Quantitative Management

Document Sample
Quantitative Management Powered By Docstoc
					  Quantitative Management

    Master Analyse et Calcul Economiques -
         e                    e
Universit´ Paris-Dauphine Ing´nierie Economique
                            e
                  et Financi`re
                 Florian Ielpo
   Centre d’Economie de la Sorbonne – CERMSEM
      DEXIA SA – Asset Liability Management

                      2008



                        1
                       What is QM ?
– Most of asset manager : rule of the thumb, experience and luck...
– Now, lots of models applied to decide on winning trades :
  quantitative managers.
Globally, asset management = methods to handle funds and portfolio
based on various assets :
 1. Equity assets :
    – technical analysis, now used for market timing. Originally :
         ”Technical analysis is the study of market action, primarily
         through the use of charts, for the purpose of forecasting
         future price trends.”
    – Factor based models : classical Capital Asset Pricing Model and
      α trading ; more complicated versions and Roll’s Arbitrage
      Pricing Theory using a multifactor approach...

                                   2
    – Portfolio insurance : e.g. Stop-Loss, Option Based Portfolio
      Insurance (OBPI) and Constant Portfolio Protection Insurance
      (CPPI).
    – Black & Litterman’s approach to Markowitz problem.
    – ...
 2. Bond Markets and FX :
    – ”No arbitrage” Taylor rules : linking monetary policy and the
      yield curve.
    – Principal component analysis for market scenarii building. PCA
      factors. Bond pricing with a limited number of factors.
    – Cointegration : trading on mean reverting spreads.
    – Macroeconomic prediction in a data-rich environment : trading
      on economics.
    – ...
Each of these approaches try to deal with the Markowitz’s enormous
lack : how do I deal with the expected return ?

                                 3
                 Purposes of this class
Not to review each of these elements (most of them may have been
presented during other courses), but to focus on several :

 1. Portfolio insurance : SL, OBPI and CPPI.
 2. Review the PCA methodology and its various applications
 3. Use basic cointegration results to trade on mean reverting spreads.




                                  4
   Why care about strategic allocation ?
Now, a few words about Markowitz. Basically, the problem is :

                             max Et [U (w rt+1 )].              (1)

Whenever :
 1. returns are gaussian

                                rt+1 , t ∼ N (µ, Σ)             (2)

 2. utility is exponential

                               U (x) = − exp{−ax}               (3)

This problem is easy to solve, because :
                                                 µ+a2 1 w Σw
                 Et [e−aw     rt+1
                                     ] = −e−aw        2         (4)

                                         5
which is equivalent to solving :
                                  a
                         max w µ − w Σw                               (5)
                                  2
Usual solution to Markowitz problem and to mean-variance efficient
frontier that is supposed to guide us for asset allocation... But several
problems remain :

 1. Usual critics :
   (a) returns are not gaussian (depends on sampling frequency)
   (b) with exponential utility, constant risk aversion :
                                     U (x)
                              RA = −       =a                         (6)
                                     U (x)
 2. but the main problem is linked to the measurement of the
    parameters :
   (a) variances ; one period framework : which model to measure

                                    6
    σt+1 |t ? In a multiperiod framework : how to predict volatility
    on that period ?
(b) correlation ; correlation strongly vary over time ! For assets :
    positive. During crisis → goes up. How to account for this ?
    How to predict it ? Impact on the convexity of efficient frontier.
(c) expectation ; what really makes money is the exected return
    forecast, because the earning of the strategy is worth :
                               T                 T
                      PT              Pi
                  log    =       log      =       ri              (7)
                      Pt   i=t+1
                                     Pi−1   i=t+1

This last quantity : hard to compute/predict (at least with simple
models). With ARMA-like : impossible (EMH). Notable
expections :
(a) SETAR-like models...
(b) or MS models

                                   7
    But : computationally intensive and not stable though time...

For these reasons : the first attempt is simply to build hedging
portfolio strategies.




                                   8
1   Portfolio Insurance




           9
                   Portfolio Insurance
Main idea : hedging strategy with equities, allowing to make the most
of favorable evolutions of the markets.
A few words of vocabulary :
– Long position in a security means
  • the holder of the position owns the security
  • will profit if the price of the security goes up.
– Short position / shorting / short selling in a security means
  • the holder does not own the asset but sold it anyway
  • expect a drop of the security price.
From now on, two types of assets :
– a risky asset whose spot price at time t is Pt
– a riskless asset whose price is Bt = e−rf (T −t) , with T the maturity
  of the riskless asset.

                                   10
Mainly, two types of strategies :
– Buy stock as they rise and sell stock when they drop are known as
  convex strategies (yield convex payoff)
– Buy stock as they drop and sell stock when they rise are knwon as
  concave strategies (yield concave payoff)
Yield different results, depending on market conditions :
– for oscillating markets, concave strategies are OK : strategies
  building on reversals
– for trendy markets, convex strategies are OK : make the most of
  directional evolutions
Convex strat. still require to have a general idea about the upcoming
evolution of assets, BUT include insurance to hedge from big losses.


