on the calculation of the solvency capital re- quirement

Seminarreihe Energy & Finance – May 26th, 2010Lehrstuhl für Energiehandel und Finanzdienstleistungen

150

200

250

300

350

400

0 0.5 1 1.5 2 2.5 3

On the Calculation of the Solvency Capital Re-quirement based on Internal Models

Daniel Bauer (gsu rmi)(with Daniela Bergmann (uni ulm) & Andreas Reuß (ifa ulm))

Seminarreihe Energy & Finance @ UNI DUE | May 26th, 2010 | Daniel Bauer

Background on Solvency II

Nested Simulations

Optimal Allocation of Computational Budget

Confidence Intervals for the SCR

Screening Procedures

Least-Squares Monte Carlo Approach

Conclusion

Seminarreihe Energy & Finance @ UNI DUE | May 26th, 2010 | Daniel Bauer Background on Solvency II


Nested Simulations





Conclusion


Solvency II Capital Requirement in a Nutshell

→ Consideration over a one-year horizon – now: t = 0 / then: t = 1

Solvency – intuitionAn insurer is solvent if

P ([MV Liabilities @ t=1] > [MV Assets @ t=1]) ≤ .5%

I What is the MV Liabilities? Implicit Definition (CFO Forum, 2008):

[MV Liabilities @ t=1] = [MV Assets @ t=1]− [MCEV],

where MCEV ≈ present value of the firm (from shareholders’ persp.)≈ Available Capital (AC):

[AC] = [Equity] + EQ [Disc. Fut. Cash-Flows from the Ins. Business]

⇒ An insurer is solvent if (i one-year risk-free rate @ t = 0)

.5% ≥ P ([AC]1 < 0) = P(−[AC]1

1 + i> 0

)= P

[AC]0 −

[AC]11 + i︸︷︷︸

=:L

> [AC]0


So...... the company is considered fine if there is more available capital at time

zero [AC]0 than the Value at Risk (V@R) of L at the 99.5% levelI Caveat : The Definition of L depends on [AC]0

... let’s call the [AC]0 on the right hand side the Solvency CapitalRequirement (SCR) – then that’s approximately what [AC]0 should be

Solvency Capital Requirement

SCR = argminx {P(L > x) ≤ 0.5%} = V @R99.5%

... We have

SCR = argminx {P(−[AC]1 > (1 + i) · (x − [AC]0) ≤ 0.5%} ,

so we only need to determine the 99.5%-quantile of −[AC]1, where

[AC]1 = [Equity]1 + [CF @ 1] + EQ [Disc. Fut. CFs from Ins. Bus.| F1]

is a random variable known at time t = 1, i.e. we need to assess itsdistribution.

Seminarreihe Energy & Finance @ UNI DUE | May 26th, 2010 | Daniel Bauer Nested Simulations


Nested Simulations





Conclusion


Model FrameworkI d-dimensional Markov process Y = (Yt )t∈[0,T ] drives financial assetsI Market-value balance sheet:

Assets LiabilitiesAt = A(t ,Yt ) Lt ("technical res."/insureds’ money)

Rt (equity/"free reserves")∑At =

∑A(t ,Yt )

∑At =

∑A(t ,Yt )

I Q risk-neutral measure with respect to Numéraire Bt = exp{∫ t

0 rs ds}I Cash-flows from insurance business (from insurer’s point of view→ profits):X = (X1, . . . ,XT ) where Xt = ft (Ys, s ∈ [0, t ])(Limit focus to market risk – non-financial risk factors could be incorporated by appropriatechoice of Y , f·, and Q. Default put option should not be considered.)

I [AC]1 =∑

R1 + X1 + EQ[

T∑

t=2

exp

{−∫ t

1rs ds

}Xt

∣∣∣∣∣Ys, s ∈ [0,1]

]

︸︷︷︸=:V1


Determination of V1 on per-policy level

Bauer, Bergmann & Kiesel (2010, ASTINBull.):Due to the way insurance contracts are administrated in a company,it is sufficient to know the values of certain state variables/accountsD (e.g. death benefit account, net survivor value,...) to determine thevalue of a contract→ (possibly high, but) finite dimensional Markov state space

⇒ V1 = EQ[

T∑

t=2

exp

{−∫ t

1rs ds

}Xt

∣∣∣∣∣ (Ys)0≤s≤1

]

= EQ[

T∑

t=2

exp

{−∫ t

1rs ds

}Xt

∣∣∣∣∣Y1, D1

]

Generalizes to company level:I Aggregation of portfolio of policies: Consider "representative" contractsI Aggregated ALM approach – state variables are e.g. reserve quota,

balance sheet positions,...


