Time-Consistent and Market-Consistent Actuarial Valuations2010/02/24 · Time-Consistent and...

Time-Consistent and Market-Consistent ActuarialValuations

Antoon Pelsser1

1Maastricht University & NetsparEmail: [email protected]

30 April 2010DGVFM Scientific Day – Bremen

A. Pelsser (Maastricht U) TC & MC Valuations 30 Apr 2010 – DGVFM 1 / 34

Introduction

Motivation

Standard actuarial premium principles usually consider “static”premium calculation:

What is price today of insurance contract with payoff at time T ?

Actuarial premium principles typically “ignore” financial markets

Financial pricing considers “dynamic” pricing problem:

How does price evolve over time until time T ?

Financial pricing typically “ignores” unhedgeable risks

Examples:

Pricing very long-dated cash flows T ∼ 30− 100 yearsPricing long-dated options T > 5 yearsPricing pension & insurance liabilitiesPricing employee stock-options


Introduction

Ambitions

In this paper I want to combine1 Time-Consistent pricing operators, see [Jobert and Rogers, 2008]2 Market-Consistent pricing operators, see [Malamud et al., 2008]

Both references concentrate on discrete-time algorithms

I will be interested in continuous-time limits of these discretealgorithms for different actuarial premium principles:

1 Variance Principle2 Mean Value Principle3 Standard-Deviation Principle4 Cost-of-Capital Principle


Introduction

Content of This Talk

1 Pure Insurance Risk

Diffusion Model for Insurance RiskVariance Principle (→ exponential indiff. pricing)Standard-Dev. Principle (→ E[] under new measure)Cost-of-Capital Principle (→ St.Dev price)Davis Price, see [Davis, 1997]

St.Dev is “small perturbation” of Variance price

2 Financial & Insurance Risk

Diffusion Model for Financial RiskMarket-Consistent PricingVariance PrincipleNumerical Illustration

3 Conclusions


Pure Insurance Risk Model for Insurance Risk

Pure Insurance Model

Consider unhedgeable insurance process y :

dy = a(t, y) dt + b(t, y) dW

To keep math simple, concentrate on diffusion setting

Discretisation scheme as binomial tree:

y(t + ∆t) = y(t) + a∆t +

{+b√

∆t with prob. 12

−b√

∆t with prob. 12


Pure Insurance Risk Time Consistency

Time Consistency

Time Consistent price π(t, y) satisfies property

π[f (y(T ))|t, y ] = π[π[f (y(T ))|s, y(s)]|t, y ] ∀t < s < T

Price of today of holding claim until T is the same as buying claimhalf-way at time s for price π(s, y(s))

“Semi-group property”

Similar idea as “tower property” of conditional expectation


Pure Insurance Risk Variance Principle

Variance Principle

Actuarial Variance Principle Πv:

Πvt [f (y(T ))] = Et [f (y(T ))] + 1

2αVart [f (y(T ))]

α is Absolute Risk Aversion

Apply Πv to one binomial time-step to obtain price πv:

πv(t, y(t)

)= Et [π

v(t + ∆t, y(t + ∆t)

)] +

12αVart [π

v(t + ∆t, y(t + ∆t)

)]

Note: we omit discounting for now



Pricing PDE

Assume πv(t, y) admits Taylor approximation in y

Evaluate Var.Princ. for binomial step & take limit for ∆t → 0

Same as derivation of Feynman-Kac, but for E[] + 12αVar[]

This leads to pde for πv:

πvt + aπv

y + 12 b2πv

yy + 12α(bπv

y )2 = 0

Note, non-linear term = “local unhedgeable variance” b2(πvy )2

Find general solution to this non-linear pde via log-transform:

πv(t, y) =1

αlnEt

[eαf (y(T ))

∣∣∣y(t) = y].

Exponential indifference price, see [Henderson, 2002] or[Musiela and Zariphopoulou, 2004]



Include Discounting

We should include discounting into our pricing

Absolute Risk Aversion α is not “unit-free”, but has unit 1/eThis conveniently compensates the unit (e)2 of Var[]. . .