                                    11
                           Strategies
Several types of basic convex strategies :

 1. Buy & Hold : linear strategy hold untill the fixed time horizon : no
    hedging (proceed with finger crossed).
 2. Stop Loss strategy : swiching from riksy to riskless asset
    depending a target threshold.
 3. Constant Proportional Portfolio Insurance : constant proportional
    risk exposure that is a function of a cushion.
 4. Option Based Portfolio Insurance : replicate the payoff of
    underlying+put to ensure a terminal value K.




                                   12
A concave strategy : constant mix strategies : keep a constant
exposure to risky asset.


Each of these strategies :
– is based on a self-financing portfolio : no cash input on the fly
– have a fixed time horizon T
– are often represented as algorithm




                                  13
                        1.1    Buy & Hold

– Not really a strategy.
– α number of shares, N quantity in riskfree rate
– Portfolio value at time 0 :

                              Q0 = αP0 + N                           (8)

– Final value of the portfolio :

                         QT = αPT + N erf (T −1)                     (9)

– Strategy makes the most of trends.
– Unaffected by volatility or correlation.
– Still : you have to know the future evolution of the risky stock




                                   14
                   1.2    Stop Loss strategy

– Full investment in risky asset
– Define

                             M0 = xP0 e−rf T ,                      (10)

  the minimum target required by the investor.
– x depends on risk appetite.
– Then : Mt evolves as

                          Mt = xP0 e−rf (T −t) ,                    (11)
                                   fixed Time varying

– Strategy :
  1. Start with full investment in risky asset
  2. If Pt < Mt , the portfolio is invested in the riskless asset
  3. If Pt > Mt , the portfolio is invested in the risky asset

                                     15
– With small daily variations, the minimum value of the portfolio is
  expected to be M0 .
– Unfortunalty, with jumps, the hedging constraint is often violated.
– Beware transaction costs / bid-ask spreads




                                  16
                     cbind(x, strat, matrix(mini_ini, n, 1))

           70   75           80          85           90       95




     0
     100
     200




17
     300
     400
     500
                cbind(x, strat, matrix(mini_ini, n, 1))

           80        90                 100               110




     0
     100
     200




18
     300
     400
     500
                 1.3     Constant Mix Policy

– Concave strategies : make the most of price reversals. OK for High
  Volatility markets
– Main idea : keep constant risk exposure to the risky asset.
Define St the absolute position in the risky asset at time t :

                           RAPt = NtP × Pt                        (12)

Define Vt the value of the portfolio with factor loadings chosen at time
t:

                         Vt = RAPt + RF Pt ,                      (13)

with RF Pt the risk free asset position.



                                   19
Purpose of the method :
                             RAPt
                      Keep    Vt    = α, α ∈ [0, 1]
α is the risk exposure.


Algorithm :

 1. Set V0 such that the exposure is α
 2. Compute Vt+1|t the portfolio at time t + 1 given loadings chosen
    at time t :
                                    Pt+1
                Vt+1|t = RAPt ×          + RF Pt × erf ×∆t       (14)
                                     Pt
    Now ⇒ the constaint stands a good chance to be violated :
    – need to change the loadings to ensure it again
    – with a self financing portfolio

                                    20
 3. Rebalance positions such that :
              RAPt+1 = αVt+1|t                                   (15)
              RF Pt+1 = Vt+1|t − RAPt+1 = (1 − α)Vt+1|t          (16)

    And go back to step 2.
A few comments :
– Unlike BH and SL → a concave strategy
– Define the number of stocks in portfolio :
                                   RAPt
                              Nt =                               (17)
                                    Pt
– Number of stock evolve contrary to the value of the stock since :
                         Nt Pt
                      α=                                         (18)
                         Vt|t−1
                                    N t Pt
                         =                                       (19)
                             Nt−1 Pt−1 + RF Pt−1
                                  21
                Pt
⇒ the higher   Pt−1   and the lower Nt

With constant mix strategies :

– when the stock rise : sell because hoping on reversal
– when the stock drop : buy because hoping on reversal
  ⇒ Locally the expected payoff of the strategy is concave. Higher
  when a lot of volatility expected




                                 22
                          Q[2:n]                cbind(x[2:n], port[2:n])

                   0.60   0.65     0.70            70     80    90    100




             0
                                          0




             100
                                          100




             200
                                          200




23
     Index
             300
                                          300




             400
                                          400




             500
                                          500
                          Q[2:n]               cbind(x[2:n], port[2:n])

                   0.46 0.50 0.54 0.58         100    140      180




             0
                                         0




             100
                                         100




             200
                                         200




24
     Index
             300
                                         300




             400
                                         400




             500
                                         500
                            1.4    CPPI

A more general approach to controling risk exposure in the portfolio :
the constant proportion portfolio insurance
Define :
–   The floor F : minimum amount of money to be recovered in the end
–   The cushion Ct = Vt − F
–   The multiple m that controls risk exposure
–   Position in the risky asset : RAPt = m × Ct

m controls the risk exposure of the portfolio accross dates.




