On the Market Value of Safety Loadings APRIA Conference Sidney 2008 - - PowerPoint PPT Presentation

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

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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

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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] ➠ Jeanblanc, Yor and Chesney [Springer, 2006] ➠ Bernard, Le Courtois and Quittard-Pinon [IME, 2005] ➠ Bernard, Le Courtois and Quittard-Pinon [NAAJ, 2006]

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Standard Participating Contracts and the Extended Fortet Method

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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)

To simplify matters, we price the representative term : I = EQT

 AT1

AT >

Lg T α

, τ<T

 

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Contract Valuation We show that :

I = ergT

T

• ds

+∞

• −∞

drs g(rs, s)

+∞

• −∞

drT fr(rT | rs, s, ls) Φ1

• µs,T;

Σs,T; L0 α

• where :

g is the density of (rτ, τ) fr is the Gaussian transition function Φ1 is a Gaussian function l is a return defined by lt = ln (At) − rgt

• µs,T and

Σs,T are conditional moments of l

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Contract Valuation The previous expression can be discretized as follows :

I = ergT

nT

• j=1

nr

• i=0

nr

• k=0

δrfr(rk | ri, tj, ltj) Φ1

• µtj,T;

Σtj,T; L0 α

• q(i, j)

where δr is the interest rate step and q(i, j) is the joint probability of τ ∈ [tj, tj+1] and r ∈ [ri, ri+1] Finally, one has to solve the recursive equation : q(i, j) = Φ( ri, tj ) −

j−1

• v=1

nr

• u=0

q( u, v ) Ψ( ri, tj | ru, tv ) where Φ and Ψ are completely known.

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Modified Participating Contracts Priced with more Standard Methods

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A Modified Contract ➠ differing slightly from the standard one ➠ in a totally identical framework for A and r ➠

• nly the Guaranteed Amount is modified

➠ Now Indexed on a Risk-Free Zero-Coupon Bond

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A New Guarantee Worth at any time t : lg

t =

βL0 P(0, T) P(t, T) = lg

T P(t, T)

where in particular : lg

0 = βL0

and lg

T =

βL0 P(0, T)

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Contract Valuation The default time becomes : τ = inf

• t < T / At < lg

t

• and the contract is priced in market value as :

V ′(0) = EQ

• e− T

0 rsds

• lg

T + δ

• αAT − lg

T

+ −

• lg

T − AT

+

1τ≥T + e− τ

0 rsds lg

τ 1τ<T

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Contract Valuation Illustration A typical expression to compute in this setting is : F(T) = QT (τ < T) It can be readily shown that : F(T) = QT

• inf

u∈[0,T[

• Au

P(u, T)

• < lg

T

• where lg

T is a constant

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Contract Valuation The solution of this problem lies in the fact that : Au P(u, T) = A0 P(0, T)eNu−1

2ξ(u)

where the martingale N is defined by : dNs = (σP(s, T) + ρσ) dZQT

1

(s) + σ

• 1 − ρ2 dZQT

2

(s) and its quadratic variation is : ξ(u) = < N >u =

u

0 [(σP(s, T) + ρσ)2 + σ2(1 − ρ2)]ds

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Contract Valuation F(T) = QT

• inf

u∈[0,T[

• Au

P(u, T)

• < lg

T

• = QT
• min

u∈[0,T]

• A0

P(0, T)eNu−1

2ξ(u)

• < lg

T

• = QT
• min

u∈[0,T]

• eBξ(u)−1

2ξ(u)

• < P(0, T) lg

T

A0

• = QT
• min

s∈[0,ξ(T)]

• Bs − 1

2s

• < ln (βα)
• where Dubins-Schwarz is the Key Theorem

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An Overall Picture of Guarantees

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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

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What the “optional" theory says Since Merton [1974], for a company like : Assets Liabilities A0 E0 (Equity) D0 (ZC Debt) Equity is a long call on the assets Debt is a risk-free ZC bond and a short put on the assets

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What the literature mimics For the last thirty years : Assets Liabilities A0 E0 (equity) L0 (policy) has been associated to a payoff like : Lg

T + δ(αAT − Lg T)+ − (Lg T − AT)+

(minimum guarantee & bonus & short default put on assets)

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What an Insurance Regulator would Want Such a structure : Assets Liabilities A0 E0 (equity) L0 (policy) should guarantee to the policyholder the payoff : Lg

T + δ(αAT − Lg T)+

(minimum guarantee & bonus )

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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

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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)

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Framework in Practice We write for the participating contract’s payoff : Lg

T + δ(αAT − Lg T)+ − (1 − ψ) × (Lg T − AT)+

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

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Valuing Policies –> Assume at this level no early default The contract can be valued as : Vψ = EQ

• e− T

0 rsds

• Lg

T + δ (αAT − Lg T)+ − (1 − ψ)

• Lg

T − AT

+

which can be computed for deterministic guarantees (Lg

t = Lg Te−rg(T−t))

• r for stochastic guarantees

(Lg

t = Lg TP(t, T))

using the methods described beforehand

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Valuing Guarantees The fully risky contract being worth : Vψ=0 = EQ

• e− T

0 rsds

• Lg

T + δ (αAT − Lg T)+ −

• Lg

T − AT

+

The fair price of the guarantee is : Vψ − Vψ=0 = ψ EQ

• e− T

0 rsds

Lg

T − AT

+

–> Similar Analyses can be done assuming early default/monitoring

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Conclusion A Discussion on the true Nature of Guarantees A bridge between financial and actuarial theories Computations done using the Extended Fortet

• r Change of Time techniques

Possible Extension to : Static or Dynamic Management of the underlying Assets

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