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Chapter 7
Why Diversif ication I s a Good I dea
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The most important lesson learned
is an old truth ratified.
- General Maxwell R. Thurman
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Outline Introduction
Carrying your eggs in more than one basket
Role of uncorrelated securities
Lessons from Evans and Archer
Diversification and beta
Capital asset pricing model
Equity risk premium
Using a scatter diagram to measure beta
Arbitrage pricing theory
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IntroductionDiversification of a portfolio is logically a
good idea
Virtually all stock portfolios seek to
diversify in one respect or another
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Carrying Your Eggs in More
Than One BasketInvestments in your own ego
The concept of risk aversion revisited
Multiple investment objectives
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Investments in Your Own EgoNever put a large percentage of investment
funds into a single security
If the security appreciates, the ego is strokedand this may plant a speculative seed
If the security never moves, the ego views this
as neutral rather than an opportunity cost
If the security declines, your ego has a very
difficult time letting go
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The Concept of
Risk Aversion RevisitedDiversification is logical
If you drop the basket, all eggs break
Diversification is mathematically sound
Most people are risk averse
People take risks only if they believe they willbe rewarded for taking them
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The Concept of Risk
Aversion Revisited (contd)Diversification is more important now
Journal of Finance article shows that volatility
of individual firms has increased
Investors need more stocks to adequately diversify
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Multiple Investment ObjectivesMultiple objectives justify carrying your
eggs in more than one basket
Some people find mutual funds unexciting Many investors hold their investment funds in
more than one account so that they can play
with part of the total
E.g., a retirement account and a separate brokerage
account for trading individual securities
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Role of Uncorrelated SecuritiesVariance of a linear combination: the
practical meaning
Portfolio programming in a nutshell
Concept of dominance
Harry Markowitz: the founder of portfolio
theory
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Variance of A Linear
CombinationOne measure of risk is the variance of
return
The variance of an n-security portfolio is:
2
1 1
where proportion of total investment in Security
correlation coefficient between
Security and Security
n n
p i j ij i j
i j
i
ij
x x
x i
i j
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Variance of A Linear
Combination (contd)The variance of a two-security portfolio is:
2 2 2 2 2 2p A A B B A B AB A Bx x x x
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Variance of A Linear
Combination (contd)Return variance is a securitys total r isk
Most investors want portfolio variance to be
as low as possible without having to give up
any return
2 2 2 2 2
2p A A B B A B AB A Bx x x x
Total Risk Risk from A Risk from B Interactive Risk
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Variance of A Linear
Combination (contd)If two securities have low correlation, the
interactive risk will be small
If two securities are uncorrelated, theinteractive risk drops out
If two securities are negatively correlated,
interactive risk would be negative andwould reduce total risk
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Portfolio Programming
in A NutshellVarious portfolio combinations may result
in a given return
The investor wants to choose the portfolio
combination that provides the least amount
of variance
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Portfolio Programming
in A Nutshell (contd)Example
Assume the following statistics for Stocks A, B, and C:
Stock A Stock B Stock C
Expected return .20 .14 .10Standard deviation .232 .136 .195
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Portfolio Programming
in A Nutshell (contd)Example (contd)
The correlation coefficients between the three stocks are:
Stock A Stock B Stock C
Stock A 1.000Stock B 0.286 1.000
Stock C 0.132 -0.605 1.000
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Portfolio Programming
in A Nutshell (contd)Example (contd)
An investor seeks a portfolio return of 12%.
Which combinations of the three stocks accomplish this
objective? Which of those combinations achieves the least
amount of risk?
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Portfolio Programming
in A Nutshell (contd)Example (contd)
Solution: Two combinations achieve a 12% return:
1) 50% in B, 50% in C: (.5)(14%) + (.5)(10%) = 12%
2) 20% in A, 80% in C: (.2)(20%) + (.8)(10%) = 12%
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Portfolio Programming
in A Nutshell (contd)Example (contd)
Solution (contd): Calculate the variance of the B/Ccombination:
2 2 2 2 2
2 2
2
(.50) (.0185) (.50) (.0380)
2(.50)(.50)( .605)(.136)(.195)
.0046 .0095 .0080
.0061
p A A B B A B AB A Bx x x x
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Portfolio Programming
in A Nutshell (contd)Example (contd)
Solution (contd): Calculate the variance of the A/Ccombination:
2 2 2 2 2
2 2
2
(.20) (.0538) (.80) (.0380)
2(.20)(.80)(.132)(.232)(.195)
.0022 .0243 .0019
.0284
p A A B B A B AB A Bx x x x
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Portfolio Programming
in A Nutshell (contd)Example (contd)
Solution (contd): Investing 50% in Stock B and 50% inStock C achieves an expected return of 12% with the
lower portfolio variance. Thus, the investor will likely
prefer this combination to the alternative of investing
20% in Stock A and 80% in Stock C.