Nested Simulations Approach

t=0 t=1 … t=T

RF1

Y(1), D(1)

Y(i), D(i)

Y(N), D(N)

P Q

Simulate N first-year paths "under P"Simulate K1 paths "under Q" starting in Y i

1, Di1 to determine V i

1N × K1 paths

→ Determine quantile via empirical distribution function

I Gordy & Juneja (2010,ManSci): Diversification of error in inner step whenestimating portfolio V@R

→ Not true in insurance context – different contracts evaluated based onsame (asset & liability) scenarios


Example: Participating contract model from Bauer et al.(2006, IME) (Must-Case)

I Term-fix insurance, interest rate guarantee (cliquet-style), participationscheme (can serve as model for company, cf. Kling et al. (2007,IME))

I Underlying portfolio St . Generalized Black-Scholes model for (St , rt ) withOrnstein-Uhlenbeck interest rates (Vasicek model)

I Guaranteed rate g, accounting par. y , participation rate δ:I A−i+1 = A+

i × (Si+1/Si),I Li+1 = Li (1 + g) +

[δ y(A−i+1 − A+

i

)− g Li

]+,I di+1 = (1− δ) y (A−i+1 − A+

i ) 1{δ y(A−i+1−A+i )>g Li}

+(y (A−i+1 − A+i )− g Li)

+ 1{δ y(A−i+1−A+i )≤g Li}

I ci+1 = (Li+1 − A−i+1)+ (no defaults!)

I A+i+1 = A−i+1 − di+1 + ci+1

⇒ X0 = 0, Xi = di − ci , i = 0, . . . ,T − 1, XT = (A−T − LT )− cT = RT − cT ,

−→ Di =(

Li , xi =A+

i −Li

Li

), Yi = (Si , ri )

I For details see Bauer, Kling, Kiesel & Russ (2006, IME) and Zaglauer &Bauer (2008, IME)


Bias in Nested Simulations (K0 = 250,000; N = 100,000)On the Calculation of the SCR based on Internal Models 15

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.001

-6000 -4000 -2000 0 2000 4000 6000

Loss (L)

K1 = 1K1 = 5

K1 = 10K1 = 100

K1 = 1000

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.001

-6000 -4000 -2000 0 2000 4000 6000

Loss (L)

K1 = 1K1 = 5

K1 = 10K1 = 100

K1 = 1000

Figure 2: Empirical density function for different choices of K1 for the estimator based onthe policyholders’ cash flows (left) and the shareholders’ cash flows (right), N = 100, 000,K0 = 250, 000

4.2 Results

In Sections 3 and ??, we introduced different methods on how to estimate the SCR in ourframework. In what follows, we implement them in the setup described in Section 4.1.In particular, we focus on contemplating pitfalls, drawbacks, as well as advantages of thedifferent methods.

4.2.1 Nested Simulations Approach

As indicated in Section 3.3, the estimation of the SCR using Nested Simulations is biased.This bias mainly depends on the choice of the estimator and the number of inner simulations.Hence, in order to develop an idea for the magnitude of this bias, we analyze the results for theestimator based on cash flows from the policyholders’ and from the shareholders’ perspective(see Section 4.1.2) and choose different numbers of inner simulations. If not noted otherwise,we fix K0 = 250, 000 sample paths for the estimation of V0, N = 100, 000 realizations for the

simulation over the first year, and choose K(i)1 = K1 ∀1 ≤ i ≤ N .

In Figure 2, the empirical density functions for both estimators and different choices ofK1 are plotted. As expected, for both estimators the distribution is more dispersed for smallK1, which has a tremendous impact on our problem of estimating a quantile in the tail:We significantly overestimate the SCR for small choices of K1. This can also be noticed inTable 2, where the estimated SCR for different choices of K1 is displayed. Moreover, weobserve that the distribution given by the estimator based on shareholders’ cash flows ismore dispersed than the estimator for the policyholders’ cashflows for the same K1. Since

the bias mainly depends on the variance of V(i)1 (K

(i)1 ), 1 ≤ i ≤ N , this indicates that this

estimator has higher variances and thus, we need more inner simulations to obtain reliableresults. This can also be seen in Table 2, where the SCR estimated via shareholder cashflows always exceeds the SCR derived from policyholders’ cash flows. Further analyses showthat in our setting, the estimator based on cash flows from the policyholders’ perspectiveis always superior to that based on shareholders’ cash flows except for some very extreme(and unrealistic) parameter choices in the contract model. Therefore, we will rely on theestimator based on cash flows from the policyholders’ perspective in the remainder of this

K1 SCR [AC]0/SCR1 3,432.5 55%5 1,874.6.2 100%10 1,606.5 117%

100 1,279.1 147%1,000 1,254.6 149%

→ Choice of K1 significantly affects SCR!

Seminarreihe Energy & Finance @ UNI DUE | May 26th, 2010 | Daniel Bauer Optimal Allocation of Computational Budget


Nested Simulations





Conclusion


Bias (adapted from Gordy & Juneja (2010, ManSci))

L(Y1,D1) = [AC]0 −∑

R1 + X1

1 + i− 1

1 + iV1,

L(Y1,D1) = [AC](K0)

0 −∑

R1 + X1

1 + i− 1

1 + i1K1

K1∑k=1

T∑t=2

exp{−∫ t

1r (k)s ds

}X (k)

t ,

and L(b99.5%×Nc+1) the (b99.5%× Nc+ 1)-st order statistics of our nestedsimulation samples L(Y (i)

1 ,D(i)1 ), 1 ≤ i ≤ N

Then, under some regularity conditions

BIAS = E[L(b99.5%·Nc+1)

]− SCR

=−0.5 ∂

∂u

[f (u)E

[Var(

L− L∣∣∣Y1,D1

)∣∣∣ L = u]]

u=SCR

f (SCR)︸︷︷︸= θα

K1 f (SCR)

+ON (1/N) + . . .

where f is density of LI θα > 0⇒ Bias positive!


