550.447
Quantitative Portfolio Theory
& Performance Analysis
Assignment

For Week of March 25th
 Read: A&L, Chapter 4 (Capital Asset Pricing Model and
its Application to Performance Measurement)
 Supplemental on Standard CAPM: E&G Chapter 13 (on Website)
 Supplemental on Modified CAPM: E&G Chapter 14 (on Website)
 Supplemental on Performance: E&G Chapter 25 (on Website)
Week of March 25, 2013
Capital Asset Pricing Model
(CAPM) and Application to
Performance Measurement




Problems: EG7: 1, 2, 4 (Due Mar 25th )
Problems: EG9: 1, 2 (Due Mar 25th )
Problems: EG13: 1, 2, 4, 8 (Due April 1st)
Problems: EG25: 1, 2, 3, 4 (Due April 1st)
1.1
Assignment
1.2
Where We Are – Recap

Mid Term: April 3rd

Last Day of Class: Wednesday, May 1st
Final: Monday, May 13th ; 9:00am – Noon

 In the classroom: Whitehead 203



Markowitz, Efficient Frontier of Risky Assets, & Sharpe’s
Insight using a 1-factor model – the Fast Algorithm
 Ad hoc, the combination of securities into efficient PFs and with a
riskless asset
Then embed these ideas into a theory of markets and
security pricing - CAPM
Capital Asset Pricing Model (CAPM) – Standard Form
 Capital Market Line
 Security Market Line
1.3
1.4
1
Plan

CAPM & Performance
Capital Asset Pricing Model (CAPM) – Standard Form

 Capital Market Line
 Security Market Line


 Markowitz: a rationale for the efficient frontier
and an investor’s choice for an optimal PF
CAPM – Black’s Zero Beta Model
Performance Measures






 PFs from a universe of only risky assets; constructed
from expected returns, variance, and covariance
 Optimal PF reflects a risk averse investor’s unique
choice to maximize expected utility of their wealth
Treynor Measure
Sharpe Ratio
Jensen Measure
Tracking Error
Information Ratio
Sortino Ratio
 Sharpe: an empirical simplification
 1-factor model allows reduction of unknown
parameters in an appealing fashion; results in
 Fast Algorithm for the Efficient Frontier w/RF asset1.6
1.5
CAPM & Performance

Markowitz and Sharpe
CAPM & Performance
CAPM

 Next evolution: incorporated the influence of the
behavior of investors on asset prices
 Resulted in a theory of asset valuation in an
equilibrium setting, drawing together risk & return
CAPM
 PF P is x of RF and (1-x) of A
 Has Expected Return: E ( RP )  x  RF  1  x   E  RA 
 P  1  x    A
 With Risk:
 Eliminating x from these gives
 Assume a Risk-Free Asset Exists
 And it has return RF and risk zero
 Let an investor form a portfolio, P , as a combination of
x of the RF asset and (1-x) of a PF, A , of risky assets
– selected from the efficient frontier
1.7
 E  RA   RF 
E  RP   RF  
 P
A


 The equation of a straight line linking the RF asset &
the PF A , assumed to be on the Markowitz frontier
1.8
2
CAPM & Performance
CAPM & Performance

CAPM
 Among the set of all such lines, there is one that
dominates all the others and dominates the frontier of
risky assets at every point
 This is the only line that forms a tangent with the efficient
frontier and does so at M
 The RF-M line represents all the linear combinations of
the efficient PF of risky assets, M, with the risk-free
investment
 This is the efficient frontier in the case where there also exists
a risk free asset
 An investor’s choice of a particular PF is a function of
1.10
risk aversion – a choice for x
1.9
CAPM & Performance

CAPM
 The whole market vs. isolated investor, assume
 All have the same expectations about return and risk
(variance and covariance), and therefore
 Construct the same efficient frontier of risky assets
 With a risk-free asset, and using separation principles,
all choose to divide between the risk-free asst and the
same risky asset PF , M
 For market equilibrium, all assets must be held in PF,
the risky asset PF (M); in which all have a share; and
therefore M must contain all risky assets in proportion
to their market capitalization – the market portfolio1.11
CAPM & Performance

