Time-Consistent and Market-Consistent Actuarial
Valuations
Antoon Pelsser1
Mitja Stadje2
1 Maastricht
University & Kleynen Consultants & Netspar
Email: [email protected]
2 Tilburg
University & Netspar
23 May 2013
ASTIN Conference – Den Haag
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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
Pricing
Pricing
Pricing
very long-dated cash flows T ∼ 30 − 100 years
long-dated options T > 5 years
pension & insurance liabilities
employee stock-options
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Introduction
Ambitions
In this joint paper with Mitja Stadje: [Pelsser and Stadje, 2013] we
want to combine
1
2
Time-Consistent pricing operators, see [Jobert and Rogers, 2008]
Market-Consistent pricing operators, see [Malamud et al., 2008]
We will be interested in continuous-time limits of these discrete
algorithms for different actuarial premium principles:
1
2
3
4
Variance Principle
Mean Value Principle
Standard-Deviation Principle
Cost-of-Capital Principle
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Introduction
Content of This Talk
1
Pure Insurance Risk
Diffusion Model for Insurance Risk
Variance 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 Risk
Market-Consistent Pricing
Variance Principle
Numerical Illustration
3
Conclusions
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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:
√
+b √∆t with prob.
y (t + ∆t) = y (t) + a∆t +
−b ∆t with prob.
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1
2
1
2
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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 claim
half-way at time s for price π(s, y (s))
“Semi-group property”
Similar idea as “tower property” of conditional expectation
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Pure Insurance Risk
Variance Principle
Variance Principle
Actuarial Variance Principle Πv :
Πvt [f (y (T ))] = Et [f (y (T ))] + 21 αVart [f (y (T ))]
α is Absolute Risk Aversion
Apply Πv to one binomial time-step to obtain price π v :
π v t, y (t) = Et [π v t + ∆t, y (t + ∆t) ] +
v
1
2 αVart [π
t + ∆t, y (t + ∆t) ]
Note: we omit discounting for now
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Pure Insurance Risk
Variance Principle
Pricing PDE
Assume π v (t, y ) admits Taylor approximation in y
Evaluate Var.Princ. for binomial step & take limit for ∆t → 0
Same as derivation of Feynman-Kaç, but for E[] + 12 αVar[]
This leads to pde for π v :
v
+ 12 α(bπyv )2 = 0
πtv + aπyv + 21 b 2 πyy
Note, non-linear term = “local unhedgeable variance” b 2 (πyv )2
Find general solution to this non-linear pde via log-transform:
h
i
1
v
αf (y (T )) π (t, y ) = ln Et e
y (t) = y .
α
Exponential indifference price, see [Henderson, 2002] or
[Musiela and Zariphopoulou, 2004]
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Pure Insurance Risk
Variance Principle
Include Discounting
We should include discounting into our pricing
Absolute Risk Aversion α is not “unit-free”, but has unit 1/e
This 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” X0 e rT
Explicit notation: α → γ/X0 e rT leads to pde:
v
πtv + aπyv + 12 b 2 πyy
+
v 2
1 γ
2 X e rt (bπy )
0
− r πv = 0
γ
f (y (T )) X0 e rt
X0 e rT
π (t, y ) =
ln E e
y (t) = y
γ
v
Note: express all prices in discounted terms
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Pure Insurance Risk
Variance Principle
Backward Stochastic Differential Equations
Pricing PDE:
v
πtv + aπyv + 12 b 2 πyy
+
v 2
1 γ
2 X e rt (bπy )
0
− r πv = 0
This non-linear PDE, represents the solution to a so-called BSDE for
the 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 ) = 21 X0γe rt Z 2 − rY .
Recent literature studies uniqueness & existence of solutions to
BSDE’s, see [El Karoui et al., 1997]
Via BSDE’s we can study time-consistent pricing operators in a much
more general stochastic setting. But we will not pursue this here.
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Pure Insurance Risk
Mean Value Principle
Mean Value Principle
Generalise to Mean Value Principle
−1
Πm
(Et [v (f (y (T )))])
t [f (y (T ))] = v
for any function v () which is a convex and increasing
Exponential pricing is special case with v (x) = e αx
Do Taylor-expansion & limit ∆t → 0:
mf
+
πtmf + aπymf + 12 b 2 πyy
00 mf
mf 2
1 v (π )
2 v 0 (π mf ) (bπy )
=0
Note: π mf (t, y ) := π m (t, y )/e rt is price expressed in discounted terms
Interpretation as generalised Variance Principle with “local risk
aversion” term: v 00 ()/v 0 ()
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Pure Insurance Risk
Standard-Deviation Principle
Standard-Deviation Principle
Actuarial Standard-Deviation Principle:
Πst [f (y (T ))] = Et [f (y (T ))] + β
p
Vart [f (y (T ))].