Choice of N vs. KMinimize Mean square error (neglecting higher order terms):

σ20

K0︸︷︷︸Var

[[AC]

(K0)

0

]

+θ2α

K 21 × f 2(SCR)

+99.5%× 0.5%

(N + 2)× f 2(SCR)︸︷︷︸Var

[L(b99.5%×Nc+1)−[AC]

(K0)

0

]

→ min

subject to K0 + N × K1 = const (available computational time)Result: Given K1, the optimal K0 and N are

N = K 21 ×

0.5%× 99.5%

2 θ2α︸︷︷︸

=c1

and K1 ×√

N K1

2× σ0f (SCR)

θα︸︷︷︸c2

I c1 and c2 need to be estimated in pilot simulationI Problem: To make bias small, for 99.5% level K1 may not be chosen "too

small"→ Immense computational effort!


Example (cont.): N × K1 + K0 = 9.75× 108

Estimation of θα via pilot simulation with N = 100,000, K1 = 100 andregression/finite difference approximation:

θα ≈ 0.027⇒ (K1; N; K0) = (300 ; 320,000 ; 1,500,000) approx. optimal

On the Calculation of the SCR based on Internal Models 17

from the 150 estimation procedures as explained above. The MSE is then estimated by thesum of the empirical variance and the squared estimated bias. This allows us to correct themean by the estimated bias. Figure 3 and Table 3 show our results.

1210

1220

1230

1240

1250

1260

1270

1280

N = 160, 000K1 = 600

N = 320, 000K1 = 300

N = 640, 000K1 = 150

N = 1, 280, 000K1 = 75

SC

R

Figure 3: 150 simulations for different choices of N and K1, K0 = 1, 500, 000, Nested Simu-lations Approach

N K1 Mean Empirical Estimated Estimated Corrected

(SCR) Variance Bias MSE Mean

160,000 600 1247.7 24.6 1.4 26.6 1246.3320,000 300 1249.3 15.8 2.9 24.0 1246.4640,000 150 1251.3 7.9 5.7 40.6 1245.6

1,280,000 75 1257.4 4.2 11.4 133.1 1246.1

Table 3: Choice of N and K1 for the Nested Simulations Approach Approach, K0 =1, 500, 000

As expected, the mean of the estimated SCRs increases as K1 decreases due to theincreased bias. In contrast to this, the empirical variance obviously decreases as N increases.Furthermore, we find that our choice of N and K1 yields the smallest estimated MSE fromthe combinations given in Table 3. Therefore, our choice appears reasonable within ourframework. Moreover, it is remarkable that if we correct the means in Table 3 by thecorresponding bias, the difference between the results for the different combinations is almostnegligible.

Therefore, we will use N = 320, 000 and K1 = 300 in the remaining part of this paper ifnot stated otherwise. With this parameter combination it takes about 16 minutes to carry

→ Based on 120 runs of sims with (N; K1) = (320,000 ; 300)(approx. 30 min each)

SCR Emp. Var. Est. Bias Est. MSE Corr. Estim.1,249.3 15.8 2.9 24.0 1,246.4

Seminarreihe Energy & Finance @ UNI DUE | May 26th, 2010 | Daniel Bauer Confidence Intervals for the SCR


Nested Simulations





Conclusion


I Idea: (cf. Lan, Nelson, and Staum (2007, IEEE))

I{L(j)≤SCR} ∼ Bernoulli(99.5%)⇒∑Nj=1 I{L(j)≤SCR} ∼ Binomial(N,99.5%)

⇒ P(L(n) > SCR

)= P

(∑Nj=1 I{L(j)≤SCR} < n

)=∑n−1

j=0

(Nj

).95j .05N−j

→ Determine r and s such that:

P (Lr ≤ SCR ≤ Ls) =∑s−1

j=r

(Nj

).95j .05N−j ≥ 1− αOUT

I Problem: Error resulting from inner simulation not considered.I Idea: Instead of order statistics of L, for upper and lower bound consider

r th and sth order statistics of

L(j) ± tK0−1,1−

αAC02

σ0√K0± tK1−1,1− ε2

σ(j)1

(1+i)√

K1,

whereε = (1− αAC1 )1/N , αIN = αAC0 + αAC1 − αAC0 × αAC1 , αIN + αOUT = αTOT .

I Then these present a confidence interval for the level αTOT .


Example (cont.): Confidence level αTOT = 90%I As before: (K1; N; K0) = (300 ; 320,000 ; 1,500,000) (optimal choice for point est.)

I αAC0 = 0.1%.


αAC0 αAC1 LB UB UB − LB UB−LB˜SCR

4.0% 1.04% 1062.6 1438.2 375.6 30%2.5% 2.56% 1068.2 1432.8 364.6 29%1.0% 4.04% 1070.8 1430.3 359.4 29%0.1% 4.90% 1071.7 1429.4 357.7 29%0.01% 4.99% 1071.5 1429.6 358.1 29%

Table 4: Confidence intervals for different combinations of αAC0 and αAC1 , αout = αin = 5%

interval is not very pronounced. More precisely, for all cases the length is approximately29-30% of the point estimate.

