slides-9-december-2014-11-17-14-15-21

Portfolio Construction

Ugo Pomante

AgendaAgenda

1. Introduction to the Portfolio Construction2. Analysis of Financial Marketsy3. Strategic Asset Allocation: Nave Portfolio Formation Rule4. Strategic Asset Allocation: A Quantitative Approachg Q pp5. Nave versus Markowitz 6. Putting Markowitz at workg7. Heuristic techniques8. Bayesian techniques8. Bayesian techniques

2

AgendaAgenda

1. Introduction to the Portfolio Construction

3

Portfolio construction: Is it easy?

At a first glimpse it might appear thatg p g ppbuilding a portfolio is easy:

h j t t t diff t you have just to aggregate differentassets

Asset 1Asset 1Asset 2Asset 3Asset 4Asset 5Asset 6

4

Portfolio construction: Is it easy? The answer is NOThe answer is NO

..Unfortunately building a GOOD portfoliothat is able to satisfy the needs/expectations ofan investor is HARD.

In fact, in order to construct a good portfolio,you have to make many difficult decisions.

5

Portfolio Construction: Problems to be solved Selection of Financial Markets where to invest the

money (Liquidity, Bonds, Equities, etc) : Which?y ( q y, , q , )How many?

Estimation of future trend of the markets selected. Construction of an optimization model that returns

the optimal portfolio (that returns the optimalweights of the Financial Market) in the long runweights of the Financial Market) in the long run.

Development of an evaluation model that it is ableto verify that the portfolio selected satisfies they pneeds of the investor.

Development of a market timing model, useful ind t k t ti l h t th tf liorder to make tactical changes to the portfolio

composition, in order to anticipate bull/bear trends. Selection of the best financial products for every

6

Selection of the best financial products for everyfinancial market.

A mistake? Very dangerous! Adverse selection of Financial Markets (too much risky

or poorly performing markets)

Error in Estimation of future trend of the marketsselected or overconfidence in the ability of prediction.

Construction of a weak optimization model. Incapacity to verify that the portfolio selected satisfiesp y y p

the needs of the investor.

Errors in market timing decisions (increase of thee uity ei ht at the be i i of a bea t e d)equity weight at the beginning of a bear trend).

Adverse selection of products for every financial market(poor performance or high costs)(poor performance or high costs).

7

Dont under-estimate the process of portfolio construction

Mistakes can be deadly Mistakes can be deadly . So, it is necessary to have:

o skilled human resources;

o good IT procedures;o good IT procedures;

o consistent models of Portfolio Construction.

8

Well-Organized proceduresWell Organized procedures

Institutional investors (pension funds mutual Institutional investors (pension funds, mutualfunds, etc) organize the process of portfolioconstruction on stagesconstruction on stages...

where every phase is able either to create(extra-performance) or destroy value (under-performance);

There are three main stages.

9

Stages of the Portfolio ConstructionStages of the Portfolio Construction

1. Strategic Asset Allocation

+2 Tactical Asset Allocation2. Tactical Asset Allocation

+3. Stock-Bond/Fund Selection

10

Stage 1: Strategic Asset Allocation (SAA)

Strategic Asset Allocation is: the portfolio composed by financial markets (or

asset classes)..

.. that the investor must hold in the long run (allthe investment horizon).

11

Strategic Asset Allocation: Example

The investor has a 5-years investment horizon.

The Asset Manager builds a portfolio, composed byfinancial markets (that is supposed to be coherent with thefinancial markets (that is supposed to be coherent with therisk tolerance of the investor).

10%32%32%

Domestic Money Market

Domestic Bond Market

5 years50%8%Foreign Bond Market

Equity Market

12On average the portfolio composition is expected to be this one in the next five years.

Stage 2: Tactical Asset Allocation (TAA)

Tactical Asset Allocation is: The change made to the strategic composition in

order to anticipate bull/bear trends.

.. that the investor must hold in the short run(next 13 months).

13

Tactical Asset Allocation: Example The SAA is the following:

10%32%

50%8%



Foreign Bond Market

Equity Market

But the Asset Manager has the expectation that in thenext 3 months the Equity Market will decrease

8% Equity Market

next 3 months the Equity Market will decrease. So, for the next 3 months he suggests the following

changes in the portfolio composition:changes in the portfolio composition:20%22%


10%32%


50%

8%Domestic Bond Market

Foreign Bond Market

Equity Market50%8%


Foreign Bond Market

Equity Market

14After three months, the tactical portfolio will be dismantled (and the strategic portfolio will be resumed)..may be we will create e new tactical solution.

Stage 3: Stock-Bond/Fund Selection

Stock-Bond/Fund Selection is: the process to select the best product for every

market in the portfolio.

You can (alternatively):o directly select stocks & bonds (stock bondo directly select stocks & bonds (stockbondselection);

i di tl l t t k & b d id tif i tho indirectly select stocks & bonds, identifying thebest fund managers (fund selection).

15

Stock-Bond/Fund Selection : Example (1/2) AF h P i F d h th f ll i SAA AFrench Pension Fund has the following SAA:

10%32%


Th b d f di t d t h th bilit f50%8%


Foreign Bond Market

Equity Market

The board of directors does not have the ability ofdirectly selecting the stocks/bonds.

so the Pension Fund identifies for every market the

F d l t d

so the Pension Fund identifies, for every market, thefund managers that are supposed to be the best ones:

M k t (A t Cl ) Funds selectedMS Euro Liquidity Fund

Markets (Asset Classes)


Parvest Euro Gov. BondsJPM Global Bonds


Foreign Bond Market16

JPM Global BondsFidelity International

o e g o d a et

Equity Market

Stock-Bond/Fund Selection : Example (2/2) 10%

32%

Domestic Money Market From

50%8%


Foreign Bond Market

Equity Market

Markets....

Equity Market

10%32%

MS Euro Liquidity Fund toMS Euro Liquidity FundParvest Euro Gov. Bonds

JPM Global Bonds

..to Products.

50%8% Fidelity International

17

The pillars of Asset Allocation

In estor: A t M

Investors preferences Asset Managers expectations about the future trend of

Investor: Asset Manager:

Financial Markets

Optimization ModelOptimization Model

OPTIMAL PORTFOLIOOPTIMAL PORTFOLIO

18

AgendaAgenda

2. Analysis of Financial Markets y

19

From Investors Preferences to Fi i l M k E l iFinancial Markets Evaluation

It i ell k o that I e to It is well known that Investors:o love return;o hate risk (are risk adverse ).

So, if we want to build a portfolio that best suitpthe investors preferences, we need to knowthe riskreturn profile of Financial Markets andpMarket Portfolio.

Ri k R l i f Fi i l M k20

RiskReturnanalysisofFinancialMarkets

The Financial Markets

Analysis of the following Asset Classes.

