capm (rohit)
TRANSCRIPT
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1999 Thomas A. Rietz 1
Diversification and the CAPMDiversification and the CAPMThe relationship between riskand expected returns
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1999 Thomas A. Rietz 2
IntroductionIntroduction Investors are concerned with
Risk
Returns
What determines the requiredcompensation for risk?
It will depend on The risks faced by investors
The tradeoff between risk and return they face
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1999 Thomas A. Rietz 3
AgendaAgenda Concepts of risk for
A single stock
Portfolios of stocks
Risk for the diversified investor: Beta
Calculating Beta
The relationship between Beta and Return:The Capital Asset Pricing Model (CAPM)
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1999 Thomas A. Rietz 4
Overview
Overview
Investors demand compensation for risk
If investors hold diversified portfolios, risk
can be defined through the interaction of asingle investment with the rest of the
portfolios through a concept called beta
The CAPM gives the required relationshipbetween beta and the return demandedon the investment!
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1999 Thomas A. Rietz 5
Vocabulary
Vocabulary
Expected return: What we expect to receive
on average
Standard deviation ofreturns: A measure of dispersion
of actual returns
Correlation The tendency for two
returns to fall above orbelow the expected returna the same or differenttimes
Beta A measure of risk
appropriate for diversified
investors Diversified investors
Investors who hold aportfolio of manyinvestments
The Capital AssetPricing Model (CAPM) The relationship between
risk and return fordiversified investors
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i
i
irprE !)(
Measuring Expected ReturnMeasuring Expected Return We describe what we expect to receive or
the expected return:
Often estimated using historical averages
(excel function: average).
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Example: Die ThrowExample: Die Throw Suppose you pay $300 to throw a fair die.
You will be paid $100x(The Number rolled)
The probability of each outcome is 1/6. The returns are:
(100-300)/300 = -66.67%
(200-300)/300 = -33.33% etc.
The expected return E(r) is:
1/6x(-66.67%) + 1/6x(-33.33%) + 1/6x0% +
1/6x33.33% + 1/6x66.67% + 1/6x100% = 16.67%!
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Example: IEMExample: IEM Suppose
You buy and AAPLi contract on the IEM for $0.85
You think the probability of a $1 payoff is 90% The returns are:
(1-0.85)/0.85 = 17.65%
(0-0.85)/0.85 = -100%
The expected return E(r) is:
0.9x17.65% - 0.1x100% = 5.88%
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Example: Market ReturnsExample: Market Returns Recent data from the IEM shows the following
average monthly returns from 5/95 to 10/99:
(http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html)
AAPL IBM MSFT SP500 T-Bills
Average Return 2.42% 3.64% 4.72% 1.75% 0.35%
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$-
$2,000
$4,000
$6,000
$8,000
$10,000
$12,000
$14,000
Apr-95
Jul-95
Oct-95
Jan-96
Apr-96
Jul-96
Oct-96
Jan-97
Apr-97
Jul-97
Oct-97
Jan-98
Apr-98
Jul-98
Oct-98
Jan-99
Apr-99
Mon
th
Value of Investment
G
rowthof$100
0Inve
stme
G
rowthof$100
0Inve
stme
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2222 )()( iii
ii
i
i Va WW !!!
Often estimated using historical averages(excel function: stddev)
Measuring Risk: StandardMeasuring Risk: Standard
Deviation and VarianceDeviation and Variance Standard Deviation in Returns:
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Example: Die ThrowExample: Die Throw Recall the dice roll example:
You pay $300 to throw a fair die.
You will be paid $100x(The Number rolled)
The probability of each outcome is 1/6.
The expected return E(r) is 16.67%.
