Risk ManagementOptimizing copula structure

Richard Schneider, Northwood Actuarial

Questions and Disclaimer

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➢ Disclaimer: Any views expressed in this presentation are those of the presenter and not necessarily of the presenter’s employer(s) or NoCA. The information contained in this presentation is of a general nature and whilst it is intended to be accurate there is no guarantee that such information is accurate. No representation or warranty is given as to the accuracy or completeness of the information contained in this presentation.

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Contents

➢ Copula 101

➢ Flexible structures

➢ Financial crisis

➢ Solvency II

➢ Hedging

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Copula 101➢ Mathematically: function from [0,1]n → [0,1] with various properties

➢ Statistically: multivariate distribution with Uniform marginals

➢ Marginals are in turn distribution functions of underlying RVs

⇒ Marginal cdf’s + copula → joint cdf

➢ C(u1,u2,...,un) = Pr (X1 <= F1-1(u1) , X2 <= F2

-1(u2) , … , Xn <= Fn-1(un))

➢ Defines dependency structure between RVs

➢ Sklar, 1959: Can always decompose multivariate into marginals + copula

C is unique when marginal are continuous

➢ First finance applications circa 2000, e.g. CDO pricing. Nowadays:

Banks Insurers

Derivative pricing ✓ -

Portfolio optimisation & hedging ✓

Regulatory capital ✓ ✓

ERM ✓ ✓

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Copula 101 – simple example

➢ How prudent is prudent?

➢ Insurance policy, valuation based on mortality and expense assumptions, increasing in both

➢ Historically valuation margins have been set at 75th percentile per assumption, say

Val = h (mort, exp); mort ~ F; exp ~ G

Val_base = h (m, e); F(m) = G(e) = 0.5

Val_prud = h (m’, e’); F(m’) = G(e’) = 0.75

… but how prudent is the combined stress? Need to know dependency structure, i.e. C(F,G). Then:

Option 1: Simulate the distribution of V using c, read off V-1(h (0.75, 0.75))

Option 2: Calculate C(0.75,0.75) directly → relies on (positive) linearity of h

➢ Comonotonic: C(u1, u2) = min (u1, u2)

➢ Countermonotonic: C(u1, u2) = max (u1+u2–1, 0)

➢ Now Pr (m>0.75, e>0.75) = 1 – C(0.75, 1) - C(1, 0.75) + C(0.75, 0.75) = C(0.75, 0.75) - 0.5

Comonotonic: 0.25

Gaussian (.25): 0.0893

Independent: 0.0625

Countermonotonic: 0

FH Bounds

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Copula 101 - “traditional” families

Elliptical (implicit)

➢ Gaussian: C(u1,u2,…,un) = Φ(Φ-1(u1), Φ-1(u2),…, Φ-1(un))

➢ t: C(u1,u2,…,un) = t(t-1(u1), t-1(u2),…, t-1(un))

Archimedean (explicit)

➢ C(u1,u2,…,un) = ψ(ψ-1(u1)+ψ-1(u2)+…+ψ-1(un))

➢ ψ : generator function, e.g. Gumbel: exp(-x1/α), Clayton: (1+ αx)-1/ α

Generic properties (bivariate case)

➢ Radial symmetry: C(u1,u2) = Cs(u1,u2) = C(1-u1,1-u2) + u1 + u2 –1

➢ Exchangeability: C(u1,u2) = C(u2,u1)

➢ Lower tail dependence: lim 𝑢 ↓ 0 C(u, u) / u

➢ Upper tail dependence: lim 𝑢 ↑ 1 (C(u, u) – 2u + 1) / (1-u)

➢ “Arachnitude” (Shaw et al, 2010): ⍴((2F1-1)2, (2F2-1)2)

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Aside: multivariate time series

➢ Conditional distribution: F(X1t, X2t, …, Xnt | ℱt-1) = C(F1(X1t | ℱt-1), F2(X2t | ℱt-1), …, Fn(Xnt | ℱt-1))

