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Outline of the Talk
- 1. Bibliography
- 2. Example of Contract
- 3. Asset Management
- 4. Illustration
Asset Risk Management of Participating Contracts Presented by : O. - - PowerPoint PPT Presentation
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
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Bibliography ➠ Brennan and Schwartz [JOF, 1976] ➠ Briys and de Varenne [Wiley, 2001] ➠ Grosen and Jørgensen [JRI, 2002] ➠ Ballotta [IME, 2005] ➠ Bacinello [ASTIN Bulletin, 2001]
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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]
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Life Office Assets Liabilities A0 E0 = (1 − α)A0 L0 = αA0 – E0 = initial equity value – L0 = initial policyholder investment
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Participating Contracts –> Minimum Guarantee Existence of a minimum guaranteed rate rg : Lg
T = L0 ergT
at T ➠ Solvency at time T : AT ≥ Lg
T
Policyholders receive Lg
T
➠ Default at time T : AT < Lg
T
Policyholders receive AT
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Participating Contracts –> Participation Bonus Bonus = δ times Benefits of the Company, when : AT > Lg
T
α > Lg
T
- α = A0
L0 < 1
- Assuming no prior bankruptcy, policyholders receive at T :
ΘL(T) =
AT if AT < Lg
T
Lg
T
if Lg
T ≤ AT ≤ Lg
T
α
Lg
T + δ(αAT − Lg T)
if AT > Lg
T
α
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Company Early Default The firm pursues its activities until T iff : ∀t ∈ [0, T[ , At > L0ergt Bt Let τ be the default time τ = inf{t ∈ [0, T] / At < Bt} In case of prior insolvency, policyholders receive : ΘL(τ) = L0ergτ
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Asset Dynamics The asset dynamics under the risk-neutral probability Q are : dAt At = rtdt + σdZQ(t) Because a big proportion of the assets are made of bonds, an interest rate model is necessary. ZQ of the assets will be correlated to ZQ
1 of the interest rates
(dZQ.dZQ
1 = ρdt).
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Stochastic Interest Rates The dynamics under Q of the interest rate r and the zero-coupon bonds P(t, T) are : drt = a(θ − rt)dt + νdZQ
1 (t)
and : dP(t, T) P(t, T) = rtdt − σP(t, T)dZQ
1 (t)
We Assume an Exponential Volatility for the Zero-Coupons : σP(t, T) = ν a
- 1 − e−a(T−t)
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Contract Valuation The market value of a standard participating contract is :
VL(0) = EQ
- e− T
0 rsds
Lg
T + δ(αAT − Lg T)+ − (Lg T − AT)+
1τ≥T + e− τ
0 rsds L0 ergτ 1τ<T
- This is typically a 2D interest rate/default problem in (r, τ)
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Buy and Hold Strategy θS is invested in risky securities θr is invested in the risk-free asset A0 = θS + θr AT = θS ST S0 + θrerT
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Constant Proportion Portfolio Insurance Floor : dF = F r dt Cushion (difference between assets value and floor level) : ∀t ∈ [0, T], Ct = At − Ft Investment in equity (with multiplier m) : ∀t ∈ [0, T], et = m Ct
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Constant Proportion Portfolio Insurance Investment in risk-free asset : ∀t ∈ [0, T], bt = At − et To sum up : At = Ft + Ct = et + bt
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Constant Proportion Portfolio Insurance
100 200 300 400 500 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 AT Probability Distribution m = 0.5 m = 1.5 m = 2.5
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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.
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Equity Default Swaps
100 200 300 400 500 0.005 0.01 0.015 0.02 0.025 0.03 AT Probability Distribution 1−x = 0.7 1−x = 0.5 1−x = 0.3
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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 t0 > 0 with an expiration T further in the future.
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Forward Start Put Options
100 200 300 400 500 0.005 0.01 0.015 0.02 0.025 AT Probability Distribution θ = 0.2 (OTM) θ = 1 (ATM) θ = 1.2 (ITM)
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