When choosing αout = 5% as before, we obtain L(r) = 1243.4 and L(s) = 1258.1, i.e. ifthe values of the inner simulation were exact, we would only have a confidence interval oflength 14.7, which corresponds to less than 1.2% of ˜SCR. Thus the remaining part of theconfidence intervals in Table 4 can be attributed to the uncertainty resulting from the innersimulation and the estimation of AC0. Hence, it appears advisable, to reduce αout and toincrease αin. Since in Table 4, we found that the results were best when αAC0 = 0.1%, weuse this value for our experiments. Our results are displayed in Table 5.

αout αin L(r) L(s) L(s)− L(s)−L(r)

SCRLB UB UB− UB−LB

˜SCR

L(r) LB

5% 5% 1243.4 1258.1 14.7 1% 1071.7 1429.4 357.7 29%4% 6% 1243.2 1258.6 15.4 1% 1072.5 1428.3 355.8 28%3% 7% 1242.1 1258.9 16.8 1% 1073.1 1428.0 354.9 28%2% 8% 1241.6 1259.4 17.8 1% 1073.4 1427.6 354.2 28%1% 9% 1240.8 1260.6 19.8 2% 1073.1 1427.7 354.5 28%

Table 5: Confidence intervals for different combinations of αout and αin, αAC1 = 0.1%

As expected the rth order statistic of the losses increases and the sth order statisticdecreases in αout. However, we only observe a slight improvement of the combined confidenceinterval when increasing αin to 8% and decreasing αout to 2%. Similar to above, we findthat the uncertainty arising from the inner simulation is relatively large in comparison tothe uncertainty arising from the outer simulation.

Therefore, it may be conductive to reconsider the choice of N and K1 given fixed αin, αout

and αAC0 . So far, we have relied on the parameters resulting from the optimization in Section??, i.e. these parameters are approximately optimal for the point estimator (5). However,this does not necessarily imply optimality when the objective is to minimize the length ofour confidence interval. Specifically, in Table 5, we observe that the difference in the rth

and the sth order statistic is quite small whereas the uncertainty from the inner simulationis enormous. Whence, in order to obtain shorter confidence intervals, it may be advisableto allocate less budget to the outer simulation and more budget to the inner simulation. Todetermine an appropriate choice of N , K1 and K0, we rely on the optimization approachpresented in Section 4.2.

⇒ CI VERY WIDE. Try to adjust N and K1: (cf. Lan, Nelson, and Staum (2007, IEEE))

I Approximate t quantiles by Normal quantiles and sample standarddeviations by values from pilot simulation

I Assume that "outer" CI decreases in length with the square root of numberof outer samples. (approx. correct for Normal losses)

→ Minimize length based on estimates from pilot simulation


Example (cont.): Confidence level αTOT = 90%,N × K1 + K0 = 9.75× 108

→ (N; K1; K0) = (20,000; 4,732; 2,860,000) approx. optimal

On

the

Calculatio

nof

the

SC

Rbase

don

Internal

Models

25

N K1 K0˜SCR L(r) L(s)

L(s)−L(r)

˜SCRLB UB UB− UB−LB

˜SCR

LB

15,000 6,291 3,135,000 1,265.9 1,220.4 1,314.9 7% 1,186.8 1,346.4 159.6 13%20,000 4,732 2,860,000 1,249.7 1,218.0 1,291.9 6% 1,179.9 1,329.2 149.3 12%25,000 3,794 2,650,000 1,235.5 1,205.6 1,273.8 6% 1,163.8 1,316.4 152.6 12%30,000 3,167 2,490,000 1,228.1 1,208.9 1,263.3 4% 1,161.8 1,310.4 148.7 12%35,000 2,718 2,370,000 1,254.5 1,221.0 1,279.9 5% 1,169.7 1,329.8 160.0 13%

Table 6: Confidence intervals for different combinations of N , K1 and K0, Γ = 97, 500, 000

I Length of CI decreases by more than 50%. "Optimal" choice based onassumptions close to numerical optimum.

I Can further decrease lengths if we additionally rely on variance reductiontechnique within inner simulation (antithetic variates): Lengths of 110.7corresponding to approx. 9% of SCR

⇒ Considerable improvements possible, but CI still too wide for practicalpurposes...

Seminarreihe Energy & Finance @ UNI DUE | May 26th, 2010 | Daniel Bauer Screening Procedures


Nested Simulations





Conclusion


Basic ideaI Run inner procedure twice (based on same outer samples), first based

on fewer (K1,1) inner simulations then based on more (K1,2) innersimulations.

I However, based on the first step, "screen" out samples that are veryunlikely to be in the tail, i.e. only run inner simulations for samples thathave not been screened out in first step.

I Approach: (cf. Lan, Nelson, and Staum (2010, OperRes)

Only keep scenarios in following set

I =

n : #

j : L(n) < L(j) − tf (n,j),1−δ

√(σ

(j)1 )2 + (σ

(n)1 )2

(1 + i)2 K1,1,

< N1 − r + 1

where δ = αSCREEN(N1−r+1)(r−1) and f (n,j) (degrees of freedom) determined by

Welch-Satterthwaite equation.I To reduce number of comparisons, pre-screening based on "maxima"

t-quantile and "maximal" stdev can be applied


Example (cont.): Confidence level αTOT = 90%,N × K1,1 + N × K1,2 + K0 = 9.75× 108

I (K1,1; N; K0) = (300/2 ; 320,000 ; 1,500,000) (K1,2 endogenous)

I Total confidence level (1− αIN − αOUT ), where αOUT = 2%, αIN = 8%I αIN = 1− (1− αSCREEN)(1− αAC0 )(1− αAC1 ) with αAC0 = 0.1%.


the distribution. We do not need the exact loss in those scenarios that do not belong tothe tail. In the next section, we discuss screening procedures to find the tail scenarios ofthe distribution. These procedures allow us to perform more inner simulations for the tailand less for the remaining scenarios to obtain better result with a reasonable computationaleffort.

