On the Market Value of Safety Loadings APRIA Conference Sidney 2008 Presented by : O. LE COURTOIS Joint Work with : C. BERNARD and F. QUITTARD-PINON 1
Outline of the Talk 1. Bibliography 2. Standard Participating Contracts and the Extended Fortet Method 3. Modified Participating Contracts Priced with more Standard Methods 4. An Overall Picture of Guarantees 5. Conclusion 2
Bibliography ➠ Brennan and Schwartz [JOF, 1976] ➠ Briys and de Varenne [Wiley, 2001] ➠ Grosen and Jørgensen [JRI, 2002] ➠ Ballotta [IME, 2005] ➠ Bacinello [ASTIN Bulletin, 2001] 3
Bibliography ➠ Longstaff and Schwartz [JOF, 1995] ➠ Collin-Dufresne and Goldstein [JOF, 2001] ➠ Jeanblanc, Yor and Chesney [Springer, 2006] ➠ Bernard, Le Courtois and Quittard-Pinon [IME, 2005] ➠ Bernard, Le Courtois and Quittard-Pinon [NAAJ, 2006] 4
Standard Participating Contracts and the Extended Fortet Method 5
Life Office Assets Liabilities E 0 = (1 − α ) A 0 A 0 L 0 = αA 0 – E 0 = initial equity value – L 0 = initial policyholder investment 6
Participating Contracts –> Minimum Guarantee Existence of a minimum guaranteed rate r g : L g T = L 0 e r g T at T ➠ Solvency at time T : A T ≥ L g T Policyholders receive L g T ➠ Default at time T : A T < L g T Policyholders receive A T 7
Participating Contracts –> Participation Bonus Bonus = δ times Benefits of the Company, when : � � A T > L g α = A 0 α > L g T < 1 T L 0 Assuming no prior bankruptcy , policyholders receive at T : if A T < L g A T T T ≤ A T ≤ L g L g if L g T Θ L ( T ) = T α if A T > L g L g T + δ ( αA T − L g T T ) α 8
Company Early Default The firm pursues its activities until T iff : A t > L 0 e r g t � B t ∀ t ∈ [0 , T [ , Let τ be the default time τ = inf { t ∈ [0 , T ] / A t < B t } In case of prior insolvency, policyholders receive : Θ L ( τ ) = L 0 e r g τ 9
Asset Dynamics The asset dynamics under the risk-neutral probability Q are : dA t = r t dt + σdZ Q ( t ) A t Because a big proportion of the assets are made of bonds, an interest rate model is necessary. Z Q of the assets will be correlated to Z Q 1 of the interest rates ( dZ Q .dZ Q 1 = ρdt ). 10
Stochastic Interest Rates The dynamics under Q of the interest rate r and the zero-coupon bonds P ( t, T ) are : = a ( θ − r t ) dt + νdZ Q 1 ( t ) dr t and : dP ( t, T ) P ( t, T ) = r t dt − σ P ( t, T ) dZ Q 1 ( t ) We Assume an Exponential Volatility for the Zero-Coupons : � 1 − e − a ( T − t ) � σ P ( t, T ) = ν a 11
Contract Valuation The market value of a standard participating contract is : � e − � T 0 r s ds � T − A T ) + � T ) + − ( L g L g T + δ ( αA T − L g V L (0) = E Q 1 τ ≥ T � + e − � τ 0 r s ds L 0 e r g τ 1 τ<T This is typically a 2D interest rate/default problem in ( r, τ ) To simplify matters, we price the representative term : A T 1 I = E Q T Lg T A T > , τ<T α 12
Contract Valuation We show that : � T � + ∞ � + ∞ � � Σ s,T ; L 0 I = e r g T µ s,T ; � ds dr s g ( r s , s ) dr T f r ( r T | r s , s, l s ) Φ 1 � α 0 −∞ −∞ where : g is the density of ( r τ , τ ) f r is the Gaussian transition function Φ 1 is a Gaussian function l is a return defined by l t = ln ( A t ) − r g t µ s,T and � Σ s,T are conditional moments of l � 13
Contract Valuation The previous expression can be discretized as follows : � � n T n r n r � � � Σ t j ,T ; L 0 I = e r g T µ t j ,T ; � δ r f r ( r k | r i , t j , l t j ) Φ 1 � q ( i, j ) α j =1 i =0 k =0 where δ r is the interest rate step and q ( i, j ) is the joint probability of τ ∈ [ t j , t j +1 ] and r ∈ [ r i , r i +1 ] Finally, one has to solve the recursive equation : j − 1 n r � � q ( i, j ) = Φ( r i , t j ) − q ( u, v ) Ψ( r i , t j | r u , t v ) v =1 u =0 where Φ and Ψ are completely known. 14
Modified Participating Contracts Priced with more Standard Methods 15
A Modified Contract ➠ differing slightly from the standard one ➠ in a totally identical framework for A and r ➠ only the Guaranteed Amount is modified ➠ Now Indexed on a Risk-Free Zero-Coupon Bond 16
A New Guarantee Worth at any time t : βL 0 l g P (0 , T ) P ( t, T ) = l g t = T P ( t, T ) where in particular : l g 0 = βL 0 and βL 0 l g T = P (0 , T ) 17
