ACTUARIAL DERIVATION OF THE COST OF COLLATERALISED LIABILITIES – WITH A MODEL OF EQUILIBRIUM IN A GENERALISED CLASS OF SECURED FUNDING AGREEMENTS

Fernando MIERZEJEWSKI, PhD Risk Officer AG Insurance

4th European Actuarial Journal Conference Leuven – 10 September 2018

Disclaimer

  The views and opinions presented hereafter are those of the author – and presenter – and do not necessarily reflect the official policy or position of AG INSURANCE Belgium.!

  The analysis and results next presented follow from theoretical investigations developed independently by the author – with no involvement or collaboration of any employee of AG INSURANCE. The author thus takes full liability and responsibility for the content of this presentation.!

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FINANCIAL TIMES – 31 AUG 2018

Have we learnt the Lessons of the Financial Crisis?!

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FINANCIAL TIMES – 31 AUG 2018 Gillian Tett – US Managing Editor

  By the autumn of 2008, a slow-burn crunch had turned into a full-blown global crisis, epitomised by the dramatic collapse of Lehman Brothers and rescue of AIG.!

  The US 2007-08 crisis was so big that it raised public debt by 24 per cent of gross domestic product.!

  " Ten years since the Lehman Collapse, the questions are still pressing: !  Why do we appear destined to suffer crises over and over

again? Why canʼt we learn from the past?!

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FINANCIAL TIMES – 31 AUG 2018 Gillian Tett – US Managing Editor

  " Financial crises share two things:!  The pre-crisis period is marked by hubris, greed, opacity

— and a tunnel vision among financiers that makes it impossible for them to assess risks. !

  When the crisis hits, there is a sudden loss of trust, among investors, governments, institutions or all three. !

  If you want to understand financial crises, then, it pays to remember that the roots of the word “credit” comes from the Latin “credere”, meaning “to believe”: !

  Finance does not work without faith!

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US FED Authorities

  " Alan Greenspan – chairman of the US Federal Reserve from AUG 1987 until JAN 2006: "  Fundamentally monetarist.!  Pursued a series of cuts in the US FED Fund Rate from a

level of about 6% in 2001/Q1 until an all-time minimum of 1% by 2004/Q4.!

  Ben Bernanke – chairman of the US Federal Reserve from FEB 2006 until JAN 2014:"  Problems in the subprime mortgage market can be regarded

as “limited”, and thus unlikely to create any “significant spillovers”.!

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US FED Fund Rate – 1989/Q1 to 2004/Q4

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Alan Greenspan – OCT 2008 Testimony to the Committee of Government Oversight and Reform

  We are in the midst of a once-in-a century credit tsunami. !  Central banks and governments are being required to take

unprecedented measures.!  In 2005, I raised concerns that the protracted period of

underpricing of risk […]. !  This crisis, however, has turned out to be much broader than

anything I could have imagined.!

  … those of us who have looked to the self-interest of lending institutions to protect shareholderʼs equity are in a state of shocked disbelief. !  Such counterparty surveillance is a central pillar of our financial

marketsʼ state of balance. !  If it fails, as occurred this year, market stability is undermined.!

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Classical Financial Economics

  Efficient Market Hypothesis (Samuelson, Fama) :"  Under conditions of perfect competition, financial assets are always

traded at fair value, making it impossible for investors to make profits by either buying undervalued stocks, or selling overpriced securities.!

  In harmony with the Chicago Schoolʼs faith in free and efficient markets.!

  Paradigm affirming – even if fundamentally unverifiable. !

  Option Pricing Theory (Black & Scholes, 1973; Merton 1974)."

  Irrelevance of the Capital Structure (Modigliani & Miller, 1954)."

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Market Imbalances as the source of Market Instability

  Firms can rely on secured funding in markets where the supply of and the demand for assets are perfectly coordinated."  Where borrowers and lenders share expectations about the future

performance of the portfolio of assets held at the aggregate level.!  Discrepancies in the arrival of buying and selling orders

may cause undesired accumulations of assets holdings – leading to lowering assets prices and collateral value."  Creditors will likely increase the levels of credit premiums and

margin requirements under such circumstances, making it more difficult for firms to issue liabilities, and ultimately giving rise to sudden contractions in the level of assets spending.!

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Structured Finance Dynamic Portfolio Strategies

  Structured finance comprises a broad class of financial products allowing firms to fund their investing portfolios by combining assets holdings, cash stocks, and option contracts. !