                                   25
The strategy is then :


 1. start with C0 = VO − F
 2. then RAP0 = m × C0 and RF P0 = V0 − RAP0
 3. Again,
                                 P1
                 V1,0   = RAP0 ×    + RF P0 × erf ×∆t    (20)
                                 P0
 4. The new cushion : C1 = V1,0 − F
 5. New position in the RA : RAP1 = m × C1
 6. And the new portfolio : V1 = RAP1 + (V0,1 − RAP1 )
 7. Go back to step 3



                                 26
Remarks :


– The CPPI is a convex strat. → m > 1 : when Pt goes up, position
  in the risky asset increases.
– When 0 < m < 1 and F = 0, constant mix strategy :
                                     RAPt
                               m=                                 (21)
                                     Vt,t−1
– When m = 1, BH strat. with F in the riskless asset.
The strategy allows the investor to take part in local trends :
– Buy when prices rise
– Sell when prices fall




                                   27
                                                                Asset/Port                                                     Cushion                                             RAP/Asset




                                                                                                                                                                   0.5
                                                                                                           10
                                                100




                                                                                                                                                                   0.4
                                                                                                           8
                                                                                         coussin[2:n]




                                                                                                                                                           Q
                                                                                                           6




                                                                                                                                                                   0.3
                                                                                                           4




                                                                                                                                                                   0.2
cbind(x[2:n], port[2:n], matrix(F, n − 1, 1))

                                                90




                                                                                                           2




                                                                                                                                                                   0.1
                                                                                                                0   100    200           300   400   500                 0   100   200           300   400   500

                                                                                                                                 Index                                                   Index



                                                                                                                                 RAP                                                 RAP/V
                                                80




                                                                                                                                                                   0.5
                                                                                                           50




                                                                                                                                                                   0.4
                                                                                                           40
                                                                                         position_action




                                                                                                                                                           alpha
                                                70




                                                                                                                                                                   0.3
                                                                                                           30




                                                                                                                                                                   0.2
                                                                                                           20
                                                                                                           10




                                                                                                                                                                   0.1
                                                      0   100   200    300   400   500                          0   100    200           300   400   500                 0   100   200           300   400   500

                                                                                                                                 Index                                                   Index




                                                                                                                          28
                                                                Asset/Port                                                       Cushion                                            RAP/Asset




                                                                                                           200




                                                                                                                                                                     15
                                                250




                                                                                                           150
                                                                                         coussin[2:n]




                                                                                                                                                                     10
                                                                                                           100




                                                                                                                                                             Q

                                                                                                                                                                     5
                                                200




                                                                                                           50
cbind(x[2:n], port[2:n], matrix(F, n − 1, 1))




                                                                                                           0




                                                                                                                                                                     0
                                                                                                                  0   100    200           300   400   500                0   100   200           300   400   500

                                                                                                                                   Index                                                  Index
                                                150




                                                                                                                                   RAP                                                RAP/V




                                                                                                           2000




                                                                                                                                                                     8
                                                                                                           1500




                                                                                                                                                                     6
                                                100




                                                                                         position_action




                                                                                                                                                             alpha
                                                                                                           1000




                                                                                                                                                                     4
                                                                                                           500




                                                                                                                                                                     2
                                                50




                                                                                                           0




                                                                                                                                                                     0
                                                      0   100   200    300   400   500                            0   100    200           300   400   500                0   100   200           300   400   500

                                                                                                                                   Index                                                  Index




                                                                                                                            29
                                                          cbind(x[2:n], port[2:n], matrix(F, n − 1, 1))

                        0                                 50                                      100                                      150




             0
             100
             200
             300
                                                                                                                                                 Asset/Port




             400
             500
                                   position_action                                                                   coussin[2:n]

                   60       65           70          75        80                               100     105   110       115   120   125    130




             0
                                                                                          0




             100
                                                                                          100




30
             200
                                                                                          200




     Index
                                                                                  Index



                                                                       RAP
                                                                                                                                                 Cushion




             300
                                                                                          300




             400
                                                                                          400




             500
                                                                                          500




                                         alpha                                                                            Q

                   0.604         0.608           0.612         0.616                             0.54         0.56         0.58     0.60
             0
                                                                                          0




             100
                                                                                          100




             200
                                                                                          200




     Index
                                                                                  Index



                                                                       RAP/V




             300
                                                                                          300
                                                                                                                                                 RAP/Asset




             400
                                                                                          400




             500
                                                                                          500
                                                     cbind(x[2:n], port[2:n], matrix(F, n − 1, 1))

                          60                    80                        100                     120                   140                       160




             0
             100
             200
             300
                                                                                                                                                        Asset/Port




             400
             500
                           position_action                                                                       coussin[2:n]

                   50          60          70           80                                              50        60            70           80




             0
                                                                                        0




             100
                                                                                        100




31
             200
                                                                                        200




     Index
                                                                                Index



                                                                  RAP
                                                                                                                                                        Cushion




             300
                                                                                        300




             400
                                                                                        400




             500
                                                                                        500




                                alpha                                                                                   Q

                   0.50             0.55         0.60                                         0.500     0.505   0.510       0.515    0.520   0.525
             0
                                                                                        0




             100
                                                                                        100




             200
                                                                                        200




     Index
                                                                                Index



                                                                  RAP/V




             300
                                                                                        300
                                                                                                                                                        RAP/Asset




             400
                                                                                        400




             500
                                                                                        500
                           1.5    OBPI

From Leland and Rubinstein (1976). Main idea here :
– again, investing in risky and riskless asset
– to mimic a put payoff max(K − St , 0) plus the risky asset.
Basically :
                  Put + Underlying = Portfolio hedge

Easy to check :
                               
                                K when P < K
                                         T
             PT + (K − PT )+ =                                 (22)
                                PT when PT > K

Insurance to have at least K in the end.