ASSET CLASSES:- Money Market EMU

B d M k t EMU

MARKET INDEXES:- JPM Euro 3 Months

Cit EMU A ll t iti- Bond Market EMU- Bond Market World- Equity Market Europe

- Citygroup EMU Aggr. all maturities- JPM Global- MSCI Europe

- Equity Market North America- Equity Market Japan- Equity Market Pacific ex Japan

- MSCI North America- MSCI Japan- MSCI Pacific ex Japanq y p

- Equity Emerging Marketsp

- MSCI Emerging Markets

21

Historical series of annual returns

JPM Euro 3 Citygroup EMU All MSCI North MSCI Pacific ex MSCI Emerging

Thanks to the market indexes we have a set of historical returns of financial markets:

JPM Euro 3 months

Citygroup EMU All Maturities JPM Global MSCI Europe

MSCI North America MSCI Japan

MSCI Pacific ex Japan

MSCI Emerging Markets

1988 7,28% 4,30% 17,89% 26,59% 25,56% 51,41% 40,98% 51,45%1989 9,16% 1,40% 1,84% 19,57% 20,59% -3,38% 6,05% 51,78%1990 11,55% 3,10% -0,96% -17,12% -17,05% -43,67% -24,16% -23,58%1991 10,41% 11,37% 17,13% 11,58% 27,60% 9,86% 32,86% 58,26%1992 11 11% 12 80% 11 33% -1 35% 9 64% -17 04% 10 01% 16 11%1992 11,11% 12,80% 11,33% -1,35% 9,64% -17,04% 10,01% 16,11%1993 9,03% 14,44% 20,41% 35,53% 15,27% 33,65% 88,14% 83,69%1994 6,30% -1,84% -9,60% -10,63% -11,72% 7,76% -25,11% -18,48%1995 6,58% 16,27% 10,18% 9,77% 23,41% -7,64% 1,77% -14,07%1996 4,83% 7,29% 12,41% 27,59% 30,93% -9,54% 26,84% 11,89%1997 4,42% 6,16% 18,31% 41,85% 52,36% -11,52% -21,47% 1,03%1998 4,46% 10,94% 6,82% 17,21% 17,75% -3,41% -16,21% -32,85%998 4,46% 10,94% 6,82% 17,21% 17,75% 3,41% 16,21% 32,85%1999 3,15% -2,97% 10,67% 33,06% 42,14% 87,20% 62,47% 90,86%2000 4,32% 8,39% 10,49% -2,46% -5,84% -22,85% -10,91% -26,37%2001 4,74% 6,25% 4,75% -16,83% -8,77% -25,97% -7,26% 0,40%2002 3,53% 8,49% 0,31% -32,86% -35,81% -25,17% -23,53% -22,66%2003 2,54% 3,77% -4,92% 11,92% 6,09% 11,76% 17,29% 25,87%2004 2,18% 7,56% 2,09% 9,28% 1,40% 6,38% 15,57% 13,54%2005 2,20% 5,67% 7,93% 23,01% 21,23% 43,29% 27,27% 50,45%2006 3,02% -0,28% -5,11% 16,65% 1,53% -5,87% 14,68% 15,72%2007 4,42% 0,97% -0,86% -0,73% -5,46% -15,38% 13,77% 22,10%2008 5,75% 9,97% 18,47% -45,21% -35,73% -26,51% -49,45% -51,85%

(in Euro)( )

22

Investors aim to maximise the return (the Investors aim to maximise the return (thefinal value) of their investments

23

Average Return (1/2)

RRRR Investors love financial markets with higher average returns:

T

RRRRR TT 121 .....

RTT

RR t

t 1

lExcel:=average(Historical series)

JPM Euro 3 months

Citygroup EMU All Maturities

JPM Global

MSCI Europe

MSCI North America

MSCI Japan



A f A l

(1988-2008)

24

Average of Annual Returns 5,76% 6,38% 7,12% 7,45% 8,34% 1,59% 8,55% 14,44%

Average of Annual Returns (2/2)Average of Annual Returns (1988-2008)

16 00%

12,00%

14,00%

16,00%

8,00%

10,00%

4,00%

6,00%

0,00%

2,00%

JPM Euro Citygroup JPM MSCI MSCI MSCI MSCI MSCIJPM Euro3 months

CitygroupEMU All

Maturities

JPMGlobal

MSCIEurope

MSCINorth

America

MSCIJapan

MSCIPacific ex

Japan

MSCIEmergingMarkets

25

Average Return of a Portfolio (1/3)

If we known:

h P f li W i h f h k ( )- the Portfolio Weight of each market (wi);

- the Average Returns of each market ( )iR

The estimation of the Portfolio Average Return is straightforward:

ki

iiPort RwR1

Excel:d t(W i ht A R t )

i 1

=sumproduct(Weights, Average Returns)

26


Using Matrices:R

R

R

2

1

kiPort

RwwwwR

21

i

R

R kR

27


JPM Euro 3 months


JPM Global

MSCI Europe

MSCI North America

MSCI Japan



W i ht 5 00% 40 00% 5 00% 17 00% 23 00% 3 00% 2 00% 5 00%

5 00%

Weights 5,00% 40,00% 5,00% 17,00% 23,00% 3,00% 2,00% 5,00%Average of Annual

Returns 5,76% 6,38% 7,12% 7,45% 8,34% 1,59% 8,55% 14,44%

k

5,00%3,00%

2,00%

5,00%

JPM Euro 3 monthsCitygroup EMU All Maturities

%32.71

i

iiPort RwR40,00%23,00% JPM Global

MSCI EuropeMSCI North AmericaMSCI JapanMSCI Pacific ex JapanMSCI Emerging Markets

5,00%17,00%

Investors want to maximise the average (or expected) return of the portfolio.

28

Investors are risk adverse: given a targeted Investors are risk adverse : given a targetedreturn they try to minimise the risk.

29

What is risk? (1/2)What is risk ? (1/2)The financial literature has formulated many mathematical &

t ti ti l i di t f l i d t t i kstatistical indicators useful in order to capture risk.Examples: Standard Deviation; Standard Deviation; Semi Standard Deviation; Downside risk; Beta; Duration/Modified Duration; V l Ri k (V R) Value at Risk (VaR); Shortfall probability; Tracking Error Volatility (TEV) Tracking Error Volatility (TEV).

Many indicators maybe a phenomenon30

Manyindicators.maybeaphenomenondifficulttomeasure.

What is risk? (2/2)What is risk ? (2/2)

Co o ly i the fi a ial e i o e t i k i Commonly in the financial environment risk isinterpreted as the uncertainty of returns;

So markets with volatile, unstable returns areconsidered risky.y

Graphical evidencesGraphicalevidences

31

Very Low volatility: Money Market EMU

Annual Return of JPM Euro 3 months (Money Market EMU) [1988-2008]

100%

80%

100%

40%

60%

20%

5.76%

-20%

0%1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

5.76%

-60%

-40%

- Low Interest Rate Risk

32

Low volatility: Bond Market EMU

Annual Return of JCitygroup EMU All Maturities (Bond Market EMU) [1988-2008]

100%

80%

100%

40%

60%

0%

20%

6.38%

-20%

0%1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

-60%

-40% - Higher Interest Rate Risk ( maturity)

- Credit Risk (corporate bonds)33

( p )

Middle volatility: International Bond MarketAnnual Return of JPM Global (International Bond Market) [1988-2008]

100%

60%

80%

40%

0%

20%

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

7.12%

-40%

-20%

- High Interest Rate Risk ( maturity)-60%

High Interest Rate Risk ( maturity)

- Credit Risk (corporate bonds)

34- Exchange Risk

High volatility: European Equity Market

Annual Return of MSCI Europe (European Equity Market) [1988-2008]

80%

100%

40%

60%

20%

7.45%

-20%

0%1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

-60%

-40%

- High Equity Risk

35- Very Low Exchange Risk (if domestic currency is )

Very High volatility: Emerg. Mkts EquityAnnual Return of MSCI EM (Emerg. Mkts Equity) [1988-2008]

100%

60%

80%

40%

60%

0%

20%

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

7.45%

-20%

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

V Hi h E it Ri k-60%

-40% - Very High Equity Risk

- High Exchange Risk (if domestic currency is )

36

Standard Deviation of Returns Fi ally e eed a tati ti al i di ato able to Finally, we need a statistical indicator able tosynthesise the volatility.