Therefore, “α-today” is different than “α-tomorrow”

Relative Risk Aversion γ is unit-free

Express ARA relative to “benchmark wealth” X0erT

Explicit notation: α→ γ/X0erT leads to pde:

πvt + aπv

y + 12 b2πv

yy + 12

γ

X0ert(bπv

y )2 − rπv = 0

πv(t, y) =X0ert

γlnE

[e

γ

X0erT f (y(T ))

∣∣∣∣y(t) = y

]Note: express all prices in discounted terms



Backward Stochastic Differential Equations

Pricing PDE:

πvt + aπv

y + 12 b2πv

yy + 12

γ

X0ert(bπv

y )2 − rπv = 0

This non-linear PDE, represents the solution to a so-called BSDE forthe triplet of processes (yt ,Yt ,Zt)

dyt = a(t, yt) dt + b(t, yt) dWt

dYt = −g(t, yt ,Yt ,Zt) dt + Zt dWt

YT = f(y(T )

),

with “generator” g(t, y ,Y ,Z ) = 12

γX0ert

Z 2 − rY .

Recent literature studies uniqueness & existence of solutions toBSDE’s, see [El Karoui et al., 1997]

Via BSDE’s we can study time-consistent pricing operators in a muchmore general stochastic setting. But we will not pursue this here.


Pure Insurance Risk Mean Value Principle

Mean Value Principle

Generalise to Mean Value Principle

Πmt [f (y(T ))] = v−1 (Et [v(f (y(T )))])

for any function v() which is a convex and increasing

Exponential pricing is special case with v(x) = eαx

Do Taylor-expansion & limit ∆t → 0:

πmft + aπmf

y + 12 b2πmf

yy + 12

v ′′(πmf)

v ′(πmf)(bπmf

y )2 = 0

Note: πmf(t, y) := πm(t, y)/ert is price expressed in discounted terms

Interpretation as generalised Variance Principle with “local riskaversion” term: v ′′()/v ′()


Pure Insurance Risk Standard-Deviation Principle

Standard-Deviation Principle

Actuarial Standard-Deviation Principle:

Πst [f (y(T ))] = Et [f (y(T ))] + β

√Vart [f (y(T ))].

Pay attention to “time-scales”:

Expectation scales with ∆tSt.Dev. scales with

√∆t

Thus, we should take β√

∆t to get well-defined limit

Note: β has unit 1/√

time


Pure Insurance Risk Standard-Deviation Principle

Pricing PDE

Do Taylor-expansion & limit ∆t → 0:

πst + aπs

y + 12 b2πs

yy + β√

(bπsy )2 − rπs = 0

Again, non-linear pde. But if πs is monotone in y then

πst + (a± βb)πs

y + 12 b2πs

yy − rπs = 0

πs(t, y) = ESt [f (y(T ))|y(t) = y ]

“Upwind” drift-adjustment into direction of risk


Pure Insurance Risk Cost-of-Capital Principle

Cost-of-Capital Principle

Cost-of-Capital principle, popular by practitioners

Used in QIS4-study conducted by CEIOPS

Idea: hold buffer-capital against unhedgeable risks. Borrow fromshareholders by giving “excess return” δ

Define buffer via Value-at-Risk measure:

Πct [f (y(T ))] = Et [f (y(T ))] + δVaRq,t

[f (y(T ))− Et [f (y(T ))]

].


Pure Insurance Risk Cost-of-Capital Principle

Scaling & PDE

Again, pay attention to “time-scaling”:

First, scale VaR back to per annum basis with 1/√

∆tThen, δ is like interest rate, so multiply with ∆tNet scaling: δ∆t/

√∆t = δ

√∆t.

Limit: for small ∆t the VaR behaves as Φ−1(q)×St.Dev. Hence,limiting pde is same as πs but with β = Φ−1(q)δ.

Conclusion: In the limit for ∆t → 0, CoC pricing is the same asst.dev. pricing


Pure Insurance Risk Davis Price

Davis Price

The variance price πv is “hard” to calculate, the st.dev. price πs is“easy” to calculate

Can we make a connection between these two concepts?