                                  32
                        payoff

               0   50    100     150   200




         0
         50




33
     S
         100
         150
         200
Question : portfolio loadings for today, with the previous final payoff ?


Under no arbitrage arguments :

 e−rf (T −t) EQ PT + K − PT )+ Ft = (Pt + P ut(K, t, T, Pt )) (23)
                 present value


Useless to by a put, just have to replicate the BS formula :

    P ut(K, P, t, T ) =e−rf (T −t) EQ [(K − P )+ ]                  (24)
                           = −N (d1 ) Pt + KN (−d2 ) e−rf (T −t)    (25)
                                 short position   Long position


The position :

  Pt + P ut(K, P, t, T ) = 1 − N (d1 ) Pt + KN (−d2 ) e−rf (T −t)   (26)
                                  Long position     Long position



                                             34
         Problem : the portfolio has to be self-financed !
⇒ Using the loadings as weights instead of quantities.


Algorithm :
– Start with V0 = (1 + N (d1 ))S0 + Ke−rf T N (−d2) Then for any
  date :
– as usually : Vt+1,t = RAPt × PPt + RF Pt × erf ∆t
                                 t+1


– ”Optimal” portfolio :
             ∗
           Vt+1 = (1 − N (d∗ ))Pt+1 + KN (−d∗ )e−rf (T −t−1)
                           1                2                  (27)




                                  35
– Exposure to risky asset α comes from :
                         ∗
                       Vt+1
     Vt+1   = Vt+1,t × ∗                                          (28)
                       Vt+1
                                                                
                      (1 − N (d∗ ))P       KN (−d∗ )e−rf (T −t−1) 
                                 1    t+1         2
            = Vt+1,t                     +
                                                                  
                              ∗
                             Vt+1                    ∗
                                                   Vt+1
                                                                   
                                                                   
                       Risky asset loading=α        1−α
                                                                  (29)

– Thus, RAPt+1 = αVt+1,t and RF Pt+1 = (1 − α)Vt+1,t
– The new portfolio : Vt+1 = RAPt+1 + RF Pt+1




                                      36
Loadings have to be computed and the port. rebalanced to ensure the
option-like hedge.


Convex strategy


Important : the hedge holds on average !




                                37
                                                        cbind(x, port, matrix(K, n, 1))

                         60                  80                                100                       120                140




             0
             100
             200
                                                                                                                                        Port/P




             300
             400
             500




38
                              position_action/x                                                                 alpha_

                   0.4        0.5     0.6         0.7                                              0.3   0.4   0.5   0.6   0.7    0.8
             0
                                                                                             0




             100
                                                                                             100




             200
                                                                                             200




     Index
                                                                                     Index
                                                                                                                                        alpha




             300
                                                                                             300




                                                              Quantity of RA




             400
                                                                                             400




             500
                                                                                             500
                                              cbind(x, port, matrix(K, n, 1))

                          60           70          80                        90           100            110




             0
             100
             200
                                                                                                                     Port/P




             300
             400
             500




39
                          position_action/x                                                     alpha_

                   0.40        0.50   0.60                                          0.3   0.4   0.5      0.6   0.7
             0
                                                                              0




             100
                                                                              100




             200
                                                                              200




     Index
                                                                     Index
                                                                                                                     alpha




             300
                                                                              300




                                                    Quantity of RA




             400
                                                                              400




             500
                                                                              500
                              Histogram of SL                                                                    Histogram of CPPI



            3500




                                                                                           6000
            2500




                                                                                           4000
Frequency




                                                                               Frequency
            1500




                                                                                           2000
            0 500




                                                                                           0
                    0   100      200              300         400                                 0   50         100    150    200         250   300

                                       SL                                                                              CPPI



                              Histogram of CM                                                                    Histogram of OBPI
            2500




                                                                                           3000
Frequency




                                                                               Frequency

                                                                                           2000
            1500




                                                                                           1000
            500
            0




                                                                                           0
                    0   50     100          150         200         250                           0        100          200          300         400

                                       CM                                                                              OBPI




                                                                          40
                       SL                                             OBPI



     400




                                                    400
     300




                                                    300
                                             OBPI
SL

     200




                                                    200
     100




                                                    100
     0




                                                    0
           100   200        300   400                     100   200          300   400

                       S                                               S



                       CM                                             CPPI
     400




                                                    400
     300




                                                    300
                                             CPPI
CM

     200




                                                    200
     100




                                                    100
     0




                                                    0
           100   200        300   400                     100   200          300   400

                       S                                               S




                                        41
2   Factor Based Method Analysis




               42
                Purposes of the Section
This section is devoted to dimension reduction in financial markets :
– Globally : a lot of information to be dealt with that may be
  redundant
– Arbitrage tracking mostly based on factor based model of the
  financial markets (CAPM,APT...)
⇒ But these factors are unobservable or observed in a noisy way
     Need for methods to estimate these factors in order to :
–   control for risk for large portfolios
–   find minimum variance portfolios
–   improve the understanding of the fine financial market dynamics
–   ...




