Statistical Inference for Mortality Model and Risk Allocation

Statistical Inference for Mortality Model and RiskAllocation

Liang Peng

RMI at GSU

March 14, 2019

Liang Peng (RMI at GSU) Statistical Inference for Mortality Model and Risk AllocationMarch 14, 2019 1 / 103

This talk is based on

Q. Liu, C. Ling and L. Peng (2019). Statistical inference forLee-Carter mortality model and corresponding forecasts. NorthAmerican Actuarial Journal. To appear.

Q. Liu, C. Liu, D. Li and L. Peng (2019). Bias corrected inferencefor a modified Lee-Carter Mortality model. ASTIN Bulletin. Toappear.

V. Aismit, L. Peng, R. Wang and A. Yu (2019). An efficientapproach to quantile capital allocation and sensitivity analysis.Mathematical Finance. To appear.


Mortality – Introduction

Mortality rate or death rate: a measure of the number of deaths (ingeneral, or due to a specific cause) in a particular population, scaled tothe size of that population, per unit of time. Mortality rate is typicallyexpressed in units of deaths per 1,000 individuals per year; thus, amortality rate of 9.5 (out of 1,000) in a population of 1,000 would mean9.5 deaths per year in that entire population, or 0.95% out of the total.Insurance applications: In order to price insurance products, andensure the solvency of insurance companies through adequate reserves,actuaries must develop projections of future insured events (such asdeath, sickness, and disability). To do this, actuaries developmathematical models of the rates and timing of the events.



Longevity risk: any potential risk attached to the increasing lifeexpectancy of pensioners and policy holders, which can eventuallytranslate in higher than expected pay-out-ratios for many pensionfunds and insurance companies. One important risk to individuals whoare spending down savings, is that they will live longer than expectedand thus exhaust their savings, dying in poverty or burdening relatives.Challenges: In the last few decades, life expectancy has considerablyincreased, which calls for more advanced/accurate techniques inmodeling, securitization, regulation etc for insurance companies andpension funds for well managing the longevity risk. Understanding andforecasting mortality trend is of importance in hedging longevity risk.



Lee-Carter Model: Lee-Carter model has become a benchmark inmodeling mortality rates, forecasting mortality risk and hedginglongevity risk.Let m(x, t) denote the observed central death rate for age (or agegroup) x in year t, where x = 1, · · · ,M and t = 1, · · · , T . To model thelogarithms of the central death rates, Lee and Carter (1992) proposedthe following simple linear regression model

logm(x, t) = αx + βxkt + εx,t,

M∑x=1

βx = 1,

T∑t=1

kt = 0, (1)

where εx,t’s are random errors with mean zero and finite variance, andthe unobserved kt’s are called mortality index.



Identification issue: conditions∑M

x=1 βx = 1 and∑T

t=1 kt = 0 areimposed in finding the least squares estimators, which are obtained bythe singular value decomposition method.SVD: Mm×n = Um×mΣm×nV

τn×n, where U and V are orthogonal

matrices and Σ is a diagonal matrix with nonnegative values on thediagonal. Write U = (U1, · · · , Um) and V = (V1, · · · , Vn). Denote thepositive values on the diagonal of Σ as λ1, · · · , λr. ThenM =

∑ri=1 λiUiV

τi

Effective prediction: As an important task of modeling mortality ratesis to forecast future mortality pattern so as to better forecast mortalityrisk and hedge longevity risk, Lee and Carter (1992) further proposedto model the estimated mortality index by a simple time series model.As a matter of fact, Lee and Carter (1992) assumed that {kt} followsfrom an ARIMA(p,d,q) model defined as

(1−

p∑i=1

φiBi

)(1− B)

dkt = µ +

(1 +

q∑i=1

θiBi

)et, (2)

where et’s are white noises.Liang Peng (RMI at GSU) Statistical Inference for Mortality Model and Risk AllocationMarch 14, 2019 6 / 103


Summary. The popular Lee-Carter mortality model is a combination of(1) and (2), and a proposed two-step inference procedure is to firstestimate parameters in (1) by the singular value decomposition methodand then to use the estimated kt’s to fit model (2). Many papers inactuarial science have claimed that an application of this model andthis two-step inference procedure to mortality data prefers a unit roottime series model, i.e., d = 1 in (2).