“Yes, we can!” using small perturbation expansion

Consider existing insurance portfolio with price πv(t, y), now add“small” position with price επD(t, y). Subst. into pde:

(πvt + επD

t ) + a(πvy + επD

y ) + 12 b2(πv

yy + επDyy ) +

12

γ

X0ertb2(

(πvy )2 + 2επv

yπDy + ε2(πD

y )2)− r(πv + επD) = 0

πv() solves the pde, cancel πv-terms


Pure Insurance Risk Davis Price

Pricing PDE

Simplify pde, and divide by ε:

πDt + aπD

y + 12 b2πD

yy + 12

γ

X0ertb2(

2πvyπ

Dy + ε(πD

y )2)− rπD = 0

Approximation: ignore “small” ε-term

πDt +

(a +

γ

X0ertb2πv

y

)πDy + 1

2 b2πDyy + rπD = 0

πD(t, y) = EDt [f (y(T ))|y(t) = y ]

Davis price πD is defined only “relative” to existing price πv ofinsurance portfolio

Note, drift-adjustment of st.dev. price scales with b


Financial & Insurance Risk


Investigate environment with financial risk that can be traded (andhedged!) in financial market and non-traded insurance risk

Model financial risk as [Black and Scholes, 1973] economy. Modelreturn process xt = ln St under real-world measure P:

dx =(µ(t, x)− 1

2σ2(t, x)

)dt + σ(t, x) dWf

Binomial time-step:

x(t + ∆t) = x(t) + (µ− 12σ

2)∆t +

{+σ√

∆t with P-prob. 12

−σ√

∆t with P-prob. 12



No-arbitrage pricing

BS-economy is arbitrage-free and complete ⇔ unique martingalemeasure Q.

No-arbitrage pricing operator for financial derivative F (x(T )):

πQ(t, x) = e−r(T−t)EQt [F (x(T ))]

Binomial step for x under measure Q:

x(t + ∆t) = x(t) + (µ− 12σ

2)∆t+ +σ√

∆t with Q-prob. 12

(1− µ−r

σ

√∆t)

−σ√

∆t with Q-prob. 12

(1 + µ−r

σ

√∆t)

Quantity (µ− r)/σ is Radon-Nikodym exponent of dQ/dP

Quantity (µ− r)/σ is also known as market-price of financial risk.



Quadrinomial Tree

Joint discretisation for processes x and y using “quadrinomial” treewith correlation ρ under measure P:

State: y + ∆y y −∆y

x + ∆x

(1 + ρ

4

) (1− ρ

4

)↖↗•↙↘

x −∆x

(1− ρ

4

) (1 + ρ

4

)

Positive correlation increases probability of joint “++” or “−−”co-movement


Financial & Insurance Risk Market-Consistent Pricing

Market-Consistent Pricing

We are looking for market-consistent pricing operators, seee.g. [Malamud et al., 2008]

Definition

A pricing operator π() is market-consistent if for any financial derivativeF (x(T )) and any other claim G (t, x , y) we have

πF+G (t, x , y) = e−r(T−t)EQt [F (x(T ))] + πG (t, x , y).

Observation: generalised notion of “translation invariance” for allfinancial risks


Financial & Insurance Risk MC Variance Pricing

MC Variance Pricing

Intuition: construct MC pricing in two steps using conditionalexpectations, see also: [Carmona, 2008], Chap. 1

First: condition on financial risk & use actuarial pricing for “pureinsurance” risk

πv(t +∆t|x±) := E[πv(t +∆t)|x±]+ 12

γ

X0er(t+∆t)Var[πv(t +∆t)|x±]

E[πv(t + ∆t)|x+] =(

1+ρ2

)πv

++ +(

1−ρ2

)πv

+−

Var[πv(t + ∆t)|x+] =(

1−ρ2

4

) (πv

++ − πv+−)2

E[πv(t + ∆t)|x−] =(

1−ρ2

)πv−+ +

(1+ρ

2

)πv−−

Var[πv(t + ∆t)|x−] =(

1−ρ2

4

) (πv−+ − πv

−−)2.

