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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