Extensions and Applications: Since the seminal paper of Lee andCarter (1992), many extensions and applications have appeared in theliterature with an open R package (’demography’). For example,a) Lee (2000): Before proceeding directly to modeling the parameter ktas a time series process, the k′ts are adjusted (taking αx and βxestimates as given) to reproduce the observed number of deaths∑

xDxt, i.e., the k′ts solve∑x

Dxt =∑x

Ext exp{αx + βxkt},

where Ext is the actual risk exposure. So for better prediction, a timeseries model is fitted to k′ts.



b) Brouhns, Denuit and Vermunt (2002). First model{Dxt = Poisson(Extµx(t))µx(t) = exp{αx + βxkt}

Use the same constraints in Lee and Carter (1992) to estimateαx, βx, kt by maximizing∑

x,t

{Dxt(αx + βxkt)− Ext exp(αx + βxkt)}.

Denote the obtained estimators by αx, βx, kt. Next step is to fit a timeseries to the mortality index k′ts as Lee and Carter (1992).c) Li and Lee (2005). Extended the Lee-Carter model to model agroup of population m(x, t, i), where x, t, i denote the age group, timeand ith population, respectively.d) Girosi and King (2007). Cited more than a dozen of papers toconfirm the wide implementation of the Lee-Carter model by policyanalysts around the world.



e) Cairns et al. (2011). Compared six different stochastic mortality

models. The functions β(i)x , k

(i)t , and γ

(i)t−x are age, period and cohort

effects, respectively; x is the mean age over the range of ages beingused in the analysis; σ2x is the mean value of (x− x)2; na is the numberof ages.

(1) logm(t, x) = β(1)x + β

(2)x k

(2)t ;

(2) logm(t, x) = β(1)x + β

(2)x k

(2)t + β

(3)x γ

(3)t−x;

(3) logm(t, x) = β(1)x + n−1a k

(2)t + n−1a γ

(3)t−x;

(4) logitq(t, x) = k(1)t + k

(2)t (x− x);

(5) logitq(t, x) = k(1)t + k

(2)t (x− x) + k

(3)t ((x− x)2 − σ2x) + γ4t−x;

(6) logitq(t, x) = k(1)t + k

(2)t (x− x) + γ

(3)t−x(xc − x).



f) D’amato et al (2014). Employed the Lee-Carter model to detectcommon longevity trends. The model is

logmxt,i = αx,i + βx,ikt,i + εxt,i,

where i denotes the ith population.g) Lin et al (2014). Employed the extended Lee-Carter model in Liand Lee (2005) to study the risk management of a defined benefit plan.h) Bisetti and Favero (2014). Applied the Lee-Carter model tomeasure the impact of longevity risk on pension systems in Italy.



i) Bootstrap methods for quantifying uncertainty in mortality models.For interval estimation and/or projection errors, bootstrap methodshave been proposed. Haberman and Renshaw (2009) proposed threedifferent bootstrap methods to construct confidence intervals forinteresting quantities based on the Lee-Carter framework and ageneralized linear Poisson model. Li (2010) used parametric bootstrap.D’Amato et al (2012) proposed sieve bootstrap method based onestimated errors in logmxt = αx + βxkt + εxt, where εxt follows from anAR(∞) model.j) Continuous SDE for modeling mortality. Dahl (2004) selected anextended Cox-Ingersoll-Ross process; Biffis (2005) chose two differentspecifications for the intensity process; Schrager (2006) proposes anM-factor affine stochastic intensity; Elisa, Spreeuw and Vigna (2008)modeled stochastic mortality for dependent lives.



