Time-consistent and Market-consistent Actuarial Valuations Salahnejh… · Time-consistent...

Time-consistent and Market-consistent ActuarialValuations

Ahmad Salahnejhad

1Maastricht University & EU HPCFinance Projecta.salahnejhad@maastrichtuniversity.nl

ahmad.salahnejhad@gmail.com

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

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 4 / 23

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)

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 5 / 23

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.

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 7 / 23

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!

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 9 / 23

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.

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 10 / 23

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)

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 11 / 23

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

qQ

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).

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 13 / 23

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

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 14 / 23

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

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 15 / 23

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!!!

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 16 / 23

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

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 18 / 23

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)

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 19 / 23

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)

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 20 / 23

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%

A. Salahnejhad (Maastricht U) TC & MC Actuarial Valuations 14-15 March 2016 22 / 23

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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