Pay attention to “time-scales”:
Expectation scales with
√ ∆t
St.Dev. scales with ∆t
√
Thus, we should take β ∆t to get well-defined limit
√
Note: β has unit 1/ time
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Pure Insurance Risk
Standard-Deviation Principle
Pricing PDE
Do Taylor-expansion & limit ∆t → 0:
q
s
πts + aπys + 12 b 2 πyy
+ β (bπys )2 − r π s = 0
Again, non-linear pde. But if π s is monotone in y then
s
πts + (a ± βb)πys + 21 b 2 πyy
− r πs = 0
π s (t, y ) = ESt [f (y (T ))|y (t) = y ]
“Upwind” drift-adjustment into direction of risk
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Pure Insurance Risk
Cost-of-Capital Principle
Cost-of-Capital Principle
Cost-of-Capital principle, popular by practitioners
Used in QIS5-study conducted by EIOPA
Idea: hold buffer-capital against unhedgeable risks. Borrow from
shareholders by giving “excess return” δ
Define buffer via Value-at-Risk measure:
h
i
Πct [f (y (T ))] = Et [f (y (T ))] + δVaRq,t f (y (T )) − Et [f (y (T ))] .
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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/ ∆t
Then, δ is like interest
√ rate,√so multiply with ∆t
Net 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 π s but with β = Φ−1 (q)δ.
Conclusion: In the limit for ∆t → 0, CoC pricing is the same as
st.dev. pricing (for a diffusion process!)
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Pure Insurance Risk
Davis Price
Davis Price
The variance price π v is “hard” to calculate, the st.dev. price π s 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 π v (t, y ), now add
“small” position with price επ D (t, y ). Subst. into pde:
v
D
(πtv + επtD ) + a(πyv + επyD ) + 12 b 2 (πyy
+ επyy
)+
2
1 γ
(πyv )2 + 2επyv πyD + ε2 (πyD )2 − r (π v + επ D ) = 0
2 X e rt b
0
π v () solves the pde, cancel π v -terms
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Pure Insurance Risk
Davis Price
Pricing PDE
Simplify pde, and divide by ε:
D
+
πtD + aπyD + 12 b 2 πyy
2
1 γ
2 X e rt b
0
2πyv πyD + ε(πyD )2 − r π D = 0
Approximation: ignore “small” ε-term
γ
D
2 v
D
πt + a +
b πy πyD + 12 b 2 πyy
+ r πD = 0
X0 e rt
π D (t, y ) = ED
t [f (y (T ))|y (t) = y ]
Davis price π D is defined only “relative” to existing price π v of
insurance portfolio
Note, drift-adjustment of st.dev. price scales with b
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Financial & Insurance Risk
Financial & Insurance Risk
Investigate environment with financial risk that can be traded (and
hedged!) in financial market and non-traded insurance risk
Model financial risk as [Black and Scholes, 1973] economy. Model
return process xt = ln St under real-world measure P:
dx = µ(t, x) − 21 σ 2 (t, x) dt + σ(t, x) dWf
Binomial time-step:
x(t + ∆t) = x(t) + (µ −
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2 σ )∆t
+
TC & MC Valuations
√
+σ √∆t with P-prob.
−σ ∆t with P-prob.
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2
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Financial & Insurance Risk
No-arbitrage pricing
BS-economy is arbitrage-free and complete ⇔ unique martingale
measure Q.
No-arbitrage pricing operator for financial derivative F (x(T )):
π Q (t, x) = e −r (T −t) EQ
t [F (x(T ))]
Binomial step for x under measure Q:
x(t + ∆t) = x(t) + (µ − 12 σ 2 )∆t+

√
 +σ ∆t with Q-prob.
√
 −σ ∆t with Q-prob.
1
2
1
2
1−
1+
√
µ−r
∆t
σ
√
µ−r
∆t
σ
Quantity (µ − r )/σ is Radon-Nikodym exponent of dQ/dP
Quantity (µ − r )/σ is also known as market-price of financial risk.
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Financial & Insurance Risk
Quadrinomial Tree
Joint discretisation for processes x and y using “quadrinomial” tree
with correlation ρ under measure P:
State:
x + ∆x
x − ∆x
y − ∆y
1−ρ
4
-%
•
. &
1−ρ
1+ρ
4
4
y + ∆y
1+ρ
4
Positive correlation increases probability of joint “++” or “−−”
co-movement
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Financial & Insurance Risk
Market-Consistent Pricing
Market-Consistent Pricing
We are looking for market-consistent pricing operators, see
e.g. [Malamud et al., 2008]
Definition
A pricing operator π() is market-consistent if for any financial derivative
F (x(T )) and any other claim G (t, x, y ) we have
πF +G (t, x, y ) = e −r (T −t) EQ
t [F (x(T ))] + πG (t, x, y ).