An issue on model assumption. Since {kt} in (2) is random, theconstraint

∑Tt=1 kt = 0 in (1) becomes unrealistic and restrictive. For

example, if one fits an AR(1) model to {kt}, say kt = µ+ φkt−1 + et,

then we have T−1∑T

t=1 ktp→ µ/(1− φ) as T →∞ when |φ| < 1

independent of T . On the other hand, we havekT /T

p→ µ{limx→γ1−ex−x } as T →∞ when µ 6= 0 and φ = 1 + γ/T for

some constant γ ∈ R. That is, the constraint∑T

t=1 kt = 0 in (1)basically says µ in (2) must be zero. Hence, a modified model withoutany direct constraint on kt’s is more appropriate.



An issue on inference. Another difficulty in using the singular valuedecomposition method in Lee and Carter (1992) is that no asymptoticresults are available for the derived estimators. That is, there is no wayto quantify the inference uncertainty. When all βx are the same (i.e.,β1 = · · · = βM = 1/M), Leng and Peng (2016) proved that theproposed two-step inference procedure in Lee and Carter (1992) isinconsistent when the model (2) is not an exact ARIMA(0,1,0) model.



Misunderstandings. It also happens that some papers in actuarialscience understand the model in a wrong way. For example, by definingm0(x, t) as the true central death rate for age x in year t, Dowd et al.(2010), Cairns et al (2011), Enchev, Kleinow and Cairns (2017) andothers interpreted the Lee-Carter model as

logm0(x, t) = αx + βxkt, kt = µ + kt−1 + et,M∑x=1

βx = 1,T∑t=1

kt = 0. (3)

Obviously this is quite confusing because model (3) basically says thetrue mortality rate m0(x, t) is random due to the randomness of kt’s.Another misinterpretation appears in Li (2010) and Li, Chan and Zhou(2015), where the Lee-Carter model is treated aslogm(x, t) = αx + βxkt without the random error εx,t in (1). This isquite problematic because it simply says that logm(x, t) andlogm(y, t) are completely dependent as both are determined by thesame random variable kt. That is, central death rates are completelydependent across ages.



Conclusions. The random error term εx,t in (1) is necessary in order toavoid the unrealistic implication that the central death rates arecompletely dependent across ages. Due to the presence of theserandom errors εx,t’s, the two-step inference procedure proposed by Leeand Carter (1992) may be inconsistent. Hence it is questionable for theexisting claim on unit root mortality index and the use of bootstrapmethods to quantify the forecast uncertainty of future mortality rates.


Mortality – Methodology

Modified Lee-Carter Model. First we propose to replace (1) by

logm(x, t) = αx + βxkt + εx,t,

M∑x=1

βx = 1,

M∑x=1

αx = 0, (4)

kt = µ+ φkt−1 + et, (5)

where et’s and εx,t’s are random errors with zero mean and finitevariance for each x. It is clear that we do not directly impose aconstraint on the unobserved random mortality index kt to ensure thatthe proposed model is identifiable. We also remark that theassumption of

∑Mx=1 αx = 0 is not restrictive at all as we can simply

move the sum to kt if∑M

x=1 αx 6= 0.



Estimation. We need a different statistical inference without using thesingular value decomposition method.Put Zt =

∑Mx=1 logm(x, t) and ηt =

∑Mx=1 εx,t for t = 1, · · · , T . Then,

by noting that∑M

x=1 αx = 0 and∑M

x=1 βx = 1, we have

Zt = kt + ηt for t = 1, · · · , T. (6)

When {kt} is nonstationary such as unit root (i.e., φ = 1 in (5)) ornear unit root (i.e., φ = 1 + γ/T for some constant γ 6= 0 in (5)), ktdominates ηt as t large enough, and so Zt behaves like kt in this case.



