

we

find

that

inflation

wasactually

0.5

percent

during

the

month.

During

any

month,Actual

inflation

=

Predicted

inflation

+

Surprise

inflationIn

this

case,

actual

inflation

was

0.5

percent

and

predicted

inflation

was

0.4percent.

Therefore,

the

surprise

in

inflation

was

0.5

-

0.4

=

0.1

percent.13-80Macroecon【o梦m轩考i资cwwwF.mxakaoczit.coomr】

1o064d548e42l

–专业f提a供cCFtAoFRrM全程s高清en视s频i+t讲i义vity,

error

term

Slope

coefficients

are

naturally

interpreted

as

the

factor

sensitivities

of

theasset.

A

factor

sensitivity

is

a

measure

of

the

response

of

return

to

each

unit

ofincrease

in

a

factor,

holding

all

other

factors

constant.The

term

εi

is

the

part

of

return

that

is

unexplained

by

expected

return

or

thefactor

surprises.

If

we

have

adequa y

represented

the

sources

of

common

risk(the

factors),

then

εi

must

represent

an

asset-specific

risk.

For

a

stock,

it

mightrepresent

the

return

from

an

unanticipatedcompany-specific

event.14-80Factor

Sensitivities

for

a

Two-Stock

Portfolio

(example)Suppose

that

stock

returns

are

affected

by

two

common

factors:

surprises

in

inflation

andsurprises

in

GDP

growth.

A

portfolio

manager

is yzing

the

returns

on

a

portfolio

oftwo

stocks,

Manumatic

(MANM)

and

Nextech

(NXT),

The

following

equations

describethe

returns

for

those

stocks,where

the

factors

FINFL.

and

FGDP,

represent

the

surprise

ininflation

and

GDP

growth,respectively:One-third

of

the

portfolio

is

invested

in

Manumatic

stock,

and

two-thirds

is

investedinNextech

stock.Formulate

an

expression

for

the

return

on

the

portfolio.State

the

expected

return

on

the

portfolio.Calculate

the

return

on

the

portfolio

given

that

the

surprises

in

inflation

and

GDP

growthare

1

percent

and

0

percent,

respectively,

assuming

that

the

error

terms

for

MANM

andNXT

both

equal

0.5

percent.

0.09

1FINFL

1FGDPRMANM

0.12

2FINFL

4FGDP

MANM

NXTRNXT15-80Factor

Sensitivities

for

a

Two-Stock

Portfolio

(answer)Solution

to

1:The

portfolio's

return

is

the

following

weighted

average

of

the

returns

to

the

twostocks:

Rp

=

(1/3)(0.09)

+

(2/3)(0

.12)

+

[(1/3)(-

I)

+

(2/3)(2)]

FINFL+

[(1/3)(1)+

(2/3)(4)]FGDP

+

(1/3)

εMANM

+

(2/3)

εNXT

=

0.11

+

1

FINFL+

3FGDP

+

(1/3)εMANM

+

(2/3)

εNXTSolution

to

2:The

expected

return

on

the

portfolio

is

11

percent,

the

value

of

the

intercept

inthe

expression

obtained

in

Part

1.Solution

to

3:Rp

=

0.11

+

1

FINFL+

3FGDP

+

(1/3)

εMANM

+

(2/3)

εNXT

=

0.11

+

1(0.01)

+

3(0)

+(1/3)(0.005)

+

(2/3)(0.005)

=

0.125

or

12.5

percent16-80Fundamental

factor

modelsthe

factors

are

attributes

of

stocks

or

companies

that

are

important

inexplaining

cross-sectional

differences

in

stock

pricesexactly

formula【梦轩考资

】Fundamental

Factor

(P/E)1

-

P/Ei1P

/

E

(attribute

value)e.g.

bijRi

ai

bi1FP/E

bi2FSIZE

i

求出FP/E,Fsizee.g.