The standard deviation is:
56.93%
%67.16%)100(6
1%)67.66(
6
1
%)33.33(6
1%)0(
6
1
%)33.33(6
1
%)67.66(6
1
222
22
22
!
vv
vv
vv
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Example: IEMExample: IEM Suppose
You buy and AAPLi contract on the IEM for $0.85
You think the probability of a $1 payoff is 90% The returns are:
(1-0.85)/0.85 = 17.65%
(0-0.85)/0.85 = -100%
The expected return E(r) is:
0.9x17.65% - 0.1x100% = 5.88%
The standard deviation is:
[0.9x(17.65%)2 + 0.1x(-100%)2 - 5.88%2]0.5 = 35.29%
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Example: Market ReturnsExample: Market Returns Recent data from the IEM shows the following
average monthly returns & standard deviations
from 5/95 to 10/99: (http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html)
AAPL IBM MSFT SP500 T-Bills
Average Return 2.42% 3.64% 4.72% 1.75% 0.35%
Std. Dev14.84% 10.31% 8.22% 3.82% 0.06%
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$-
$2,000
$4,000
$6,000
$8,000
$10,000
$12,000
$14,000
Apr-95
Jul-95
Oct-95
Jan-96
Apr-96
Jul-96
Oct-96
Jan-97
Apr-97
Jul-97
Oct-97
Jan-98
Apr-98
Jul-98
Oct-98
Jan-99
Apr-99
Mon
th
Value of Investment
G
rowthof$100
0Inve
stme
G
rowthof$100
0Inve
stme
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Risk and Average ReturnRisk and Average Return
T-Bill
S&P
MSFT
IBM
PL
. %
. %
. %
. %
. %
. %
3. %
3. %
. %
. %
. %
. % . % . % . % . % . % . % . % . %
S d rd D iation
AverageReturn
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Measures of AssociationMeasures of Association Correlation shows the association across
random variables
Variables withPositive correlation: tend to move in the
same direction
Negative correlation: tend to move inopposite directions
Zero correlation: no particular tendencies to
move in particular directions relative to each
other
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VAB is in the range [-1,1]
Often estimated using historical averages
(excel function: correl)
Covariance in returns, WAB, is defined as:
)()()()( BABiAii
iBBiAAi
i
iAB rErErrprErrErp !!
W
BA
AB
AB
WW
WV !
Covariance and CorrelationCovariance and Correlation
The correlation, VAB, is defined as:
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Notation for Two Asset andNotation for Two Asset and
Portfolio ReturnsPortfolio ReturnsItem Asset A Asset B Portfolio
Actual Return r Ai rBi rPi
Expected Return E(rA) E(rB) E(rP)Variance WA
2 WB2 WP
2
Std. Dev. WA WB WP
Correlation in Returns VABCovariance in Returns WAB = WAWBVAB
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Example: Market ReturnsExample: Market Returns Recent data from the IEM shows the following
monthly return correlations from 5/95 to 10/99: (http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html)
AAPL IBM MSFT SP500 T-Bills
AAPL 1.000 0.262 0.102 0.046 -0.103
IBM
1.000 0.240 0.362 -0.169
M 1.000 0.550 -0.073
SP500 1.000 -0.003
T-Bill 1.000
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y = 0.3777x + 0.0105
Correl = 0.262
$(0)
$(0)
$(0)
$(0)
$-
$0
$0
$0
$0
$1
-20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%
AAPL Return
IBMR
eturn
Correlation of AAPL & I MCorrelation of AAPL & I M
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Risk and Average ReturnRisk and Average Return
T-B ll
S&P500
S T
IBM
AAP
0 0
0 5
1 0
1 5
2 0
2 5
0
5
4 0
4 5
5 0
0 0 2 0 4 0 6 0 8 0 10 0 12 0 14 0 16 0
Sta a De iati
A
e
ageRet
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The standard deviation is not a linearcombination of the individual asset standarddeviations
Instead, it is given by:
)w(12w)w(1+w ABBAAA22
A
22
Ap VWWWWW ! BA
%08.10262.01031.0.148405.5x0.2x0
1031.05.01484.05.0 22222p !
vvv
vv!W
Two Asset Portfolios: RiskTwo Asset Portfolios: Risk
The standard deviation a the 50%/50%, AAPL &
IBM portfolio is:
The portfolio risk is lower than either individual
assets because of diversification.