This motivates us to minimize the following least squares

T∑t=2

(Zt − µ− φZt−1)2 ,

and obtain the least squares estimator for µ and φ asµ =

∑Ts=2 Zs

∑Tt=2 Z

2t−1−

∑Ts=2 Zs−1

∑Tt=2 ZtZt−1

(T−1)∑Tt=2 Z

2t−1−(

∑Tt=2 Zt−1)2

,

φ =(T−1)

∑Tt=2 ZtZt−1−

∑Ts=2 Zs

∑Tt=2 Zt−1

(T−1)∑Tt=2 Z

2t−1−(

∑Tt=2 Zt−1)2

.



Similarly, by minimizing the following least squares

T∑t=1

(logm(x, t)− αx − βxZt)2 ,

we obtain the least squares estimator for αx and βx as αx =∑Ts=1 logm(x,s)

∑Tt=1 Z

2t−

∑Ts=1 logm(x,s)Zs

∑Tt=1 Zt

T∑Tt=1 Z

2t−(

∑Tt=1 Zt)

2,

βx =T∑Ts=1 logm(x,s)Zs−

∑Ts=1 logm(x,s)

∑Tt=1 Zt

T∑Tt=1 Z

2t−(

∑Tt=1 Zt)

2.



In order to derive the asymptotic properties of the above proposedestimators, we assume the following regularity conditions for thestationary errors in (4) and (5).

C1) E(et) = 0, E(εx,t) = 0 for t = 1, · · · , T and x = 1, · · · ,M ;

C2) there exist β > 2 and δ > 0 such that suptE|et|β+δ <∞,suptE|εx,t|β+δ <∞ for x = 1, · · · ,M ;

C3) σ2e = limT→∞E{T−1(∑T

t=1 et)2} ∈ (0,∞) and

σ2x = limT→∞E{T−1(∑T

t=1 εx,T )2} ∈ (0,∞) for x = 1, · · · ,M ;



C4) sequence {(et, ε1,t, · · · , εM,t)τ} is strong mixing with mixing

coefficients

αm = supk≥1

supA∈Fk1 ,B∈F∞k+m

|P (A ∩B)− P (A)P (B)|

such that∑∞

m=1 α1−2/βm <∞, where Fk+mk denotes the σ-field

generated by {(et, ε1,t, · · · , εM,t)τ : k ≤ t ≤ k +m} and Aτ denotes

the transpose of the matrix or vector A.



Under the above regularity conditions, it is known that forr = (r1, · · · , rM+1)

τ ∈ [0, 1]M+1 there is a (M + 1)-dimensionalGaussian Process {W (r)} such that(∑[Tr1]

t=1 et

σe√T

,

∑[Tr2]t=1 ε1,t

σ1√T

, · · · ,∑[TrM+1]

t=1 εM,t

σM√T

)τD→W (r), (7)

in the space D([0, 1]M+1), where [x] is the floor function, D([0, 1]M+1)denotes the space of real-valued functions on [0, 1]M+1 that are right

continuous and have finite left limits, ”D→” denotes the weak

convergence of the associated probability measures.



Throughout we will use ”d→” and ”

p→” to denote the convergence indistribution and in probability, respectively. We also use Wi(ri) todenote the marginal distribution of W (r) for the ith variable and define

W∗(s) =

M+1∑i=2

σi−1Wi(s) for s ∈ [0, 1].

As µ in (5) is often nonzero in real mortality rates, the followingtheorem provides the asymptotic theory for the proposed estimatorswhen (5) holds with µ 6= 0 and φ = 1 + γ/T for some constant γ ∈ R.Similar results can be derived with a different rate of convergence for φwhen µ = 0 and φ = 1 + γ/T .



Theorem

Assume (4) and (5) hold with conditions C1)–C4), µ 6= 0, φ = 1 + γ/Tfor some constant γ ∈ R and k0 is a constant. Definefγ(s) = (1− eγs)/(−γ) for s ∈ [0, 1]. Then the following convergencesare true as T →∞.(i)

T 3/2(φ− φ)d−→ σe

µ

∫ 10 fγ(s) dW1(s)−W1(1)

∫ 10 fγ(s) ds∫ 1

0 f2γ (s) ds− (

∫ 10 fγ(s) ds)2

.

