Lecture 1

EDHEC Business School 1

Analysis of Risk and Performance

Gideon Ozik, PhD, CFA [email protected]

Teaching Assistant: Altan Pazarbasi

[email protected]

EDHEC Business School

Financial Economics Track M1 Fall Semester 2014


IntroducNon : the role of porPolio management

What is porPolio management? Por5olio management refers to the act of managing one’s (individual, company, and insBtuBon) investments in the form of bonds, shares, cash, mutual funds etc. so that he/she earns the “maximum” profits within the sBpulated Bme frame Why is porPolio management useful to individuals/customers ? •  Individuals invest now for a future date •  If we knew the future, it would have been an easy task but we have to

face with uncertainty for many financial investments

What is the task of the porPolio manager? •  OpBmize investor’s por5olio based on available informaBon •  The best informaBon: esBmates of distribuBon of future returns


A three steps job

1st Step : gather informaNon and make assumpNons on future returns •  Returns are independently and idenBcally distributed ? •  Returns predictability ? •  What is the probability distribuBon of returns ?

2nd Step : opNmize your porPolio based on informaNon •  Por5olio models •  Asset pricing models •  Buy and Hold, AcBve or Dynamic Por5olio management

3rd Step : Evaluate PorPolio performance (ex ante and ex post) •  Performance measures •  Performance persistence


Course Outline Part I : PorPolio Choice (3h lectures / 3h of tutorial)

•  Markowitz Model (1952) •  Por5olio OpBmizaBon in pracBce (case study)

Part II : Asset Pricing models (3h lecture / 1.5h tutorial)

•  The CAPM : theory and empirical tests •  Arbitrage Pricing Theory •  CAPM test (case study)

Part III : The efficient Market hypothesis and its consequences (3h lecture / 3h tutorial)

•  Weak, Semi strong and Strong form of EMH •  Can we beat the market ? (case study)

Part IV : Performance evaluaNon (3 h tutorial)

•  Risk Adjusted Measures


Course Outline Reference Books 1)  Por'olio theory and investment analysis (8th edi6on), Authors : Elton, Gruber, Brown

and Goetzmann

2)  Modern Investment Management, an Equilibrium Approach, Bob LiIerman and Quan6ta6ve Resources Group GSAM

3) Ac6ve Por'olio Management, Grinold and Kahn Grading system: 20% homework 10% con6nuous evalua6on 70% final exam

EDHEC Business School 6 6

Por5olio Choice : The Markowitz model (1952)


Markowitz model (1952) 1)  IntroducBon : basic concepts of staBsBcs

2)  The Markowitz model (1952) 2.1 The Mean variance criterion 2.2 The Normality condiBon 2.3 The 2 asset case 2.4 The N asset case 2.5 OpBmal por5olio weights

3)  Por5olio choice in PracBce – an empirical test 3.1 Weakness of the Markowitz model 3.2 The real performance of the Markowitz Model 3.3 Por5olio strategies 3.4 EvaluaBng the performance of each strategy


1. IntroducNon, some prerequisites staNsNcs

ArithmeBc Returns : Logarithmic Returns LRt = ln

Pt +Div(t−1;t )Pt−1

"

#$

%

&'

Expected return:

1

);1(1

−

−− +−=

t

ttttt P

DivPPAR

1)1()( −+= ∏kt

tARRE

Raw prices Adjusted Prices

ARt =Pt −Pt−1Pt−1

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−1

lnt

tt P

PLR

∏ −+=t

pt

tARRE 1)1()(

Assume a Bme series of K+1 (closing) prices: P1, P2, P3,…….. Pk

Probability distribuBon: P1, P2, P3,…….. Pk

E(R) = 1k

LRtt=1

k

∑

E(R) = ptLRtt=1

k

∑


different formulas for the same reality. Same conclusions? Yes theoreBcally and empirically. Example : a por5olio replicaBng the CAC 40 Period: 1 day Closing Adjusted Prices: Pt = 4,510 Pt-‐1 = 4,500. ARt = Pt/Pt -‐1 – 1 = 4,510/4,500 -‐1 = 0.2222% LRt= ln(Pt/Pt t-‐1) = ln(4,510) – ln(4,500) = 0.2220%

IntroducNon (2)


Good approximaBon for “small” returns (a minute, an hour, a day…)

IntroducNon (3), arithmeNc and logarithmic returns

ARt =Pt +Div[t−1,t ]

Pt−1−1

Pt = Pt−1(1+ ARt )−Div[t−1,t ]

LRt = ln(Pt +Div[t−1,t ]

Pt−1`)

eLRt =Pt +Div[t−1,t ]

Pt−1Pt = e

LRtPt−1 −Div[t−1,t ]

Pt−1(1+ ARt )−Div[t−1,t ] = eLRtPt−1 −Div[t−1,t ]

(1+ ARt ) = eLRt

ARt = eLRt −1

LRt = ln(1+ ARt )


Variance and covariance : how far a set of numbers is spread out

IntroducNon (4)

Variance: The expectaBon, or mean, of the squared deviaBon of that variable from its expected value or mean.