The most common parameter is the:

Standard deviation ()Standarddeviation()

2T Excel:

1

2

RRi

i

Excel:=stdev(Historical series)

371T


an easy interpretation (1/2)Annual Return of MSCI Europe (European Equity Market) [1988 2008]

100%

60%

80%

20%

40%

0%1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

7.45%

-40%

-20%

-60% Standarddeviationcanbeseenastheaverageofgapsbetweentheaveragereturn

38

g gap gandeveryannualreturn.


an easy interpretation (2/2)Annual Return of MSCI Europe (European Equity Market) [1988 2008]

100%

60%

80%

20%

40%

0%1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

7.45%

-40%

-20%

The standard deviation of MSCI Europe annual-60%

The standard deviation of MSCI Europe annualreturn is 22.01%;

We can say that the annual return is likely to have an39

y yaverage deviation from the average return of 22.01%.

Standard Deviations of Asset Classes JPM Euro 3

monthsCitygroup EMU All Maturities

JPM Global

MSCI Europe

MSCI North America

MSCI Japan



of Annual Returns 2,88% 5,11% 8,55% 22,01% 22,53% 29,95% 31,29% 37,93%

(1988-2008)

Returns , , , , , , , ,RANKING 1 2 3 4 5 6 7 8

Standard Deviation of Annual Returns (1988-2008)

35,00%

40,00%

20 00%

25,00%

30,00%

10,00%

15,00%

20,00%

0,00%

5,00%

JPM Euro Citygroup JPM MSCI MSCI MSCI MSCI MSCI

40

JPM Euro3 months

CitygroupEMU All

Maturities

JPMGlobal

MSCIEurope

MSCINorth

America

MSCIJapan

MSCIPacific ex

Japan

MSCIEmergingMarkets

Standard deviation of a Portfolio: NOT a weighted average (1/2)

If we known:

h P f li W i h f h k ( )

weighted average (1/2)

- the Portfolio Weight of each market (wi)

- the standard deviations of each market (i)

The estimation of the Portfolio standard deviation is NOT the following: k

iiiPort w

1

That is, The portfolio standard deviation is NOT the

i 1

weighted average of the standard deviation of the markets.

41

Standard deviation of a Portfolio: NOT a weighted average (2/2)weighted average (2/2)

JPM Euro 3 months


JPM Global

MSCI Europe

MSCI North America

MSCI Japan



of Annual 2 88% 5 11% 8 55% 22 01% 22 53% 29 95% 31 29% 37 93%

5 00%

Returns 2,88% 5,11% 8,55% 22,01% 22,53% 29,95% 31,29% 37,93%Weights 5,00% 40,00% 5,00% 17,00% 23,00% 3,00% 2,00% 5,00%

k

5,00%3,00%

2,00%

5,00%

JPM Euro 3 monthsCitygroup EMU All Maturities

%96.141

i

iiPort w 40,00%23,00% JPM GlobalMSCI EuropeMSCI North AmericaMSCI JapanMSCI Pacific ex JapanMSCI Emerging Markets

5,00%17,00%

42

NOT a weighted average: 1st empirical evidence

Using historical series of MSCI market indices on the timehorizon 2001-2008, we measure the standard deviation of the,following equity market sectors:

- MSCI Europe Pharmaceutical, Ph =12,54%;MSCI Europe Pharmaceutical, Pharm 12,54%;- MSCI Europe Biotechnology, Biotech= 30,32%;

WhichisthestandarddeviationoftheMSCI Europe Pharma/Biotech?MSCIEuropePharma/Biotech?

%4412 It cant be the %44.12/ BiotechPharma ca be eweightedaverage!

43

NOT a weighted average: 2nd empirical evidenceSt d d D i ti f A l R t (1988 2008)Standard Deviation of Annual Returns (1988-2008)

MSCI World

MSCI EmergingMarkets

MSCI World


MSCI N th A i

MSCI Japan

MSCI Europe

MSCI North America

Th ld it k t h t d d d i ti f t th t i

0,00% 5,00% 10,00% 15,00% 20,00% 25,00% 30,00% 35,00% 40,00%

44

The world equity market has a standard deviation of returns that is lower than the standard deviation of all the country markets.

Again, risk cant be the weighted average.

The diversification effect Since 1952 is well known that it is possible to reduce risk

avoiding concentration. P b D i h b k Proverb: Dont put your eggs in the same basket

Fi i l hi t h th t k t h th t d t Financial history shows that markets have the tendency tomove one each other in a different way:

year 1998: MSCI Europe (+17 21%) vs MSCI EM (-year 1998: MSCI Europe (+17.21%) vs MSCI EM (32.85%)

year 1995: MSCI North America (+23.41%) vs MSCIy ( )Japan (+1.77%)

Th k t th di ifi d b h i f fi i l k t thThanks to the diversified behaviour of financial markets, theportfolio standard deviation is lower than the weightedaverage.

45

g

We need to Capture the diversification effect

In order to measure the diversification effect (that is, the( ,power of diversification in reducing risk) we mustmeasure:

The Correlation ()

46

Correlation (): characteristics (1/2)o The correlation is calculated for a couple of markets;o -1 +1o If > 0, markets move in the same direction (both gain or

both lose)If +1 k t f tl i th di tio If = +1, markets perfectly move in the same direction

o If < 0, markets move in opposite direction (one gains, theother loses)other loses)

o If = -1, markets perfectly move in opposite direction (theymove perfectly synchronised, but in opposite direction)p f y y , pp )

o If = 0, markets are independent (no tendency to move inthe same or in the opposite direction) (follows)

47

Correlation (): characteristics (2/2)o If =+1, no diversificationo If

Correlation: the scatter graph

60,00%MSCI North America

Case 1: Positive correlation1997

40,00%

50,00%1997

10 00%

20,00%

30,00%

-10,00%

0,00%

10,00%

-50,00% -40,00% -30,00% -20,00% -10,00% 0,00% 10,00% 20,00% 30,00% 40,00% 50,00%

MSCI Europe

-30,00%

-20,00%

2008 = +0.919

-50,00%

-40,00%

St t d t i th di ti (20 i 21)49

Strong tendency to move in the same direction (20 times on 21)


60,00%MSCI North America

Case 2: Zero correlation = +0 012

40,00%

50,00%

+0.012

20,00%

30,00%

10 00%

0,00%

10,00%

-20,00% -15,00% -10,00% -5,00% 0,00% 5,00% 10,00% 15,00% 20,00%


-30,00%

-20,00%

-10,00%

-50,00%

-40,00%

No tendency (12 times in the same direction 9 times in opposite direction)50

No tendency (12 times in the same direction - 9 times in opposite direction)


100 00%MSCI Japan

Case 3: Negative correlation

80,00%

100,00%

= -0.26

40,00%

60,00%

0 00%

20,00%


-20,00%

0,00%-20,00% -15,00% -10,00% -5,00% 0,00% 5,00% 10,00% 15,00% 20,00%

-60,00%

-40,00%

(7 times in the same direction 14 times in opposite direction)51

(7 times in the same direction - 14 times in opposite direction)

Correlation Matrix The Correlation Matrix shows the correlations between all

the couples of markets:

(1988-2008)

CorrelationsJPM Euro 3

monthsCitygroup EMU All Maturities

JPM Global

MSCI Europe

MSCI North America

MSCI Japan



JPM Euro 3 months 1

Citygroup EMU All Maturities 0,30 1 JPM Global 0,27 0,58 1

MSCI Europe -0,09 -0,07 0,31 1 MSCI North

America 0,02 0,01 0,44 0,92 1 MSCI Japan 0 21 0 26 0 24 0 63 0 57 1MSCI Japan -0,21 -0,26 0,24 0,63 0,57 1

MSCI Pacific ex Japan 0,05 0,01 0,31 0,70 0,55 0,75 1

MSCI Emerging Markets 0,10 -0,20 0,25 0,68 0,60 0,78 0,91 1

52

Markets , , , , , , ,

Correlation Matrix with Excel

Insert here the return series of all

the markets

53

TheGiftofglobalization As showed by the correlation matrix, globalization has

strongly increased the correlation amoung equity markets.