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Concept of DominanceDominanceis a situation in which investors
universally prefer one alternative over
another All rational investors will clearly prefer one
alternative
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Concept of Dominance (contd)A portfolio dominates all others if:
For its level of expected return, there is no
other portfolio with less risk
For its level of risk, there is no other portfolio
with a higher expected return
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Concept of Dominance (contd)Example (contd)
In the previous example, the B/C combination dominates the A/C
combination:
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.005 0.01 0.015 0.02 0.025 0.03
Risk
Exp
ec
tedRe
turn
B/C combination
dominates A/C
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Harry Markowitz: Founder of
Portfolio TheoryIntroduction
Terminology
Quadratic programming
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Introduction Harry Markowitzs Portfolio SelectionJournal
of Finance article (1952) set the stage for modernportfolio theory
The first major publication indicating the important ofsecurity return correlation in the construction of stock
portfolios
Markowitz showed that for a given level of expectedreturn and for a given security universe, knowledge ofthe covariance and correlation matrices are required
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TerminologySecurity Universe
Efficient frontier
Capital market line and the market portfolio
Security market line
Expansion of the SML to four quadrants
Corner portfolio
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Security UniverseThe secur ity universeis the collection of all
possible investments
For some institutions, only certain investmentsmay be eligible
E.g., the manager of a small cap stock mutual fund
would not include large cap stocks
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Efficient FrontierConstruct a risk/return plot of all possible
portfolios
Those portfolios that are not dominatedconstitute the eff icient frontier
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Efficient Frontier (contd)
Standard Deviation
Expected Return100% investment in security
with highest E(R)
100% investment in minimumvariance portfolio
Points below the efficient
frontier are dominated
No points plot above
the line
All portfolios
on the line
are efficient
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Efficient Frontier (contd)The farther you move to the left on the
efficient frontier, the greater the number of
securities in the portfolio
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Efficient Frontier (contd)When a risk-free investment is available,
the shape of the efficient frontier changes
The expected return and variance of a risk-freerate/stock return combination are simply a
weighted average of the two expected returns
and variance
The risk-free rate has a variance of zero
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Efficient Frontier (contd)
Standard Deviation
Expected Return
Rf
A
B
C
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Efficient Frontier (contd)The efficient frontier with a risk-free rate:
Extends from the risk-free rate to point B
The line is tangent to the risky securities efficientfrontier
Follows the curve from point B to point C
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Capital Market Line and the
Market PortfolioThe tangent line passing from the risk-free
rate through point B is the capital marketline (CML)
When the security universe includes all possibleinvestments, point B is the market portfol io
It contains every risky assets in the proportion of itsmarket value to the aggregate market value of allassets
It is the only risky assets risk-averse investors willhold
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Capital Market Line and the
Market Portfolio (contd)Implication for investors:
Regardless of the level of risk-aversion, allinvestors should hold only two securities:
The market portfolio
The risk-free rate
Conservative investors will choose a point near
the lower left of the CML Growth-oriented investors will stay near themarket portfolio
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Capital Market Line and the
Market Portfolio (contd)Any risky portfolio that is partially invested
in the risk-free asset is a lending portfol io
Investors can achieve portfolio returns
greater than the market portfolio by
constructing a borrowing portfol io
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Capital Market Line and the
Market Portfolio (contd)
Standard Deviation
Expected Return
Rf
A
B
C
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Security Market LineThe graphical relationship between
expected return and beta is the securitymarket line (SML)
The slope of the SML is the market price ofrisk
The slope of the SML changes periodically asthe risk-free rate and the markets expectedreturn change
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Security Market Line (contd)
Beta
Expected Return
Rf
Market Portfolio
1.0
E(R)
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Expansion of the SML to
Four QuadrantsThere are securities with negative betas and
negative expected returns
A reason for purchasing these securities is theirrisk-reduction potential
E.g., buy car insurance without expecting an
accident
E.g., buy fire insurance without expecting a fire
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Security Market Line (contd)
Beta
Expected Return
Securities with NegativeExpected Returns
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Corner PortfolioA corner portfol iooccurs every time a new
security enters an efficient portfolio or an