2

1

211 )]([)( RErpRVar

k

iii −== ∑

=

σ

))] ( ))( ( [( ) , ( 2 2 1 1 2 1 R E R R E R E R R Cov - - =

] )) ( [( ) ( 2 R E R E R Var - =

Covariance:

Corr(R1,R2 )`=Cov(R1,R2 )

σ1σ 2

CorrelaBon:


IntroducNon – a few examples

DistribuBon of returns: example: Stock XYZ

Expected Return and Variance calcula:ons:

n Rate of Return Probability of Occurrence

1 12% 0.18

2 10% 0.24

3 8% 0.29

4 4% 0.16

5 -4% 0.13

Total 1.00


IntroducNon – a few examples

DistribuBon of returns: example: Stock XYZ

Expected Return and Variance calcula:ons:

E(RXYZ) = 0.18*12% +0.24*10% + 0.29*8% + 0.16*4% + 0.13*(-‐4%) = 7%

Var (RXYZ) = 0.18*(12% -‐ 7%)2 +0.24*(10% -‐ 7%)2 + 0.29*(8% -‐ 7%)2 + + 0.16*(4% -‐ 7%)2 + 0.13*(-‐4% -‐ 7%)2 = 24.1(%%)

n Rate of Return Probability of Occurrence

1 12% 0.18

2 10% 0.24

3 8% 0.29

4 4% 0.16

5 -4% 0.13

Total 1.00


IntroducNon – a few examples PorPolio theory (cont.) Variance is squared-‐unit therefore it is easy to convert to standard deviaBon according to the formula:

In the XYZ example SD (XYZ) = sqrt(24.1 %%) = 4.9%

(Ri))sqrt(Var )Var(R )SD(R iiRi ===σ


Review of probability

Covariance – the degree in which returns of two assets co-‐vary:

Example: consider stocks XYZ and ABC

n Rate of Return XYZ

Rate of Return ABC

Probability of

Occurrence

1 12% 21% 0.18

2 10% 14% 0.24

3 8% 9% 0.29

4 4% 4% 0.16

5 -4% -3% 0.13

Total 1.00

E(RXYZ) = 7%

E(RABC) = 10% Var (RXYZ)? Var (RABC)? 24.1%%; 53.6%%


Review of probability

Covariance – the degree in which returns of two assets co-‐vary:

Example: consider stocks XYZ and ABC Cov (XYZ, ABC) = 0.18(12%-‐7%)(21%-‐10%) + 0.24(10%-‐7%)(14%-‐10%)+

+ 0.29(8%-‐7%)(9%-‐10%) + 0.16(4%-‐7%)(4%-‐10%) + 0.13(-‐4%-‐7%)(-‐3%-‐10%) = 34

n Rate of Return XYZ

Rate of Return ABC

Probability of

Occurrence

1 12% 21% 0.18

2 10% 14% 0.24

3 8% 9% 0.29

4 4% 4% 0.16

5 -4% -3% 0.13

Total 1.00

E(RXYZ) = 7%

E(RABC) = 10% Var (RXYZ)? Var (RABC)? 24.1%%; 53.6%%


The Markowitz Model (1952) Hypothesis Investors are raBonal and risk averse PorPolio Choice Investors choose por5olios based on their expected returns and expected risk Model Uses: historical mean returns – expected returns historical (co) variances– expected (co) variances


The Markowitz Model (1952) The general model Min(σ 2

p )s.tE(Rp ) = µ

and xii=1

N

∑ =1

Target value for the expected return


Normality condiNon Returns are normally distributed

The distribuBon of returns can be fully described by two parameters


The Two-‐asset case

Expected returns of the por5olio:

Variance : V (RP ) = x2V (R1) +(1− x)

2V (R2 )+ 2x (1− x) Cov(R1, R2 )