Today traditional risky assets are not able to produce bigbenefits of diversification.

This is the main reason why many institutional investorssuggest not to limit the investment to the classical asset classes(bonds and listed stocks)They suggest to invest money also in alternative investments:They suggest to invest money also in alternative investments :

Hedge funds; Commodities; Pay attention: They do not

54 Private Equity; Real Estate.

show negative correlation!

Standard Deviation of a Portfolio (1/2)

If we known:- the portfolio weight of each market (wi)the portfolio weight of each market (wi)- the standard deviations of each market (i)- the correlations between couples of markets (i j)p (i,j)

The estimation of the Portfolio standard deviation is the following:

k k jijijiPort ww , i j1 1

A two markets portfolio:22

55122121

222

211 2)()( wwwwport

Standard Deviation of a Portfolio (2/2)

Using Matrices:

kj

kj

ww

22

11

,2,23,22,21,2

,1,13,12,11,1

...............

C

iikijiiii

kkiiPort

wwwww

,,3,2,1,

2211 Corr

kkkkjkkkk w ,,3,2,1,

=SQRT(MMULT(MMULT(rw,corrM),cw))

56cw=transpose(rw)

Standard deviation of a Portfolio: Numerical exampleNumerical example

JPM Euro 3 Citygroup EMU JPM MSCI MSCI North MSCI MSCI Pacific MSCI Emergingmonths All Maturities Global Europe America Japan ex Japan Emerging Markets of Annual

Returns 2,88% 5,11% 8,55% 22,01% 22,53% 29,95% 31,29% 37,93%Weights 5,00% 40,00% 5,00% 17,00% 23,00% 3,00% 2,00% 5,00%

%44.111 1

,

k

i

k

jjijijiPort ww

1 1 i j

k%96.14

1

k

iiiPort w

57

Standard deviation: is it a good measure of risk?Annual Return of MSCI Europe (European Equity Market) [1988-2008]

100%

80%

40%

60%

0%

20%

7.45%

-20%

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

-60%

-40%

58Risk means bad returns, so we should focus only on volatility that has

negative consequences.

Semi-standard deviation (semi-)This statistical indicator is perfect when you want to measurethe downside risk of a marketB t thi i l dBut this measure is rarely used.

Why?

It is difficult to measure the semi of a portfolio It is difficult to build a model of portfolio optimization in

hi h h i k i d i iwhich the risk is measured using semi-

Refusing the semi is a convenient solution!

59

From volatility to potential lossIs it easy to interpret a measure of volatility?

Financial experience suggests that investors are notable to interpret the meaning of standard deviation.

For investors risk means losses, not volatility.. so it can be useful to capture risk estimating the

f li i l lportfolio potential loss.

Value at Risk (VaR)

60

Introduction to VaR modelsAn investor want to invest money in the European EquityMarket. His holding period is 1 year.H k h h h k f h kHe wants to know which is the risk of this equity market.The standard deviation of annual returns is 22.01%.Therefore the annual return is likely to have an averageTherefore, the annual return is likely to have an averagedeviation from the average return of 22.01%.

This statistical indicator is not able to tell him which is the risk he would incur in case of sizeable and exceptional losses (year:2008)

If he want to explore the darkest side of the risk61

If he want to explore the darkest side of the risk, he need a VaR methodology.

VaR models: the aim

VaR models are able to estimate the potential losses.

For example, thanks to them we can say:Do you want to invest 100,000 on European Equity Market?Well, you have to know that in case of terrible financial eventsyou can lose 35,000!

A capital loss of 35% is very easy to understand!A capital loss of 35% is very easy to understand!

62

VaR models: definition

Given a time horizon (=1 year), Value atRi k i h i l l ( 35%) hRisk is the potential loss (=-35%) wherethe confidence level (=98%) means that the

b bili f hi h l i f l lprobability of higher losses is 1-conf.level(=2%).

Key elements:

1 Time horizon;1. Time horizon;

2. Potential loss (not maximum loss);

3. Confidence level (1-c.l. is the prob. of higher losses).

63

VaR models: Calculation (1/2)

We analyse a parametrical methodology named variance-covariancecovariance .The statistical assumption of this method is the following:

Returns are normally distributedReturns are normally distributed

160180

120140160

6080

100

204060

640

VaR models: Calculation (2/2)

Given this assumption, VaR is estimated as follows:Given this assumption, VaR is estimated as follows:

kRVaR kVaAVERAGE RETURN

STANDARD DEVIATION

THE K VALUE IS RELATED TO THE CONFIDENCE LEVEL WE CHOOSE:

S N V ON

- IF c.l = 95% k = 1.65- IF c.l = 98% k = 2.05- IF c.l = 99% k = 2.33 Statistical Tables

65

VaR models: Example

%457R EE1 year VaR of European Equity Market

%01.22

%45.7

.

.

R

EuropeEqu

EuropeEqu

05.2%98.. kcl

pq

%7,37%01.2205.2%45.7 kRVaRGiven a 1 year time horizon, the potential loss is -37.7%. The probability of higher losses is 2%.p y g %If an investor wants to invest money on European Equity Market he has to tolerate a -37.7% annual loss.

66

In the following analysis we make theg yassumption that we are an Asset ManagementCommittee involved in a Strategic AssetCommittee involved in a Strategic AssetAllocation process.

Our objective is to identify a good model toj y gbuild a SAA.

67

AgendaAgenda

3. Strategic Asset Allocation: Nave Portfolio Formation Rule

68

This first solution follows aqualitative approachqualitative approachNave portfolios refusespmathematical solutions.

69

NavePortfolioFormationRule:A primitive approachAprimitiveapproach

Nave strategies: th ti / t ti ti f are mathematics/statistics free; dont need optimization models; dont need numericalquantitative estimations; estimations don t need numericalquantitative estimations; estimationscan be qualitativejudgemental (European Equity Market willbeat the Japanese Equity Market).

Nave strategies:Nave strategies: are easy to put into practice; can generate good solution, never optimal ones; generate portfolios that are usually diversified andreasonable.

70

NavePortfolioFormationRule:Example (1/7)Example(1/7)

We need to identify the SAA of a pension fund.As members al the Asset Allocation Committee we need toAs members al the Asset Allocation Committee, we need to

manage a procedure able to identify the portfolio of assetclasses.

