Optimal Securitization via Impulse Control

Optimal Securitization via Impulse Control R¨ udiger Frey (joint work with Roland C. Seydel) Mathematisches Institut Universit¨ at Leipzig and MPI MIS Leipzig

Bachelier Finance Society, June 2010

Bachelier Finance Society, June 2010 (1)

Optimal Securitization via Impulse Control

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Introduction

Overview

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Introduction

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Optimal Securitization Strategy The model Specific model inputs Solution of the Optimization Problem

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Numerical Results



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Introduction

Securitization Consider a commercial bank lending to customers. In a securitization transaction the bank sells part of its loan portfolio to investors in a bond-like form, passing on part of the risk and return. Investor A

Investor B

Customer

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CDO

Investor C

Potential benefits of securitization On the macro level: possibly mitigation of concentration risk and easier refinancing for banks On the micro level securitization can be an important risk management tool for commercial banks (reduction of leverage) Securitization is of course not problem-free but this is not our focus here. Bachelier Finance Society, June 2010 (3)


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Introduction

Securitization ctd. It is not costless to securitize (sell) loans: An ABS transaction typically entails fixed costs: Rating agency fees, legal costs, time spent . . . The more you sell, the lower the price investors want to pay (liquidity and agency problems) Finding a good (optimal) securitization strategy is non-trivial . . . it is not optimal to sell all at once, but rather distributed over time; . . . it is not optimal to securitize all the time, but rather at discrete points in time. Conclusion. Determination of an optimal securitization strategy (for the bank) leads to a a dynamic optimization problem under fixed and variable transaction costs. ⇒ apply impulse control methods. Bachelier Finance Society, June 2010 (4)


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Optimal Securitization Strategy

Overview

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Introduction

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Numerical Results



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The model

The Agents Involved

Bank’s creditor

Customers

Bank

ABS Investors

Bank’s shareholder



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The model

Modeling a commercial a bank ctd Consider a commercial bank solely engaged in lending business. Loan portfolio is homogeneous and loans to customers have maturity ∞ (perpetuities) with nominal 1. Starting point is the fundamental balance sheet equation assets = liabilities cash · 1cash>0 + loans = equity − cash · 1cash 0

Loans

Cash

−Cash

Cash < 0

Equity



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The model

Model dynamics State variables of the model: Nominal loan exposure Lt =: Xt1 . Cash balance Ct =: Xt2 (positive or negative). Note that exposure t = Lt L+C . leverage equals πt = loanequity t Economic state variable Mt =: Xt3 that affects default rates Dynamics of L: dLt = −dNt + βt dPt , where Nt is a point process with state dependent intensity λ(Mt− )Lt− , representing the timing of defaults. (Bottom-up view: defaults are M with intensity λ(M ).) conditionally independent given F∞ t− The stochastic control βt ∈ {0, 1} allows to increase the loan exposure at the advent of a potential customer. Advents of customers are modelled by jumps of the standard Poisson process P. Bachelier Finance Society, June 2010 (8)


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The model

Model dynamics (2) Cash C . Cash position is affected by interest payed or earned on cash position given by rB Ct dt (rB refinancing cost) interest rL Lt dt earned on loan position recovery payments in case of default (1 − δ)dNt (here LGD δ = 1). cash-reduction because of issuing of new loans −βt dPt . Hence we have the following cash-dynamics dCt = (rB (Xt )Ct + rL Lt ) dt + dNt − βt dPt Economic state M. Markov switching process (continuous-time Markov chain) between two economic states, Mt ∈ {0, 1}. Bachelier Finance Society, June 2010 (9)


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The model

The Securitization strategy (impulses) Each securitization of ζi ≥ 0 loans at a stopping time τi has the following effects: ˇ 1 Reduce loan exposure: L τ = Lτ − − ζi . i

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Increase cash by market value η(·) of the amount sold minus fixed costs cf > 0: ˇτ − + η(M ˇ τ − , ζi ) − cf C τi = C i i

In summary, bring the process X = (L, C , M)T from an old state x to the new state Γ(x, ζ) with Γ(x, ζ) = (x1 − ζ, x2 + η(x3 , ζ) − cf , x3 )T A sequence γ = (τi , ζi )i≥1 is called an impulse control strategy.



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The model

The optimization problem We consider the model on the state space S = {x ∈ R3 : x1 > −1, x1 + x2 > 0} (as long as the bank does not default and Lt ≥ 0.) Fix horizon date T > t and some concave increasing utility function U. Let τ = τS ∧ T . We assume that the shareholders want the bank to maximize the expected utility of its liquidation value at τS , J (α) (t, x) = E(t,x) [U (max(η(Mτ , Lατ ) + Cτα , 0))]

(1)

by an choosing optimal stochastic and impulse control strategy α = (β, γ). Define the value function v (t, x) = sup{J (α) (t, x) : α admissible}. Bachelier Finance Society, June 2010 (11)