                                  43
                Organization of this subsection :

1. Crash introduction to Principal Component Analysis
2. Use for minimum variance portfolio building
3. Use for term structure analysis in swap rates




                                 44
            2.1     A Crash Introduction to PCA

 Principal Component Analysis = a way around dimension reduction
                            problems
Basics on PCA
Let Xt = (x1,t , x2,t , . . . , xn,t ) , t = 1, ..., T be the matrix of
observations to be modeled
Remarks :
 1. Xt is covariance stationary : the covariance process does not
    depend upon t, but only on the sampling frequency :

                                  Σt = Σt+1 = ...                         (30)

     – E.g. GARCH-like models are not covariance stationary, just like
       financial time series (in general)
     – GARCH models are conditionally covariance stationary ⇒

                                        45
      filtered series are stationary
 2. Xt can be such that T << n and PCA still works
 3. PCA are anyway applied to financial time series with important
    consequences on asset allocation.
 4. Most of the time, the time series are considered to be (time)
    locally stationary .
 5. In Meucci’s book : ”time invariance” for stationarity


Assume that :

 1. Xt has zero mean
 2. Xt has unit variance




                                      46
Correlation matrix :
                            C = T −1 X X                         (31)

PCA is based on eigenvalue/vector decomposition of C
Purposes of PCA
            Xt ⇒ Ft = (F1,t , ..., Fk,t ) , such that k << n     (32)

Requirements of PCA
– Form factors Fi named principal component
– That best summarize (possibly redundant) information in the
  dataset
– Quantity of information = variability of the dataset
– Factors must be ordered : first factor = explain most of the
  information
– Principal component are uncorrelated with each other
Used in many fields (chemiometrics, econometrics, finance, biology...)
                                   47
The spectral theorem in short
Let S be a symmetric positive definite matrix, i.e. :

                  1. S = S                                    (33)
                  2. |S| > 0 or ∃v ∈ Rn : v Sv > 0            (34)

then S can be decomposed as :

                              S = DΛD ,                       (35)

with

               Λ = diag(λ1 , ..., λn ) with λ1 > . . . > λn   (36)
               D = D−1 ⇔ DD = In                              (37)
               T r(S) = T r(DΛD ) = T r(Λ) = n                (38)
               D is a rotation matrix                         (39)


                                    48
Solution to the PCA problem
– C fulfill the spectral theorem requirements, thus :

                                 C = DΛD                            (40)

– The ith optimal factor is given by :

                                 Fi = XDi ,                         (41)

  with Di the ith column of D.
– V (Fi ) = T −1 Fi Fi = Di X T X Di = Di CDi = λi
                        F F
– Similarly : ρ(F ) =    T    = Λ, which is diagonal ⇒ no correlation
  between factors

Remark
                    Since D is a rotation matrix :

            ⇒ Factors are rotations of the original dataset

                                     49
Applications
Factor analysis is usually applied to :
– Interest Rates Swap (IRS) to recover the usual three factors of the
  yield curves
– Implied volatilities accross moneynesses to recover the three factors
  in implied volatility dynamics
– Equity stocks to build riskless portfolios
– ...
Most of the time :
– better understanding of financial asset dynamics
– better understanding of risk level implicit in positions
– help build strategies
– help build hedge against specific movements
– help build funds with a controled risk exposure
– ...

                                   50
               2.2    PCA applied to swap rates

rt (τ ) = swap rate with time to maturity τ on date t
Comments
–    Swap rates used to build term structure of interest rates
–    Higher risk than for bills (no collateral)
–    But consistent with interbank loans and CB target rate
–    No ZC yield, but coupon paying par yield
Important point :
    1. Swap rates are not second order stationary
    2. Daily variations can be assumed to be second order stationary in a
       first approach
    3. But they are not ! Second order moments and comoments vary
       through time...

                                     51
First case : the EURO curve
– New market
– German dominance
– Progressive learning of the market
– Different periods of vol. due to diff. central bankers + market
  conditions
– ...




                                 52
                                          Term structure of interest rates



    6
    5
x

    4
    3
    2




        2000−01−02 2000−12−20 2001−12−08 2002−11−26 2003−11−14 2004−11−01 2005−10−20 2006−10−08 2007−09−26

                                                   1:length(dates)




                                                   53
                Eigenvalues



      0.8
      0.6
val

      0.4
      0.2
      0.0




            5                 10   15

                     Index




                54
                                            Euro rates eigenvectors

vec[, 1:3]

             0.2
             −0.2




                              0   5    10              15             20      25   30

                                                       mat



                                      Correlation between factors and rates
             0.2
             −1.0 −0.6 −0.2
D




                              0   5    10              15             20      25   30

                                                       mat




                                                 55
                                                                           Cum. first three fact.

scale(apply((F), 2, cumsum))

                               2
                               1
                               0
                               −1
                               −2




                                    2000−01−02 2000−12−20 2001−12−08 2002−11−26 2003−11−14 2004−11−01 2005−10−20 2006−10−08 2007−09−26

                                                                               1:length(dates)



                                                                          1Y−5Y−10Y swap rates
                               6
x[, c(4, 8, 13)]

                               5
                               4
                               3
                               2




                                    2000−01−02 2000−12−20 2001−12−08 2002−11−26 2003−11−14 2004−11−01 2005−10−20 2006−10−08 2007−09−26

                                                                               1:length(dates)