) ( ) 1 ( ) (

) ) 1 ( ( ) (

2 1

2 1

R E x R E x

R x R x E R E P

- + = - + =

The correlaBon effect

+ 1.0 > ρ> -1.0

If ρ = 1.0, assets are perfectly posiBvely correlated

Si ρ = -‐ 1.0, assets are perfectly negaBvely correlated

V (RP ) = x2V (R1) +(1− x)

2V (R2 )+ 2x (1− x)σ1σ 2ρ1,2


The two asset case -‐ example

Assume we have a two asset por5olio: Asset C: E(RC) = 12% SD(RC) = 30% WC = 50% Asset D: E(RD) = 18% SD(RD) = 40% WD = 50%

What is the expected return of the por5olio? E(Rp) = WC*E(RC) + WD*E(RD) = 0.50*12% + 0.5*18% = 15%

What is the variance of the por5olio?

Var(Rp) = (WC)2 Var(RC) + (WD)2 Var(RD) + 2*WC*WD Cov(RC,RD)= = 0.52*(30%)2 + 0.52*(40%)2 + 2*0.5*0.5*Cov(RC,RD)

But we know that, Cov(RC,RD) = Cor(RC,RD) * SD(RC) * SD(RD)

= Cor(RC,RD) * 30% * 40% SubsBtuBng we get, Var(Rp) = 0.52*(30%)2 + 0.52*(40%)2 + 2*0.5*0.5*30%*40%* Cor(RC,RD)


The two asset case -‐ example what is then the standard deviaBon of the por5olio? σp= sqrt(Var(Rp)) =

= sqrt [0.52*(30%)2 + 0.52*(40%)2 + 2*0.5*0.5*30%*40%* Cor(RC,RD)]= = sqrt [625 + 600*Cor(RC,RD)]

How does the correlaNon changes the volaNlity of the porPolio?

Portfolio Volatility as Functin of Asset Correlation

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

-1.00-0.90-0.80-0.70-0.60-0.50-0.40-0.30-0.20-0.10 0.0

00.100.200.300.400.500.600.700.800.901.00

Assumed Correlation Between C and D

SD (R

p)


The two asset case -‐ example Another way to look at the effect of correlaBon: Consider two assets: A: Rt = 3% Vol = 5% B: Rt = 7% Vol = 12%

Correlation Effect On the Portfolio

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00%

SD (Rp)

E (R

p)

Corr = 0.9 Corr = 0 Corr = -0.9


The effect of diversificaBon by correlaBon


Minimum variance por5olio

),()()(2)()(),()()()(

212121

21212*

RRCorrRRRVRVRRCorrRRRVx

σσσσ

−+

−=

OpBmal weights :

0 ) , ( ) ( ) ( 4 ) , ( ) ( ) ( 2 ) ( 2 ) ( 2 ) ( 2

0 ) (

2 1 2 1 2 1 2 1 2 2 1

= - + - + ⇔

=

R R Corr R R x R R Corr R R R V R xV R V x

dx R dV p

σ σ σ σ

),()1(2)()1()()( 2122

12 RRCovxxRVxRVxRV P −+−+=


DiversificaNon effect as a funcNon of number of stocks in the porPolio


Matrix notaNon, expected returns Expected Returns E(Rp) = xE(R1) + (1-‐x)E(R2) Let’s define two vectors: Weights: X = [x1, x2,…. xN,] and in the two asset case, X = [x1, 1-‐x1] Expected returns: R = [E(r1), E(r2),… E(rN)] and in the two asset case, R = [E(r2), E(r2)] Then, E(Rp) = RXT


Matrix notaNon, porPolio risk

22

12

1221

σ σ

2Stock

σ

σ1Stock

2Stock 1Stock

The por5olio variance is the sum of the variances and the covariances.

Let’s define the covariance matrix of returns Ʃ

And the vector X = [x ,1-‐x]

Then XƩXT is

σ p2 = x2σ1

2 + (1− x)2σ 22 + 2x(1− x)σ12 = XΣX

T


Matrix notaNon, expected returns General model Where X = [x1, x2,…. xN,] a vector of asset weights R = [E(r1), E(r2),… E(rN)] the vector of asset expected returns, and 1 = [1, 1,….1] the vector of 1s length N and

Min(σ 2p )

s.t

E(Rp ) = µ, xii=1

N

∑ =1

Min(X∑XT )s.tRXT = µ,X1T =1

∑ =

σ 21,1........σ1,N

σ 2,1.......σ 2,N

.σ N ,1........σ

2N

"