the

return

differencebetween

low

and

high

P/EstocksRegression

(crosssectional

data)Returnbi1bi2………………………………………不同公司的R和对应的bi1，bi2No

economic

interpretationasset

return

can

be

explained

by

the

price-earnings

ratio,

market

capitalizationb

Asset

i's

attribut

value

-

average

attribute

value

专业提供CFA

FRM全清+讲义17-80【梦轩考资

】Standardized

betaage

dividend

yield

will

have

aDividend

yield

example:after

standardization

a

stock

with

afactor

sensitivity

of

0,a

stock

with

a

dividend

yield

one

standard

deviation

above

the

average

willhave

a

factor

sensitivity

of

1,and

a

stock

with

a

dividend

yield

one

standard

deviation

below

the

averagewill

have

a

factor

sensitivity

of

-1.Suppose,

for

example,

that

an

investment

has

a

dividend

yield

of

3.5

percent

andthat

the

average

dividend

yield

across

all

stocks

being

considered

is

2.5

percent.Further,

suppose

that

the

standard

deviation

of

dividend

yields

across

all

stocks

is

2

percent.The

investment's

sensitivity

to

dividend

yield

is

(3.5%

-

2.5%)/2%

=

0.50,or

one-half

standard

deviation

above

average.专业提供CFA

FRM全清+讲义18-80【梦轩考资

】Standardized

betaThe

scaling

permits

all

factor

sensitivities

to

be

interpreted

similarly,despitedifferences

in

units

of

measure

and

scale

in

the

variables.The

exception

to

this

interpretation

is

factors

for

binary

variables

such

asindustry

membership.

A

company

either

participates

in

an

industry

or

it

doesnot.The

industry

factor

sensitivities

would

be

0

-

1

dummy

variables;in

models

that

recognize

that

companies

frequently

operate

in

multipleindustries,

the

value

of

the

sensitivity

would

be

1

for

each

industry

in

whicha

company

operated.专业提供CFA

FRM全清+讲义19-80Statistical

Factor

modelsStatistical

factor

modelsuses

multivariate

statistics

(factor ysis

or

principal

components)toidentify

multiple

statistical

factors

that

explain

the

covariance

amongasset

returnsmajor

weakness:

the

statistical

factors

do

not

lend

themselves

well

toeconomic

interpretation20-80Arbitrage

Pricing

Theory

(APT)APTMultifactor

modelsCharacteristicscross-sectional

equilibrium

pricingmodel

that

explains

the

variationacross

assets’

expected

returnstime-series

regression

that

explainsthevariation

over

time

in

returns

for

oneassetAssumptionsequilibrium-pricing

model

thatassumes

no

arbitrage

opportunitiesad

hoc

(i.e.,

rather

than

beingderiveddirectly

froman

equilibrium

theory,

thefactors

are

identified

empirically

bylooking

for

macroeconomic

variablesthat

best

fit

the

data)Interceptionrisk-free

rateexpected

return

derived

from

the

APTequation

in

macroeconomic

factormodelThe

relation

between

APT

and

multifactor

models21-80Arbitrage

Pricing

Theory

(APT)CAPMAPTAssumptionsAllinvestors

should

hold

somecombination

of

the

market

portfolioand

the

risk-free

asset.

To

control

risk,less

risk

averse

investors

simplyholdmore

of

the

market

portfolio

and

less

ofthe

risk-free

asst.APT

gives

no

special

role

to

the

marketportfolio,

and

is

far

more

flexible

thanCAPM.