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Correlations andCorrelations and
DiversificationDiversification Suppose
E(r)A = 16% and WA = 30%
E(r)B = 10% and WB = 16%
Consider the E(r)P and WP of securities Aand B as wA and V vary...
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Case 1: Perfect positive correlationCase 1: Perfect positive correlation
between securities, i.e.,between securities, i.e., VVABAB = +1= +1
8%
9%
10%
11%
12%
13%
14%15%
16%
17%
0% 10% 20% 30% 40%
Std. Dev.
Exp.
Ret
(10% 16%)
(16% 30%)
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Case 2: Zero correlation betweenCase 2: Zero correlation between
securities, i.e.,securities, i.e., VVABAB = .= .
2%
13%
14%
15%
16%
17%
0% 10% 20% 30% 40%
Std. v.
p.
t
(1 %,16%)
(16%,3 %)V
(11.33%,14.1 %)
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Case 3: Perfect negative correlationCase 3: Perfect negative correlation
between securities, i.e.,between securities, i.e., VVABAB == --11
8%
9%
10%
11%
12%
13%
14%15%
16%
17%
0% 10% 20% 30% 40%
Std. Dev.
Exp.
Ret
(10%,16%)
(16%,30%).
(11.33%,14.12%)
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8%
9%
10%
11%
12%
13%
14%15%
16%
17%
0% 10% 20% 30% 40%
Std. Dev.
Exp.
Ret
1
r=0
r=-1
(10%,16%)
(16%,30%)
ComparisonComparison
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w2w+
w2w+
w2w+
www
M FTI M,M FTI MM FTI M
M FTAAPL,M FTAAPLM FTAAPL
I MAAPL,I MAAPLI MAAPL
2
M FT
2
M FT
2
I M
2
I M
2
AAPL
2
AAPL
p
V
V
V
!
3 Asset Portfolios: Expected3 Asset Portfolios: Expected
Returns and Standard DeviationsReturns and Standard Deviations Suppose the fractions of the portfolio are given
by wAAPL, wIBM and wMSFT.
The expected return is: E(rP) = wAAPLE(rAAPL) + wIBME(rIBM) + wMSFTE(rMSFT)
The standard deviation is:
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%59.30 72.03
1036.0
3
102 2.0
3
1)( !vvv!
PRE
%75.7
240.00822.01031.03
1
3
12+
102.00822.01484.031
312+
262.01031.01484.03
1
3
12+
0822.0311031.0
311484.0
31 2
2
2
2
2
2
2
p
!
!W
For the Naively DiversifiedFor the Naively Diversified
Portfolio, this gives:Portfolio, this gives:
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For the Naively DiversifiedFor the Naively Diversified
Portfolio, this gives:Portfolio, this gives:
T-Bill
S&P500
SFT
IB
PL
Naive
P rtf li
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
4.5%
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Sta dard Deviati
vera
eRet
r
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The Concept of Risk With NThe Concept of Risk With N
Risky AssetsRisky Assets As you increase the number of assets in a
portfolio:
the variance rapidly approaches a limit, the variance of the individual assets contributes less
and less to the portfolio variance, and
the interaction terms contribute more and more.
Eventually, an asset contributes to the risk of aportfolio not through its standard deviation butthrough its correlation with other assets in theportfolio.
This will form the basis forCAPM.
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Portfolio variance consists of two parts:
1. Non-systematic (or idiosyncratic) risk and
2. Systematic (or covariance) risk
The market rewards only systematic riskbecause diversification can get rid of non-systematic risk
riskSystematic
ij
risksystematicNon
ipnn
WWW
!