CAPM
 The whole market vs. isolated investor
 This result is from Fama (1970)
 So in the presence of a risk-free asset, the efficient
frontier is the line
 E  RM   RF 
E  RP   RF  
 P
M


 For the whole market, M
 Known as the capital market line
 This result, associated with equilibrium, allows the
establishment of a relationship for individual securities
1.12
3
CAPM & Performance

CAPM & Performance
CAPM

 Let us Assume the Following
 Demonstration of Sharpe (Security Market Line)
 Invest x in asset i and (1-x) in the market PF
 The resulting PF, P, has expected return and associated risk given by
Investors are risk averse, seek to maximize expected utility of wealth at period end
Assume normal distribution of returns: only expected return & variance matter
The one investment period is the same for all
No limit on borrowing & lending at the risk-free rate
Information is cost-free & available to all simultaneously: all share THE view on
forecast returns, variance and covariance
 Markets are perfect: no taxes, no transaction costs, all assets traded and divisible





(1)
(2)

 And with (1) & (2)
In practice, we use the return on a broad market index as a surrogate for the market
 Consider any risky asset i below the market line of all efficient PFs
 Invest x in asset i and (1-x) in the market PF
1.13
CAPM & Performance

 Demonstration of Sharpe (Security Market Line)
 Therefore, comparing the slopes – which must be the same
 E  Ri   E  RM    M
M   
E  RM   RF
M
 We find
 E  Ri   E  RM    M E  RM   RF

 i,M   M 2
M
E  Ri   E  RM  
E  RM   RF
M2

i,M


  M 2   E  RM    E  RM   RF   i , M2  1
M

 E  RM   RF   i , M
 RF  
2
M
 i,M   M 2
 As we already saw, the slope of the market line when there is a risk-free asset is
b
CAPM
(3)
 The equilibrium market PF M already contains asset i (it contains all assets), P is
made up with an excess of asset i – w/proportion x – over the market PF; since
this excess must be nil at equilibrium, point M is characterized by x = 0 &  P   M
 With the market at equilibrium, the slope of the tangent to the efficient frontier at M
is
 P
1.14
 Demonstration of Sharpe (Security Market Line)
 So we get
E  RP 
E  RP  E  RP  x

 P
 P x
E  RP 
 E  Ri   E  RM 
x
2
2
 P x i  1  x   M  1  2 x   i , M

P
x
CAPM & Performance
CAPM
E  RP 
 E  Ri   E  RM    P

 P
x  i 2   M 2  2 i , M    i , M   M 2

 By varying x we construct all PFs obtained by combining i and PF M – the curve
through the points i and M above
 The coefficient of the tangent to this curve at any point is the slope
 Consider the risk-free asset & the market PF – which define the capital market line
 When equilibrium exists, prices adjust so that all assets are held by investors:
supply is equal to demand – & the market PF is made up of all traded assets

E  RP   xE  Ri   1  x  E  RM 
 P 2   x 2 i 2  1  x   M 2  2 x 1  x   i , M 
2
 Demonstration of Sharpe (to price Individual Securities – CAPM)

CAPM
1.15
 Thus from the CAPM, at equilibrium, we see that all assets are on the line which is
often called the security market line – in that, the return of every asset is equal to
the rate of return of the risk free asset plus a risk premium
1.16
 This premium equals (price of risk) x (quantity of risk) …
4
CAPM & Performance

CAPM
CAPM & Performance
 E  RM   RF   i , M
E  Ri   RF  
2
M

 Demonstration of Sharpe (Security Market Line)
 The contribution of the CAPM
 The price of risk is the difference between the expected return on the market PF
and the risk-free rate: E  RM   RF
 i,M
 The quantity of risk, called beta, is defined by the ratio:    2
M
 Beta is the covariance between the return on asset i and the return on M divided
by the variance of the market PF
 By using beta the CAPM relationship is then
 Established a theory for valuing individual securities
 Contributed to a better understanding of market
behavior and how asset prices were set
 Highlighted the relationship between risk & return
E  Ri   RF    E  RM   RF 
 And the importance of taking risk into account
 The CAPM provides that at equilibrium the returns on assets, less
the risk-free rate, have a linear link to the return on the market PF –
when the market PF is built according to Markowitz’s principles
 This original version of the CAPM is based on assumptions that the
financial markets do not completely respect; subsequently other
versions were formed to better allow market realities (later)
1.17
CAPM & Performance

