Time-Consistent and Market-Consistent Actuarial
Valuations
Antoon Pelsser1
1 Maastricht University & Netspar
Email: [email protected]
30 April 2010
DGVFM Scientific Day – Bremen
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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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Ambitions
In this paper I want to combine
1
2
Time-Consistent pricing operators, see [Jobert and Rogers, 2008]
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 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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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 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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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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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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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
αf (y (T )) v
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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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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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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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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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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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 + 12 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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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 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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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
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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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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
2 v
D
D
1 2 D
b
π
πtD + a +
y πy + 2 b πyy + r π = 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
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
+
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√
+σ √∆t with P-prob.
−σ ∆t with P-prob.
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2
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2
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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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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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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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MC Variance Pricing
Intuition: construct MC pricing in two steps using conditional
expectations, see also: [Carmona, 2008], Chap. 1
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
v
E[π (t + ∆t)|x−] = 2 π−+
+ 1+ρ
π−−
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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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
= e −r ∆t 21 1 − µ−r
∆t π (t + ∆t|x+) +
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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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
πtv + a − ρb µ−r
πy + 2 b πyy +
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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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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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
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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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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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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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MC Davis Price
We can again consider Davis price, by “small perturbation” expansion
This leads to pricing pde:
πtD + (r − 21 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
+ 21 b 2 πyy
+ ρσbπxy
− 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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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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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
s
1
s0
0 −1
s
2 δ πy (B − P Σ P)πy − r π = 0
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
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 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 Derivative
Securities, 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.
A. Pelsser (Maastricht U)
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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 Wüthrich, 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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