Observation: generalised notion of “translation invariance” for all
financial risks
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Financial & Insurance Risk
MC Variance Pricing
MC Variance Pricing
Intuition: construct MC pricing in two steps using conditional
expectations. For full results we refer to [Pelsser and Stadje, 2013].
First: condition on financial risk & use actuarial pricing for “pure
insurance” risk
γ
π v (t +∆t|x±) := E[π v (t +∆t)|x±]+ 12
Var[π v (t +∆t)|x±]
r
(t+∆t)
X0 e
v
v
π++
+ 1−ρ
π+−
2
2
2
v
v
Var[π v (t + ∆t)|x+] = 1−ρ
π++
− π+−
4
1+ρ
v
v
E[π v (t + ∆t)|x−] = 1−ρ
π
+
π−−
−+
2
2
2
2
v
v
Var[π v (t + ∆t)|x−] = 1−ρ
π−+
− π−−
.
4
E[π v (t + ∆t)|x+] =
1+ρ
2
For ρ = 1 or ρ = −1 no unhedgeable risk left ⇒ Var = 0
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Financial & Insurance Risk
MC Variance Pricing
Pricing PDE
Second: use no-arbitrage pricing for “artificial” financial derivative
π v (t, x, y ) = e −r ∆t EQ [π v (t + ∆t|x±)]
√ v
∆t π (t + ∆t|x+) +
= e −r ∆t 21 1 − µ−r
σ
√
µ−r
v
1
1
+
∆t
π
(t
+
∆t|x−)
2
σ
Do Taylor-expansion & limit ∆t → 0:
v
πtv + (r − 12 σ 2 )πxv + a − ρb µ−r
πy +
σ
v
2
v 2
v
1 2 v
1 2 v
1 γ
2 σ πxx + ρσbπxy + 2 b πyy + 2 X e rt (1 − ρ )(bπy ) − r π = 0
0
Impact on x: “Q-drift” (r − 12 σ 2 )
Impact on y : adjusted drift a − ρb µ−r
σ
Non-linear term for “locally unhedgeable variance” (1 − ρ2 )(bπyv )2
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Financial & Insurance Risk
MC Variance Pricing
Pure Insurance Payoff
Unfortunately, we cannot solve the non-linear pde in general
Special case: consider pure insurance payoff (and constant
MPR µ−r
σ ), then no (explicit) dependence on x
v 1 2 v
πy + 2 b πyy +
πtv + a − ρb µ−r
σ
1 γ
2 X e rt (1
0
− ρ2 )(bπyv )2 − r π v = 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 the
insurance risk
Note: incomplete market ⇒ no martingale representation, therefore
delta-hedge is not πxv
In fact: hold ρbπyv /σ in x as hedge
Economic explanation for drift-adjustment in y , a kind of
“quanto-adjustment”
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Financial & Insurance Risk
MC Variance Pricing
Pure Insurance Payoff (2)
General solution via log-transform:
#
"
γ(1−ρ2 )
rt
f
(y
(T
))
X
e
0
π v (t, y ) =
ln EP̃ e X0 e rT
y (t) = y
γ(1 − ρ2 )
Measure P̃ induces drift-adjusted process for y
See also, [Henderson, 2002] and [Musiela and Zariphopoulou, 2004]
who derived this solution in the context of exponential indifference
pricing
We can generalise to Mean Value Principle for any convex function
v ()
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Financial & Insurance Risk
MC Variance Pricing
Numerical Illustration
Consider “unit-linked” insurance contract with payoff:
y (T )S(T ) = y (T )e x(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 πxv + ρbπyv /σ as hedge
MC Variance hedge also builds additional reserve as “buffer” against
unhedgeable risk
Fin Delta t=4
Payoff at T=5
1.00
150.00
0.80
100.00
50.00
0.00
0.50
0.30
0
30
0.10
-0.10
-50.00
-100.00
-150.00
51.88
70.03
-0.30
94.53
127.60
172.25
-0.50
Insurance Risk "y"
0.60
0.40
0.20
0.00
-0.20
-0.40
-0.60
-0.80
-1.00
232.51
Share Price
A. Pelsser (Maastricht U)
59.16
0.40
0
40
0.20
0.00
Insurance Risk "y"
-0.20
79.85
107.79
Share Price
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145.50
-0.40
196.40
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Financial & Insurance Risk
MC Variance Pricing
Numerical Illustration (2)
Investigate impact of correlation ρ
Compare ρ = 0.50 (left) and ρ = 0 (right)
Fin Delta t=4
Fin Delta t=4
1.00
1.00
0.80
0.80
0.60
0.40
0.20
0.00
-0.20
-0.40
-0.60
-0.80
-1.00
0.60
0.40
0.20