Asset

returns

follow

amultifactor

process,

allowing

investorsto

manageseveral

risk

factors,ratherthan

just

one.conclusionsThe

risk

of

the

investor’s

portfolio

isdetermined

solely

by

theresultingportfolio

beta.Investor’s

unique

circumstancesmaydrivethe

investor

tohold

portfoliostitled

away

from

the

market

portfolio

inorder

to

hedge

or

speculate

on

multiple

risk

factors.Comparison

CAPM

and

APT22-80【梦轩考资

】Active

Risk(R

R

)2PtBtn

1Active

riskActive

returnDefinition:

the

differences

in

returns

between

a

managed

portfolioand

its

ben

arkExactly

formula:

active

return

RP

RBActive

risk

(tracking

error)Definition:

the

standard

deviation

of

active

returnsExactly

formula:(

RP

RB

)active

risk

s专业提供CFA

FRM全清+讲义23-80【梦轩考资】Information

RiskInformation

RiskDefinition:

the

ratio

of

mean

active

return

to

active

riskPurpose:

a

tool

for

evaluating

mean

active

returns

per

unit

of

active

riskExactly

formulaP

B

RBs(R

R

)IR

RP专业提供CFA

FRM全清+讲义24-80Information

ratio

-

exampleTo

illustrate

the

calculation,

if

a

portfolio

achieved

a

mean

return

of

9percentduring

the

same

period

that

its

ben ark

earned

a

meanreturn

of

7.5

percent,and

the

portfolio's

tracking

risk

was

6

percent,

we

would

calculate

aninformation

ratio

of

(9%

-

7.5%)/6%

=

0.25.Setting

guidelines

for

acceptable

active

risk

or

tracking

risk

is

one

of

the

waysthat

some

institutional

investors

attempt

to

assure

that

the

overall

risk

and

stylecharacteristics

of

their

investmentsare

in

line

with

those

desired.25-80【梦轩考资

】Active

risk

squaredWe

can

separate

a

portfolio's

active

risk

squared

into

two

components:Active

risk

squared

=

s2

(R

R

)P

BActive

factor

risk

is

the

contribution

to

active

risk

squared

resulting

from

theportfolio's

different-than-ben

ark

exposures

relative

to

factors

specified

inthe

risk

model.Active

specific

risk

or

asset

selection

risk

is

the

contribution

to

active

risksquared

resulting

from

the

portfolio's

active

weights

on

individual

assets

asthose

weights

interact

with

assets'

residual

risk."Active

risk

squared

=

Active

factor

risk

+

Active

specific

risk专业提供CFA

FRM全清+讲义26-80Portfolio

ManagementSS18R53

An

introduction

to

multifactor

modelsR54 ysis

of

active

portfolio

managementR55

Economics

and

investment

marketsR56

The

portfolio

Management

Process

and

the

InvestmentPolicy

Statement27-80【梦轩考资

】Value

addedNThe

value

added

or

active

return

is

defined

as

the

difference

between

the

returnon

the

manage

portfolio

and

the

return

on sive

ben ark

portfolio.RA

RP

RBValue

added

is

related

to

active

weights

in

the

portfolio,

defined

as

differencesbetween

the

various

asset

weights

in

the

managed

portfolio

and

their

weights

inthe

ben ark

portfolio.

Individual

assets

can

be

overweighed

(have

positiveactive

weights)

or

underweighted(have

negative

active

weights),

but

thecomplete

set

of

active

weights

sums

to

zero.RA

wi

Rii1NRA

wi

RAii1专业提供CFA

FRM全清+讲义28-80position

of

value

addedThe

common position:

value

added

due

to

asset

allocation

and

valueadded

due

to

security

selection.The

total

value

added

is

the

difference

between

the

actual

portfolio

and

theben arkreturn:M

MMMRA

wP,

j

RP,

jj

1RA

wj

RB,

j

wB,

j

RB,

jj

1

wP,

j

RA,

jj

1

j

1RA

(wstocks

RB,stocks

wbonds

RB,bonds

)

(wP,stocktocks

wP,bonds

RA,bonds)29-80【梦轩考资

】The

Sharpe

ratioThe

sharpe

ratio

measures

reward

per

unit

of

risk

in

absolute

returns.An

important

property

is

that

the

Sharpe

ratio

is

unaffected

bytheaddition

of

cash

or

leverage

in

aportfolio.

