Session 15, Flexible Probability Stress Testing. Moderator: Dan dibartolomeo. Presenter: Attilio Meucci, CFA, Ph.D.
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1 Session 15, Flexible Probability Stress Testing Moderator: Dan dibartolomeo Presenter: Attilio Meucci, CFA, Ph.D.
2 Attilio Meucci Entropy Pooling STUDY IT: (white papers and code) DO IT: Advanced Risk and Portfolio Management Bootcamp
3 ENTROPY POOLING Theory ENTROPY POOLING Implementation ENTROPY POOLING Applications REFERENCES CONCLUSIONS
4 Market distr. not returns, not normal, not equilibrium e.g. 2-yr swap rate 5-yr swap rate 5yr swap 2yr swap
5 Market distr. not returns, not normal, not equilibrium e.g. 2-yr swap rate 5-yr swap rate 5yr swap 2yr swap Pricing delta/gamma/vega, full pricing, e.g. duration + convexity
6 Market distr. not returns, not normal, not equilibrium e.g. 2-yr swap rate 5-yr swap rate 5yr swap 2yr swap Pricing delta/gamma/vega, full pricing, e.g. duration + convexity Optimization utility, mean-cvar, mean-variance
7 Market distr. not returns, not normal, not equilibrium 1 Focus non-linear functions and external factors X X 2 e.g. 2-yr swap rate 5-yr swap rate V X + X convexity factor
8 Market distr. Focus not returns, not normal, not equilibrium non-linear functions and external factors e.g. 2-yr swap rate 5-yr swap rate 5yr swap 2yr swap
9 Market distr. Focus not returns, not normal, not equilibrium non-linear functions and external factors e.g. 2-yr swap rate 5-yr swap rate Views full distribution specification 2yr swap
10 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification view on expectations (BL), medians e.g. 2-yr swap rate 5-yr swap rate 2yr swap
11 Market distr. not returns, not normal, not equilibrium 1 Focus Views non-linear functions and external factors full distribution specification view on expectations (BL), medians ranking X X 2 e.g. 2-yr swap rate 5-yr swap rate V X 1
12 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification e.g. 2-yr swap rate 5-yr swap rate view on expectations (BL), medians ranking views on volatilities 2yr swap
13 Market distr. not returns, not normal, not equilibrium 1 Focus Views non-linear functions and external factors full distribution specification view on expectations (BL), medians ranking views on volatilities correlation stress-testing X X 2 e.g. 2-yr swap rate 5-yr swap rate V X 1
14 Market distr. not returns, not normal, not equilibrium 1 Focus Views non-linear functions and external factors full distribution specification view on expectations (BL), medians ranking views on volatilities correlation stress-testing view on tail behavior X X 2 e.g. 2-yr swap rate 5-yr swap rate V X 1
15 Market distr. not returns, not normal, not equilibrium 1 Focus Views non-linear functions and external factors full distribution specification X X 2 e.g. 2-yr swap rate 5-yr swap rate V X 1 partial distribution specification
16 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification e.g. 2-yr swap rate 5-yr swap rate partial distribution specification 2yr swap
17 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors e.g. 2-yr swap rate 5-yr swap rate 5yr swap?? 2yr swap
18 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors e.g. 2-yr swap rate 5-yr swap rate 5yr swap?? Posterior? 2yr swap
19 Market distr. not returns, not normal, not equilibrium 1 Focus Views non-linear functions and external factors full distribution specification X X 2 e.g. 2-yr swap rate 5-yr swap rate V X 1 partial distribution specification Posterior? relative entropy distance btw. distributions
20 Market distr. not returns, not normal, not equilibrium 1 Focus Views non-linear functions and external factors full distribution specification X X 2 e.g. 2-yr swap rate 5-yr swap rate V X 1 partial distribution specification Posterior least distance from prior relative entropy distance btw. distributions
21 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification partial distribution specification Posterior least distance from prior, views satisfied relative entropy distance btw. distributions