(ii)

T 1/2(µ− µ)d−→σeW1(1)

∫ 10 f

2γ (s) ds− σe

∫ 10 fγ(s) ds

∫ 10 fγ(t) dW1(t)∫ 1

0 f2γ (s) ds− (

∫ 10 fγ(s) ds)2

.



Theorem (Continue)(iii) For x = 1, 2, . . . ,M

T1/2

(αx − αx)d−→ βxY∗ − Yx,

where

Y∗ =

∫ 10 fγ(s) ds

∫ 10 fγ(t) dW∗(t)−W∗(1)

∫ 10 f

2γ(s) ds∫ 1

0 f2γ(s) ds− (

∫ 10 fγ(s) ds)2

and

Yx =σx∫ 10 fγ(s) ds

∫ 10 fγ(t) dWx+1(t)− σxWx+1(1)

∫ 10 f

2γ(s) ds∫ 1

0 f2γ(s) ds− (

∫ 10 fγ(s) ds)2

.

(iv) For x = 1, 2, . . . ,M

T3/2

(βx − βx)d−→

βxZ∗ − Zxµ

where

Z∗ =W∗(1)

∫ 10 fγ(s) ds−

∫ 10 fγ(s) dW∗(s)∫ 1

0 f2γ(s) ds− (

∫ 10 fγ(s) ds)2

and

Zx =σxWx+1(1)

∫ 10 fγ(s) ds− σx

∫ 10 fγ(s) dWx+1(s)∫ 1

0 f2γ(s) ds− (

∫ 10 fγ(s) ds)2

.



Unit root test. Under H0, we have fγ(s) = s and so it follows from theabove Theorem (i) that

T 3/2(φ− φ)d→ 12σe

µ{W1(1)

2−∫ 1

0W1(s) ds} ∼ N(0,

12σ2eµ2

). (8)

In order to employ the above limit to test H0 : φ = 1, one has toestimate σ2e . Unfortunately the estimators proposed by Phillips andPerron (1988) are not applicable due to the involved ”measurementerrors” ηt’s. Here we employ the idea of block sample varianceestimation in Carlstein (1986) and Politis and Romano (1993).



For integer L, define et = Zt − µ− φZt−1 for t = 2, · · · , T , andUi = L−1

∑Lj=1 ei+j for i = 1, · · · , T − L. Then for estimating

σ2e = limT→∞E(∑Tt=2 et√T

)2, we consider

σ2e = L

1

T − L

T−L∑i=1

U2i − (

1

T − L

T−L∑j=1

Uj)2

.

We reject the null hypothesis H0 : φ = 1 at level a ifµ2T 3(φ−1)2

12σ2e

≥ χ21,1−a, where χ2

1,1−a denotes the (1− a)-th quantile of the

chi-squared distribution with one degree of freedom.



Forecast. When the null hypothesis H0 : φ = 1 is not rejected, we couldforecast the future d mortality rates based on models (4) and (5) withφ = 1, which are

logm(x, T + r) = αx + βx{ZT + rµ} for r = 1, 2, · · · , d.

Note that

logm(x, T + r)− logm(x, T + r)

= αx + βx{ZT + rµ} − αx − βx{rµ+ kT +∑r

s=1 eT+s} − εx,T+r= αx − αx + r{βxµ− βxµ}+ (βx − βx)ZT

+βxηT − βx∑r

s=1 eT+s − εx,T+r= βxηT − βx

∑rs=1 eT+s − εx,T+r + op(1).

(9)In order to quantify the uncertainties of the above forecasts, one has toestimate the distribution function



Gr(y) = P (βxηT − βxr∑s=1

eT+s − εx,T+r ≤ y).