  Constant Proportion Portfolio Insurance (CPPI)."  Guarantee a minimum level of terminal wealth at the end of a

pre-determined time horizon.!  Constant Proportion Debt Obligations (CPDO). "

  Guarantee high yields to investors.!  Investors are committed to fully afford the risk of default.!

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CPDO Structure FITCH Report – Linden et al. (2007)

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CPPI & CPDO as Opposite Dynamic Strategies FITCH Report – Linden et al. (2007)

  CPPI & CPDO Credit Notes:"  Principal is protected by a low-risk portfolio of cash and cash

equivalents.!  Return is increased by investing in a risky CDS portfolio.!

  Managers of CPPI contracts must diminish leverage in response to widening CDS spreads: "  Ensures that principal is returned at maturity.!

  Managers of CPDO notes increase leverage in response to widening CDS spreads:"  Leverage is subject to an upper bound.!  Discounted or no principal may be returned at maturity.!

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  CONSTANT PROPORTION PORTFOLIO INSURANCE (CPPI)

  BLACK MONDAY – 19 OCT 1987

Black Monday – 19 OCT 1987 Matt Maley ex-Salomon (CNBC 16 OCT 2017)

  Portfolio insurance became quite popular that year [1987] […] thus, the "hedge" was popularized.!  Institutions who bought the product engaged in an agreement to sell

short S&P 500 futures if the stock market fell by a certain amount.!  Before 1987, if investors began selling aggressively "into a

falling market," it's because they had no choice. !  They were getting margin calls and they had to sell. !

  With portfolio insurance, these people did not have to "sell" to raise money. !  They were simply contractually obligated to "sell into a falling

market" due to their portfolio insurance agreements.!  The problem came when investors from several different

areas "had to sell" at the same time, with each obligation further exacerbating the situation.!

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Utility-Maximising CPPI Black & Perold (1992)

  The best portfolio insurance strategy can be found by solving for the intertemporal investment-consumption rules that maximise expected utility."  This can be done under fairly standard assumptions of

frictionless markets and no borrowing restrictions."  CPPI invests a constant multiple of a cushion in risky

assets up to a borrowing limit – where the cushion is the difference between wealth and the specified floor."  In the absence of borrowing constrains and transaction costs,

CPPI is a special case of HARA utility-maximising rules."  Black & Perold (1987) demonstrate that the borrowing-

constrained rule is also utility-maximising."

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Effects of Portfolio Insurance Utility-maximisation approach

  A general equilibrium approach is adopted to determine the level of assets prices, both under the presence and the absence of portfolio insurance."

  Thus, both the level of risk premiums and the volatility of assets prices can be proved to be affected by portfolio insurance – i.e. by the levels of assets spending and terminal wealth."

  No consensus regarding the effect of portfolio insurance:"  Increasing assets volatilities and credit premiums (Brennan &

Schwartz, 1989)."  Decreasing assets volatilities and credit premiums (Basak,

1995)."

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CPPI Optimal Leverage Ratio Actuarial Risk Theory

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€

Maxλ

E u X − λ( )+[ ] = u x − λ( )⋅ dxλ

+∞

∫s.t. r0 +CDS( )⋅ λ > E X−[ ]

€

F .O.C. :∂ E u X − λ( )+[ ]

∂λ⋅ u'

⎫ ⎬ ⎪

⎭ ⎪ λ =λ*

− r0 +CDS( ) = 0

€

X = ΔA /A : random proceeds A : assets spendingA⋅ X − λ( ) : cushion L = A⋅ λ : leverage

CPPI Optimal Leverage Ratio Dual Theory of Choice (Yaari, 1987)

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€

Maxλ

Eϕ X − λ( )+[ ] = x − λ( )⋅ ϕ dx( )λ

+∞

∫s.t. r0 +CDS( )⋅ λ > Eϕ X−[ ]

€

F .O.C. :∂ Eϕ X − λ( )+[ ]

∂λ

⎫ ⎬ ⎪

⎭ ⎪ λ =λ*

− r0 +CDS( ) = 0

€

X = ΔA /A : random proceeds A : assets spendingA⋅ X − λ( ) : cushion L = A⋅ λ : leverage

CPPI Optimal Leverage Ratio

  The utility-maximisation approach replicates the classical approach of James Tobin (1958), where the preference for cash balances – i.e. the preference for liquidity – is characterised as behaviour towards risk."

  Yet, utility functions are reflect of the attitude towards payments – rather than probabilities."

  Distorted probabilities under the dual theory of choice explicitly represent the attitude towards risk."