                                                                               56
                   lambda[, i]                      lambda[, i]                     lambda[, i]

            0.02      0.06       0.10          0.12 0.16 0.20 0.24          0.50   0.60   0.70    0.80




     2002
                                        2002
                                                                     2002




57
     2004
                                        2004
                                                                     2004




     2006
                                        2006
                                                                     2006




     2008
                                        2008
                                                                     2008
                              Contribution to fact. 1 of 1 Y




         0.2
v[, i]

         0.0
         −0.2




                2000   2002              2004                  2006   2008




                              Contribution to fact. 1 of 5 Y
         0.3
         0.1
v[, i]

         −0.1
         −0.3




                2000   2002              2004                  2006   2008




                              Contribution to fact. 1 of 9 Y
         0.3
         0.1
v[, i]

         −0.1
         −0.3




                2000   2002              2004                  2006   2008




                                   58
Second case : the US case
–   Old market
–   No particular dominance
–   from 2000 to 2007 : one change in CB
–   ...




                                 59
                                          Term structure of interest rates



    8
    7
    6
    5
x

    4
    3
    2
    1




        1999−01−03 2000−02−05 2001−03−09 2002−04−11 2003−05−14 2004−06−15 2005−07−18 2006−08−20 2007−09−22

                                                   1:length(dates)




                                                   60
                Eigenvalues



      0.6
      0.4
val

      0.2
      0.0




            5                 10   15

                     Index




                61
                                        US rates eigenvectors



             0.2 0.4
vec[, 1:3]

             −0.2




                         0   5    10              15             20      25   30

                                                  mat



                                 Correlation between factors and rates
             0.5
             0.0
D

             −1.0 −0.5




                         0   5    10              15             20      25   30

                                                  mat




                                            62
                                                                            Cum. first three fact.



                   2
                   1
scale(F)

                   −1 0
                   −3




                                     1999−01−03 2000−02−05 2001−03−09 2002−04−11 2003−05−14 2004−06−15 2005−07−18 2006−08−20 2007−09−22

                                                                                1:length(dates)



                                                                           1Y−5Y−10Y swap rates
                   1 2 3 4 5 6 7 8
x[, c(4, 8, 13)]




                                     1999−01−03 2000−02−05 2001−03−09 2002−04−11 2003−05−14 2004−06−15 2005−07−18 2006−08−20 2007−09−22

                                                                                1:length(dates)




                                                                                63
                   lambda[, i]                  lambda[, i]                      lambda[, i]

            0.00       0.04             0.0   0.1   0.2   0.3   0.4          0.6 0.7 0.8 0.9 1.0




     2000
                                 2000
                                                                      2000




     2002
                                 2002
                                                                      2002




64
     2004
                                 2004
                                                                      2004




     2006
                                 2006
                                                                      2006




     2008
                                 2008
                                                                      2008
                              Contribution to fact. 1 of 1 Y




         0.3
         0.1
v[, i]

         −0.1
         −0.3




                2000   2002                       2004         2006   2008




                              Contribution to fact. 1 of 5 Y
         0.3
         0.1
v[, i]

         −0.1
         −0.3




                2000   2002                       2004         2006   2008




                              Contribution to fact. 1 of 9 Y
         0.3
         0.1
v[, i]

         −0.1
         −0.3




                2000   2002                       2004         2006   2008




                                   65
    Lessons from the US and Euro rates
– Three factors in the yield curve :
  1. Level (85%)
  2. Slope (10%)
  3. Convexity (5%)
– Factors are not stable :
  • importance varies through time
  • composition seems to switch depending on dates
  Why ? mainly : time varying correlation that defines the
  eigenvectors and eigenvalues
– Important for risk management of bond portfolio : main blow comes
  from the level
– Importance of strategic allocation : butterlfy strategies yield the less
  volatile payoff. Beware of level position → will go though most of
  the market moves.
                                    66
Steps further
Possible to summarize much more information :
–   Global risk factor and the contribution of the US
–   Enhance the global risk perception of the bank
–   Allow for fine understanding of these factors
–   Help build asset pricing models close to reality :
    • Vasicek, Cox Ingersoll Ross = one factor models (only level)
    • other affine models (Piazzesi (2001)) allow for more factors




                                   67
Remarks
Similar results obtained for Implied Volatilities :
 1. Alexander : departure from at the money implied volatility with a
    fixed time to maturity :
    – With volatilities : need to deal with additional difficulties :
      • No fixed time to maturity products avail.
      • The underlying varies a lot : distance from strike to spot
        unstable. ⇒ cannot compare raw implied volatilities through
        time
      • Alexander compares departure from ATM vol : series closer to
        stationarity
    – Results : again three factors : parallel shifts, asymmetry and
      curvature of the smile.


                                    68
2. Da Fonseca and Cont :
   – another approach taking explicitely the dimensional problem
     into account :

      Time to marity   × Calendar time + Fixed TTM + Fixed moneyness

                     e
  – Use Karhunen-Lo`ve decomposition to take this dimensionality
    into account.
  – Again, three factors : level of the smile term structure, potential
    asymmetry and curvature.