1. First, we select the asset classes where putting money:

Money Market EMU Bond Market EMU

Safe Assets Bond Market EMU

Equity Market Europe Equity Market North America Risky

Assets

q y Equity Market Japan Equity Market Pacific ex Japan

E it E i M k t

Risky Assets

71

Equity Emerging Markets


2. We identify the risk profile of the portfolio selected:

Given the expected risk tolerance of investors that will putmoney in the pension fund we make the following decision:money in the pension fund we make the following decision:

M M k t EMU Safe Money Market EMU Bond Market EMU Equity Market Europe

Safe Assets = 70%

Equity Market Europe Equity Market North America Equity Market Japan

Risky Assets = 30%

Equity Market Pacific ex Japan Equity Emerging Markets

72Risk profile can be captured with quantitative methdology (seeCucurachi). But Nave portfolios refuses quantitative approach.


3. We select the weights inside the SafeAssets Group

In the time horizon of the investment we forecast a generalincrease of interest rates in EMU areaincrease of interest rates in EMU area.So, in order to maximise the expected return, we need toreduce the maturity of the Safe-Assets Group.reduce the maturity of the Safe Assets Group.

WeightsSafe-Assets Weights55%

Safe Assets

Money Market EMU15%70%

Bond Market EMUSum

73


4. We select the weights inside the RiskyAssets Group

If we dont have a view about the future trends of the StockMarkets, we should replicate the composition of the World Market.This solution is named Market Neutral: it is loyal to the Market.

Risky-Assets Equity Markets Portfolio Weightsy

EquityMarketEurope

Equity Market North America

q yCapitalisation

g

31% (31%30%)=9.3%48% (48% 30%) 14 4%EquityMarketNorthAmerica

EquityMarketJapan

Equity Market Pacific ex Japan

48% (48% 30%)=14.4%10% (10% 30%)=3.0%4% (4% 30%)= 1 2%EquityMarketPacificexJapan

EquityEmergingMarkets

Sum

4% (4% 30%) 1.2%7% (7% 30%)=2.1%

100% 30%

74No valueadded for investors: we just replicate the market!


4. We select the weights inside the RiskyAssets Group

But we try to beat the market, so we depict the future: Europe will over perform North Americap p EM will over perform Japan Pacific ex Japan NEUTRAL

Risky-Assets

E

Equity Markets Capitalisation

New Group Weights

Portfolio Weights

31% 40% (40% 30%) 12 0%EuropeNorthAmerica

Japan

31% 40% (40%30%)=12.0%48% 39% (39% 30%)=11.7%10% 5% (5% 30%)= 1 5%

d

o

n

e

d

Japan

PacificexJapan

Em.Mkts

10% 5% (5% 30%)=1.5%4% 4% (4% 30%)=1.2%7% 12% (12% 30%)=3.6%a

b

a

n

d

75Sum

( )100% 30%

NavePortfolioFormationRule:Example (6/7)

Th fi l tf li

Example(6/7)

The final portfolioAssets Portfolio

WeightsMoneyMarketEMU 55.0%BondMarketEMU 15.0%EquityMarketEurope 12.0%EquityMarketNorthAmerica 11.7%Equity Market Japan 1 5%

1,5%

1,2%

3,6% Money Market EMU

Bond Market EMU

EquityMarketJapan 1.5%EquityMarketPacificexJapan 1.2%EquityEmergingMarkets 3.6%

55,0%

15,0%

12,0%

11,7%

,

Equity Market Europe

Equity Market North America

Equity Market Japan

Equity Market Pacific exJ

q y g g 3.6%Sum 100.0%

76

JapanEquity Emerging Markets

NavePortfolioFormationRule:Example (7/7)

Th fi l tf li

Example(7/7)

The final nave portfolio: is diversified; has a reasonable composition.

but at its best: it is a good solution. it is a good solution. it is not the optimal one.

Ifyouwantmore,youneed

77MODERNPORTFOLIOTHEORY(MPT)

AgendaAgenda

4. Strategic Asset Allocation: A Quantitative Approach g Q pp

78

Quantitative Approach:The Markowitz ModelThe Markowitz Model

Harry Markowitzs Portfolio selection is they ffather of portfolio optimization

d hi d l ( if it i 50 ld) i..and his model (even if it is 50 years old) iswidely used in portfolio construction.

N d bt th th th ti lNo doubt, there are other mathematicalapproach. But no one has the Markowitzs

d l tit d t bmodel aptitude to be: rigorous from a mathematical point of view; b i l d easy to be implemented.

79

The Markowitz Model: The hypothesises

Given a unique time horizonq

..investors want to maximise the expectedt (They lo e etu )return (They love returns).

Investors are risk adverse (They hate risk)( y )

The statistical parameter used to measure risk isthe ta da d de iatiothe standard deviation.

One of the measures considered, the semi-standard deviation, producesefficient portfolios some what preferable to those of the standard deviation.Those produced by the standard deviation are satisfactory, however, and thestandard deviation itself is easier to use, more familiar to many, and perhaps

i i h h i d d d i i ( k i )80

easier to interpret than the semi-standard deviation. (Markowitz, 1959).

TheExpectedReturn StandardDeviationPrinciplePrinciple

Risk is bad variable:Therefore, investors are willing to increase risk only if higherTherefore, investors are willing to increase risk only if higherrisk produces higher return.

E(R)E(R) .DA. . C

B.

Solutions B & C are inefficient

81Solutions D is efficient: it is an optimal solution for high risk tolerance investors

The Markowitz Model: A very scheduled processA very scheduled process

Asset ClassesMKT1MKT2

MKT1MKT2

E(R)MKT1MKT2MKT2

MKT3MKT4MKT5MKT6MKT7

MKT2MKT3MKT4MKT5MKT6MKT7





Optimization1

1

R

e

n

d

i

m

e

n

t

o

11

11

11

Rischio

11

11

1

82

The Markowitz Model: A few remarksA few remarks

1. 5 stages to be performed.2 Th i ti i d t ti t2. The process is timeexpensive: you need to estimate many

parameters.3. For example: with 8 asset classes selected, it is necessary to3. For example: with 8 asset classes selected, it is necessary to

estimate: 8 expected return; 8 standard deviation; 28 correlation.

4 Unfortunately asset managers dont like to produce4. Unfortunately asset managers don t like to producequantitative and numerical estimation (European equitymarket is expected to perform 7.0%).they prefer toproduce qualitative estimation (European equity marketwill beat North American equity market ).

5 N if h M k i d l83

5. No way, if we want to use the Markowitz model,quantitative estimations are required.

Stage 1: Selection of Asset Classes (1/2)

Asset ClassesAsset Classes From a theoretical point of view, we shouldnot narrow the number of investment

MKT1MKT2MKT3MKT4

MKT1MKT2MKT3MKT4

not narrow the number of investmentopportunities.

Therefore we could select dozens of assetl ( h b h d d )MKT5

MKT6MKT7MKT8

MKT5MKT6MKT7MKT8

classes (they may be hundreds). But this theoretical position cannot be

performed because of practical problems:MKT9MKT10MKT11MKT12

MKT9MKT10MKT11MKT12

p p p Increase of parameters to be estimated; Reduction of Asset Under Management(AUM) for every asset class Increase(AUM) for every asset class Increaseof management fees

84

Stage 1: Selection of Asset Classes (2/2)

Asset ClassesAsset Classes Asset Managers usually select not more than10 12 asset classes

MKT1MKT2MKT3MKT4

MKT1MKT2MKT3MKT4

1012 asset classes Marginal players (ex: Japanese Money

Market ) are ignored;MKT5MKT6MKT7MKT8

MKT5MKT6MKT7MKT8

Similar (highly positively correlated)markets are aggregated.