RF

wP

(RP

RF）

SRP(Rc)

P

PSTD w

STD(R

)SR

RCPPSTD(R

)

RFSR

RP

SR专业提供CFA

FRM全清+讲义30-80【梦轩考资

】Information

ratioark

relativeThe

information

ratio

measures

reward

per

unit

of

risk

in

benreturns.An

important

property

is

that

the

Information

ratio

is

unaffected

by

theaddition

of

ben arkportfolio

in

a

portfolio.RP

RB

RAIR

STD(RP

RB

)

STD(RA

)RC

RBSTD

RC

RB

=

wRP

1

w

RB

RBwSTD(RP

RB

)wRA

RAwSTD(RA

)

STD(RA

)IR

专业提供CFA

FRM全清+讲义31-80Constructing

Optimal

PortfoliosGiven

the

opportunity

to

adjust

absolute

risk

and

return

with

cash

or

leverage,the

overriding

objective

is

to

find

the

single

risky

asset

portfolio

with

theumSharpe

ratio,

whatever

the

investor’s

risk

aversion.A

similarly

important

property

in

active

management

theory

is

that,

given

theopportunity

to

adjust

active

risk

and

return

by

investing

in

both

the

activelymanaged

and

ben ark

portfolios,

the

squared

Sharpe

ratio

of

an

activelymanaged

portfolio

is

equal

to

the

squared

Sharpe

ratio

of

the

ben ark

plus

theinformation

ratio

squared:SR2

SR2

IR2P

B32-80The

preceding

discussion

on

adjusting

active

risk

raises

the

issue

of

determiningthe

optimal

amount

of

active

risk,

without

resorting

to

utility

functions

thatmeasurerisk

aversion.

For

unconstrained

portfolios,

the

level

of

active

risk

thatleads

to

the

optimal

portfolio

is:By

definition,

the

total

risk

of

the

actively

managed

portfolio

is

the

sum

of

theben ark

return

variance

and

active

return

variance.STD

RP

STD

RB

STDRA

22

2Constructing

Optimal

PortfoliosA

BBSRSTDR

IR

STDR

33-80【梦轩考资

】Examples-1Suppose

that

the

historical

performance

of

the

Fidelity

Magellan

and

VanguardWindsor

mutual

funds

in

Exhibits

2

and

3

are

indicative

of

the

futureperformance

of

hypothetical

funds

“Fund

I”

and

“Fund

II.”

In

addition,

supposethat

the

historical

performance

of

the

S&P

500

ben ark

portfolio

showninExhibit

1

is

indicative

of

expected

returns

and

risk

going

forward,

as

shownbelow.

We

use

historical

values

in

this

problem

for

convenience,

but

in

practicethe

forecasted,

or

expected,

values

for

both

the

ben

ark

portfolio

and

theactive

funds

would

be

subjectively

determined

by

the

investor.专业提供CFA

FRM全清+讲义34-80【梦轩考资

】Examples-2Excerpted

from

Exhibits

1

and

2

(based

on

a

risk-free

rate

of

2.8%)S&P

500Fidelity

Magellan

(Fund

I)Vanguard

Windsor

(Fund

II)Average

annual

return10.0%8.6%10.4%Return

standarddev.15.2%17.9%17.3%Sharpe

ratio0.470.320.44Excerpted

from

Exhibit

3Fidelity

Magellan

(Fund

I)Vanguard

Windsor

(Fund

II)Active

return–1.5%0.4%Active

risk6.1%7.4%Informationratio−0.250.05Ben

arkS&P

500S&P

500专业提供CFA

FRM全清+讲义35-80【梦轩考资

】Examples-3State

which

of

the

two

actively

managed

funds,

Fund

I

or

Fund

II,

would

bebetter

to

combine

withthe

passive

ben

ark

portfolio

and

why.Calculate

thepossible

improvement

over

the

S&P

500

Sharperatio

from

theoptimal

deployment

of

a

new

fund,

called

“Fund

III,”

which

has

an

expectedinformation

ratio

of

0.20.Suppose

Fund

III

comes

with

an

active

(i.e.,

ben ark

relative)

risk

of

5.0%but

the

investor

wants

to

adjust

the

active

risk

to

6.5%.