#

$$$$$

%

&

'''''

σ i, j =σ j,i

∀i, j


The case of N assets

We can find the opBmal weights analyBcally by the Lagrangian funcBon, but only without restricBons on por5olio weights (short-‐sales are allowed, turnover is not limited etc.) Inclusion of addiBonal (and complex) constraints someBmes requires opBmizaBon souware



0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

E (R )

SD

Eff. Front A1

A2

A3

A4

A5

Efficient fronNer

Varying μ, we can form the efficient fronBer



Efficient fronNer – back to the two asset example

Discussion about the correlaBon…and expected returns Efficient por5olio: A) highest level of return for a given level of risk B) lowest level of risk for a given level of return Consider two assets: Asset C: E(RC) = 12% SD(RC) = 30% Asset D: E(RD) = 18% SD(RD) = 40% CorrelaBon between RC and RD is – 0.5 Considering the following five por5olios, is there one which you would never take?

Portfolio Wc WD E(Rp) SD(Rp)

1 100% 0% 12.0% 30.0%

2 75% 25% 13.5% 19.5%

3 50% 50% 15.0% 18.0%

4 25% 75% 16.5% 27.0%

5 0% 100% 18.0% 40.0%


Por5olio ConstrucBon

o  P1 and P2 are feasible but only P3, P4 and P5 are on the efficient fronBer

Feasible and Effiecient Frontier for Assets C and D

0.0%

2.0%

4.0%6.0%

8.0%

10.0%

12.0%

14.0%16.0%

18.0%

20.0%

0% 5% 10% 15% 20% 25% 30% 35% 40% 45%

SD (Rp)

E (R

p)

P1 P2

P4

P3 P5


dual problems

We can transform the opBmizaBon problem into these ones :

AlternaBvely, we can consider investor

Where γ is the risk aversion coefficient of the investor

Target value for the variance

Risk Aversion parameter

Target expected return minVar(Rp )

s.t..E(Rp ) = µ __ and ___ xi =1i=1

N

∑

maxE(Rp )

s.t..Var(Rp ) =σ2 __ and ___ xi =1

i=1

N

∑

maxE(Rp )−γ var(Rp )

s.t.. xi =1i=1

N

∑


A few notes on investors preferences •  How do people choose their por5olios? Expected returns?

1

-‐1

p = 0.5

q = 0.5

E( R ) = 0

Std = 1

2

2

p = 0.5

q = 0.5

E( R ) = 2

Std = 0

A

B

B is clearly bezer than A


A few notes on investors preferences •  How do people choose their por5olios?

5

-‐1

p = 0.5

q = 0.5

E( R ) = 2

Std = 3

2

2

p = 0.5

q = 0.5

E( R ) = 2

Std = 0

A

B

B sBll seems bezer but maybe not for everyone


A few notes on investors preferences •  How do people choose their por5olios?

7

-‐1

p = 0.5

q = 0.5

E( R ) = 3

Std = 4

2

2

p = 0.5

q = 0.5

E( R ) = 2

Std = 0

A

B

???


A few notes on investors preferences

How do people make their por5olio choices? Risk / Standard deviaBons? We need to consider the a{tude of the investor toward risk

Exp. Return St. Dev.

Portfolio A 16.0% 18% Portfolio B 12.8% 14% Portfolio C 9.6% 11%

Which investment is preferable?


A few notes on investors preferences, uNlity

What does uNlity funcNon include:

A.  More apples is always bezer than less apples! B.  The next apple is not as tasty as the previous (but it is sBll bezer to have it) C.  Some addiBonal mathemaBcal properBes

Consider this uBlity funcBon:

Up = E(Rp) – 0.5 γ (σp)2 Where RA is the risk aversion parameter which captures the amtude of the investor

toward risk such that: RA = 0 risk neutrality RA = 1 -‐ 3 low risk aversion RA > 5 high risk aversion


Investment preferences, uNlity

Consider this uBlity funcBon: Up = E(Rp) – 0.5 γ (σp)2

The uBlity funcBon allows us to rank the por5olios. We can now compare between two investors with different a{tude toward risk: A risk neutral investor with RA = 0 and a risk averse investor with RA = 10

Exp. Return

St. Dev. Utility of γ = 0

Rank for γ=0

Utility of γ = 1

Rank for γ = 1

Portfolio A 16.0% 18% 16.0% 1 5.20% 3

Portfolio B 12.8% 14% 12.8% 2 5.81% 2

Portfolio C 9.6% 11% 9.6% 3 6.39% 1



•  Let’s look at two investors: –  Paul is neutral toward risk neutral (ex. RA = 0) –  Danna hates risk (ex. RA = 8)