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Specific model inputs

The players revisited

Bank’s creditor

Customers

Bank

ABS Investors

Bank’s shareholder



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Modeling market value of loans η(·) The loan portfolio of our bank consists of perpetuities (loans with maturity ∞, i.e., the nominal is never paid back). The risk-neutral value of one such perpetual loan is Z τ −ρs −ρτ ∞ e rL ds + e (1 − δ(Mτ )) M0 = m , pm := E 0

for τ the default time of the loan, and ρ risk-free interest rate. The vector p ∞ can be obtained by a simple inversion of the generator matrix of M. With constant default intensity λ: p ∞ = (rL + (1 − δ)λ) (ρ + λ). To account for risk aversion, one possible choice for the market value of ζ loans is: ∞ ∞ η(m, ζ) := ζ · min (1, pm · (1 − δλ(m)) ) < ζpm

(2)



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Modeling refinancing cost rb (·) Basic rule of thumb: On average, the bank’s creditors want to earn the risk-free interest ρ, so they will demand a refinancing rate rB according to 1 + ρ = (1 − PD) · (1 + rB ),

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where PD = probability of default of the bank over one year. Equation (3) leads to ρ + PD . (4) rB := 1 − PD Now, for a given loan amount ` and cash position c we define ∆L `+c PD := P(∆L > ` + c) = P > (5) ` ` ⇒ model the distribution of the [0, 1]-valued relative loss ∆L/`. Bachelier Finance Society, June 2010 (14)


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Modeling refinancing cost rb (·) ctd. Goal: Model distribution of relative loss ∆L/` as seen by creditors Losses in our loan portfolio follow a Bernoulli mixture distribution (for fixed t) with path of M as common factor. Large portfolio approximation: typically a Bernoulli mixture distribution converges for granularity going to ∞ to a limiting distribution depending only on the common factor (see [McNeil et al., 2005]) Choosing a probit-normal factor leads to the well-known continuous Vasicek loss distribution Vp,% , h √ i p Vp,% (x) = N 1/ %(N −1 (x) 1 − % − N −1 (p)) (p ∈ (0, 1) ≈ average default rate, % ∈ (0, 1) ≈ correlation) We take p ≈ λ(Mt ); the parameter ρ can be used as additional risk-aversion parameter on behalf of creditors Bachelier Finance Society, June 2010 (15)


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Refinancing cost: examples

Vasicek loss dist. for ms=0.000000, rel. VolaL=0.016125, Uncertainty time T=1.500000

Refin. cost rB for ms=0.000000, rel. VolaL=0.016125, Uncertainty time T=1.500000

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Loss distribution Vp,ρ (left) and refinancing rate rB (right) as function of the inverse leverage (` + c)/` . Bachelier Finance Society, June 2010 (16)


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Solution of the Optimization Problem

Numerical considerations

We want to find an optimal impulse control for the bank It seems impossible to find an analytical solution Usual approach: solve numerically the HJBQVI1 (partial integro-differential equation) by iterated optimal stopping and thus obtain the value function v = supα E[U(wealth)] From the value function, derive an (approximately) optimal strategy

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Hamilton-Jacobi-Bellman quasi-variational inequality (17)


Bachelier Finance Society, June 2010

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Value function v and HJBQVI Our aim is to find v as solution of the HJBQVI (here partial difference equation) min(− sup {ut + Lβ u}, u − Mu) = 0

in [0, T ) × S

(6)

β∈{0,1}

for Lβ the infinitesimal generator of the state variable process X = (L, C , M) where x˜ := (x1 , x2 ) = (`, c): −1 β L u(x) = u(˜ x+ , x3 ) − u(x) λ(x3 )x1 0 β + u(˜ x+ , x3 ) − u(x) λP −β + (u(˜ x , 1 − x3 ) − u(x)) λx3 ,(1−x3 ) + (rB (x)x2 + rL x1 )ux2 and M the intervention operator selecting the momentarily best impulse, Mu(t, x) = supζ {u(t, Γ(x, ζ))}. Bachelier Finance Society, June 2010 (18)


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HJBQVI: Existence and Uniqueness

We can prove using results from [Seydel, 2008]: Theorem (Parabolic viscosity solution) Assume that c 7→ rB (`, c, m) is continuous, and U continuous and bounded from below. Further assume that lim inf c↓−` rB (`, c, ·) > rL for ` > 0, and η(·, ζ) ≤ ζ. Then the value function v is the unique viscosity solution of (6), and it is continuous on [0, T ] × Z × R × {0, 1} (i.e., continuous in time and in cash). Proof: See [Frey and Seydel, 2009].



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Numerical Results

Overview

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Introduction

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Numerical Results



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Numerical Results

Basic data √ Power law utility U(x) = x Symmetric Markov chain transition intensities of 0.3 for M Default intensities per loan: λ(0) = 2.6% in expansion and λ(1) = 4.7% in contraction, no loan default recovery (δ ≡ 1) Risk-free rate ρ = 0.04, loan interest rate rL = 0.08 Market value η: A form slightly more procyclical than (2) ⇒ proportional transaction costs of 0% (≈ 6.5%) in expansion (contraction) Refinancing cost rB is based on Vasicek loss distribution with with p = 1.5λ, and correlation % = 0.2 (0.4) in expansion (contraction). Fixed transaction costs cf = 0.5 State equations: dLt = −dNt

dCt = (rB (Xt )Ct + rL Lt ) dt Γ(t, x, ζ) = (x1 − ζ, x2 + η(x3 , ζ) − cf , x3 )T Bachelier Finance Society, June 2010