Contract Valuation The default time becomes : � � t < T / A t < l g τ = inf t and the contract is priced in market value as : � � � + � e − � T � � + − � l g αA T − l g l g V ′ (0) = E Q 0 r s ds T + δ T − A T 1 τ ≥ T T � + e − � τ 0 r s ds l g τ 1 τ<T 18
Contract Valuation Illustration A typical expression to compute in this setting is : F ( T ) = Q T ( τ < T ) It can be readily shown that : � � � � A u < l g F ( T ) = Q T inf T P ( u, T ) u ∈ [0 ,T [ where l g T is a constant 19
Contract Valuation The solution of this problem lies in the fact that : A u A 0 P (0 , T ) e N u − 1 2 ξ ( u ) P ( u, T ) = where the martingale N is defined by : � dN s = ( σ P ( s, T ) + ρσ ) dZ Q T 1 − ρ 2 dZ Q T ( s ) + σ ( s ) 1 2 and its quadratic variation is : � u 0 [( σ P ( s, T ) + ρσ ) 2 + σ 2 (1 − ρ 2 )] ds ξ ( u ) = < N > u = 20
Contract Valuation � � � � A u < l g F ( T ) = Q T inf T P ( u, T ) u ∈ [0 ,T [ � � � � A 0 P (0 , T ) e N u − 1 < l g 2 ξ ( u ) = Q T min T u ∈ [0 ,T ] � � � � < P (0 , T ) l g e B ξ ( u ) − 1 2 ξ ( u ) T = Q T min A 0 u ∈ [0 ,T ] � � � � B s − 1 = Q T min < ln ( βα ) 2 s s ∈ [0 ,ξ ( T )] where Dubins-Schwarz is the Key Theorem 21
An Overall Picture of Guarantees 22
Market Value of Safety Loadings Current literature does not pay attention to safety loadings Safety Loading is an Actuarial Practice to Protect an Insurance Company Simply : the higher the immobilized capital per policy, the lower the ruin probability Actuaries and insurance regulators want safe companies We want to tackle this problem from a Finance viewpoint 23
What the “optional" theory says Since Merton [1974], for a company like : Assets Liabilities E 0 (Equity) A 0 D 0 (ZC Debt) Equity is a long call on the assets Debt is a risk-free ZC bond and a short put on the assets 24
What the literature mimics For the last thirty years : Assets Liabilities A 0 E 0 (equity) L 0 (policy) has been associated to a payoff like : T ) + − ( L g L g T + δ ( αA T − L g T − A T ) + (minimum guarantee & bonus & short default put on assets) 25
What an Insurance Regulator would Want Such a structure : Assets Liabilities A 0 E 0 (equity) L 0 (policy) should guarantee to the policyholder the payoff : L g T + δ ( αA T − L g T ) + (minimum guarantee & bonus ) 26
Now, the Reality Companies happen to bankrupt Life Insurance companies do also bankrupt –> Perfect guarantees do not exist Trying to make contracts perfectly safe is expensive to policyholders... ... furthermore some commercial risk would rise –> Perfect guarantees would be difficult to create 27
A Mixed Framework But : Policyholders are not (potentially junk-) Bondholders ...Different levels of Risk Aversion, Different Regulations... –> We construct a mixed framework, where LI policies are more protected than bonds, but not fully protected though (leading to a description of LI policies as hybrid debt) 28
Framework in Practice We write for the participating contract’s payoff : T ) + − (1 − ψ ) × ( L g L g T + δ ( αA T − L g T − A T ) + where ψ is the Policyholders’S Immunization degree ψ is the level of safety : –> ψ = 0 is the Mertonian case (policyholder = bondholder) –> ψ = 1 is for fully safe contracts –> obviously 0 < ψ < 1 Also : the higher ψ , the higher protection costs to policyholders 29
Valuing Policies –> Assume at this level no early default The contract can be valued as : � � � + � � e − � T � T ) + − (1 − ψ ) L g T + δ ( αA T − L g L g 0 r s ds V ψ = E Q T − A T which can be computed for deterministic guarantees ( L g t = L g T e − r g ( T − t ) ) or for stochastic guarantees ( L g t = L g T P ( t, T )) using the methods described beforehand 30
Valuing Guarantees The fully risky contract being worth : � � � + � � e − � T � T ) + − L g T + δ ( αA T − L g L g 0 r s ds V ψ =0 = E Q T − A T The fair price of the guarantee is : � � + � e − � T 0 r s ds � L g V ψ − V ψ =0 = ψ E Q T − A T –> Similar Analyses can be done assuming early default/monitoring 31
Conclusion A Discussion on the true Nature of Guarantees A bridge between financial and actuarial theories Computations done using the Extended Fortet or Change of Time techniques Possible Extension to : Static or Dynamic Management of the underlying Assets 32
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