6.3 Numerical Experiments

As before, we aim for two-sided 90%- confidence intervals for the SCR. In a first step, weanalyze the results of the screening procedure for our base case from Chapter ??, i.e. we fixN1 = 320, 000, K0 = 1, 500, 000, and we use a total budget of Γ = 97, 500, 000. Furthermore,we choose αout = 2%, αin = 8%, and αAC0 = 0.1% as in the previous section about confidenceintervals without screening. The remaining budget in the second run is allocated equally toall surviving scenarios.

To get a first idea, how to choose αscreen, we fix K1,1 = 150, i.e. we use half of the maximalnumber of inner simulations for the first run, and compute the length of the confidenceinterval for different choices of αscreen. Table 7 shows our results. We observe that there is

αscreen αAC1˜SCR LB UB UB − LB UB−LB

˜SCR

1% 6.98% 1,246.2 1,189.7 1,304.2 114.4 9.18%2% 6.03% 1,248.3 1,191.8 1,305.8 113.9 9.13%3% 5.06% 1,246.8 1,190.1 1,305.4 115.3 9.25%4% 4.07% 1,247.8 1,191.5 1,305.9 114.4 9.17%5% 3.06% 1,248.4 1,191.1 1,306.2 115.2 9.23%6% 2.03% 1,247.8 1,189.5 1,306.3 116.7 9.36%7% 0.98% 1,248.0 1,189.9 1,305.7 115.9 9.28%

Table 7: Confidence intervals for different choices of αscreen

hardly any difference for the different choices of αscreen. Thus, we choose αscreen = 4% such

that the error due to screening is similar to the error arising from the estimation of AC(i)1 .

Furthermore, we find in Table 7 that the length of the confidence interval is already reducedfrom 12% for the optimal parameter choice without screening (cf. Table 6 in Section ??) to9% when the screening procedure is applied, although we did not optimize N1 so far. Ofcourse, when comparing the two results, we need to keep in mind that the computation ofthe confidence interval takes longer when screening is applied due to the increased number ofoperations. However, in practical applications, the effort for the projection of the insurer’sassets and liabilities will in general be the primary source of the numerical complexity suchthat the additional effort for screening will be negligible.

Having chosen the different error levels, we now optimize N1 for given K1,1 = 150 andK0 = 1, 500, 000 according to the algorithm described in the previous subsection. Based ona pilot simulation with N1 = 10, 000 samples, we find that N1 ≈ 75, 000 is optimal. In thiscase, we obtain a confidence interval length of 68.8 which corresponds to 6% of ˜SCR.25

25Here and in the remainder of this chapter, ˜SCR is the point estimate resulting from the screening procedureas described at the end of Section 5.1.

I Similar to before: Optimize allocation based on similar assumptions(addtl.: Percentage of survivors is approximately the same in pilot andactual simulation)

I trade-off: Accuracy in screening versus inner simulations in second run


Example (cont.): Confidence level αTOT = 90%,N × K1,1 + N × K1,2 + K0 = 9.75× 108

I αOUT = 2%, αIN = 8%, αAC0 = 0.1%.I (K1,1; K0) = (150; 1,500,000) (N, K1,2 endogenous)

⇒ N1 = 75,000 approx. optimal, Result:[LB; UB] = [1,212.2; 1,281.0], i.e. UB−LB

SCR ≈ 5.53%I Influence of K1,1:


80%

85%

90%

95%

100%

100 200 300 400 500 600 700 800 900 1,000

K1,1

0

10,000

20,000

30,000

40,000

50,000

100 200 300 400 500 600 700 800 900 1,000

K1,2

K1,1

Figure 9: Percentage of scenarios that are (pre-) screened out (left), Number of inner sim-ulations in the second run (K1,2) (right), N1 = 75, 000, K0 = 1, 500, 000, total budgetΓ = 97, 500, 000

in the first four columns of Table 9. We find that the length of the confidence intervalis roughly decreasing up to K1,1 = 500). For larger choices, it slightly increases, but forK1,1 = 1, 000 it decreases again which seems to be a random effect. The optimal N1 givenby the optimization algorithm is rather instable for smaller K1,1. Presumably, this is dueto the rather inaccurate pilot simulation. However, one may guess that there is an upwardstrend up to K1,1 = 300. Here, the gain from screening out more scenarios with a higherK1,1 dominates the drawback of using more computational budget for the first run. Thisinterpretation is also supported by the fifth column of Table 9 where we display the numberof inner simulations in the second run: K1,2 increases although N1 and K1,2 - and hencethe budget used for the first run - increase. For K1,1 ≥ 500, we find that the optimal N1 isdecreasing since the gain from screening out more scenarios with a larger K1,1 does no longercompensate for the higher computational costs within the first run. Therefore, N1 needs tobe decreased. In these cases, the number of inner simulations still increases which partiallycompensates for the higher uncertainty arising from the outer simulation.