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Numerical Results

Optimal impulses impulses in time 7.000000, ms=1.000000 40

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Figure: Impulses in expansion (left) and contraction (right) for T = 7. The light areas mark the impulse departure points (with the lightness indicating how far to the left the impulses goes, i.e., how many loans are sold), the cyan circles represent the corresponding impulse arrival points



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Numerical Results

Further results of numerical analysis Even with substantial transaction costs, securitization is a useful risk management tool for the bank: utility indifference argument shows that value of bank is increased substantially by the possibility of securitization Risk-dependent refinancing cost creates a major incentive to securitize Optimal securitization strategy is largely influenced by form of transaction cost - Low fixed cost cf ⇒ more transactions - Strongly procyclical market value of loans (high transaction costs in contraction), ⇒ only little securitization in contraction, but more loans are securitized in expansion Weakly procyclical market value ⇒ High securitization activity in contraction

Additional control of loan exposure (β) had only small effect Bachelier Finance Society, June 2010 (23)


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Numerical Results

Cash value of securitization Utility indifference iter3 − iter1 (time 7.000000), ms=0.000000

Utility indifference iter3 − iter1 (time 7.000000), ms=1.000000

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Figure: Cash value of securitization in expansion (M = 0) and in contraction (right), for T = 7. For every x we display the cash amount a such that v3 (x1 , x2 − a) = v1 (x1 , x2 ) (v3 being the value function with impulses, v1 without



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Numerical Results

Impulse and stochastic control Utility indifference v−fct MSSC=0 − MS=0 (time 7.000000)

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Figure: Impulse and stochastic control: Cash value of additional stochastic control (top row), and optimal strategy (bottom row) in expansion (left) and contraction (right), for T = 7. Business arrival intensity λP = 2 Bachelier Finance Society, June 2010 (25)


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Numerical Results

Literature Frey, R. and Seydel, R. C. (2009). Optimal securitization of credit portfolios via impulse control. Working paper. McNeil, A. J., Frey, R., and Embrechts, P. (2005). Quantitative risk management. Princeton Series in Finance. Princeton University Press, Princeton, NJ. Concepts, techniques and tools. Øksendal, B. and Sulem, A. (2005). Applied stochastic control of jump diffusions. Universitext. Springer-Verlag, Berlin. Seydel, R. C. (2008). General existence and uniqueness of viscosity solutions for impulse control of jump-diffusions. MPI MIS Preprint 37/2008. Bachelier Finance Society, June 2010 (26)


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Viscosity solutions

Viscosity solutions (1) Definition (Viscosity solution) A function u ∈ PB([0, T ] × Rd ) is a (viscosity) subsolution of (6) if for all (t0 , x0 ) ∈ [0, T ] × Rd and ϕ ∈ PB ∩ C 1,2 ([0, T ) × Rd ) with ϕ(t0 , x0 ) = u ∗ (t0 , x0 ), ϕ ≥ u ∗ on [0, T ) × Rd , ! ∂ϕ min − sup + Lβ ϕ + f β , u ∗ − Mu ∗ ≤ 0 ∂t β∈B in (t0 , x0 ) ∈ ST , and min (u ∗ − g , u ∗ − Mu ∗ ) ≤ 0 in (t0 , x0 ) ∈ ∂ + ST (the parabolic boundary). [...] Bachelier Finance Society, June 2010 (27)


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Viscosity solutions

Viscosity solutions (2) Definition (Viscosity solution (cont’d)) A function u ∈ PB([0, T ] × Rd ) is a (viscosity) supersolution of (6) if for all (t0 , x0 ) ∈ [0, T ] × Rd and ϕ ∈ PB ∩ C 1,2 ([0, T ) × Rd ) with ϕ(t0 , x0 ) = u∗ (t0 , x0 ), ϕ ≤ u∗ on [0, T ) × Rd , ! ∂ϕ β β min − sup +L ϕ+f , u∗ − Mu∗ ≥ 0 ∂t β∈B in (t0 , x0 ) ∈ ST , and min (u∗ − g , u∗ − Mu∗ ) ≥ 0 in (t0 , x0 ) ∈ ∂ + ST . A function u is a viscosity solution if it is sub- and supersolution. Bachelier Finance Society, June 2010 (28)


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Further numerical results

Impulses in time impulses in time 3.000000, ms=0.000000 40

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Figure: Impulses in expansion for different T : top left T = 1, top right T = 3, bottom left T = 5 and bottom right T = 7 Bachelier Finance Society, June 2010 (29)


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Further numerical results

Optimal impulses, no Markov-switching impulses in time 7.000000 40

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Figure: Impulses without Markov switching, for only expansion (left) and only contraction (right) for T = 7. Market value according to procyclical form (a), corresponding to 0% (about 17%) proportional transaction costs in expansion (contraction). For the colour code, see the explanations in Figure 1. Otherwise, same data as in Figure ?? Bachelier Finance Society, June 2010 (30)


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