CAPM
 Provided an operational theory that allowed the return
on an asset to be evaluated relative to its risk (beta)
1.18
CAPM & Performance
CAPM

 The contribution of the CAPM
CAPM
 The contribution of the CAPM
 Total risk is shown to be broken into 2 parts
 systematic risk, beta, which measures the variation of the
asset relative to market movements; and
 an idiosyncratic component which is unique for each asset
 This breakdown was already established earlier with
Sharpe’s empirical market model, but now comes with
a theory (recall: var  Rit   i2 var  RMt   var   it 
)
 Idiosyncratic risk is not rewarded by the market and is
diversified away
1.19
 The correct measure of risk for an individual asset is
therefore the beta and its reward is the risk premium
 Asset betas as a linear combination for a portfolio
 CAPM underlies relative value analysis – the actual
market price vs. its equilibrium value
 Also it shows the merit of a combination of the market
PF with the risk-free asset in PF decision making and
underlies the efficacy of passive management and
index funds
1.20
5
Questions?
CAPM & Performance

CAPM
 Issues with the CAPM
 Particularly, the perfect markets assumption of strong
form efficiency is problematic;
 But the demonstration of CAPM is based on efficiency
at equilibrium
 This efficiency is a consequence of the assumption that all
investors make the same forecasts concerning assets
 All construct the same efficient frontier of risky assets and
choose to invest only in the efficient PF on the frontier
 Since the market is the aggregation of the individual investors
PFs (a set of efficient PFs) the market PF is efficient
1.21
CAPM & Performance

1.22
CAPM & Performance
CAPM

 Issues with the CAPM
CAPM
 Modified Versions of the CAPM
 In the absence of the assumption of homogeneous investor
forecasts, we can no longer assure the efficiency of the market
PF and consequently of the validity of the equilibrium model
 The theory of market efficiency is closely linked to the CAPM
 It is not possible to test the validity of one without the other
 An important point in a noted criticism of the model by Rolle (of
which more later)
 Empirical tests of the CAPM involve verifying, from the
empirical formulation of the market model (which we
looked at previously), that ex post: alpha is nil
1.23
 Several authors have investigated relaxing certain
constraining elements of the assumptions embedded
in CAPM





Some are presented in the text (and E&G 14)
The first, by Black, an approach using a zero-beta model
The second by Brennan, takes taxes into account
Merton’s continuous time version allows stochastic RF rate
Also, w/inflation and based on consumption
 Apart from the original, Black’s model is the most
frequently used
1.24
6
CAPM & Performance

CAPM & Performance
CAPM

 Black’s Zero-Beta Model
CAPM
 Black’s Zero-Beta Model
 Accommodates (relaxation of) two assumptions
 The model
 The existence of a risk-free rate in practice
 The assumption of a single rate for borrowing and lending
 Assume we can determine zero-beta portfolios – those
uncorrelated with the market PF (market-risk-hedged)
 These PF all have the same return E(RZ) – as they have the
same systematic risk: beta = zero
 Among all these PF, only one is on the efficient frontier
 This is the PF with minimum risk
 We consider 2 PFs on the efficient frontier – the market, M,
and the zero-beta, Z
 Form a PF of x in Z and (1-x) in M
 Black showed that
 CAPM was still valid w/o a risk-free asset, and developed a
 Version of the model by replacing it with a zero beta PF
1.25
CAPM & Performance

CAPM & Performance
CAPM

 Black’s Zero-Beta Model
 The expected return E  RP   xE  RZ   1  x  E  RM 
2
2
2
2
2
 With risk  P   x  Z  1  x   M  2 x 1  x   Z , M 
 But beta is zero (    Z , M  M 2  0 )  Z , M  0 , so
2
 Thus  P 2   x 2 Z 2  1  x   M 2 
 The slope of the tangent at point M that intersects the y-axis at
point E(RZ) is given by E  RP  E  RP  x
 P
CAPM
 Black’s Zero-Beta Model
 The model
 Where
1.26