  However, as the stop-loss premium is a convex function of its argument, under the dual formulation of the CPPI Optimal Guarantee problem, the optimal cash balance actually minimises the expected excess of return (Kaas et al., 2008)."

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  CONSTANT PROPORTION DEBT OBLIGATIONS (CPDO)

  GLOBAL FINANCIAL CRISIS 2008

Global Financial Crisis 2008 Gordy & Willemann (2012)

  The global financial crisis of 2008 started in 2007 as a crisis in the subprime mortgage market in the United States, and propagated to the international banking system finishing with the collapse of the investment bank Lehman Brothers in September 2008.!  The issuance of such products as Mortgage-Backed Securities (MBS),

Credit Default Swaps (CDS), Collateralized Debt Obligations (CDO), and Constant Proportion Debt Obligations (CPDO) dramatically increased during the years previous to the crisis.!

  Major weaknesses of CPDO contracts!  High sensitivity to the volatility of the underlying indexes.!  Vulnerable to inversions of the structure of credit spreads – leading to

downward slope.!  Severe spikes in credit spreads can trigger cash-out events – when

leverage reaches the specified upper bound and the contract is defaulted.!

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Spreads for selected financial names 2007 Gordy & Willemann (2012)

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CPDO Optimal Leverage Ratio Dual Theory of Choice (Yaari, 1987)

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€

Minλ

Eϕ X + λ( )−[ ] = x + λ( )⋅ ϕ dx( )−∞

−λ

∫s.t. CDS⋅ λ < Eϕ X+[ ] − NPV CPDO NOTE( )

€

F .O.C. :∂ Eϕ X + λ( )+[ ]

∂λ

⎫ ⎬ ⎪

⎭ ⎪ λ =λ*

− CDS = 0

€

X = ΔA /A : random proceeds A : assets spendingA⋅ X − λ( ) : cushion L = A⋅ λ : leverage

CPDO Optimal Leverage Ratio Proportional Hazard Distortion

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€

λ* CDS( ) = Tϕ ,−X−1 CDS( ) with Tϕ ,−X x( ) = Pϕ −X > x{ }

€

X = ΔA /A : random proceeds A : assets spendingA⋅ X − λ( ) : cushion L = A⋅ λ : leverage€

λ* CDS( ) = T−X−1 CDS( )[ ]θ with ϕθ p( ) = p

1θ ∀p∈ 0,1[ ]

θ : distortion ≡ perception of probabilities

Equilibrium Credit Spread 28

€

L = A⋅ λ s*( ) = A⋅ Tθ ,−X−1 s*( ) = A⋅ T−X

−1 s*( )[ ]θ

L : aggregate sup ply of credit

€

Xi = ΔAi /Ai : random proceeds Ai : assets spending

Ai ⋅ Xi − λi( ) : cushion Li = Ai ⋅ λi : leverage

€

X = ω i ⋅ Xii=1

N

∑ comonotonic dependence structure

X1 ,… ,XN : reference portfolio , ω i = Ai A

Sensitivity of the Market Equilibrium

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€

s* = Tθ ,−X α( ) = P −X > α{ }1θ = P X ≤ −α{ }

1θ = Fθ ,X −α( )

α = L A : fixed leverage policy

€

Δs* ≈ − fθ ,X −α( )⋅ Δα ≈ P −α − c⋅ Δα ≤ X ≤ −α + c⋅ Δα{ }1θ

fθ ,X = dFθ ,X dX : probability density

€

Δs* = −δθLA⎛

⎝ ⎜

⎞

⎠ ⎟ ⋅

ΔLL−ΔAA

⎛

⎝ ⎜

⎞

⎠ ⎟

δθ L A( ) = L A( )⋅ fθ ,X −L A( ) : sensitivity factor

Dependency on Assets Volatility Elliptical Probability Distributions

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Dependency on Assets Volatility Gaussian Distribution – Risk Lovers (th<1)

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Description of Financial Crisis

  Target Leverage Ratio = 1% (one day horizon)."  Assets volatility range:"

  Low daily volatility : approx. 0.50% (normal)"  High daily volatility : approx. 5.00% (crisis)"

  Credit spreads jumps observed in crises:"  From 100 BPS to 900 BPS"

  Such transitions are possible in the model:"  Firms behave as risk-lovers (theta<1.00)."  Diminished sensitivity at higher volatility levels."  Markets show more resistance to adjust in response to

variations in the levels of leverage and assets spending."

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  Thank you for your attention – comments & questions are welcome!!

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