To show that our optimization algorithm provides appropriate parameter combinations,we perform some further numerical experiments. For all choices of K1,1, we compute theconfidence interval for the same N1 as in Table 8 (provided the computational budget is highenough). Subsequently, we choose that N1 that leads to the smallest confidence interval.Our results are displayed in the last four columns of Table 9. We find similar effects as forthe parameter combinations given by the optimization algorithm. The good performance ofthe optimization algorithm is also underlined by Figure 10, where we display the length ofthe confidence interval for the parameters from the optimization algorithm as well as for thebest combination from our numerical experiments. We observe that there are only slightdifferences which are in part due to the inaccuracies of the optimization approach. But theoptimal combinations from our numerical experiments are also imprecise because we onlyperformed the numerical experiment for 12 different choices of N1.

Furthermore, Figure 10 shows that the differences for different choices of K1,1 are rathersmall, i.e. for almost all choices of K1,1, we can find an appropriate N1 such that the con-fidence interval is close to minimal. Only, when choosing K1,1 very small, the resultingconfidence interval is wider because in this case, the screening algorithm does not eliminate

I Optimal choice of K1,1 decreases length to about 4.7%I When including variance reduction technique (antithetic variates) length

further decreased to approx. 2% of SCR


Soo...I GOOD NEWS:

I Simulation design matters! Length of confidence interval was decreased bymore than a factor of 14!

I Recent working paper “Efficient Risk Estimation via Nested SequentialSimulation" by M. Broadie, Y. Du, and C. Moallemi (Columbia University)provides adaptive approach that may further increase efficiency

I NOT SO GOOD NEWS:I Our example contract is very simple, and we rely on 9.75× 108 scenarios,

but still the length of the confidence interval is 2% of the SCR⇔ 30bps ofbalance sum

→ Questionable if even these advanced approaches will present practicableresults

I Indication that the confidence intervals (which are rather conservative) are"useless"?

I Important differences for estimation of SCR vs. estimating VaR of aportfolio of financial derivatives:

I Positions are calculated based on same scenarios.→ No diversification of error!I Main complexity is the generation of (asset & liability) scenarios, rendering

nested approach not practicable (?) – Solution???

Seminarreihe Energy & Finance @ UNI DUE | May 26th, 2010 | Daniel Bauer Least-Squares Monte Carlo Approach


Nested Simulations





Conclusion


Least-Squares Monte Carlo

→ Approach popular for valuation of non-European options (fast!)Longstaff & Schwartz (2001,RFS), Clément, Lamberton & Protter (2002,Fin&Stoch)

→ Idea: Use 2 approximations1. Continuation value replaced by finite linear combination of certain basis

functions2. Use MC simulations and LS regression to approximate linear comb. in 1

⇒ Opt. stopping rule by comparing exerc. val. to "synthetic" contin. val.

Idea here:I Simulate N paths (first year P, then Q)I For i = 1, . . . ,N, determine

PV i1 :=

T∑

t=2

exp

{−∫ T

1rs(ωi ) ds

}Xt (ωi ) = EQ

[PV i

1

∣∣∣F1

]

︸︷︷︸=h(Y i

1,Di1)

+εi

1∗ Replace h by finite linear combination of certain basis functions, say h2∗ Use MC simulations and LS regression to approximate linear comb. in 1∗

⇒ Use given emp. cdf of (h + R1 + X1) to determine quantile


Does it work?(a) Setup?

I T = EQ [ ·|Y1,D1] is orthogonal projection L2Q(F) −→ L2

Q(σ(Y1,D1))

I T⊥ = I − EQ [ ·|Y1,D1] is orthogonal projection on L2Q(σ(Y1,D1))⊥

→ EQ [ εi |Y1,D1] = EQ[

T⊥ PV i1

∣∣∣Y1,D1

]= 0 and EQ

[εi εj

∣∣Y1,D1]

= 0, i 6= j (MC real.)

→ Regression model!

(b) Approximation?

I L2Q(σ(Y1,D1)) can be identified with

L2(Rm+d ,B,FQ) =

{f : Rm+d → R

∣∣∫

Rm+df 2 dFQ <∞

}.

Now FQ is finite and thus regular, so L2Q(σ(Y1,D1)) is separable.

→ Every element can be approximated by finite linear combination ofarbitrary linearly independent system.


Example (cont.): Average of 150 runs with N = 320,000,K0 = 1,500,000

A LSM Approach to the Calculation of Capital Requirements 12

The initial value of the short rate is r0 = 4.19%. The estimated correlation is ρ = −0.0597and the market price of risk is λ = −0.5061.

For the insurance contract, similarly to Bauer et al. (2006), we assume a guaranteedminimum interest rate of g = 3.5%, a minimum participation rate of δ = 90%, an initialpremium of L0 = 10, 000 and a maturity of T = 10. Moreover, we assume that y = 50%of earnings on market values are declared as earnings on book values and that the initialreserve quota equals x0 = R0/L0 = 10%, i.e. R0 = x0 · L0 = 1, 000.

4.1.4 Least-Squares Monte Carlo Approach

As we have illustrated in the previous paragraph, in order to obtain accurate results, theNested Simulations Approach requires a large number of simulations and is hence very time-consuming. As a consequence, this approach may not be feasible for more complex specifi-cations. For the Least-Squares Monte Carlo Approach, on the other hand, considerably lesssimulations are needed to obtain accurate results. However, the drawback of this methodlies in the choice of the regression function.