0.00
-0.20
-0.40
-0.60
-0.80
-1.00
59.16
0.40
0
40
0.20
0.00
Insurance Risk "y"
-0.20
79.85
107.79
Share Price
145.50
-0.40
196.40
59.16
0.40
0
40
0.20
0.00
Insurance Risk "y"
-0.20
79.85
107.79
Share Price
145.50
-0.40
196.40
Positive correlation leads to higher delta, as this also hedges part of
insurance risk: hold πxv + ρbπyv /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
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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 “pure
insurance” risk
√ p
π s (t + ∆t|x±) := E[π s (t + ∆t)|x±] + δ ∆t Var[π s (t + ∆t)|x±]
Second: do no-arbitrage valuation under Q
This leads to linear pricing pde (if π s (t, x, y ) monotone in y ):
p
2 b πs +
±
δ
1
−
ρ
πts + (r − 21 σ 2 )πxs + a − ρb µ−r
y
σ
1 2 s
2 σ πxx
s
s
+ ρσbπxy
+ 12 b 2 πyy
− r πs = 0
Drift adjustment for y is now p
combination of “hedge cost” plus
“upwind” risk-adjustment ±δ 1 − ρ2 b
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Financial & Insurance Risk
MC Davis Price
MC Davis Price
We can again consider Davis price, by “small perturbation” expansion
This leads to pricing pde:
πtD + (r − 12 σ 2 )πxD + a − ρb µ−r
σ +
1 2 D
2 σ πxx
γ
X0 e rt (1
− ρ2 )b 2 πyv πyD +
D
D
+ ρσbπxy
+ 21 b 2 πyy
− r πD = 0
Drift adjustment for y is now combination of “hedge cost” plus
risk-adjustment X0γe rt (1 − ρ2 )b 2 πyv
Davis price defined relative to existing price π v (t, x, y )
p
Note, st.dev. pricing depends on (1 − ρ2 ) b
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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)
1
dx = µ dt + Σ 2 · dWf
1
dy = a dt + B 2 · dW
Partitioned covariance matrix C (n + m) × (n + m)
P is n × m matrix of financial & insurance
covariances
Σ P
C=
P0 B
The market-price of financial risks is an n-vector Σ−1 (µ − r )
Multi-dim pricing pde for π v :
0
πtv + r 0 πxv + a − P 0 Σ−1 (µ − r ) πyv + 12 Cij πijv +
v0
0 −1
v
v
1 γ
π
(B
−
P
Σ
P)π
y
y − rπ = 0
2 X e rt
0
Note: (B − P 0 Σ−1 P) is conditional covariance matrix of y |x
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Financial & Insurance Risk
Multi-dimensional MC StDev Price
Multi-dimensional MC StDev Price
Multi-dim pricing pde for π s :
0
πts + r 0 πxs + a − P 0 Σ−1 (µ − r ) πys + 12 Cij πijs +
q
1
δ
πys 0 (B − P 0 Σ−1 P)πys − r π s = 0
2
Note: unlike 1-dim case, does not simplify to linear pde
Simplification only possible if (B − P 0 Σ−1 P) has rank 1 and all πis
have same sign
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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 expectations
MC Variance Principle
Numerical Illustration for “unit-linked” contract
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References
References I
Black, F. and Scholes, M. (1973).
The pricing of options and corporate liabilities.
Journal of Political Economy, 81:637–659.
Davis, M. (1997).
Option pricing in incomplete markets.
In Dempster, M. and Pliska, S., editors, Mathematics of Derivative
Securities, pages 216–226. Cambridge University Press.
El Karoui, N., Peng, S., and Quenez, M. (1997).
Backward stochastic differential equations in finance.
Mathematical finance, 7(1):1–71.
Henderson, V. (2002).
Valuation of claims on nontraded assets using utility maximization.
Mathematical Finance, 12(4):351–373.
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References
References II
Jobert, A. and Rogers, L. (2008).
Valuations and dynamic convex risk measures.
Mathematical Finance, 18(1):1–22.
Malamud, S., Trubowitz, E., and Wüthrich, M. (2008).
Market consistent pricing of insurance products.
ASTIN Bulletin, 38(2):483–526.
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