MKT9MKT10MKT11MKT12

MKT9MKT10MKT11MKT12

85

Stage 2: Expected Returns [E(R)] (1/2)

Exp. ReturnsExp. Returns MY SUGGESTION:MKT1MKT2MKT3MKT4

Exp. ReturnsMKT1MKT2MKT3MKT4

MKT1MKT2MKT3MKT4


Expected Returns shouldnt be thehistorical average returns. Empirical studies say that the Rear viewMKT4

MKT5MKT6MKT7MKT8




Empirical studies say that the Rearviewmirror strategy doesnt work.

Future is different from the past (returnsb bili di ib iMKT8MKT9

MKT10MKT11MKT12




probability distribution are notstationary).

MKT12MKT12MKT12MKT12

Awrongbelief:Among financial practitioners is widely spread the idea that Harryg p y p yMarkowitz suggested to use historical estimators. This is wrong:The procedures, I believe, should combine statistical techniques and thejudgment of practical men. [] One suggestion is to use the observed

86

judgment of practical men. [] One suggestion is to use the observedparameters for some period of the past. I believe that better methods, whichtake into account more information, can be found (Markowitz, 1952)

Stage 2: Expected Returns [E(R)] (2/2)

Exp. ReturnsExp. Returns Empirical studies suggest that historicalaverage return are not good predictor ofMKT1MKT2MKT3MKT4


MKT1MKT2MKT3MKT4


average return are not good predictor ofthe future return.

Empirical studies suggest that estimationi E(R) d dl MKT4MKT5

MKT6MKT7MKT8




error in E(R) are deadly. Asset Managers must forecast the future,

not trust the predictive power of the past.MKT8MKT9MKT10MKT11MKT12




p p p Expected returns must be forward looking,

not backward looking.MKT12MKT12MKT12MKT12

Statisticaltechniquescanbeuseful: Macroeconomic models: based on the connection between

future return and macroeconomics factor; Autoregressive models: based on the study of trend of the

87

Autoregressive models: based on the study of trend of thehistorical series of returns.

Stage 3: Standard Deviations () (1/2) Empirical studies suggest that historicalstandard deviations are good predictor of the standard deviations are good predictor of thefuture standard deviation.Estimation error in standard deviation are

d dl

MKT1MKT2MKT3MKT4

MKT1MKT2MKT3MKT4

not deadly.MKT5MKT6MKT7MKT8

MKT5MKT6MKT7MKT8MKT9MKT10MKT11MKT12

MKT9MKT10MKT11MKT12

MY SUGGESTION:You can apply the classical rule using the observed forYou can apply the classical rule , using the observed forsome period of the past.We save time and focus our efforts on Expected Return

88

We save time and focus our efforts on Expected Returnprediction.

Stage 3: Standard Deviations () (2/2) If you want, you can use more sophisticatedtechnical models: technical models:MKT1

MKT2MKT3MKT4



MKT9MKT10MKT11MKT12

Implied volatility; E i d l (ARCH GARCH) Econometric models (ARCH, GARCH).

89

Stage 4: Correlations () (1/2) Empirical studies suggest that historicalcorrelations are good predictor of the future1

1

correlations are good predictor of the futurecorrelation.Estimation error in correlation are notd dl

11

11

11

1

11

11

11

1 deadly.1 11

11

1

11

11

11

MY SUGGESTION:You can apply the classical rule using the observed forYou can apply the classical rule , using the observed forsome period of the past.

90

Stage 5: Correlations () (2/2) If you want, you can use more sophisticatedtechnical models:1

1

technical models:1

11

11

11

11

11

11

111

11

11

11

11

11

Econometric models (ARCH, GARCH).

91

Final Stage: Optimization (1/3) If we have: Asset Classes, E(r), and We can optimize (Quadratic Programming).

OptimizationOptimizationR

e

n

d

i

m

e

n

t

o

E

(

R

)

Fi d th i ht ( ) bl tRischioRisk

Objective function MIN Portfoglio

Find the weights (wi) able to:

Constraints:1st constraint: Exp. Return = E(R)*p ( )

w1 ++ .. wi + ..wn =12nd constraint: 3rd i

92

wi 03rd constraint:

Final Stage: Optimization (2/3)Mathematical structure of the Markowitzoptimization:

OptimizationOptimizationR

e

n

d

i

m

e

n

t

o

E

(

R

)

Mi 2RischioRisk

Min PortW:sConstraint

k

REREw ii

1

*)()(n

k

1i

kiw

w

i

i

,...,1 with 0

11i

93

Final Stage: Optimization (3/3)Mi 2

w

REREw

Min

i

ii

PortW

1

*)()(

:sConstraint

n

k

1i

2

We run this optimization for a targeted expected return[E(R)*]

The optimization returns:kiw

w

i

i

,...,1 with 0

11i o the portfolio composition

o that is efficient as, given the targeted E(R), it is ableto minimise the standard deviation

Running the optimization for different targeted E(R) weobtain a range of efficient portfolio

Optimization

t

o

)

Efficient Frontier

R

e

n

d

i

m

e

n

t

E

(

R

)

94Rischio

MarkowitzOptimization:An application (1/8)Anapplication(1/8)

Asset Classes selected:Asset Classes selected:

95


Expected Returns estimated:Expected Returns estimated:

96


Standard deviations estimated:Standard deviations estimated:

97


Correlations estimated:Correlations estimated:

98


Output: Efficient FrontierOutput: Efficient Frontier

E(R)

99


Output: Portfolio composition

100


All the other portfolios are inefficientAll the other portfolios are inefficient

N P f liNave Portfolio

101


A better portfolio exists:A better portfolio exists:

102

MarkowitzOptimization:Excel(1/2)

Markowitz Optimization can be easily processed using Excel

=SUMPRODUCT(B2:B8;D2:D8)

SQRT(MMULT(MMULT(TRANSPOSE(N2 N8) F2 L8) (N2 N8)))

=SUM (D2:D8)

Then click all together:

103

=SQRT(MMULT(MMULT(TRANSPOSE(N2:N8);F2:L8);(N2:N8)))g

Ctrl+Shift+Enter

MarkowitzOptimization:Excel(2/2)

104ConstraintsWeights Risk

Markowitzversus Nave

Which is your choice to build a SAA?Which is your choice to build a SAA?

MARKOWITZ NAVE

105

AgendaAgenda

5. Nave versus Markowitz

106

It glitters but. g e s bu .Markowitz optimization seems to be the best solution.N h l fi i l li h h d h hi d l hNevertheless financial literature has showed that this model hassome problems:1 Effi i t tf li ft bl (P tf li1. Efficient portfolios are often unreasonable (Portfolios

highly concentrated and/or big weights to marginalmarkets)markets ).

2. Efficient Portfolios are unstable (small changes inexpected returns can strongly affect the portfolioexpected returns can strongly affect the portfoliocomposition).

3 Estimations are supposed to be perfect (Asset managers3. Estimations are supposed to be perfect (Asset managersare clairvoyant! Estimation error doesnt exist).

4 Efficient portfolios are estimation error maximizers107

4. Efficient portfolios are estimation error maximizers

Extra-argument 1:Efficient Portfolios are unstableEfficient Portfolios are unstable

108

Extra-argument 1:Efficient Portfolios are unstable (1/5)

Asset Management Committee Alfa has the following estimationfollowing estimation

109


Optimal Portfolio are the following:

Only European E itEquity

110

Extra-argument 1:Efficient Portfolios are unstable (3/5)Asset Management Committee Beta has the same expectations. The only difference is the following:

7.4%7%7%

Very homogenous forecast..similar views 111

about the future trend of the markets, but


The portfolio composition is the opposite:

Only North American EquityEquity

112


It is not encouraging/reassuring to realize that very small changes in expected returns canvery small changes in expected returns can strongly affect the portfolio composition.