Due to the construction of our contract and the asset model, the following variablesare natural choices for the regressors:7 A+

1 , r1, L1 and x1 = R1/L1. Since we already havea good approximation of the desired distribution from the Nested Simulations Approach,we first choose the regression function on the basis of this knowledge. We use a bottom-upscheme starting with only one regressor; by analyzing the residuals, we successively add more

regressors. Since clearly, lower variances σ(i)1 , 1 ≤ i ≤ N , result in a better least-squares

estimate, we again use the estimator based on cash flows from the policyholders’s perspective.Furthermore, we use N = 320, 000 real-world scenarios and K0 = 1, 500, 000. We perform150 estimates of the SCR for each regression function. Subsequently, we compute the averageof the 150 estimates. Table 2 shows our results for different regression functions.

# Regression Function Mean

(SCR)1 α

(N)0 + α

(N)1 · A1 1007.3

2 α(N)0 + α

(N)1 · A1 + α

(N)2 · A2

1 1165.5

3 α(N)0 + α

(N)1 · A1 + α

(N)2 · A2

1 + α(N)3 · r1 1272.6

4 α(N)0 + α

(N)1 · A1 + α

(N)2 · A2

1 + α(N)3 · r1 + α

(N)4 · r2

1 1276.5

5 α(N)0 + α

(N)1 · A1 + α

(N)2 · A2

1 + α(N)3 · r1 + α

(N)4 · r2

1 + α(N)5 · L1 1233.2

6 α(N)0 + α

(N)1 · A1 + α

(N)2 · A2

1 + α(N)3 · r1 + α

(N)4 · r2

1 + α(N)5 · L1 + α

(N)6 · x1 1233.9

7 α(N)0 + α

(N)1 · A1 + α

(N)2 · A2

1 + α(N)3 · r1 + α

(N)4 · r2

1 + α(N)5 · L1 + α

(N)6 · x1 + α

(N)7 · A1 · er1 1241.3

8 α(N)0 + α

(N)1 · A1 + α

(N)2 · A2

1 + α(N)3 · r1 + α

(N)4 · r2

1 + α(N)5 · L1 + α

(N)6 · x1 + α

(N)7 · A1 · er1

+α(N)8 · L1 · er1 1244.5

9 α(N)0 + α

(N)1 · A1 + α

(N)2 · A2

1 + α(N)3 · r1 + α

(N)4 · r2

1 + α(N)5 · L1 + α

(N)6 · x1 + α

(N)7 · A1 · er1

+α(N)8 · L1 · er1 + α

(N)9 · eA1/10000 1245.9

Table 2: Estimated SCR for different choices of the regression function, K0 = 1, 500, 000,N = 320, 000, LSM Approach

7While at time t=1, the state of the contract is entirely described by Yt = (A+t , rt), this is not the case for

later dates, where Dt = (Lt) is necessary to represent the state of the contract. However, our first analysesshow that the Least Squares algorithm performs similarly well in these situations.

I Result from Nested Sims (benchmark): 1,246.4 (2.5 days). Now in < 10min!

I HOWEVER: Influence of basis function quite pronounced. Usedknowledge from Nested Sims approach for derivation of regression fct. –cheating

→ Functional analytic bias. Hard to asses...


Choice of Regression Function (applied)I Problem: Common criteria for variable selection (Mallows Cp, AIC,. . .)

rely on homoskedasticityI Generalization:

(ε = (e1, . . . , em) , ei = (ei (Y1,D1), . . . , ei (YN ,DN ))′ , ¯V i1 =

∑Mk=1 αk · ek (Y i

1,Di1))

E1

[ N∑

i=1

(PV i

1 − ¯V i1

)2]

=N∑

i=1

E1

[(V i

1 − ¯V i1

)2]

︸︷︷︸=SMSE

+N∑

i=1

σi1−2 tr

(ε(ε′ε)−1

ε′diag(σ1

1 , . . . , σN1

))

⇒ SMSE =N∑

i=1

(PV i

1 − ¯V i1

)2−

N∑

i=1

σi1 + 2 tr

(ε (ε′ε)

−1ε′diag

(σ1

1 , . . . , σN1))

I Problem: Need nested simulations to estimate σi ...I Some authors propose variable selection models for heteroskedastic data

(see e.g. Baek, Kraman & Ahn (2005,CommStatist)). Work well inapplications, but no theoretical results available.

→ Idea: Use homoskedasticity assumption to derive choice based on"simple criterion"


Example (cont.): Choice of Regression Function

Choice via Mallow’s CP :