Unfortunately it seems that Gr(y) can not be estimatednonparametrically without imposing more conditions on εx,t’s.However, we have

1M

∑Mx=1

logm(x, T + r)− 1M

∑Mx=1 logm(x, T + r)

= 1M {ηT −

∑rs=1 eT+s − ηT+r}+ op(1)



and the distribution function of

Hr(y) = P( 1

M(ηT −

r∑s=1

eT+s + ηT+r) ≤ y)

can be estimated nonparametrically by

Hr(y) =1

T − r

T−r∑t=1

I(− 1

M

r∑s=1

et+s ≤ y).

Define

cl,a = sup{y : Hr(y) ≤ a/2} and cu,a = sup{y : Hr(y) ≤ 1− a/2}.



Then an interval forecast for 1M

∑Mx=1 logm(x, T + r) with level a is

Ia =

(1

M

M∑x=1

logm(x, T + r)− cu,a,1

M

M∑x=1

logm(x, T + r)− cl,a

).

Then for any fixed integer r ≥ 1 and a ∈ (0, 1),

P (1

M

M∑x=1

logm(x, T + r) ∈ Ia)→ a as T →∞.


Mortality – Bias Corrected Inference

The above estimation solves the following score equations forx = 1, . . . ,M :

∑Tt=2{Zt − µ− φZt−1} = 0,∑Tt=2{Zt − µ− φZt−1}Zt−1 = 0,∑Tt=2{logm(x, t)− αx − βxZt} = 0,∑Tt=2{logm(x, t)− αx − βxZt}Zt = 0.

(10)



By noting that the inconsistency of the least squares estimators viasolving (10) is due to the correlation betweenZt− µ0− φ0Zt−1 = et + ηt− φ0ηt−1 and Zt−1 = kt−1 + ηt−1, we proposethe simple bias corrected estimators via solving the following modifiedscore equations:

∑Tt=3{Zt − µ− φZt−1} = 0,∑Tt=3{Zt − µ− φZt−1}Zt−2 = 0,∑Tt=3{logm(x, t)− αx − βxZt} = 0,∑Tt=3{logm(x, t)− αx − βxZt}Zt−1 = 0,

(11)



which give

µ =

∑Ts=3 Zs

∑Tt=3 Zt−1Zt−2 −

∑Ts=3 Zs−1

∑Tt=3 ZtZt−2

(T − 2)∑T

t=3 Zt−1Zt−2 −∑T

s=3 Zs−1∑T

t=3 Zt−2,

φ =(T − 2)

∑Tt=3 ZtZt−2 −

∑Ts=3 Zs

∑Tt=3 Zt−2

(T − 2)∑T

t=3 Zt−1Zt−2 −∑T

s=3 Zs−1∑T

t=3 Zt−2,

αx =

∑Ts=3 logm(x, s)

∑Tt=3 ZtZt−1 −

∑Ts=3 logm(x, s)Zs−1

∑Tt=3 Zt

(T − 2)∑T

t=3 ZtZt−1 −∑T

s=3 Zs∑T

t=3 Zt−1,

βx =(T − 2)

∑Tt=3 logm(x, t)Zt−1 −

∑Ts=3 logm(x, s)

∑Tt=3 Zt−1

(T − 2)∑T

t=3 ZtZt−1 −∑T

s=3 Zs∑T

t=3 Zt−1

for x = 1, . . . ,M . We remark that the estimator φ is the same as themodified Yule-Walker estimator in Staudenmayer and Buonaccorsi(2005) for a time series model with measurement errors, and obviouslywe have

∑Mx=1 αx = 0 and

∑Mx=1 βx = 1.



To present the asymptotic distribution of the proposed bias-correctedestimators, we need some notations. Putθ = (µ, φ, α1, β1, · · · , αM−1, βM−1)τ ,θ = (µ, φ, α1, β1, · · · , αM−1, βM−1)τ and letθ0 = (µ0, φ0, α1,0, β1,0, · · · , αM−1,0, βM−1,0)τ denote the true value of θ.Note that we exclude αM and βM in the above definitions due to theconstraints

∑Mx=1 αx,0 = 0 and

∑Mx=1 βx,0 = 1.