Question: Can I trust a model that Qgive importance to basis points?

113

Efficient portfolios areestimation error maximizersestimation error maximizers

Because of the estimation error, efficient portfolios arelik l t h b d flikely to have very bad performance.Example:- The Return of Emerg.Mkts Equity is expected to be8.0% (the highest) ( g )- Efficient portfolios with high risk are concentratedon Emerging Market Eq it on Emerging Market Equity - Ex-post we discover that the estimation is wrong, asthis market collapses - The concentration produces a blood bath.

114

p

Efficient portfolios areestimation error maximizersestimation error maximizers

Since estimation error is often large,Since estimation error is often large,portfolios selected according to theMarkowitz criterion are likely not moreMarkowitz criterion are likely not moreefficient than a Nave portfolio.

MARKOWITZ NAVE

115

AgendaAgenda

6. Putting Markowitz at workg

116

PuttingMarkowitzatwork

InordertoputMarkowitzatwork,weneedtoe o e the e fe t e ti atio hy othe iremovetheperfectestimationhypothesis

Asset Manager are not clairvoyant. They make mistakes.Estimation error must be managedEstimation error must be managed.It is better to have portfolios with lower expected return,but with lower exposition to estimation error.but with lower exposition to estimation error.The problem is the portfolio concentration.

We need to modify the model, in order to 117

fy ,promote a greater diversification

Twotechniques

Heuristic Approaches Bayesian ApproachesHeuristic Approaches Bayesian Approaches

They adjust the inputs They adjust the y j pestimated (above all

expected returns)Optimization Process

118

AgendaAgenda

7. Heuristic techniques

119

TwoHeuristicApproach

Constrained Optimization ResamplingTMConstrained Optimization Resampling

DifficultEasy

120

ConstrainedOptimization

It is necessary to add supplementaryconstraints to the Markowitz optimization

Find the weights (wi) able to:constraints to the Markowitz optimization

Objective function MIN PortfoglioConstraints:

1st constraint: Exp. Return = E(R)* w1 + + wi + w =12nd constraint: w1 ++ .. wi + ..wn 12 constraint: wi 03rd constraint: wiKi4th constraint: wiHi5th constraint: >

121These Constraints drive a larger diversification wiHi5 constraint: >

ConstrainedOptimizationExample (1/3)Example (1/3)

Asset Classes selected:Asset Classes selected: Expected Returns estimated: Standard deviations estimated:Standard deviations estimated:


Supplementary constraintsSupplementary constraints

122

ConstrainedOptimizationExample (2/3)

Output: Constrained Frontier

Example (2/3)

Output: Constrained Frontier

E(R)

C i d f i i l h h ffi i f i123

Constrained frontier is lower than the efficient frontier

ConstrainedOptimizationExample (3/3)

Output: Portfolio composition

Example (3/3)

Di ifi i iDiversification increases

124

Extra-argument 2:Improving the ConstrainedImproving the Constrained

Optimization

125

Extra-argument 2:Improving the Constrained Optimization (1/4)

We observed that using traditionalconstrained it is hard to build wellconstrained it is hard to build welldiversified portfolios;

May be a few of the possible portfolios are May-be, a few of the possible portfolios arewell diversifiedbut others are still very

t t dconcentrated; In order to well diversify all the possible

portfolios we can use different constraintscalled: : Infra-group constraints

126

Infra group constraints

Extra-argument 2:Improving the Constrained Optimization (2/4)

Examples of infra-group constrains: At best Emerging Market Equity is 18% of At best, Emerging Market Equity is 18% of

the equity composition (upper bound); North America Equity Market can not be North America Equity Market can not be

less than 25% of the equity composition.

I use together upper & lower boundsI use together upper & lower bounds, so every risky asset is free to move

127

y yinside a reasonable range.

Extra-argument 2:Improving the Constrained Optimization (3/4)Example: Input

Example of Optimization with infra-group constraints:Asset Classes selected:Asset Classes selected: Expected Returns estimated: Standard deviations estimated:Standard deviations estimated:Example of Optimization with infra-group constraints:


Correlations estimated:Correlations estimated:Correlations estimated:Correlations estimated:

Infra group constraintsInfra-group constraints

128

Extra-argument 2:Improving the Constrained Optimization (4/4)Example: Output

All portfolios are well diversified129

All portfolios are well-diversified

ResamplingTM (1/) Resampling is a methodology that force a certain

level of portfolio diversification.level of portfolio diversification. Resampling is based on:

1 The simulation of a large number of statistically1. The simulation of a large number of statisticallyconsistent investment scenarios

2 The simulated E(R) and are used as input of a2. The simulated E(R), and are used as input of anew Markowitz Optimization.

3. After repeating steps 2. thousands of time the final3. After repeating steps 2. thousands of time the finalportfolios (Resampled Portfolios) have thecomposition of the average efficient portfolio

130

Resampling:Example (1/3)



131


Output: Resampled FrontierOutput: Resampled Frontier

0.09

0.1

0.07

0.08t

u

r

n

0.05

0.06

E

x

p

e

c

t

e

d

R

e

t

0.03

0.04

UnconstrainedResampled

0 0.05 0.1 0.15 0.2 0.250.02

Standard Deviation

Resampled

R l d f i i l h h ffi i f i132

Resampled frontier is lower than the efficient frontier


Output: Portfolio compositionWeights

80%

100%

Weights

60%

80%

20%

40%

0%1 12 23 34 45 56 67 78 89 100

Diversification increasesPortfolios

133

Extra-argument 3:A deeper analysis of ResamplingTMA deeper analysis of Resampling

134

Extra-argument 3:A deeper analysis of Resampling (1/7)

In order to process the resampling technique we need:we need:

Markowitz Simulation POptimization Process

135


The need to simulate returns: We know that our expectations can be wrong; We know that our expectations can be wrong; So in order to incorporate uncertainty, we can run a simulation process that return behavioursrun a simulation process that return behaviours of market returns that are different from our

t tiexpectation.

136


Si l ti A hi l t tiSimulation: A graphical representation

Hi h fid L fid

E(R) E(R)

High confidence Low confidence

( )

137

ExpectationSimulation


Wh t d d i d t i l t ?What do we need in order to simulate? Forecasts ( E(R), , ) Confidence on estimations Random process that is able to make

deviations from the expectation.