For ρ = 1 or ρ = −1 no unhedgeable risk left ⇒ Var = 0



Pricing PDE

Second: use no-arbitrage pricing for “artificial” financial derivative

πv(t, x , y) = e−r∆tEQ[πv(t + ∆t|x±)]

= e−r∆t(

12

(1− µ−r

σ

√∆t)πv(t + ∆t|x+) +

12

(1 + µ−r

σ

√∆t)πv(t + ∆t|x−)

)Do Taylor-expansion & limit ∆t → 0:

πvt + (r − 1

2σ2)πv

x +(a− ρbµ−rσ

)πvy+

12σ

2πvxx + ρσbπv

xy + 12 b2πv

yy + 12

γ

X0ert(1− ρ2)(bπv

y )2 − rπv = 0

Impact on x : “Q-drift” (r − 12σ

2)

Impact on y : adjusted drift(a− ρb µ−rσ

)Non-linear term for “locally unhedgeable variance” (1− ρ2)(bπv

y )2



Pure Insurance Payoff

Unfortunately, we cannot solve the non-linear pde in general

Special case: consider pure insurance payoff (and constantMPR µ−r

σ ), then no (explicit) dependence on x

πvt +

(a− ρbµ−rσ

)πvy + 1

2 b2πvyy + 1

2

γ

X0ert(1− ρ2)(bπv

y )2 − rπv = 0

Note that for ρ 6= 0 the pde is different from the “pure insurance” pde

Via correlation with financial market we can still hedge part of theinsurance risk

Note: incomplete market ⇒ no martingale representation, thereforedelta-hedge is not πv

x

In fact: hold ρbπvy/σ in x as hedge

Economic explanation for drift-adjustment in y , a kind of“quanto-adjustment”



Pure Insurance Payoff (2)

General solution via log-transform:

πv(t, y) =X0ert

γ(1− ρ2)lnEP

[e

γ(1−ρ2)

X0erT f (y(T ))

∣∣∣∣∣y(t) = y

]

Measure P induces drift-adjusted process for y

See also, [Henderson, 2002] and [Musiela and Zariphopoulou, 2004]who derived this solution in the context of exponential indifferencepricing

We can generalise to Mean Value Principle for any convex functionv()



Numerical Illustration

Consider “unit-linked” insurance contract with payoff:y(T )S(T ) = y(T )ex(T )

Numerical calculation in quadrinomial tree with 5 time-steps of 1 year

“Naive” hedge is to hold y(t) units of share S(t)

In fact: hold πvx + ρbπv

y/σ as hedge

MC Variance hedge also builds additional reserve as “buffer” againstunhedgeable risk

0 30

0.50-50.00

0.00

50.00

100.00

150.00

Payoff at T=5

-0.50

-0.30

-0.10

0.10

0.30

0.50

-150.00

-100.00

-50.00

0.00

50.00

100.00

150.00

51.8870.03

94.53127.60

172.25232.51

Insurance Risk "y"

Share Price

Payoff at T=5

0 40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

Fin Delta t=4

-0.40

-0.20

0.00

0.20

0.40

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

59.1679.85

107.79

145.50

196.40

Insurance Risk "y"

Share Price

Fin Delta t=4



Numerical Illustration (2)

Investigate impact of correlation ρ

Compare ρ = 0.50 (left) and ρ = 0 (right)

0 40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

Fin Delta t=4

-0.40

-0.20

0.00

0.20

0.40

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

59.1679.85

107.79

145.50

196.40

Insurance Risk "y"

Share Price

Fin Delta t=4

0 40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

Fin Delta t=4

-0.40

-0.20

0.00

0.20

0.40

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

59.1679.85

107.79

145.50

196.40

Insurance Risk "y"

Share Price

Fin Delta t=4

Positive correlation leads to higher delta, as this also hedges part ofinsurance risk: hold πv

x + ρbπvy/σ as hedge

Price for ρ = 0.00 at t = 0 is e26.75

Price for ρ = 0.50 at t = 0 is e18.79, due to less unhedgeable risk

Price for ρ = 0.99 at t = 0 is −e1.98, due to drift-adjustment


Financial & Insurance Risk MC Standard-Deviation Pricing

MC Standard-Deviation Pricing

Again, do two-step construction

First: condition on financial risk & use actuarial pricing for “pureinsurance” risk

πs(t + ∆t|x±) := E[πs(t + ∆t)|x±] + δ√

∆t√Var[πs(t + ∆t)|x±]

