Time-consistent and Market-consistent ActuarialValuations
Ahmad Salahnejhad
1Maastricht University & EU HPCFinance [email protected]
Supervisor: Prof. Dr. Antoon PelsserMaastricht University & Kleynen Consultants & Netspar
14-15 March 2016High Performance Computing in Finance
Aberdeen Asset Management – London, UK
A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 1 / 23
Introduction
Research Outputs
3 Papers:
Time-consistent actuarial valuations, Insurance Mathematics andEconomics, Volume 66, January 2016, Pages 97-112 .
Market-consistent valuation by two-step operator and its applicationon life insurance pricing, Working paper.
Time-consistent and Market-consistent valuation of the participatingpolicy with hybrid profit-sharing, Working paper.
A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 2 / 23
Introduction
Motivation
Actuarial Pricing for Modern Life/pension Contract:
Hybrid liabilities from financial and actuarial risksLiabilities are not fully traded in the market.Financial part is usually hedgeable while actuarial part is not.Dynamic pricing needed due to path dependency & embedded options
Regulatory requirement for Market-consistent Valuation
“Re-Valuation” Insurance liabilities
Payoff in “Long-term”
What happens in the middle time?!How do we reflect this in pricing?Require time-consistency for Actuarial price operators?
A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 3 / 23
Introduction
Ambitions
Build Time-consistent Actuarial Pricing
Combine Actuarial & Financial Pricing for Market-consistentValuation
“Applied Techniques” to implement Time-consistent andMarket-consistent Pricing
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Time-Consistent Valuation
Time-Consistency
If X1 ≥ X2 at time T , then Π[t,X1] ≥ Π[t,X2] for all t < T
Note: The conditional expectation operator EQ[H | Ft ] is TC.
Extension of “tower property” to actuarial operators
Π[H(T ) | GAt
]= Π
[Π[H(T ) | GAs
]| GAt
]for t ≤ s < T
Well-known Actuarial operators are NOT time-Consistent
Variance Price: Π[H] = E[H] +1
2αVar[H], α ≥ 0
(1a)
Std-Deviation Price: Π[H] = E[H] + β√Var[H], β ≥ 0
(1b)
Cost-of-Capital Price: Π[H] = E[H] + δVaRq [H − E[H]] , δ ≥ 0(1c)
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Time-Consistent Valuation
Construct the Time-consistent Valuation
[Jobert and Rogers, 2008]: Time-Consistent valuation can be constructedby the “backward iteration” of the static one-period valuation.
Figure:A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 6 / 23
Time-Consistent Valuation Contributions & Findings
Continuous-time limit of the Time-consistent actuarialoperators
Diffusion Insurance Process: dyt = a(t, yt)dt + b(t, yt) dWt
Variance premium principle → Exponential indifference price
E[f (yT )|yt ] + 12αVar[f (yT )|yt ] ≡
1
αlnEt
[eαf (yT )
∣∣∣yt] . (2)
Standard-Deviation principle
E[f (yT )|yt ] + β√Var[f (yT )|yt ] ≡ ESt [f (yT )|yt ] (3)
with “risk-adjusted” dyS =(a(t, y)± βb(t, y)
)dt + b(t, y) dW S.
Cost-of-Capital principle
E[f (yT )|yt ] + δVaRq [f (yT )− E[f (yT )|yt ]|yt ] ≡ EC [f (yT )|yt ] (4)
with “risk-adjusted” dyC =(a(t, y)± δkb(t, y)
)dt + b(t, y) dWC.
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Time-Consistent Valuation Contributions & Findings
Continuous-time limit of the Time-consistent actuarialoperators
Jump-Diffusion Insurance Process:
We found PIDEs. There exist a convergent time-consistent price.Each operator reflects the effect of the jump differently.VaR fails to capture part of the jump effect!!!
Figure:A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 8 / 23
Market-Consistent Valuation Market-Consistency
Market-Consistency
xt : traded hedgeable financial process
yt : unhedgeable insurance process
G (xT , yT ): General hybrid claim
HS(xT ): financial derivative
ΠG(G + HS) = ΠG [G ] + EQG[HS]
(5)
Generalised notion of “translation invariance” for financial risk
Market-consistent valuation can not be “improved” by hedging
Roughly saying: If there is anything hedgeable (even inpayoff G), it must be hedged via Market-consistentvaluation!
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Market-Consistent Valuation One-period MC Valuation
Two-step Valuation
[Pelsser and Stadje, 2014]: Market-consistent valuation can beconstructed by “Two-step Market Evaluation”.