                                  69
December 2007 Contracts Implied Volatility Surface     June 2007 Contracts Implied Volatility Surface   December 2010 Contracts Implied Volatility Surface




 0.7                                                  0.6                                                 0.5




                                                                                                        Impli
Impli




    0.6




                                                     Impli
                                                        0.5                                                 0.4
      0.5




                                                                                                             e
      e




                                                          0.4




                                                          e




                                                                                                          d Vo
   d Vo




                                                                                                                 0.3




                                                       d Vo
          0.4                                                 0.3                                                0.2




                                                                                                            latilit
     latilit




           0.3                           50                                                  50                                                    50



                                                         latilit
                                        100                    0.2                          100                   0.1                             100
            0.2
                                            Time




                                                                                                Time




                                                                                                                                                      Time
                                                                                                                 y
             y




                                                                0.1

                                                                y
             0.90                       150                      0.90                       150                        0.90                       150
                    0.95                                                0.95                                                  0.95
                      Mon1.00          200                                Mon1.00          200                                  Mon1.00          200
                         eyne 1.05                                           eyne 1.05                                             eyne 1.05
                              ss                                                  ss                                                    ss
                                     1.10                                                1.10                                                  1.10




                                                                             70
  Final remarks on stationarity problems
Several way around the problem :

 1. Modelling the eigenvalues may not be the right answer :
    – need to maintain positivity
    – tricky/non stationnary dynamics of the eigenvalues
    – no a priori model
 2. Modelling the correlations = a better way, since the quest for
    definite positive dynamic correlation matrices is achieved both a
    continuous and discrete time :
    – Discrete : Dynamics Conditional Correlation Model of Engle
      (2002)
      • GARCH dynamic for volatility
      • Independent model for correlation in a autoregressive way
      • Extended in Capiello, Engle and Sheppard (2003) to accomodate
        leverage and asymmetric correlation effects.

                                   71
  • QML estimation is easy to implement (estimation yet tricky)
  • makes it possible to forecast covolatility matrices
– Wishart Affine Stochastic Correlation Model (WASC) of da
  Fonseca, Grassello and Tebaldi (2006) in a continuous time
  setting :
  • Multidim. Heston-like model
  • Ok for asymmetric correlation effect and leverage
  • Can be used for optimal portfolio choice, term structure
    modelling...
  • Closed form expression for the dynamics of the first eigenvalue of
    the correlation matrix
  • ...


Yet = attempts to model the eigenvalues proved to obtain
interesting results.



                               72
         2.3     PCA for portfolio management

Remark
In PCA analysis, the eigenvalues measure the level of risk associated to
the linear combination of the assets in the portfolio :

                             Fi = R × Di ,                         (42)

with R the return matrix, Di the ith eigenvector.

                              σ(Fi ) = λi .                        (43)

– Di = portfolio loadings
– Fi = portfolio value


                                   73
Several Trading Strategies
– For large dataset : final eigenvalues are closed to zero ⇒ ∼ riskless
  portfolio.
  In a market with no arbitrage, expected return of such portfolio
  should be riskfree rate
  Borrow risk free rate + Long position in Fi
– Some portfolios have low risk level but are long short positions ⇒
  low fund level required (asset management) ? Interesting because :
  1. factors are orthogonal : port. is hedged against the remaining
     risk factors
  2. even with positive correlation : diversification effect
  3. Long/short : limited amount on initial investement (but possible
     leverage)




                                  74
    8000
                     SPX


                     FTSE
    6000




                     DAX


                     CAC
    4000
x

    2000
    0




           1990−01−02 1992−02−29 1994−04−27 1996−06−23 1998−08−20 2000−10−16 2002−12−13 2005−02−08 2007−04−07




                                                        75
                             Eigenvalue                                        Eigenvec.


     0.7
     0.6




                                                               0.5
     0.5
     0.4
vp




                                                          vc

                                                               0.0
     0.3
     0.2




                                                               −0.5
     0.1




           1.0   1.5   2.0      2.5    3.0   3.5   4.0                SPX   FTSE           DAX   CAC

                               Index




                                                         76
                                                    Portfolio behavior


    0.5
    0.0
    −0.5
    −1.0
F

    −1.5
    −2.0
    −2.5




           1990−01−03 1992−02−29 1994−04−25 1996−06−20 1998−08−15 2000−10−10 2002−12−06 2005−01−30 2007−03−28




                                                         77
                         E/V frontier with PCA port.


          14
          12
          10
          8
abs(mu)

          6
          4
          2
          0




               10   15              20                 25   30

                                    sigma




                                  78
                               Eigenv. #1                                          Eigenv. #3




                                                              0.20
          0.8




                                                              0.15
          0.7
vp[, i]




                                                    vp[, i]

                                                              0.10
          0.6




                                                              0.05
          0.5




                 1990   1995         2000   2005                     1990   1995         2000   2005




                               Eigenv. #2                                          Eigenv. #4




                                                              0.14
          0.25




                                                              0.10
          0.20
vp[, i]




                                                    vp[, i]

                                                              0.06
          0.15




                                                              0.02
          0.10




                 1990   1995         2000   2005                     1990   1995         2000   2005




                                                   79
                         Quantity invested in SPX                                    Quantity invested in DAX



          0.4




                                                                      0.5
          0.2
vc[, i]




                                                            vc[, i]
          0.0




                                                                      0.0
          −0.2




                                                                      −0.5
          −0.4




                 1990   1995          2000          2005                     1990   1995          2000          2005