 P x
E  RP 
 P x Z  1  x   M
 E  RZ   E  RM  and

P
x
x
2
 The model
 And after forming the partials from the equations above, at
point M where x = 0 we have  P ( M )   M so with
2
2
E  RP 
 P x Z  1  x   M
 E  RZ   E  RM  and

  M
x
x
P
 We get
E  RP  E  RP  x E  RM   E  RZ 


 P
 P x
M
 In addition, this line intersects the y-axis at the point E(RZ) so
the equation can be written as  E R  E R 
E  RP   E  RZ   

2
1.27

M

M
 Z  R
  P
 Identical to original CAPM capital market line
 But with E(RZ) instead of RF

1.28
7
CAPM & Performance

CAPM & Performance
CAPM

 Black’s Zero-Beta Model
 Black’s Zero-Beta Model
 The model
 The model
 It is now possible to show that the return on any risky asset
can be written as the return on the zero-beta portfolio and the
market PF
 We proceed with the security market line as with the standard
formulation of CAPM in the presence of the risk-free asset
 As before consider all PFs made up of the risky asset and the
market PF: the slope of the tangent to this curve at M is
 But now the slope equals slope off the new capital market line
 E  Ri   E  RM    M E  RM   E  RZ 

 i,M   M 2
M
 Solving for E(Ri) , as before, this gives the security market line
E  Ri   E  RM  
E  RM   E  RZ 
M2

i,M


  M 2   E  RM    E  RM   E  RZ    i , M2  1
M

 E  RM   E  RZ    i , M
 E  RZ   
 E  RZ   i  E  RM   E  RZ  
2
 E  Ri   E  RM    M
 i,M   M 2
M

 i  i , M2
M
1.29
CAPM & Performance

CAPM
CAPM
 Black’s Zero-Beta Model
 The model
 The same as the original CAPM but with risk-free rate, RF ,
replaced by the return on the zero-beta PF, E(RZ)
 Zero beta PFs depend on short selling to hedge
 With the other variations we see similarly that
the general form of the CAPM is preserved
 The main advantage of these models is to
suggest improvements in performance
measurement indicators
1.31
1.30
CAPM & Performance

CAPM and Performance Measurement
 We have observed that return on its own was not
sufficient to appreciate performance and it was
necessary to associate a measure of the risk taken to
get that return
 Modern PF Theory and the CAPM establish the
quantitative link between risk and return
 More specifically, these theories highlight the notion
of rewarding risk
 Now we can look at indicators (performance
measures) taking both risk and return into account
1.32
8
CAPM & Performance

CAPM & Performance
CAPM and Performance Measurement
 The Treynor Measure
E  RP   RF
T 
P

P
 Measures the relationship of the expected excess over the riskfree rate to its systematic risk
 Drawn directly from the PF CAPM relationship
 Measures the excess return, or risk premium, of a PF over the
RF rate, compared with the total risk of the PF measured by σ
 Drawn from the capital market line
 E  RM   RF 
E  RP   RF  
 P
M


 Or
E  RP   RF E  RM   RF
SP 

E  RP   RF   P  E  RM   RF 
E  RP   RF
P
 E  RM   RF
P
 Works best for a diversified PF – it only accounts systematic risk
 Depends on the “market” reference chosen to find beta of PF 1.33
CAPM & Performance
M
 At equilibrium, the Sharpe ratio of the PF and the Sharpe ratio
of the market are equal
 This comparison indicates whether the expected return on the
1.34
PF is enough to compensate for the additional risk
 The LHS is the Treynor ratio for the PF where the RHS is the
ratio for the market PF

CAPM and Performance Measurement
 The Sharpe Measure
E  RP   RF
SP 
  RP 
CAPM & Performance
CAPM and Performance Measurement