➢ e.g. Copula GARCH model: ℰit = ℴit.Wit ℴ2it = µ + ∑⍺ ℰ2

i,t-j + ∑β ℴ2i,t-j

➢ Conditional copula for Wt e.g. Gaussian: C(Φ(W1t | ℱt-1 ), … , Φ(Wnt | ℱt-1 ))

… although Gaussian doesn’t capture asymmetric dependence, e.g. ‘flight to quality’ or asymmetric central bank response to currency movement

Dynamic properties of model

➢ Dynamic volatility (AR, regime-switches, full stochastic, etc), what about dynamic dependence?!

➢ Correlations can change outside of tails too! e.g. gradual increase in co-movement between GBP and USD interest rates from c 1990 onwards

➢ ESG options (copula or not): (1) ignore, or (2) model directly, or (3) model via hidden drivers, or (4) capture via cascade → purpose?

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Flexible structures for t copula

Issue 1: “shape exchangeability”, i.e. 1 df parameter controlling shape of dependence for all risk pairs

⇒ Individuated t (IT) copula, allows 1 df parameter for each RV

➢ Stochastic representation, hence simulation algorithm, similar to standard t but with n inverse Gammas:

ui = tdf(i) ( invᴦ (dfi / 2, dfi / 2)1/2 . (Az)i ) i=1…n A = Cholesky lower

➢ Halfway house is Grouped t: n → n1,…,nk

➢ e.g. n1 = {insurance & operational risks}, n2 = {market & credit risks}

→ improves regulatory risk margins / risk adjustments

Issue 2: symmetry

⇒ Skewed t copula, introduces skewness parameter

➢ Simulation again similar

+ -

External stakeholders

Calibration

Complexity

Risk management

Reporting accuracy

Flexibility

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Flexible structures for Archimedean copulae

Issue 1: Direction of tail dependence

⇒ Rotations:

0 deg: CUU α = 1 / (1 - τ) standard Gumbel

90 deg: CLU α’ = -1 / (1 + τ’)

180 deg: CLL α = 1 / (1 - τ) “survival Gumbel”

270 deg: CUL α’ = -1 / (1 + τ’)

τ: Kendall’s tau

Issue 2: Exchangeability

⇒ Hierarchical structure:

e.g. 2 layers, k+1 copulae: C(u1, … , un) = Couter (Cinner,1(u1,…, un(1)), … , Cinner,k (un(k-1)+1,…,un))

e.g. 3 layers, 7 copulae:

Credit portfolio

UK

Financials Industrials

US

Financials Industrials

NB: compatibility conditions

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Flexible structures: vinesPairwise copula construction

➢ Copula density: ∂C(u1,…,un)/∂u1… ∂un = c(u1,…,un)

⇒ f(x1,…,xn) = ∂F(x1,…,xn)/∂x1… ∂xn = ∂C(F1(x1),…,Fn(xn))/∂x1… ∂xn

= c(F1(x1),…,Fn(xn)). ∏fi(xi)

➢ fj|i = fji / fi = cji . fj . fi / fi = cji . fj

⇒ f1,…,n = f1.f2|1.f3|12…fn|1…n-1 = f1.(c21.f2).(c31|2.f3|2)……

= f1.(c21.f2).(c31|2.f3.c32)…… = ∏fi . ∏ ⊂ cji . ∏ ⊂ cji|klm…

Systemic risk example

➢ Brechmann et al (2013): interconnectedness of banks / insurers across EU / US / AP

➢ CDS spread model calibrated to 2006-11 data. Range of marginal and copula time series model tested

➢ GARCH errors, marginals fitted to normal, inverse Gaussian & GH

➢ C Vine > Gaussian > t > Gumbel ranked by lowest BIC; Vine parameters << Gaussian after independence tests

➢ Vine: Mix of Gaussian, t, Gumbel, Clayton, Frank + rotations

➢ Systemic stress tests, conditional simulations. Key findings:

• EU banks / insurers → EU insurers / banks > EU banks / insurers → US banks / insurers

• US banks / insurers → US insurers / banks ≈ US banks / insurers → EU banks / insurers

14|23

13|2

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1 2

24|3

23 34

3 4

unconditional c’s

marginals

conditional c’s

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Was the Gaussian Copula responsible for the financial crisis?!