ΠGAt [G (xT , yT )] = EQ[
ΠP[G(xT , yT
) ∣∣∣∣ (yt , xT )
]∣∣∣∣ (yt , xt)
]. (6)
First/Inner step: Fix the financial risk and apply the actuarialoperator,
ΠP[G(xT , yT
) ∣∣∣ σ (GAt ,FST
)]:= GS (xT , yt)
The output is perfectly hedgeable.
Second/Outer step: Conditional expectation under Q
Reflects the no-arbitrage argument for the hedgeable part of thegeneral position.
Gives initial capital needed to hedge the payoff/position.
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Market-Consistent Pricing Binomial Tree
Quadrinomial Discretization
At a typical time-step (t, t + ∆t), every state (xt , yt) of the payoff at timet will develop to four different states of the world at time t + ∆t,
(xt , yt)
(x−t+∆t , y−t+∆t)
(x−t+∆t , y+t+∆t)
(x+t+∆t , y
−t+∆t)
(x+t+∆t , y
+t+∆t)
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Market-Consistent Pricing Binomial Tree
Two-step Binomial Discretization
We pretend that first, xt evolves ending to two different states at t + ∆t.Only then, given each state of xt+∆t , the process yt moves:
(xt , yt)
x−t+∆t
y−t+∆t
1-p
y+t+∆tp
1−q Q
x+t+∆t
y−t+∆t
1-p
y+t+∆tp
Clean separation of financial and actuarial pricing in each “half-step”.
Tech. cond: financial info arrives more frequently than insurance infoA. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 12 / 23
Market-Consistent Pricing Multi-period/Dynamic MC Valuation
Market-Consistent & Time-Consistent
Combine Backward Iteration & Two-step Actuarial valuation. In discreteset {0,∆t, 2∆t, ...,T −∆t,T} dividing [0,T ]
Start from T over (T −∆t,T ) and value G (T , x(T ), y(T )) atT −∆t.
πGA (T −∆t, xT−∆t , yT−∆t) = EQ[
ΠP[G(T , xT , yT
) ∣∣∣∣ GAT−∆t ,FST
]∣∣∣∣ GAT−∆t ,FST−∆t
](7)
π(T −∆t, xT−∆t , yT−∆t) is the New payoff in (T − 2∆t,T −∆t).
Move Back to T − 2∆t
πGA(T − 2∆t, x , y) = EQ[ΠP(πGA(T −∆t, x , y)
∣∣ GAT−2∆t ,FST−∆t
) ∣∣ GAT−2∆t ,FST−2∆t
].
Repeat the procedure till (0,∆t).
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Market-Consistent Pricing Contributions & Findings
Contributions for the 2nd Paper
Found continuous-time limit of the Time-consistent two-stepvaluation for some Actuarial operators.
Found Some analytical solutions when the financial and actuarial risksare independent.
Implemented regression-based computation method the for backwarditeration of the two-step valuation.
Calculated“time-consistency risk premium” = Time-consistent price -One-period Price
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3rd Paper; Applied Market-consistent Valuation Product Definition & Modeling
Application: Pricing the Participating Contract
Payoff
G (PT , rT , κT ) =(e−
∫ T0 rtdt
)× P
(h)T × Nx(T ). (8)
Nx(T ): Number of Survivors at Maturity T ,
PT : Policy Reserve at Maturity
rP(t): Policy Interest rate
Pt = Pt−1 × rP(t).
Three Underlying risk Drivers
At : Investment asset (Financial) - Black-Scholes Modelrt : Interest rate (Financial) - Hull-White Modelκt : Longevity trend (Actuarial) - Lee-carter Model
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3rd Paper; Applied Market-consistent Valuation Product Definition & Modeling
Profit-Sharing Mechanism
[Grosen and Jorgensen, 2000]: Path-dependent crediting mechanism
rP(t) = max
{rG , α
(At−1
Pt−1− (1 + γ)
)}t = 1, 2, ...,T (9)
rG : Guaranteed Interest rate
γ: Target Buffer Ratio, Realistic Value: 10-20%
α: Distribution Ratio, Realistic Value: 20-50%
Similar to an Option element with strike value rG .
Pure financial mechanism: No longevity risk plays a role!!!
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3rd Paper; Applied Market-consistent Valuation Product Definition & Modeling
Hybrid Profit-Sharing Mechanism
Hybrid crediting mechanism
r(h)P (t) = max
{rG , α
(BE0 (Nx(T ))
BEt (Nx(T ))× At−1
Pt−1− (1 + γ)
)}(10)
All investment asset, interest rate and longevity play role!!!
Price calculated by Numerical Methods; No Analytical Solution.