Can we say anything about their preferences given investment A, B and C

Utility Comparison

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

0 1 2 3 4 5 6 7 8 9

Level of Risk Aversion

Utili

ty

ABC



Indifferent uBlity curves Using the “handy” mean/variance framework, we can easily draw indifference curves: Up = E(Rp) – 0.5 γ (σp)2

por5olio opportunity set

U3 U2

U1

SD ( Rp)

E ( R

p)

U3 > U2 > U1

OpNmal PorPolio


OpNmal porPolio weights

Max{E(Rp )−γVar(Rp )} =Max{XRT −γX∑XT}

s.t

X1T =1→ xii=1

N

∑ =1

X*T =1γ∑−1(RT − n1T )

where

n = B− γA,A =1∑−11T ,and

B =1∑−1 RT

OpBmal weights are derived by solving a standard quadraBc program

For which the soluBon is known,


Homework 1

Goals: 1)  match opBmal por5olios to investors with different risk aversion levels and 2)  trace the efficient fronBer. Assignment: 1) OpBmal por5olios:

a) set up and opBmizaBon problem in excel, and use the dataset named (HW1) to calculate opBmal por5olio for risk aversion parameters (0,1,2,..15) b) Store weights, expected returns and risk for each of opBmal por5olios. c) discuss the results

2) Efficient fronBer: setup an opBmizaBon problem (without risk aversion) and trace the efficient fronBer (have at least 10 points on the fronBer, by choosing feasible E(Rp) (or risk for the duality problem). 3) Prepare a presentaBon of your results: 4 groups will be randomly chosen at the beginning of next session and will be graded based on their work.


PorPolio choice in pracNce Markowitz Model : some weaknesses (1)

Are the returns normally distributed ? •  No! •  Empirical tests on various markets show that returns distribuBons are

skewed and fat tailed

•  On one hand, several authors argued that extreme returns occur too ouen to be consistent with normality (Mandelbrot, 1963, Fama, 1963, Blazberg and Gonedes, 1974, Kon, 1984, Loretan and Phillips, 1994, Longin, 1996).

•  On the other hand, crashes are found to occur more ouen than booms (Fama, 1965, Ardi{, 1971, Simkowitz and Beedles, 1978, Singleton and Wingender, 1986, Peiro, 1999).

•  Is variance a a suitable measure of risk? Not clear


Markowitz Model : some weaknesses (2)

•  Assets returns parameters are esBmated: •  from past data, •  from experts views

•  Induces esBmaBon errors in por5olio inputs: •  Mean returns, •  Variances, •  Covariances

•  If esBmaBon error has a big effects on por5olio choice, we should expect big impact on por5olio performance!


Markowitz Model : some weaknesses (2)

•  Chopra & Ziemba (1993) examine sensiBvity of the MV strategy to esBmaBon errors in the first two moments.

•  Sanfilippo (2002) perfomed a simulaBon study with por5olios of different size and for different risk measures (variance and also semi variance)

•  Results: a huge impact of esBmaBon error on por5olio performance !


Ex ante, real and ex post efficient fronBers Effects of esBmaBon error in mean-‐variance and mean-‐semi variance models Sanfilippo (2002)


The real performance of the Markowitz Model

•  We focus on the real performance of Mean-‐Variance opBmizaBon model ; •  We want to know if the Markowitz model can beat a naïve strategy •  The objecBve of this paper is to understand the condiBons under which

mean-‐variance opBmal por5olio models can be expected to perform well even in the presence of esBmaBon risk.

“OpNmal Versus Naive DiversificaNon: How Inefficient is the 1/N PorPolio Strategy?”

Victor DeMiguel Lorenzo Garlappi Raman Uppal

The Review of Financial Studies, 2009


DJU: Data used for the paper

Ø  Note that the paper does not have a dedicated data secBon…


DJU: Methodology

•  Authors compare 14 por5olio allocaBon models relaBve to that of the 1/N policy across seven empirical datasets of monthly returns.

•  If investors consider only the mean and, allocaBon problem becomes:

•  With soluBon: •  And relaBve weights:

•  The methods differ in esBmaBng


DJU: Methodology

•  Authors compare 14 por5olio allocaBon models relaBve to that of the 1/N policy across seven empirical datasets of monthly returns.