➢ Li (2000) proposed a Gaussian Copula model for pricing CDOs.

➢ Two variants: Structural models (copula captures dependency between underlying firms’ asset values) and default intensity models (models defaults times directly, by constructing cdf of marginal default times and joining with copula)

➢ “The formula that killed Wall Street” - blamed by some for leading to incorrect pricing and ratings. Counter-argument – model used more widely than it was understood and warnings not heeded.

➢ Yoshiba (2016) illustrates the risk in relation to copula selection. Large variation in tranche spreads based on a hypothetical CDO portfolio (τ = 10% in all cases):

source: T Yoshiba, B of J (2016) “Risk aggregation with copula for banking industry”

Gaussian Copula model risk

➢ Asymptotically tail-independent, so underestimates joint early defaults.

➢ Implied correlations differ by tranche (analogous to volatility smile in Black Scholes)

➢ Tail dependence does not imply that defaults occur simultaneously (intensity models).

➢ Gaussian Copula still in widespread use in banking, owing to computational simplicity and a ‘common language’ for pricing. But…

• More conservative assumptions, e.g. stressed correlations

• Modifications to the model, e.g. stochastic recovery rate

➢ Some banks now using hierarchical Archimedean or vines for ERM

Equity (0-6%) Mezzanine (6-18%) Senior (18-36%) Super senior (36–100%)

Gaussian 1147 bps 63 1 0.00

t (6 df) 900 128 9 0.04

t (3 df) 736 165 22 0.20

Rotated Gumbel 1018 59 19 2.69

Clayton 861 136 13 0.10

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Solvency IIStandard Formula

➢ Uses hierarchical “Stress Test & Correlation” approach. Based on Multivariate Normal theory, in particular linear correlations.

• Issue 1: Correlations between capital amounts, not risk drivers. Implicitly assumes linearity of loss function. Invalidated when distributions transformed,

e.g. CEIOPS L2 advice example of two independent risks, both capped at 98%ile loss, back-solved correlation of 45%.

• Issue 2: Linear correlation invalidated by skew, kurtosis in marginal

• Issue 3: Ignores tail dependence!

⇒ Correlations uplifted, to achieve best approximation to aggregate 99.5%ile VaR… but can only be industry average.

Internal Models

➢ UK survey data / SFCRs suggests widespread use of Gaussian Copula (or non-hierarchical ST&C) albeit with stressed correlations, significant minority using t-copula (possibly one firm using IT). Makes sense because:

▪ Easy to implement and simulate from (esp if only options in software package)

▪ Heterogeneous risks, correlation matrix gives control. Unlikely to model individual assets.

➢ PRA Jan 2016 letter on QI’s has the following to say about Gaussian Copulae:

The PRA has developed its QIs on the basis of a simple dependency structure based on the use of a Gaussian copula parameterised via the analysis of correlations between risks…

…The PRA’s approach should not be seen as an endorsement of a particular methodology; in practice it has reviewed and approved models that use a variety of dependency structures. The extensive modelling freedom afforded by Solvency II allows the choice of simple models of dependency provided that appropriate adjustments are made such that the ultimate capital requirement corresponds to the standard set out in ‘Solvency Capital Requirement – General Provisions 3.4’ of the PRA Rulebook. Given the difficulty in specifying the dependency relations between multiple risks across the whole distribution the PRA has viewed all dependency structures as pragmatic approximations valid for modelling in specific contexts, provided firms can demonstrate that they understand the limitations and, where appropriate, mitigate them…