Mortality – Bias Corrected InferenceFor the stationary case, i.e., |φ0| < 1 independent of T , we define the symmetric matrixΣ = (σi,j)1≤i,j≤2M with

σ1,1 = E(e1 + (1− φ0)η1)2, σ1,2 =

µ0

1− φ0

σ1,1,

σ1,2x+1 = E{(εx,1 − βx,0η1)(e1 + (1− φ0)η1)}, σ1,2x+2 =µ0

1− φ0

σ1,2x+1,

σ2,2 = {E(e1 + η1)2 + φ20E(η21)}{

µ20(1−φ0)2

+E(e21)

1−φ20+ E(η21) + 2E(e1η1)}

−2φ0{E(e1η1) + E(η21)}{µ20

(1−φ0)2+φ0E(e21)

1−φ20+ φ0E(e1η1)},

σ2,2x+2 = E{e1(εx,1 − βx,0η1)}{µ20

(1−φ0)2+φ0E(e21)

1−φ20+ φ0E(e1η1)}

+E{η1(εx,1 − βx,0η1)}{µ20

1−φ0− φ0E(e1η1)− φ0E(η21)},

σ2,2x+1 = σ1,2x+2, σ2x+1,2y+1 = E{(εx,1−βx,0η1)(εy,1−βy,0η1)}, σ2x+1,2y+2 =µ0

1− φ0

σ2x+1,2y+1,

σ2x+2,2y+2 = E{(εx,1 − βx,0η1)(εy,1 − βy,0η1)}{µ20

(1− φ0)2+E(e21)

1− φ20

+ 2E(e1η1) + E(η21)}

for 1 ≤ x, y ≤ M − 1 andΓ = diag(A0, · · · , AM−1) (12)

with

A0 = · · · = AM−1 =

1µ0

1−φ0µ0

1−φ0µ20

(1−φ0)2+φ0E(e21)

1−φ20+ φ0E(e1η1)

.Liang Peng (RMI at GSU) Statistical Inference for Mortality Model and Risk AllocationMarch 14, 2019 37 / 103

Mortality – Bias Corrected InferenceFor the nonstationary case, i.e., φ0 = 1 + ρ/T for some ρ ∈ R, we define the symmetric matrix

Σ = (σi,j)1≤i,j≤2M with

σ1,1 = E(e21), σ1,2 = E(e

21)

∫ 1

0fρ,µ0 (s) ds,

σ1,2x+1 = E{e1(εx,1 − βx,0η1)}, σ1,2x+2 = E{e1(εx,1 − βx,0η1)}∫ 1

0fρ,µ0 (s) ds,

σ2,2 = E(e21)

∫ 1

0f2ρ,µ0

(s) ds, σ2,2x+1 = σ1,2x+2,

σ2,2x+2 = E{e1(εx,1 − βx,0η1)}∫ 1

0f2ρ,µ0

(s) ds, σ2x+1,2y+1 = E{(εx,1 − βx,0η1)(εy,1 − βy,0η1)},

σ2x+1,2y+2 = E{(εx,1 − βx,0η1)(εy,1 − βy,0η1)}∫ 1

0fρ,µ0 (s) ds,

σ2x+2,2y+2 = E{(εx,1 − βx,0η1)(εy,1 − βy,0η1)}∫ 1

0f2ρ,µ0

(s) ds

for 1 ≤ x, y ≤ M − 1, where

fρ,µ0 (s) =

{µ0

esρ−1ρ

if ρ 6= 0,

µ0s if ρ = 0,(13)

andΓ = diag(A0, · · · , AM−1) (14)

with

A0 = · · · = AM−1 =

(1

∫ 10 fρ,µ0 (s)ds∫ 1

0 fρ,µ0 (s)ds∫ 10 f

2ρ,µ0

(s)ds

).