 The Sharpe Measure
CAPM and Performance Measurement
 Jensen Measure
 Since measure is based on total risk, σ , not β , it enables
performance of not very well diversified PFs to be measured;
idiosyncratic risk is included
 Sharpe is widely used – as it is drawn from (Markowitz) PF
theory, not the CAPM like Treynor and Jensen (next) indices
 It does not refer to a market index – and thereby rely on the
equilibrium assumption – so it is not subject to Rolle’s criticism
(of assuming an efficient market)
 Has been generalized by substituting a Benchmark PF in place
of the RF rate – often called the Information Ratio or the
Generalized Sharpe Ratio
1.35
 Jensen’s Alpha is defined as the differential between the return
on the PF in excess of the risk-free rate and the return explained
by the market model, or
E  RP   RF   P   P  E  RM   RF 
 Calculated by carrying out the regression
RPt  RFt   P   P  RMt  RFt    Pt
 The alpha measures the additional return generated by the
manager over the return forecast by the model for risk taken
 The statistical significance of alpha comes from the t-statistic of the regression
– the estimated value of the alpha divided by its standard deviation
 If alpha is assumed normal, then a t-statistic > 2 => luck over skill is < 5%
1.36
9
CAPM & Performance

CAPM & Performance
CAPM and Performance Measurement

 Jensen Measure
CAPM and Performance Measurement
 Relationships between the measures and their use
 Unlike Sharpe and Treynor, Jensen contains the benchmark
 As in Treynor, it considers only the systematic risk
 This method, unlike the first two, does not allow PF with different
levels of risk to be compared
 Treynor and Jensen
 From the definition of the Jensen Alpha
E  RP   RF   P   P  E  RM   RF 
 Dividing by beta (Black-Treynor)
E  RP   RF
 The value of alpha is proportional to the level of risk taken, measure by beta
P
 To compare PFs with different levels of risk, calculate the BlackTreynor Ratio defined by αP / βP
 The Jensen measure can rank PFs in a “peer group”
 It is subject to the same criticism as Treynor regarding choice of
reference index
 In addition, with market timers, beta varies
1.37
CAPM & Performance

 Gives the Treynor indicator on the LHS
 Treynor ratio and Jensen’s alpha are linked by the exact algebraic relationship
E  RP   RF
 TP 
P
P
  E  RM   RF 
P 
1.38

 Relationships between the measures and their use
CAPM and Performance Measurement
 Relationships between the measures and their use
 Sharpe and Jensen
 Treynor and Sharpe
 It is possible to establish a relationship between the Sharpe and Jensen
indicators by using the definition of beta
 PM  PM  P M

M2
M2
 Where the correlation between the portfolio and the market is close to one – as
in a well diversified PF we have from above (definition of Jensen’s alpha)
E  RP   RF   P   P  E  RM   RF    P 
 Dividing by vol of PF
E  RP   RF
P
P
  E  RM   RF 
P 
CAPM & Performance
CAPM and Performance Measurement
P 


P
 E  RM   RF 
M 
P 
TP 
SP 
1.39

E  RP   RF
P
M2

M
 E  RP   RF 

M
P
 E  RP   RF 
TP  
 M  S P M
 Hence
 And the PF’s Sharpe indicator is on the LHS
M2
 Then from the definition of the Treynor ratio
 So
 P  E  RM   RF 

P
M
 E  RM   RF 

SP  P  
P
M
 Again starting from our beta definition and assuming a well diversified PF so
the correlation is close to 1, where  PM
 PM  P M  P
TP
P
M
1.40
10
CAPM & Performance

CAPM & Performance
CAPM and Performance Measurement

 Relationships between the measures and their use
CAPM and Performance Measurement
 Comparisons Summarized
 The three allow ranking of PFs in a period: the higher, the better
 Sharpe & Treynor are based on the same principle, but different
risk measures (total risk vs. systematic risk, beta)
 Sharpe can be used for all PFs; Treynor only for well diversified
PFs
 Jensen is limited to the relative study of PFs with the same beta
 Sharpe is the widest used and simplest to interpret: the additional
return obtained is compared with a risk indicator taking into
account the additional risk taken to obtain it
 Sharpe wrote that the Sharpe ratio was a better measure of past
performance while the Treynor ratio was more suitable for
1.41
anticipating future results
1.42
CAPM & Performance