…Two particular areas where the Gaussian copula was deemed to be too simplistic and requiring adjustments were:

a. Allowance for lack of diversification in stressed conditions (also known as ‘tail dependency’ allowance). For example, where data analysis is used in the QI derivation, the QI calibration is generally stronger than would simply be obtained by calculating the mathematical rank correlation between two historical data series. The PRA’s overall data analysis has involved considering the results of a number of statistical tools such as time series plots, scatter plots, rolling correlations over the data period, historic correlation, confidence intervals, goodness of fit tests and the coefficient of finite tail dependence observed in the data.

b. Asymmetric relations: the real-world dependency relation between two risks will not generally depend on the exposure of an individual insurer to those risks. However, where simplifications are adopted to describe a complex dependency relation by means of simple dependency structures (eg a Gaussian copula) it may be necessary to adjust the correlations depending on exposures to ensure the system used to measure diversification benefits captures the right dependency characteristics (eg reduced diversification in extreme scenarios compared to benign scenarios).

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Gaussian with stressed correlations

Issue 1: Tail correlations

➢ Dependency of P1, P2 follows t copula, but modelled as Gaussian.

➢ How far into the tail to target?

L(P1,P2) = P1 + P2

Issue 2: “Asymmetry”

➢ Change loss distribution via hedge / reinsurance

L(P1,P2) = P1 + min(P2, cap) + prem

U CFTD: t (6df, 𝜌=.25)

CFTD: Gaussian (𝜌=.25)

Implied 𝜌to equate

0.9 0.238 0.193 0.34

0.95 0.186 0.122 0.40

0.995 0.116 0.028 0.54

lim 𝑢 ↑ 1 0.080 0 N/A

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Hedge effectiveness – single risk

Single risk exposure: truncate distribution through hedge / reinsurance

➢ Prem (seller) = EQ(L-min(L, cap)) or EP(L-min(L, cap)) + margin

➢ PV CoC (buyer) = CoC% x PV (VAR.995 - min(VAR.995 , cap)) x (1+respectability margin)

➢ Prem < PV CoC → Proceed to DD stage 2

➢ Arbitrages model differences and leverages counterparty diversification benefit

Multiple risk exposures: truncate single risk

➢ PV CoC benefit reduction for diversification

➢ High tail dependence improves value for money / copula determines optimal quantile of cap

➢ … and ECap could take it further into the tail!

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Hedge effectiveness – multi-risk

Truncate multiple risks, separate structures

➢ Is optimization done holistically? e.g. annuity with future investment at risk-free

➢ optimal hedge (longevity, IR) ≠ optimal hedge (longevity, 0) + optimal hedge (0, IR)

… even ignoring cross-terms

➢ “Perfect hedge”: 100% longevity swap, matched assets (physical + IR swaps) → £££

➢ Alternative: retain 100% longevity… what duration to match? Swaptions? → £££

➢ Alternative: hedge longevity tail… what duration to match? Where is ‘sweet spot’?

Truncate multiple risks, single structure. Examples:

➢ 1. Multi-name CDS. Regulatory model has full dependence within each risk node (e.g. UK, Financial, BBB), bank has multi-name copula.

→ may look good in regulatory model, but bad deal.

➢ 2. ‘Asset share’ put. Regulatory model has average tail dependency across all risks (market & insurance), bank has tailored copula.

→ may look bad in regulatory model, but missed opportunity.

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Stochastic IR

Optimal hedge ratio

Minimum variance hedge (or mean-variance utility maximization with martingale assumption)

argminh var (ds – h.df) = E(ds)2 – 2h.ds.df + E(df)2

⇒ h = covds,df / vardf = ⍴ds,df . ℴds/ℴdf

… but traditional model has constant h and based on bivariate normal returns

Stage 1

c1980’s

•ARCH / GARCH

•Constant (conditional) correlations

•Focus still on direct hedging i.e. spot with own futures

•Multivariate N or t

•Dynamic variance ✓

Stage 2

c1990’s

•Widespread use of GARCH

•Dynamic (conditional) correlation structure

•Extension to cross hedging (finance & agriculture)