old security leaves Moving along the risky efficient frontier from
right to left, securities are added and deleted
until you arrive at the minimum variance
portfolio
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Quadratic ProgrammingThe Markowitz algorithm is an application
ofquadratic programming
The objective function involves portfoliovariance
Quadratic programming is very similar to linear
programming
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Markowitz Quadratic
Programming Problem
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Lessons from
Evans and ArcherIntroduction
Methodology
Results
Implications
Words of caution
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Introduction
Evans and Archers 1968Journal ofFinance article
Very consequential research regarding portfolioconstruction
Shows how nave diversif icationreduces the
dispersion of returns in a stock portfolioNave diversification refers to the selection of
portfolio components randomly
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Methodology
Used computer simulations:
Measured the average variance of portfolios of
different sizes, up to portfolios with dozens ofcomponents
Purpose was to investigate the effects of
portfolio size on portfolio risk when securities
are randomly selected
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Results
Definitions
General results
Strength in numbers
Biggest benefits come first
Superfluous diversification
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Definitions
Systematic r iskis the risk that remains after
no further diversification benefits can be
achievedUnsystematic r iskis the part of total risk
that is unrelated to overall market
movements and can be diversified Research indicates up to 75 percent of total risk
is diversifiable
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Definitions (contd)
Investors are rewarded only for systematic
risk
Rational investors should always diversify
Explains why beta (a measure of systematic
risk) is important
Securities are priced on the basis of their beta
coefficients
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General Results
Number of Securities
Portfolio Variance
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Strength in Numbers
Portfolio variance (total risk) declines as thenumber of securities included in theportfolio increases
On average, a randomly selected ten-securityportfolio will have less risk than a randomlyselected three-security portfolio
Risk-averse investors should always diversifyto eliminate as much risk as possible
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Biggest Benefits Come First
Increasing the number of portfolio
components provides diminishing benefits
as the number of components increases Adding a security to a one-security portfolio
provides substantial risk reduction
Adding a security to a twenty-security portfolio
provides only modest additional benefits
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Superfluous Diversification
Superf luous diversif icationrefers to theaddition of unnecessary components to analready well-diversified portfolio
Deals with the diminishing marginal benefits ofadditional portfolio components
The benefits of additional diversification inlarge portfolio may be outweighed by thetransaction costs
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Implications
Very effective diversification occurs when
the investor owns only a small fraction of
the total number of available securities Institutional investors may not be able to avoid
superfluous diversification due to the dollar size
of their portfolios
Mutual funds are prohibited from holding more than
5 percent of a firms equity shares
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Implications (contd)
Owning all possible securities would
require high commission costs
It is difficult to follow every stock
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Words of Caution
Selecting securities at random usually gives
good diversification, but not always
Industry effects may prevent properdiversification
Although nave diversification reduces risk,
it can also reduce return Unlike Markowitzs efficient diversification
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Diversification and Beta
Beta measures systematic risk
Diversification does notmean to reduce beta
Investors differ in the extent to which they willtake risk, so they choose securities with
different betas
E.g., an aggressive investor could choose a portfolio
with a beta of 2.0E.g., a conservative investor could choose a
portfolio with a beta of 0.5
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Capital Asset Pricing Model
Introduction
Systematic and unsystematic risk
Fundamental risk/return relationshiprevisited
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Introduction
The Capital Asset Pricing Model (CAPM)
is a theoretical description of the way in
which the market prices investment assets The CAPM is apositive theory
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Systematic and
Unsystematic Risk
Unsystematic risk can be diversified and is
irrelevant
Systematic risk cannot be diversified and is
relevant
Measured by betaBeta determines the level of expected return on a
security or portfolio (SML)
i /
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Fundamental Risk/Return
Relationship Revisited
CAPM
SML and CAPM
Market model versus CAPM
Note on the CAPM assumptions
Stationarity of beta
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CAPM
The more risk you carry, the greater the
expected return:
( ) ( )
where ( ) expected return on security
risk-free rate of interest
beta of Security
( ) expected return on the market
i f i m f
i
f
i
m
E R R E R R
E R i
R
i
E R
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CAPM (contd)