22 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification partial distribution specification Posterior least distance from prior, views satisfied
23 Market distr. 2-yr swap rate 5-yr swap rate 5yr swap 2yr swap estimated normal fit 2yr swap 5yr swap
24 Market distr. X1 X 2 2-yr swap rate 5-yr swap rate prior USD SWAP 2Y rate prior USD SWAP 5Y rate estimated normal fit yr swap yr swap
25 Market distr. X1 X 2 Focus 2-yr swap rate 5-yr swap rate V X 1 prior USD SWAP 2Y rate prior USD SWAP 5Y rate estimated normal fit yr swap yr swap
26 Market distr. X1 X 2 Focus Views 2-yr swap rate 5-yr swap rate V X 1 { V} σ mv { } mv { } bp 2 prior USD SWAP 2Y rate prior USD SWAP 5Y rate estimated normal fit yr swap yr swap
27 Market distr. X1 X 2 Focus Views 2-yr swap rate 5-yr swap rate V X 1 { V} σ mv { } mv { } bp 2 prior USD SWAP 2Y rate prior USD SWAP 5Y rate estimated normal fit yr swap yr swap Posterior 0 posterior USD SWAP 2Y rate 5 correct normal fit yr swap
28 Market distr. X1 X 2 Focus Views 2-yr swap rate 5-yr swap rate V X 1 { V} σ mv { } mv { } bp 2 prior USD SWAP 2Y rate prior USD SWAP 5Y rate estimated normal fit yr swap yr swap Posterior 0 posterior USD SWAP 2Y rate posterior USD SWAP 5Y rate 5 correct normal fit yr swap yr swap
29 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification partial distribution specification Posterior least distance from prior, views satisfied
30 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification partial distribution specification Posterior least distance from prior, views satisfied Confidence multi-user, multi-confidence
31 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification partial distribution specification Posterior least distance from prior, views satisfied Confidence multi-user, multi-confidence 100(1-c) % of times: PRIOR
32 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification partial distribution specification Posterior least distance from prior, views satisfied Confidence multi-user, multi-confidence 100c % of times: POSTERIOR
33 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification partial distribution specification Posterior least distance from prior, views satisfied Confidence multi-user, multi-confidence Pricing delta/gamma/vega, full pricing,
34 Market distr. Focus Views not returns, not normal, not equilibrium non-linear functions and external factors full distribution specification partial distribution specification Posterior least distance from prior, views satisfied Confidence multi-user, multi-confidence Pricing delta/gamma/vega, full pricing, Optimization mean-variance, mean-cvar,
35 ENTROPY POOLING Theory ENTROPY POOLING Implementation ENTROPY POOLING Applications REFERENCES CONCLUSIONS
36 Implementation Entropy Pooling Parametric Non-Parametric (Fully Flexible Probabilities)
37 Implementation Entropy Pooling Parametric Non-Parametric (Fully Flexible Probabilities)
38 Applications Entropy Pooling Parametric Non-Parametric (Fully Flexible Probabilities) Bayesian networks Panic copulas - Sort portfolios - Implied expected returns Fuzzy Membership State Conditioning Black-Litterman ++ Kernels Crisp Conditioning Time Conditioning 86
39 Analytical implementation Market distr. Focus Views Posterior Confidence Pricing Optimization 87
40 Analytical implementation Market distr. Focus Views Posterior Confidence Pricing Optimization 88
41 Analytical implementation Market distr. Focus Views Posterior Confidence Pricing Optimization 89
42 Analytical implementation Market distr. Focus Views Posterior Confidence Pricing Optimization 90
43 Analytical implementation Market distr. Focus Views Posterior Confidence Pricing Optimization 91
44 Analytical implementation Market distr. Focus Views Posterior Confidence Pricing Optimization 92
45 Non-parametric implementation Parametric Distributions Entropy Pooling Fully Flexible Probabilities joint scenario of N risk drivers probability of joint scenario 93