Second: do no-arbitrage valuation under Q

This leads to linear pricing pde (if πs(t, x , y) monotone in y):

πst + (r − 1

2σ2)πs

x +(

a− ρbµ−rσ ± δ√

1− ρ2 b)πsy+

12σ

2πsxx + ρσbπs

xy + 12 b2πs

yy − rπs = 0

Drift adjustment for y is now combination of “hedge cost” plus“upwind” risk-adjustment ±δ

√1− ρ2 b


Financial & Insurance Risk MC Davis Price

MC Davis Price

We can again consider Davis price, by “small perturbation” expansion

This leads to pricing pde:

πDt + (r − 1

2σ2)πD

x +(

a− ρbµ−rσ + γX0ert

(1− ρ2)b2πvy

)πDy +

12σ

2πDxx + ρσbπD

xy + 12 b2πD

yy − rπD = 0

Drift adjustment for y is now combination of “hedge cost” plusrisk-adjustment γ

X0ert(1− ρ2)b2πv

y

Davis price defined relative to existing price πv(t, x , y)

Note, st.dev. pricing depends on√

(1− ρ2) b


Financial & Insurance Risk Multi-dimensional MC Variance Price

Multi-dimensional MC Variance Price

Vectors x of asset returns (n-vector), y of insurance risks (m-vector)

dx = µ dt + Σ12 · dWf

dy = a dt + B12 · dW

Partitioned covariance matrix C (n + m)× (n + m)P is n ×m matrix of financial & insurance covariances

C =

(Σ PP ′ B

)The market-price of financial risks is an n-vector Σ−1(µ− r)Multi-dim pricing pde for πv:

πvt + r ′πv

x +(

a− P ′Σ−1(µ− r))′πvy + 1

2

(Cijπ

vij

)+

12

γ

X0ert

(πvy′(B − P ′Σ−1P)πv

y

)− rπv = 0

Note: (B − P ′Σ−1P) is conditional covariance matrix of y |xA. Pelsser (Maastricht U) TC & MC Valuations 30 Apr 2010 – DGVFM 30 / 34

Financial & Insurance Risk Multi-dimensional MC StDev Price

Multi-dimensional MC StDev Price

Multi-dim pricing pde for πs:

πst + r ′πs

x +(

a− P ′Σ−1(µ− r))′πsy + 1

2

(Cijπ

sij

)+

12δ√πsy′(B − P ′Σ−1P)πs

y − rπs = 0

Note: unlike 1-dim case, does not simplify to linear pde

Simplification only possible if (B − P ′Σ−1P) has rank 1 and all πsi

have same sign


Conclusions

Conclusions

1 Pure Insurance Risk

Variance Principle (→ exponential indiff. pricing)Standard-Dev. Principle (→ E[] under new measure)Cost-of-Capital Principle (→ St.Dev price)Davis Price: St.Dev. price is “small perturbation” of Variance price

2 Financial & Insurance Risk

Market-Consistent Pricing: via two-step conditional expectationsMC Variance PrincipleNumerical Illustration for “unit-linked” contract


References

References I

Black, F. and Scholes, M. (1973).The pricing of options and corporate liabilities.Journal of Political Economy, 81:637 – 659.

Carmona, R. (2008).Indifference Pricing: Theory and Applications.Princeton Univ Press.

Davis, M. (1997).Option pricing in incomplete markets.In Dempster, M. and Pliska, S., editors, Mathematics of DerivativeSecurities, pages 216–226. Cambridge Univ Pr.

El Karoui, N., Peng, S., and Quenez, M. (1997).Backward stochastic differential equations in finance.Mathematical finance, 7(1):1–71.


References

References II

Henderson, V. (2002).Valuation of claims on nontraded assets using utility maximization.Mathematical Finance, 12(4):351–373.

Jobert, A. and Rogers, L. (2008).Valuations and dynamic convex risk measures.Mathematical Finance, 18(1):1–22.

Malamud, S., Trubowitz, E., and Wuthrich, M. (2008).Market consistent pricing of insurance products.ASTIN Bulletin, 38(2):483–526.

Musiela, M. and Zariphopoulou, T. (2004).An example of indifference prices under exponential preferences.Finance and Stochastics, 8(2):229–239.


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