                        Quantity invested in FTSE                                    Quantity invested in CAC
          0.5




                                                                      0.5
vc[, i]




                                                            vc[, i]
          0.0




                                                                      0.0
          −0.5




                                                                      −0.5
                 1990   1995          2000          2005                     1990   1995          2000          2005




                                                           80
                                                Moment #1                                              Moment #3



              200




                                                                                  1.5
              100




                                                                                  0.5
moment[, i]




                                                                    moment[, i]
              0




                                                                                  −0.5
              −200 −100




                                                                                  −1.5
                                  1990   1995        2000   2005                         1990   1995        2000   2005




                                                Moment #2                                              Moment #4
              10 15 20 25 30 35




                                                                                  1
moment[, i]




                                                                    moment[, i]

                                                                                  0
                                                                                  −1
              5




                                                                                  −2
              0




                                  1990   1995        2000   2005                         1990   1995        2000   2005




                                                                   81
                            Distrib of average return for PCA port #1                                                    Distrib of average return for PCA port #3




                                                                                                      1400
            1000




                                                                                                      1000
Frequency




                                                                                          Frequency
            600




                                                                                                      600
            200




                                                                                                      0 200
            0




                             −0.4           −0.2           0.0          0.2        0.4                        −0.15      −0.10    −0.05    0.00         0.05    0.10     0.15

                                                   moment[, i]                                                                            moment[, i]



                            Distrib of average return for PCA port #2                                                    Distrib of average return for PCA port #4
            1400




                                                                                                      1500
            1000




                                                                                                      1000
Frequency




                                                                                          Frequency
            600




                                                                                                      500
            0 200




                                                                                                      0
                    −0.20   −0.15   −0.10     −0.05      0.00    0.05     0.10   0.15                                 −0.10      −0.05    0.00           0.05     0.10          0.15

                                                   moment[, i]                                                                            moment[, i]




                                                                                         82
                       Backtested returns for PCA portfolio


          10
          8
abs(mu)

          6
          4
          2
          0




               0   5   10             15              20      25   30

                                      sigma




                                     83
Main results


–   Port.   1   :   global port.
–   Port.   2   :   SP500
–   Port.   3   :   FTSE/DAX vs. CAC
–   Port.   4   :   CAC vs. DAX (long/short portfolio)
Hard to find a group of port. that clearly beat the market but :
– No idea about the future direction : hard to predict the sign to
  associate to Di
– No risk free rate used with PCA
– Still possible to combine it with port. insurance technics
– ...




                                         84
     3     Pairs trading and cointegration

General idea
In the previous section : DAX/CAC long short port.
⇒ long in one asset and short in the other for minimum capital
requirement


General purpose of this method :
               Bet reversals in long term relationships
Ex : when you know the usual spread between DAX and CAC is x and
now it is worth y>x, you can bet on the reversal to the long term value
Problems
– how to estimate this long term value ?
– is this long term value stable ?
                                   85
1. To the first question : cointegration.

   What is cointegration ?
   – If X1 and X2 are I(1) time series (first difference are stat.time
     series)
   – Estimate with OLS X1 = α + βX2 +
   – If the estimated t is I(0) then we say
     X1 and X2 are cointegrated
   The estimated gives us clues around :
   – how far are indexes from their long term relationship ?
     answer : far if | | is big
   – what is the speed of mean reversion to it ?
     answer : estimate AR(P) process on . For AR(1) :

                              t   =θ   t−1   + νt ,              (44)

     if θ is small : fast
                                  86
  if θ is close to 1 : slow
Trading idea

  When the residual is big enough, trade on the collapse of the
indexes to their long term relation, if the mean reversion speed is
                          high enough.

Problems
–   How to choose the estimation window ?
–   How stable are long term relationship ?
–   How to choose the investment horizon ?
–   ...




                               87
A simple example on DAX and CAC again :


                 Estimate       Std. Error    t value    P-val
 (Intercept)    -3.608e+02     1.388e+01      -26.00    <2e-16
    CAC         1.231e+00       3.807e-03     323.46    <2e-16
Signif. codes : 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1


Residual standard error : 369.9 on 4563 degrees of freedom
Multiple R-Squared : 0.9582, Adjusted R-squared : 0.9582
F-statistic : 1.046e+05 on 1 and 4563 DF, p-value : < 2.2e-16



                                     88
    7000
    6500
P

    6000
    5500




           2006−11−11   2006−11−29   2006−12−17   2007−01−04   2007−01−22   2007−02−10   2007−02−28   2007−03−18




                                                               89
             Spread to the long term relation


      150
      100
      50
      0
eps

      −50
      −100
      −150
      −200




              janv.                             mars




                         90
                     ACF of DAX



      1.0
      0.6
ACF

      0.2
      −0.2




             0   5       10       15   20

                        Lag



                     ACF of CAC
      1.0
      0.6
ACF

      0.2
      −0.2




             0   5       10       15   20

                        Lag




                      91
                                   ACF

                −0.2   0.0   0.2   0.4   0.6   0.8   1.0




           0
           5




92
           10

     Lag
                                                           ACF of LT




           15
           20

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:47
posted:7/23/2011
language:French
pages:92
Description: Quantitative Management document sample