The CAPM deals with expectations about
the future
Excess returns on a particular stock are
directly related to:
The beta of the stock The expected excess return on the market
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CAPM (contd)
CAPM assumptions:
Variance of return and mean return are all
investors care about Investors are price takers
They cannot influence the market individually
All investors have equal and costless access to
information
There are no taxes or commission costs
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CAPM (contd)
CAPM assumptions (contd):
Investors look only one period ahead
Everyone is equally adept at analyzing
securities and interpreting the news
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SML and CAPM
If you show the security market line with
excess returns on the vertical axis, the
equation of the SML is the CAPM The intercept is zero
The slope of the line is beta
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Market Model Versus CAPM
The market model is an ex postmodel
It describes past price behavior
The CAPM is an ex ante model
It predicts what a value should be
M k t M d l
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Market Model
Versus CAPM (contd)
The market model is:
( )
where return on Security in period
intercept
beta for Security
return on the market in period
error term on Security in period
it i i mt it
it
i
i
mt
it
R R e
R i t
i
R t
e i t
N t th
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Note on the
CAPM Assumptions
Several assumptions are unrealistic:
People pay taxes and commissions
Many people look ahead more than one period
Not all investors forecast the same distribution
Theory is useful to the extent that it helps us learn
more about the way the world acts Empirical testing shows that the CAPM works
reasonably well
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Stationarity of Beta
Beta is not stationary
Evidence that weekly betas are less than
monthly betas, especially for high-beta stocks Evidence that the stationarity of beta increases
as the estimation period increases
The informed investment manager knows
that betas change
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Equity Risk Premium
Equi ty r isk premiumrefers to thedifference in the average return betweenstocks and some measure of the risk-free
rate The equity risk premium in the CAPM is the
excess expected return on the market
Some researchers are proposing that the size ofthe equity risk premium is shrinking
U i A S tt Di t
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Using A Scatter Diagram to
Measure Beta
Correlation of returns
Linear regression and beta
Importance of logarithmsStatistical significance
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Correlation of Returns
Much of the daily news is of a generaleconomic nature and affects all securities
Stock prices often move as a group
Some stock routinely move more than theothers regardless of whether the market
advances or declinesSome stocks are more sensitive to changes ineconomic conditions
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Linear Regression and Beta
To obtain beta with a linear regression:
Plot a stocks return against the market return
Use Excel to run a linear regression and obtainthe coefficients
The coefficient for the market return is the betastatistic
The intercept is the trend in the security pricereturns that is inexplicable by finance theory
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Importance of Logarithms
Taking the logarithm of returns reduces the
impact of outliers
Outliers distort the general relationship
Using logarithms will have more effect the
more outliers there are
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Statistical Significance
Published betas are not always useful
numbers
Individual securities have substantialunsystematic risk and will behave differently
than beta predicts
Portfolio betas are more useful since some
unsystematic risk is diversified away
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Arbitrage Pricing Theory
APT background
The APT model
Comparison of the CAPM and the APT
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APT Background
Arbitrage pr icing theory (APT)states that anumber of distinct factors determine themarket return
Roll and Ross state that a securitys long-runreturn is a function of changes in:
Inflation
Industrial production
Risk premiums
The slope of the term structure of interest rates
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APT Background (contd)
Not all analysts are concerned with the
same set of economic information
A single market measure such as beta does notcapture all the information relevant to the price
of a stock
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The APT Model
General representation of the APT model:
1 1 2 2 3 3 4 4( )where actual return on Security
( ) expected return on Security
sensitivity of Security to factor
unanticipated change in factor
A A A A A A
A
A
iA
i
R E R b F b F b F b FR A
E R A
b A i
F i
Comparison of the
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Comparison of the
CAPM and the APT
The CAPMs market portfolio is difficult to
construct:
Theoretically all assets should be included (real estate,
gold, etc.)
Practically, a proxy like the S&P 500 index is used
APT requires specification of the relevantmacroeconomic factors
Comparison of the
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Comparison of the
CAPM and the APT (contd)
The CAPM and APT complement each
other rather than compete
Both models predict that positive returns willresult from factor sensitivities that move with
the market and vice versa
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