Asset Risk Management of Participating Contracts Presented by : O. - PowerPoint PPT Presentation

Asset Risk Management of Participating Contracts Presented by : O. LE COURTOIS Joint Work with : C. BERNARD 1

Outline of the Talk 1. Bibliography 2. Example of Contract 3. Asset Management 4. Illustration 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] ➠ Bernard, Le Courtois and Quittard-Pinon [IME, 2005] ➠ Bernard, Le Courtois and Quittard-Pinon [NAAJ, 2006] ➠ Black, Perold [JEDC, 1992] 4

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 5

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 6

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 7

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 8

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 ). 9

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 : σ P ( t, T ) = ν � 1 − e − a ( T − t ) � a 10

Contract Valuation The market value of a standard participating contract is : e − � T � T ) + − ( L g 0 r s ds � L g T + δ ( αA T − L g T − A T ) + � 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 ) 11

Buy and Hold Strategy θ S is invested in risky securities θ r is invested in the risk-free asset A 0 = θ S + θ r S T + θ r e rT A T = θ S S 0 12

Constant Proportion Portfolio Insurance Floor : dF = F r dt Cushion (difference between assets value and floor level) : ∀ t ∈ [0 , T ] , C t = A t − F t Investment in equity (with multiplier m ) : ∀ t ∈ [0 , T ] , e t = m C t 13

Constant Proportion Portfolio Insurance Investment in risk-free asset : ∀ t ∈ [0 , T ] , b t = A t − e t To sum up : A t = F t + C t = e t + b t 14

Constant Proportion Portfolio Insurance 0.09 m = 0.5 m = 1.5 0.08 m = 2.5 0.07 Probability Distribution 0.06 0.05 0.04 0.03 0.02 0.01 0 0 100 200 300 400 500 A T 15

Equity Default Swaps An EDS provides protection against a dramatic decline in the price of a risky portfolio, whereas the reference asset of the CDS is a debt instrument, and protection is provided against a possible default. For example, an equity default swap might provide protection against a 70% decline in the price of the risky portfolio with regard to its initial value. If this event happens, a fixed recovery rate (typically 50% ) of the incurred loss, is paid to the investor, and, of course, installments are ceased. 16

Equity Default Swaps 0.03 1−x = 0.7 1−x = 0.5 1−x = 0.3 0.025 Probability Distribution 0.02 0.015 0.01 0.005 0 0 100 200 300 400 500 A T 17

Forward Start Put Options To guarantee a fixed rate g over the lifetime T of the contract when there exist no long-term options, it is also possible to buy forward starting put options with shorter maturities, say 1 year. Each option protects the company against a decrease in the assets value for each period. Indeed, a forward starting option is an option that starts at some specified future date t 0 > 0 with an expiration T further in the future. 18

Forward Start Put Options 0.025 θ = 0.2 (OTM) θ = 1 (ATM) θ = 1.2 (ITM) 0.02 Probability Distribution 0.015 0.01 0.005 0 0 100 200 300 400 500 A T 19

Conclusion 0.045 CPPI: m=1.5 FwPuts: θ =0.5 0.04 EDSs: 1−x=0.7 0.035 Probability Distribution 0.03 0.025 0.02 0.015 0.01 0.005 0 0 100 200 300 400 500 A T 20