# A1 L1 x1 r1 A21 A3

1 L21 L3

1 x21 x3

1 r21 r3

1 A1· A1· A1· L1· x1· A1L1· A1· L21· A1· A2

1· L1· L21· A1x1· A2

1x1 Cp

of L1 x1 er1 er1 er1 x1 x21 x1 (er1 )2 er1 (er1 )2 er1 er1

regr.1 - - - x - - - - - - - - - - - - - - - - - - - - - - 11822.992 - - - - - x - - - - - - - - - - - - - - x - - - - - 513.913 - - - x - - - - - - - - - - - - - - - - - - - x - x 52.134 - - - x - - - - - - x - - - - - - - - x - x - - - - 7.905 - - x - - x - - - - x - - - - - - - - - x x - - - - 3.726 x - - - - x - - - - x - x - - - - - - - x x - - - - 4.337 x - - - - x - - x - x - - - - - x - - - - - - x x - 4.548 - - - x - x - - - - x - - x - x - x - - x x - - - - 5.489 x x - - - x x - - - x - x - - - - - - x x x - - - - 6.4410 x x - - - - x x - - - - x - - x x - x x x - - - - - 5.3511 x x - - x - x x - - - - x - - x x - x - x - - - - x 3.9012 x x - - x - x x - - - - x x - x x - x - x - - - - x 4.9813 x x - - x - x x - x - - x - - x x x x - x - - - - x 6.1714 x x x - x - x x - - - - x x - - x x x - - x x x - - 6.9115 x x - - x - x x - x - - x x x - - - - x x x x x x - 7.5416 x x - - x - x x - x - - x x x - x - - x x x x x x - 9.0317 x x - x x x x x - x x - x - - x - - x x x x - x x - 10.5118 x x - x x x x x - x x - x - - x - - - x x x x x x x 12.0519 x x - - x - x x x x x x x x - - - - x x x x x x x x 13.9220 x x - - x x x x x x x x x x - - - - x x x x x x x x 15.8621 x x x x x x x x x - x x x x x x x x - x - x x - - x 17.5922 x x x x x x x x - x - - x x x x x x x x x x x - x x 19.3723 x x x x x x x x - x - x x x x x x x x x x x x - x x 21.2424 x x x x x x x x x - x x x x x x x x x x x - x x x x 23.1325 x x x x x x x x x x x x x x x x x - x x x x x x x x 25.04

Table 6: Choice of the regression function via Mallow’s Cp

33

I SCR for Regression Function (5): 1,245.9 (Average of 120 runs a ).Results for "close" choices similar.

I So far results in line with "generalized" variable selection criterion→ To obtain accurate results, it is important not to employ arbitrary function,

but it appears sufficient to rely on roughly coherent method to determinesuitable choice.


Example (cont.): Approximation of distribution

A LSM Approach to the Calculation of Capital Requirements 14

calculate the average of the estimated SCR. Figure 1 illustrates our results. Table 3 displaysthat the mean is quite stable and very close to the result from the Nested SimulationsApproach. The empirical variance, on the other hand, is considerably higher than in theNested Simulations Approach. However, one needs to keep in mind that we only needN sample paths for the time interval (1, T ] in the LSM Approach, whereas the NestedSimulations Approach requires N · K1 paths. Therefore, given the same computationalconstraint, we could employ far more real-world scenarios eventually yielding a significantlylower empirical variance.

N Mean Empirical Solvency

(SCR) Variance Ratio

160,000 1245.4 110.9 151%

320,000 1245.9 39.1 151%

640,000 1245.3 24.0 151%

1,280,000 1245.4 12.1 151%

Table 3: Results for the LSM estimator

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.001

-150

0

-100

0

-500 0

500

100

0

150

0

2000

Loss (L)

Nested SimulationsLSM

Figure 2: Empirical density functions for N = 320, 000 and K1 = 300

Since we might also be interested in other quantiles or further information about thedistribution such as alternative risk measures, we now analyze the quality of the approxima-tion of the whole distribution. Figure 2 shows the empirical density functions for the NestedSimulations Approach and the LSM Approach for one run with a fixed seed. We find thatthe two distributions are very similar and hence, the LSM Approach provides an efficientalternative to Nested Simulations.

→ Also fairly accurate estimation of other risk measures feasible.


Current/Future Research: Improvement of the basis functionI Question: Within model framework (choice of (Y ,D)’s), can we find basis

functions that provide good choice for any cash-flow model (choice off ’s)?

I Idea: Interpret model (f1(·, ·), . . . , fT (·, ·)) as vector in suitable Hilbertspace H and then consider operator

O : (f1(·, ·), . . . , fT (·, ·)) 7→T∑

k=1

p(1, k ; Y1) EQk [ f (Yk ,Dk )| (Y1,D1)]

with O : H → L2Q(σ(Y1,D1)).

→ Under mild conditions, O is a compact operator as Fredholm integraloperator, i.e. it exhibits approximation property

I Approaches to find approximation:I Spectral analysis of OI Representation of Hilbert-Schmidt kernel (basically transition density) via

suitable analytic representationI TBD

Seminarreihe Energy & Finance @ UNI DUE | May 26th, 2010 | Daniel Bauer Conclusion


Nested Simulations





Conclusion


Conclusion

I Nested simulations:→ Inadequate choice of (N,K0,K1) in nested simulations may yield erroneous

outcomes→ Results obtained in practical applications with only few inner simulations

have to be interpreted with care→ A lot can be achieved by an appropriate simulation design→ But still immense computational effort to achieve accurate/practicable results

I LSM:→ Fast approach to achieve relatively accurate results→ Results similarly positive when calculating SCR for later dates ("richer sigma

field") or different risk measures→ Care required in choice of regression function even though simple

algorithms yield good results in our applications→ Open questions/Future Research: General validity of approximation.

Theoretical results?→ Nevertheless: Promising alternative for practical applications


Contact

Daniel [email protected]

Georgia State UniversityUSA

www.rmi.gsu.edu

Thank you!

on the calculation of the solvency capital re- quirement

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