•  Comparison criteria: 1.  Out-‐of-‐sample Sharpe raBo 3.  Certainty-‐equivalent (CEQ) return for the expected uBlity of a

mean-‐variance investor

5.  Turnover (trading volume) for each por5olio strategy.


DJU : Methodology (cont)

𝑅↓𝑠𝑡 − 𝑅↓𝑓𝑡 = 𝛼↓𝑠 + 𝛽↓𝑠 (𝑅↓𝑚𝑡 − 𝑅↓𝑓𝑡 )+ 𝜀↓𝑠𝑡  (2)

𝑅↓𝑖𝑡 − 𝑅↓𝑠𝑡 = (𝛼↓𝑖  − 𝛼↓𝑠 )+ (𝛽↓𝑖 − 𝛽↓𝑠 )(𝑅↓𝑚𝑡 − 𝑅↓𝑓𝑡 )+ 𝜀↓𝑖𝑡  (3)

𝜀↓𝑖𝑡 =(𝑅↓𝑖𝑡 − 𝑅↓𝑠𝑡 )− (𝛽↓𝑖 − 𝛽↓𝑠 )(𝑅↓𝑚𝑡 − 𝑅↓𝑓𝑡 ) (4)


DJU Results: Table 3 (Sharpe RaNo)

•  Sharpe raBo is higher “in-‐sample” than “out-‐of-‐sample” for MV •  “Out of sample” Sharpe RaBon of 1/N is almost always bezer!


DJU Results: Table 4 (CEQ)

•  CEQ results are in line of Sharpe RaBo of Table 4


DJU Results: Table 5 (Turnover)

•  All strategies across all dataset have higher turnover than 1/N


DJU: Conclusions

Ø  The paper compares performance of 14 models of opBmal asset allocaBon, relaBve to that of the benchmark 1/N policy over seven different empirical datasets as well as simulated data.

Ø  Out-‐of-‐sample Sharpe raBo of sample-‐based mean-‐variance strategy is lower than that of 1/N strategy, indicaBng that esBmaBon errors erodes all the gains from opBmal, relaBve to naive, diversificaBon.

Ø  Extensions to the sample-‐based mean-‐variance model designed to deal

with esBmaBon errors typically do not outperform the 1/N benchmark.

Ø  No single model consistently delivers a Sharpe raBo or a CEQ higher than that of the 1/N por5olio, which also has a very low turnover.


Semi replicaNon of porPolio strategies

Naïve strategy Por5olio weight = 1/N in each of the N risky assets. It does not involve any opBmizaBon or esBmaBon and completely ignores the data.

OpNmal Mean Variance strategy

Allowing short-‐sales, opBmal weights are given as in slide 20 To restrict short-‐sales, you can use the excel solver to find the opBmal

weights

Max{E(Rp )−1Var(Rp )}s.t

xii=1

N

∑ =1


EvaluaNng the performance of each strategy

•  Our goal is to study the performance of each of the models across a dataset that have been considered In the literature on asset allocaBon.

•  Our analysis relies on a “rolling-‐sample” approach. •  For the MV strategy , given a T month-‐long dataset of asset returns, we

choose an esBmaBon window of length M = 60. •  In each month t, starBng from t = M + 1, we use the data in the previous

M months to esBmate the parameters needed to implement the MV strategy. These esBmated parameters are then used to determine the relaBve por5olio weights in the por5olio. We then use these weights to compute the return in month t + 1.

•  For the naïve strategy, we only compute the realized returns of the por5olio, mulBplying each of the assets returns of month t+1 by 1/N


EvaluaNng the performance of each strategy

•  We start our por5olio at month M+1 and re-‐esBmate the parameters to compute por5olio weights for the next month

•  This process is conBnued by adding the return for the next period in the dataset and dropping the earliest return, unBl the end of the dataset is reached.

The outcome of this rolling-‐window approach is a series of T − M monthly out-‐of-‐sample returns generated by each of the porPolio strategies

•  We can evaluate the performance of each of the two strategies by calculaBng the ex post sharpe raBo with the returns from month M+1 unBl month T.


Homework part 2 (tutorial)

•  You’ll use the dataset that is on class website

•  Make your tests on Excel and prepare a few ppt slides where you will show your methodology and interpret your results.

•  You have the rest of the current session, the week to implement your empirical test and interpret your results. During sessions Bme, I’ll be here to help you

•  4 groups will be randomly chosen at the beginning of the next session and will be graded based on their work

Documents

Lecture 1