CAPM and Performance Measurement
 Extensions to the Jensen Measure
 Jensen’s Measure draws its comparison to the security market line
E  RP   RF   P   P  E  RM   RF 
 Suppose instead, we use the capital market line
 The managed PF P is determined to have total risk σP
 Alternatively, a portfolio A on the capital market line with the same risk
 E  RM   RF 
would have return
E  RA   RF  

M
 P

 The manager’s challenge is to have constructed this PF such that its
relative performance is positive
 E  RM   RF
E  RP   E  RA   E  RP   RF  
M

 Let’s look at some data on this differential …

 P

1.43
1.44
11
CAPM & Performance

The differential data graphically

M is the Dow-Jones Market Index
Alternatively, we could use Sharpe’s Ratio (the slope) –
which he (Sharpe) called Reward-to-Variability Ratio 1.45

1.46
CAPM & Performance

The slopes

This is what we see …

Sharpe measure ranks PF B better than PF A
 Investor combining risk-free rate and PF B would get better
performance than doing the same with PF A for any level of risk

In the data, compare the Affiliated Fund and the Boston
Fund; there are two ways …
1.47

Differential measure ranks manager A better than mgr B
 The distance A-A’ is greater than B-B’
 Manager A was able to outperform a mixture of the market PF and
RF with the same risk as A’ by more than Manager B could
outperform the market PF and RF at the same risk level B’
1.48
12
CAPM & Performance

CAPM & Performance
CAPM and Performance Measurement

 Extensions to the Jensen Measure
 Tracking Error
 Jensen’s Measure draws a comparison to the security market line
E  RP   RF   P   P  E  RM   RF 
 Used to evaluate benchmarked portfolios – constructing a portfolio
with a given level of risk, but allowing for the manager to deviate
from the benchmark composition: TE    RP  RB 
 We could also turn to other variations on the CAPM such as with
Black’s zero beta portfolio and make a comparison – this uses the
zero-beta “modified” security market line
 The lower the value, the closer the risk to the benchmark; usually must
stay under a threshold, set in advance
 To respect this constraint, the PF must be reallocated regularly as the
market evolves
 It is necessary to balance the frequency of rebalancing with the
transaction costs of rebalancing
 The real test is whether the additional return is sufficient to make up for
the additional risk taken on by the PF – for this we check another
indicator, the Information Ratio
1.50
E  RP   E  RZ    P   P  E  RM   E  RZ  
1.49
CAPM & Performance

CAPM & Performance
CAPM and Performance Measurement

 Information Ratio (or Appraisal Ratio)
 Where the risk-free asset is replaced by the benchmark PF
 And it is sometimes referred to in the literature as the “generalized
Sharpe ratio”
ER   ER 

IR 
P
B
  RP  RB 
CAPM and Performance Measurement
 Information Ratio (or Appraisal Ratio)
 IR of a PF is defined by the residual return (the share of the return
not explained by the benchmark) compared with its residual risk
(the residual return variation or tracking error)
 Sharpe presented the IR as a generalization of his ratio
 The IR is defined as
CAPM and Performance Measurement

P
  P 
 Where the numerator denotes the residual PF return, as Jensen
 The denominator is the tracking error
1.51
 Managers seek to maximize IR – high residual return with a low TE
 For a given IR value, the lower the TE the higher the chance that the
managers performance will persist over time
 IR is an indicator that allows an evaluation of the manager’s level
of information compared to the public, together with his skill in
achieving a performance that is better than the average manager
 Since this measure does not take the systematic PF risk into
account, it is not appropriate for comparing well-diversified PFs
with concentrated PFs
1.52
13
CAPM & Performance

CAPM and Performance Measurement
 Sortino Ratio
 As Sharpe ratio uses variance – and we pondered semi-variance
to account for returns above/below the mean – Sortino uses a
concept of minimum acceptable return (MAR) in place of the riskfree return
 The standard deviation of returns is replaced by the standard deviation
of returns that are below the MAR
Sortino Ratio 
E  RP   MAR
1 T

T t 0
 RPt  MAR 
2
RPt  MAR
 Next we turn to recently developed risk-adjusted return
measures – most notably variations on the Sharpe Ratio
1.53
14