46 Non-parametric implementation Parametric Distributions Entropy Pooling Fully Flexible Probabilities Parametric Entropy Pooling Non-Parametric Entropy Pooling 94
47 Non-parametric implementation Parametric Distributions Entropy Pooling Fully Flexible Probabilities Parametric Entropy Pooling Non-Parametric Entropy Pooling Bayesian networks Panic copulas - Sort portfolios - Implied expected returns Fuzzy Membership State Conditioning Black-Litterman ++ Kernels Crisp Conditioning Time Conditioning 95
48 Non-parametric implementation probability = 1 / num scenarios prior 96
49 Non-parametric implementation probability < 1 / num scenarios probability = 1 / num scenarios probability > 1 / num scenarios prior bearish bullish
50 Non-parametric implementation probability = 1 / num scenarios regular market 98
51 Non-parametric implementation probability < 1 / num scenarios probability = 1 / num scenarios probability > 1 / num scenarios regular market low volatility high volatility
52 Non-parametric implementation probability = 1 / num scenarios market distribution 100
53 Non-parametric implementation probability = 0 probability = 1 scenario analysis market distribution 101
54 Market distr. Non-parametric implementation 102
55 Market distr. Non-parametric implementation joint scenario of N risk drivers Probability of joint scenario 103
56 Market distr. Non-parametric implementation joint scenario of securities prices Pricing Probability of joint scenario
57 Market distr. Non-parametric implementation Pricing Optimization 105
58 Market distr. Non-parametric implementation Focus Views Posterior Confidence Pricing Optimization 106
59 Market distr. Focus Views Non-parametric implementation scenario index Posterior Confidence Pricing Optimization 107
60 Market distr. Focus Non-parametric implementation Views X 1 e.g. 2-yr swap rate X 2 5-yr swap rate V X 1 mv µ { } J j= 1 µ V p µ j j Posterior Confidence Pricing Optimization 108
61 Market distr. Focus Views Non-parametric implementation Posterior 109
62 Market distr. Focus Views Non-parametric implementation Posterior 110
63 Market distr. Focus Views Non-parametric implementation Posterior 111
64 Non-parametric implementation Posterior Dual formulation: linearly constrained convex optimization in # variables = # views 112
65 Market distr. Focus Views Non-parametric implementation Posterior Confidence Pricing Optimization 113
66 Market distr. Focus Views Non-parametric implementation Posterior Confidence Pricing Optimization 114
67 Market distr. Focus Views Non-parametric implementation Posterior Confidence Pricing Optimization 115
68 Market distr. Focus Non-parametric implementation Views Posterior Confidence Pricing Optimization 116
69 ENTROPY POOLING Theory ENTROPY POOLING Implementation ENTROPY POOLING Applications REFERENCES CONCLUSIONS
70 Case study: portfolios from sorts S10 S9 9 S8 8 S7 7 S6 6 S5 5 S4 4 S3 3 Dell 2 IBM 1 expected returns expected returns no views expected return portfolio weights efficient frontier frontier no views S10 S7 S9 S4 S6 Dell S8 S3 S5 IBM volatility 118
71 Case study: portfolios from sorts S10 S9 9 S8 8 S7 7 S6 6 S5 5 S4 4 S3 3 Dell 2 IBM 1 expected returns expected returns no views expected return portfolio weights Dell IBM S7 view: E{ R } E{ R } frontier no views S4 S3 efficient frontier volatility S6 S5 S8 S9 S10 119
72 Case study: portfolios from sorts S10 S9 9 S8 8 S7 7 S6 6 S5 5 S4 4 S3 3 Dell 2 IBM 1 S10 S9 9 S8 8 S7 7 S6 6 S5 5 S4 4 S3 3 Dell 2 IBM 1 expected returns expected returns no views expected return expected returns expected returns after views expected return portfolio weights portfolio weights view: E{ R } E{ R } Dell IBM Dell S7 IBM S7 S4 S3 S3 efficient frontier frontier no views volatility S6 S5 efficient frontier after views frontier S6 S volatility S9 S8 S9 S8 S10 S10
73 Case study: portfolios from sorts Views: Sample means Standard ranking Centroid Entropy Pooling AA CAT BA KO DIS HD PG UTX VZ MCD KFT MMM GE DD T IBM AXP JNJ WMT CSCO XOM BAC INTC MSFT MRK CVX HPQ PFE TRV JPM Prior [% p.a.] Common [% p.a.] AC [% p.a.] EP [% p.a.]