•Dynamic dependence ✓ (less important for direct hedging)

•Applicability ✓

Stage 3

c2000’s

•Copula GARCH (Patton, 2006)

•Dynamic Copula GARCH

•Asymmetric dependence ✓

Hsu et al (2008): direct hedging of FTSE 100 & S&P 500 with own

futures. Cross-hedging of foreign (Swiss) equity index with currency

futures. Compares hedge performance across traditional, GARCH

(CCC), GARCH (DCC), Dynamic copula GARCH (Gaussian,

Gumbel, Clayton). Skewed t marginals. Of the copula models:

➢ Gaussian performed best for direct hedges

➢ Gumbel performed best for cross-hedge

➢ Copula mostly outperformed stage 1&2 models

Wu (2018): cross-hedging of sorghum (no futures market) with corn

futures. Compares traditional, GARCH (CCC), GARCH (DCC), Static

copula GARCH (Gaussian, t, Clayton, Gumbel, Frank), Dynamic

copula GARCH (Gaussian, Gumbel). Normal and t marginals.

➢ Clayton performed best.

➢ Dynamising copulas didn’t help - possibly not capturing correct

source of variability (for copula parameter).

➢ GARCH (DCC) performed relatively well.

➢ Traditional performed poorly for OOS data.

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Cross-hedging case study: Property

➢ Large increase in UK life insurer exposure to property, direct commercial & indirect residential

➢ UK: IPD index TR futures / swaps - commercial property. Traded on Eurex. Low liquidity since crisis.

➢ US: Case-Shiller price index futures & options– residential property. Traded on CME. 2995 trades (2007), 136 (2017)

➢ Mostly for efficient portfolio management, not ERM. IPF, 2018:

The application of property derivatives for the purpose of ERM has, surprisingly, been largely overlooked.

This is particularly puzzling given how well real estate has performed in recent years, relative to other asset classes.

Property futures can be used to manage one’s overall property exposure to real estate, and, by extension,

the associated regulatory capital.

So why aren’t life insurers doing more to hedge property risk?

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Life co.'s seek pickup

ERM sales (£bn) 10yr gilt (%)

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IPD outstanding notional (£bn’s)

• No-arbitrage arguments don’t work

• Volatility

• Banks’ reluctance

• Dynamic hedging (delta and / or dynamic dependence)

• Collateral

• Idiosyncratic risk dominates, large exposure (£bn’s) needed to diversify and provide hedge-effectiveness

• Regional dynamics

• UK commercial / US residential vs UK residential

• Timing differences, e.g. IPD futures 5yr max duration

• TR vs price

• IPD: appraisal values Basis Risk

Portfolio Size

PriceLiquidity

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Cross-hedging case study: Property

What if property markets were (mostly) complete?

➢ Buy and sell delta and vega, directly or through bank

➢ Arbitrage-free futures prices, unique EMM and standard option pricing techniques

➢ Capital requirement reduced to hedging error and other basis risks

… but they aren’t

➢ Indifference pricing, MPR from ‘similar’ market, probability transforms, etc to narrow price range

➢ Capital requirements highly sensitive to model

Cross-hedge using other markets?

➢ Liquidity vs basis risk.

➢ Positive HPI / CPI correlation?

➢ Depends on horizon, realized vs forecast inflation, data period, lag, granularity, etc

➢ Hinkelman & Swidler (2008) performed regression on US HPI & several traded futures

(CPI, FX, equities, commodities) to find possible hedge portfolio

→ largely ineffective, with CPI contribution insignificant (Live Hogs performed better!)

What if we found a correlated asset?

➢ Probability of hedge working when needed?

➢ Does existing copula give too little / too much benefit? Copula sensitivities as part of DD.

➢ Regulatory (and internal) approval for capital reduction.

➢ Sufficient “arachnitude”?

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Annual inflation vs house prices (UK)

UK CPI Nationwide HPI