Describe

how

thatadjustment

would

be

made.

(No

calculations

required,

give

a

qualitativedescription.)Again,

suppose

Fund

III

comes

with

an

active

risk

of

5.0%.

Determine

theweight

of

the

ben ark

portfolio

required

to

create

a

combined

portfolio

withthe

highest

possible

expected

Sharpe

ratio.专业提供CFA

FRM全清+讲义36-80【梦轩考资

】Examples-4Solution

to

1:Fund

II

has

the

potential

to

add

more

value

as

measured

by

the

Sharpe

ratio,because

Fund

II

has

the

higher

expected

information

ratio:

0.05

comparedwith

–0.25.Solution

to

2:Properly

combined

with

the

S&P

500

ben ark

portfolio,

Fund

III

hasthepotential

to

increase

the

expected

Sharpe

ratio

from

0.47

for

the

passiveben ark

portfolio

to

an

expected

Sharpe

ratio

of

(0.472

+

0.202)1/2

=

0.51.Solution

to

3:To

increase

the

active

risk

of

Fund

III

to

the

optimal

level,

the

investorwould

need

to

be

more

aggressive

in

managing

the

portfolio,

take

a

shortposition

in

the

ben ark,

or,

more

simply,invest

less

than

he

or

sheotherwise

would

have

invested

in

the

ben ark

or

other

actively

managed

fund.

37-80专业提供CFA

FRM全清+讲义【梦轩考资

】Examples-5Solution

to

4:The

optimal

amountof

active

risk

is

(0.20/0.47)15.2%

=

6.5%,

the

valueproposed

in

Question

3.

The

ben arkportfolio

weight

needed

to

adjustthe

active

risk

in

Fund

III

is

1

−

6.5%/5.0%

=

−30%.Note

that

at

the

6.5%

optimal

level

of

active

risk,

Fund

III

has

an

expectedactive

return

of

0.20(6.5%)

=

1.3%,

a

total

expected

excess

return

of

7.2%

+1.3%

=

8.5%,

and

a

total

risk

of

(15.22

+

6.52)1/2

=

16.5%,

for

an

expectedSharpe

ratio

of

8.5/16.5

=

0.52,

within

rounding

error

of

the

0.51

valuecalculated

for

Question

2.专业提供CFA

FRM全清+讲义38-80Active

Security

ReturnsThe

Correlation

Triangle39-80Active

Security

ReturnsSignal

quality

is

measured

by

the

correlation

between

the

forecasted

activereturns,

μi,

at

the

top

of

the

triangle,

and

the

realized

active

returns,

RAi,

at

theright

corner,

commonly

called

the

information

coefficient

(IC).Investors

with

higher

IC,

or

ability

to

forecast

returns,

will

add

more

valueover

time,

but

only

to

the

extent

that

those

forecasts

are

exploited

in

theconstruction

of

the

managedportfolio.The

correlation

between

any

set

of

active

weights,

Δwi,in

the

left

corner,

andforecasted

active

returns,

μi,

at

the

top

of

the

triangle,

measures

the

degree

towhich

the

investor’s

forecasts

are

translated

into

active

weights,

called

thetransfer

coefficient

(TC).40-80Information

CoefficientAssume

ICis

the

ex

ante

(i.e.,

anticipated)

cross-sectional

correlation

betweenthe

N

forecasted

active

returns,

μi,

and

the

N

realized

active

returns,

RAi.

To

bemore

accurate,

IC

is

the

ex

ante

risk-weighted

correlation.IC=COR

RAi

i

,

i

i

41-80Scale

Active

Return

Forecasts

and

Size

Active

weightsIn

addition

to

employing

mean–varianceoptimization,

proofs

of

the

fundamental

law

generally

assume

th tive

return

forecasts

are

scaled

pri