Simulation

138

Extra-argument 3:A d l i f R liA deeper analysis of Resampling (5/7)

Example:Example:

139


If we are able to simulate, we can estimate the Resampled portfoliosp p

140

Extra-argument 3:A deeper analysis of ResamplingA deeper analysis of Resampling (7/7)

Stage 1: ExpectationsMSCI

EuropeMSCI USA

MSCI Japan

MSCI EM

E(R) 7,0% 6,0% 4,5% 8,0%

20 0% 21 0% 22 8% 29 0%

MSCI Europe

MSCI USA

MSCI Japan

MSCI EM

MSCI Europe 1

MSCI USA 0,85 1

MSCI Japan 0,60 0,65 1 20,0% 21,0% 22,8% 29,0% p

MSCI EM 0,76 0,76 0,65 1

Stage 2: Confidence Very low-Low-Medium-High-Very high

Stage 2+1: 1st Simul.Path Simulations

a

p

Optimiz.MSCI

EuropeMSCI USA

MSCI Japan

MSCI EM

E(R) 8,0% 6,9% 6,2% 2,3%

16,0% 21,5% 29,2% 23,5%gTime

C

a

MSCI Europe

MSCI USA

MSCI Japan

MSCI EM

MSCI Europe 1

MSCI USA 0,91 1MSCI Japan 0,54 0,64 1

MSCI EM 0,90 0,85 0,62 1

O i iPath Simulations

MSCI Europe

MSCI USA

MSCI Japan

MSCI EM

, , , ,

Stage 2+2: 2st Simul. Optimiz.Time

C

a

p

p p

E(R) 7,1% 8,6% 10,1% 11,6%

20,9% 23,0% 23,3% 31,8%MSCI

EuropeMSCI USA

MSCI Japan

MSCI EM

MSCI Europe 1

MSCI USA 0,87 1MSCI Japan 0,76 0,77 1

MSCI EM 0,78 0,87 0,62 1

OptimizPath Simulations

MSCI Europe

MSCI USA

MSCI Japan

MSCI EM

E(R) 7,6% 10,6% 6,0% 11,0% 23 7% 25 4% 21 6% 33 3%

MSCI Europe

MSCI USA

MSCI Japan

MSCI EM

E(R) 2,7% 2,0% 4,6% 4,3% 21,5% 21,2% 20,3% 36,5%

Stage 2+3000: 3kst Simul.Optimiz.

Time

C

a

p

23,7% 25,4% 21,6% 33,3%

Average composition100%100%

0.08

0.09

1410%

20%

40%

60%

80%

1 12 23 34 45 56 67 78 89 10Portfolios

0%

20%

40%

60%

80%

1 12 23 34 45 56 67 78 89 10Portfolios

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350.02

0.03

0.04

0.05

0.06

0.07

E

(

R

)

Sigma

AgendaAgenda

8. Bayesian Techniques

142

Bayesian Techniques

The most common and widely used Bayesian technique is:

Th Bl k Litt M d lThe Black-Litterman Model

143

The Black-Litterman Model

The Black Litterman Model createsThe Black-Litterman Model createsbetter expected return forecasts to be pused withthe Markowitz optimizationthe Markowitz optimization

144

How does it work?

Start with the Market Neutral Start with the Market NeutralExpected Returns.

Apply your views of how certainmarkets are going to behavemarkets are going to behave.

The end result is a set of returnforecasts that give rise todiversified portfolios when useddiversified portfolios when usedwith the Markowitz Optimization.

145

Portfolio Market Neutral

The Market Neutral Portfolio is the The Market Neutral Portfolio is thecapitalization-weighted portfolio ofthe assets.

The Market Neutral expectedThe Market Neutral expectedReturns are the expected returnsth t i li d b th M k tthat are implied by the MarketNeutral Portfolio.

146

Add your own views

Investors generally have opinions orInvestors generally have opinions, orviews, about how certain marketswill behave in the future.

Each view includes a measure ofcertainty.

147

Example of an Absolute View

Opinion: I think that European EquityOpinion: I think that European EquityMarket is going to do well.

View: European Equity Market willhave a return of 11%have a return of 11%

Confidence of View: 55%

148

Example of an Relative View

Opinion: I believe that Europe isOpinion: I believe that Europe isgoing to outperform Japan.

View: European Equity Market willoutperform Jananese Equity Marketoutperform Jananese Equity Marketby 3%.

Confidence of View: 80%

149

Combine the Market Returns withyour Views

Market Neutral Views

MergingExpected Returns Views

Merging

Bl k LittBlack-Litterman Expected Returns

150

Final Result

Thanks to the Black-Litterman ForecastReturn efficient portfolio are muchReturn, efficient portfolio are muchmore diversified..

and the composition reflects our views.

100%

Peso

40%

60%

80%

0%

20%

40%

151

1 12 23 34 45 56 67 78 89 100Portafogli

Market Neutral Expected Returns: lexample

Asset ClassMKT1MKT2

MKT1MKT2

Rend. Market Neutral6,41%



,7,12%8,67%7,32%MKT8

MKT9MKT10MKT11MKT12


,8,95%

11

1

Ottimizzazione1

11

11

1

4%

28%12%

6%Azionario Pac ex Japan

Azionario Europa

Azionario America

152

11

1150%

Azionario Giappone

Azionario EM

Market Neutral Expected Returns: lexample

Matrice Varianze-Covarianze0,0408 0,0266 0,0215 0,0171 0,0329 0 0266 0 0278 0 0303 0 0227 0 0345

Pesi market neutralAzionario Pac ex Japan 4,0000%Azionario Europa 28 0000%

0,0266 0,0278 0,0303 0,0227 0,0345 0,0215 0,0303 0,0475 0,0321 0,0443 0,0171 0,0227 0,0321 0,0413 0,0357 0,0329 0,0345 0,0443 0,0357 0,0690

Azionario Europa 28,0000%Azionario America 50,0000%Azionario Giappone 12,0000%Azionario EM 6,0000%

Misura di avvversione al rischio

Misura di avvversione al rischio1,4310

Matrice Varianze-Covarianze0,0408 0,0266 0,0215 0,0171 0,0329 0,0266 0,0278 0,0303 0,0227 0,0345 0,0215 0,0303 0,0475 0,0321 0,0443 0,0171 0,0227 0,0321 0,0413 0,0357

Pesi market neutral

4,0000%28,0000%50,0000%12,0000% Rf

0,0329 0,0345 0,0443 0,0357 0,0690 6,0000%

Azionario Pacifico ex GiapponeRend. Market Neutral

6,41% Azionario EuropaAzionario AmericaAzionario Giappone

,7,12%8,67%7,32%

153Azionario EM

,8,95%

Black & Litterman: the views (1/4) Th t i t ti f t f th B&L d l i th t The most interesting feature of the B&L model is that

this model does not impose that the Asset Managersproduce detailed estimates for all the marketsproduce detailed estimates for all the marketsinvolved.

In practical terms the Asset Managers might simply In practical terms, the Asset Managers might simplyexpress estimates for only a few of the markets underobservation (e g Those which they know better)observation (e.g. Those which they know better).

Estimates might be either absolute or relative:

It is mandatory that every estimate is accompanied by apercentage representing the degree of confidence vis-a-vis

154

percentage representing the degree of confidence vis-a-visthe estimates (either in % or in relative terms)

Black & Litterman: views (segue)Wi h h f d i i h fi l i i With the purpose of determining the final returns, it is necessaryto build a matrix (P) so as to make possible to identify which are the asset classes involved by the viewsthe asset classes involved by the views.

1 2 3 4 5Azionario

Pacifico ex Azionario EuropaAzionario America

Azionario Giappone Azionario EMGiappone Europa America Giappone

0 0 1 1 0p1 0 0 -1 1 00 1 0 0 0 P

p1p2

155

Black & Litterman: views (segue)A t l (Q) th t id tifi th t A vector column (Q), on the contrary, identifies the returns which characterise either the absolute and/or the relative views.

4%9%9%

Q

156

Black & Litterman: views (segue)

22% Cc122% c2

157

Black & Litterman: le views (segue)Th l t i t i id tifi d b t i hi h t The last input is identified by a matrix which representsthe confidence of the analysts (Asset Managers) have intheir views.

How to build this matrix is one of the most widely debatedissues.

We determine the matrix with the following method:

T 00011

T

T

pp

ppc

00110

0001

22

111

TKK pp

c

11000000

2

KKK ppc

158Elements ci of vector C

Synthesis

QPPP TT 11111BL )()( 159

slides-9-december-2014-11-17-14-15-21

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