11
1 22
Variance of a naively diversifiedVariance of a naively diversified
portfolio ofN assetsportfolio ofN assets
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Naive DiversificationNaive Diversification
Number of Assets
V
ar.ofPortfoli
V!
V!
V!
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Sta ar
Devaiti
0
2
4
6
8
10
12
14
16
1 3 5 7 911 13 15 17 19 21 23 25
Number fSt cksi Portfolio
Expecte
Portfolio
Retur
a
S
ta
ar
Deviati
on
Avera eMont ly Return
ConsiderNaive Portfolios of 1ConsiderNaive Portfolios of 1through all 2 of these Assetsthrough all 2 of these Assets(Added in Alphabetical Order(Added in Alphabetical Order
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The Capital Asset PricingThe Capital Asset Pricing
ModelModel CAPM Characteristics:
Fi = WiWmVim/Wm2
Asset Pricing Equation:E(ri) = rf+ Fi[E(rm)-rf]
CAPM is a model of what expected returnsshould be if everyone solves the same
passive portfolio problem CAPM serves as a benchmark
Against which actual returns are compared
Against which other asset pricing models are
compared
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TheIdea Behind CAPMThe
Idea Behind CAPM
The value of an asset reflects The risk associated with that asset given
Investors own a combination of The risk free asset and
The market portfolio.
A risky asset
Has no effect on the risk free rate.Effects the portfolio through its covariancewith it.
The market price of risk is: E(Rm)-Rf
Where do these ideas come from?
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The Capital Asset PricingThe Capital Asset Pricing
ModelModel Advantages:
Simplicity
Works well on average Disadvantages:
Makes many simplifying assumptions aboutmarkets, returns and investor behavior
How do you estimate beta? Can all aspects ofrisk be summarized by beta?
What is the true market portfolio and risk freerate?
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Feasible portfolios withFeasible portfolios with
N risky assetsN risky assets
Expected
return (E i)
Std dev (Wi)
Efficientfrontier
Feasible Set
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Dominated and EfficientDominated and Efficient
PortfoliosPortfoliosExpec e
e Ei)
Std de (Wi)
AB
C
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How would you find theHow would you find the
efficient frontier?efficient frontier?1. Find all asset expected returns and
standard deviations.
2. Pick one expected return and minimizeportfolio risk.
3. Pick another expected return and minimizeportfolio risk.
4. Use these two portfolios to map out theefficient frontier.
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Exp t d
tu n (Ei)
Std d v ( Wi)
D
Utility m ximi ing
isky ss t po t olio
U
tility MaximizationU
tility Maximization
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Expected
ret r (Ei)
Std dev (Wi)
DM
E
Utility maximization withUtility maximization with
a riskfree asseta riskfree asset
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ThreeImportant
F
undsThreeImportant
F
unds The riskless asset has a standard deviation
of zero
The minimum variance portfolio lies onthe boundary of the feasible set at a pointwhere variance is minimum
The market portfolio lies on the feasibleset and on a tangent from the riskfree asset
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All isky assets
and po t oliosExpected
etu n (E i)
Std dev (Wi)
Risklessasset Minimum
Va iance
Po t olio
Ma ket
Po t olio
Efficientfrontier
A world with one risklessA world with one riskless
asset andN
risky assetsasset andN
risky assets
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e
m
fm
fe
rrErrE W
W
-
!
)()(
The Capital Market LineThe Capital Market Line
All investors face the same Capital MarketLine (CML) given by:
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Equilibrium Portfolio ReturnsEquilibrium Portfolio Returns
The CML gives the expected return-riskcombinations for efficient portfolios.
What about inefficient portfolios? Changing the expected return and/or risk of an
individual security will effect the expected return and
standard deviation of the market!
In equilibrium, what a security adds to the risk ofa portfolio must be offset by what it adds interms of expected return
Equivalent increases in risk must result in equivalent
increases in returns.
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? AE(R R E(R R
where
i f m f i
ii m im
m
im
m
) )!