A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 17 / 23
3rd Paper; Applied Market-consistent Valuation Product Definition & Modeling
Market-Consistent Std-Dev Pricing
Multi-period two-step Std-Dev price over (t, t + 1) (by Backwarditeration),
πt(At+1, rt+1, κt+1) =
EQ[EP[(
e−∫ t+1t rsds
)πt+1(At+1, rt+1, κt+1)
∣∣∣ κt ,At+1, rt+1
]+β
√VarP
[(e−
∫ t+1t rsds
)πt+1(At+1, rt+1, κt+1) | κt ,At+1, rt+1
]| κt ,At , rt
]with terminal condition
πT (AT , rT , κT ) =(e−
∫ TT−1 rsds
)× P
(h)T × Nx(T ) (11)
One-period two-step Std-Dev Price (“Traditional Actuarial Price”)
Expected Value
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Implementation Numerical Method
Numerical Techniques
Conditional operators at each time step given the state of theunderlying processes at previous step.Methods:
Nested Monte Carlo (Inefficient computation)Markov Chain Discretization / Trinomial Tree (Inefficient in higherdimension)Least-Square Monte-Carlo (Regress Now) (More EfficientComputation)
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Implementation Numerical Method
LSMC for Two-step Std-Deviation Valuation
[Longstaff & Schwartz, 2001] and [Glasserman & Yu, 2002] usedregression-based methods to valuate American options.
First/Inner Step Regression
EP[(e−(rTT−rT−1(T−1))
)P
(h)T × Nx(T )
∣∣∣ AT , rT , κT−1
]=∑K−1
k=0 ak(1,T )ek(AT , rT , κT−1)
EP[((
e−(rTT−rT−1(T−1)))P
(h)T × Nx(T )
)2∣∣∣∣ AT , rT , κT−1
]=∑K−1
k=0 ak(2,T )ek(AT , rT , κT−1)
Calculate the conditional premium πs(ST , κT−∆t),
πs (AT , rT , κT−1) = EP[f
(h)T
]+ β
√EP[(f
(h)T )2
]−(EP[f
(h)T
])2.
Second/Outer Step Regression
EQ [πs (AT , rT , κT−1)) | AT−1, rt−1, κT−1] =∑K−1
k=0 bTk eπs(AT−1, rT−1, κT−1).(13)
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Implementation Contributions & Findings
TC Two-step Std-Dev Price vs Expected Value
Figure: Time-consistent and market-consistent Standard-Deviation actuarial price vs.discounted expected value of the participating contract with 95% confidence intervaland different maturities T = 1, 2, ..., 30. Parameter set: A0 = P0 = 100, σA = 15%,σr = 1%, ρA,r = 0.25, rG = 2%, α = 0.3, γ = 0.25, n = 1, 000, N = 100.
A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 21 / 23
Implementation Contributions & Findings
Proportion of the Risk-loadings in MC Price
Expected Value < 1-period Std-Dev Price < TC Std-Dev Price
Table: Values of the participating contract with different maturities and initial cohort ofN40(0) = 1, 000 and the ratio of one-period risk loading and time-consistency riskpremium on top of the expected value of the contract. Parameter set: A0 = P0 = 100,σA = 15%, σr = 1%, ρA,r = 0.25, rG = 2%, α = 0.3, γ = 0.25, n = 1, 000, N = 100.
T
Two-step Price 5 10 15 20 25 30
Expected-value 95,558.6 92,088.3 89,138.6 86,328.2 83,600.2 79,284.6One-perod Std-Dev 95,574.6 92,123.6 89,198.5 86,422.7 83,746.6 79,513.5Time-Consistent Std-Dev 95,752.7 93,072.1 91,342.2 89,802.1 88,761.6 86,365.5One-period Risk-loading 0.02% 0.04% 0.07% 0.11% 0.19% 0.31%Time-consistency Premium 0.19% 1.03% 2.40% 3.91% 5.99% 8.62%Total Risk-loading 0.20% 1.07% 2.47% 4.02% 6.17% 8.93%Ratio of TC Premium 91.8% 96.4% 97.2% 97.2% 97.0% 96.5%
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References
References I
Grosen, A. and Jorgensen, P. L. (2000).Fair valuation of life insurance liabilities: The impact of interest rateguarantees, surrender options, and bonus policies.Insurance: Mathematics and Economics, 26:37–57.
Jobert, A. and Rogers, L. (2008).Valuations and dynamic convex risk measures.Mathematical Finance, 18(1):1–22.
Pelsser, A. and Stadje, M. (2014).Time-consistent and market-consistent evaluations.Mathematical Finance, 24(1):25–65.
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