74 Case study: implied expected returns market capital. weights sample means Black-Litterman implied exp. returns Entropy Pooling implied exp. return XOM WMT VZ UTX TRV T PG PFE MSFT MRK MMM MCD KO KFT JPM JNJ INTC IBM HPQ HD GE DIS DD CVX CSCO CAT BAC BA AXP AA W i ht [%] P i [% ] BL [% ] EP [% Meucci ] 2012
75 Case study: implied expected returns market capital. weights sample means Black-Litterman implied exp. returns Entropy Pooling implied exp. return XOM WMT VZ UTX TRV T PG PFE MSFT MRK MMM MCD KO KFT JPM JNJ INTC IBM HPQ HD GE DIS DD CVX CSCO CAT BAC BA AXP AA W i ht [%] P i [% ] BL [% ] EP [% Meucci ] 2012
76 Case study: implied expected returns market capital. weights sample means Black-Litterman implied exp. returns Entropy Pooling implied exp. return XOM WMT VZ UTX TRV T PG PFE MSFT MRK MMM MCD KO KFT JPM JNJ INTC IBM HPQ HD GE DIS DD CVX CSCO CAT BAC BA AXP AA W i ht [%] P i [% ] BL [% ] EP [% Meucci ] 2012
77 Case study: implied expected returns 10 Expectation -covariance ellipsoids Sample BA 0 DD 0-5 Black-Litterman AXP Entropy-Pooling AA 10 5 CAT BAC CSCO CAT 126
78 Case study: distributional stress-testing Prior (exponential time decay) State indicator (2yr swap rate - current level) time State indicator (1->5yr ATM implied swaption vol - current level) Membership (pseudo Gaussian kernel) Posterior (Entropy Pooling mixture) time 128
79 Case study: distributional stress-testing Prior (exponential time decay) State indicator (2yr swap rate - current level) time State indicator (1->5yr ATM implied swaption vol - current level) Membership (pseudo Gaussian kernel) time Posterior (Entropy Pooling mixture) time time 129
80 Case study: distributional stress-testing Prior (exponential time decay) State indicator (2yr swap rate - current level) State indicator (1->5yr ATM implied swaption vol - current level) Membership (pseudo Gaussian kernel) Posterior (Entropy Pooling mixture) time 130
81 Case study: distributional stress-testing P&L scenarios time P&L distribution P&L ex-ante performance statistics 131
82 Case study: panic copulas mixture market i) calm market ii) panic market homogeneous (high) panic correlations
83 Case study: panic copulas mixture market iii) panic triggers a-la Merton panic threshold i) calm market ii) panic market homogeneous (high) panic correlations
84 Case study: panic copulas mixture market iii) panic triggers a-la Merton panic threshold i) calm market ii) panic market Entropy-Pooling risk premium adjustment homogeneous (high) panic correlations
85 Case study: panic copulas
86 Case study: option trading view: G 6m G 2m 147
87 Case study: option trading view: G 6m G 2m
88 Case Study: Fully Flexible Bayesian Networks (FFBN) Spreads Country default Rates
89 FFBN prior: our best guesstimate
90 FFBN structured prior as Bayesian network
91 FFBN stress-test: conditional statements
92 FFBN consistency check for conditional statements
93 FFBN posterior: Entropy Pooling
94 FFBN posterior: Entropy Pooling
95 FFBN posterior: Entropy Pooling
96 FFBN example: the market
97 FFBN example: the risk drivers
98 FFBN example: the frequentist prior Prior probabilities
99 FFBN example: stress-testing
100 FFBN example: stress-testing
101 FFBN example: stress-testing Posterior probabilities
102 FFBN example: stress-testing Prior correlations Posterior correlations
103 ENTROPY POOLING Theory ENTROPY POOLING Implementation ENTROPY POOLING Applications REFERENCES CONCLUSIONS
104 References 151
105 References 152
106 References 153
107 References 154
108 References 155
109 References 156
110 Entropy Pooling Meucci (2008) References 157
111 Entropy Pooling Meucci (2008) References Parametric Non-Parametric (Fully Flexible Probabilities) Meucci (2012) - Sort portfolios - Implied expected returns Meucci, Nicolosi (2015) Multi-Horizon portfolio management Meucci, Ardia, Colasanteee (2011) Meucci (2010) Bayesian networks Panic copulas Fuzzy Membership State Conditioning Meucci (2011) Meucci (2011) Black-Litterman ++ Meucci (2008) Kernels Crisp Conditioning Meucci (2010) Time Conditioning 158
112 ENTROPY POOLING Theory ENTROPY POOLING Implementation ENTROPY POOLING Applications REFERENCES CONCLUSIONS
113 Conclusions Entropy Pooling: Market represented by generic non-linear risk factors, not only returns Market distribution fully general, not only normal Views/stress-testing on any function of the market, not only linear portfolios Views on any feature, not only on expectations: median, volatility, correlations, tails Views are equalities and inequalities: ranking is possible Applications span several areas of portfolio management and risk management: Fully Flexible Probabilities for stress-testing Fully Flexible causal inputs/bayesian networks for stress-testing Panic copulas True implied expected returns Market-aware portfolios from sorts and much more 159
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