! !
F
FW W V
W
W
W2 2
The CAPM Pricing Equation!The CAPM Pricing Equation!
The expected return on any asset can bewritten as:
This is simply the no arbitrage condition!
This is also known as the Security MarketLine (SML).
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? A? A%25.7)E(r
75.0035.0085.0035.0)E(rr)E(rr)E(r
IBM
IBM
ifmfi
!
!! F
Using the CAPM: FindingUsing the CAPM: Finding
E(rE(rii Suppose you have the following
information:
rf= 3.5% E(rm)=8.5% FIBM=0.75 What should E(rIBM) be?
Answer:
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? A? A? A
? A
75.0035.0085.0
035.07250.0
035.0085.0035.07250.0
)()(
IBM
D
iffi
!
!
!
!
F
F
F
Using the CAPM: FindingUsing the CAPM: Finding FFii
Suppose you have the followinginformation:
rf= 3.5% E(rm)=8.5% E(rIBM)=7.25% What should FIBMbe?
Answer:
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? A? A
%5.3
75.01
75.0085.07250.0r
75.01r75.0085.07250.075.0r75.0085.0r7250.0
75.0r085.0r7250.0
r)E(rr)E(r
f
f
ff
ff
ifmfi
!
!
!!
!
! F
Using the CAPM: Finding rUsing the CAPM: Finding rff
Suppose you have the following information:E(rm)=8.5% FDE=0.75 E(rDE)=7.25%
What should rfbe?
Answer:
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Notes on Estimating bsNotes on Estimating bs
Let rit, rmt and rft denote historical returns forthe time period t=1,2,...,T.
The are two standard ways to estimate
historical Fs using regressions: Use the Market Model: rit-rft = Ei + Fi(rmt-rft) + eit Use the Characteristic Line: rit = ai + birmt + eit
Ei = ai + (1-bi)rft and Fi = bi
Typical regression estimates: Value Line (Market Model):
5 Yrs, Weekly Data, VW NYSE as Market
Merrill Lynch (Characteristic Line): 5 Yrs, Monthly Data, S&P500 as Market
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Example Characteristic Line:Example Characteristic Line:
AAPL vs S&P500 (IEM DataAAPL vs S&P500 (IEM Data
y = 0.1844x + 0.0182
R2
= 0.0022
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
50%
-15% -10% -5% 0% 5% 10% 15%
S&P500 Premium
AAPLPremiu
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Example Characteristic Line:Example Characteristic Line:
IBM vs S&P500 (IEM DataIBM vs S&P500 (IEM Data
y = 0.9837x + 0.0191
R2 = 0.1325
-30%
-20%
-10%
0%
10%
20%
30%
40%
-15% -10% -5% 0% 5% 10% 15%
S&P500 Premium
IBMPremiu
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Example Characteristic Line:Example Characteristic Line:
MSFT vs S&P500 (IEM DataMSFT vs S&P500 (IEM Datay = 1.1867x + 0.027
R2
= 0.3032
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
25%
30%
-15% -10% -5% 0% 5% 10% 15%
S&P500 Premium
MSFTPremiu
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Notes on EstimatingNotes on Estimating FFss
Betas for our companies
AAPL IBM MSFT SP500
Raw: 0.1844 0.9838 1.1867 1
Adjusted: 0.4563 0.9891 1.1245 1
Avg. R: 2.42% 3.64% 4.72% 1.75%
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Average Returns vsAverage Returns vs
(Adjusted Betas(Adjusted BetasMSFT
IBM
S&P500
AAPl
T-Bills
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
4.50%
5.00%
- 0.20 0.40 0.60 0.80 1.00 1.20
Beta
AverageRetu
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1999 Thomas A. Rietz
64
SummarySummary
State what has been learned
Define ways to apply training
Request feedback of training session
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65
Where to get more informationWhere to get more information
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Consulting services, other sources