Theorem

Assume µ0 6= 0, and {(et, ε1,t, · · · , εM,t)τ : t = 1, . . . , T} is a sequence

of independent and identically distributed random vectors with meanszero and finite covariance matrix.i) When |φ0| < 1 independent of T (i.e., stationary case), we have

√TΓ{θ − θ0}

d→ N(0,Σ).

ii) When φ0 = 1 + ρ/T for some constant ρ ∈ R (i.e., near unit root ifρ 6= 0 and unit root if ρ = 0), we have

DT {θ − θ0}d→ N(0, Γ−1ΣΓ−1),

where DT is a diagonal matrix with T 1/2 in the odd diagonal elementsand T 3/2 in the even diagonal element.


Mortality – Data Analysis

To illustrate how the proposed model and inference can be applied tomortality data and how the new method differs from the classicLee-Carter model, we employ the mortality data from the HumanMortality Database (HMD) (seehttp://www.mortality.org/cgi-bin/hmd/country.php?cntr=USA&level=1). To gain a robust conclusion, we study the centraldeath rates of U.S. female, male and combined population between 25and 74 years old from year 1933 to year 2015, and use the mortalitydata by 5-year age groups. Hence we have M = 10 and T = 83.



First, to implement the classic Lee-Carter model, we employ thestatistical R package ‘demography’ to obtain estimates for αx’s, βx’s,kt’s, and then use the obtained estimates for kt’s to fit model (5) byusing ‘lm’ in the statistical software R. We report the estimates forαx’s, βx’s, µ and φ in the following three tables for the female, maleand combined mortality rates, respectively. As the asymptoticproperty for the estimate of φ is unknown, one can not simply use thestandard errors obtained from ’lm’ to conclude whether φ = 1 or not.Also one can not employ the commonly employed unit root tests basedon estimates of kt’s to test H0 : φ = 1.



x 1 2 3 4 5 6 7 8 9 10αx -7.011 -6.736 -6.377 -5.984 -5.572 -5.159 -4.770 -4.348 -3.929 -3.465

βx 0.135 0.128 0.119 0.106 0.095 0.090 0.083 0.080 0.081 0.083

Estimate Standard Errorµ -0.157 0.022

φ 0.975 0.006

Table 1: Female mortality rates. Parameter estimates are obtained from fittingmodels (1) and (5) based on the two-step inference in Lee and Carter (1992).



x 1 2 3 4 5 6 7 8 9 10αx -6.262 -6.125 -5.844 -5.472 -5.048 -4.612 -4.204 -3.798 -3.415 -3.015

βx 0.088 0.094 0.106 0.109 0.108 0.108 0.103 0.099 0.096 0.090


φ 0.994 0.008

Table 2: Male mortality rates. Parameter estimates are obtained from fittingmodels (1) and (5) based on the two-step inference in Lee and Carter (1992).



x 1 2 3 4 5 6 7 8 9 10αx -6.562 -6.381 -6.075 -5.697 -5.279 -4.853 -4.454 -4.045 -3.656 -3.237

βx 0.106 0.109 0.112 0.108 0.103 0.101 0.094 0.090 0.089 0.088


φ 0.983 0.007

Table 3: Combined mortality rates. Parameter estimates are obtained from fittingmodels (1) and (5) based on the two-step inference in Lee and Carter (1992).



Next we apply our proposed inference to fit models (4) and (5) to thefemale, male and combined mortality rates. Again we use ‘lm’ toobtain our proposed least squares estimates and report the estimatesfor αx’s, βx’s, µ and φ in the following three tables for the female, maleand combined mortality rates, respectively. As before, the standarderrors obtained from ‘lm’ is inaccurate since it ignores the involved ηt’sand so one can not conclude whether φ = 1 or not from these threetables. Although the estimate for φ obtained from the new method issimilar to that obtained from the Lee-Carter method, estimates for µare quite different for both methods since the new method does notassume

∑Tt=1 kt = 0.



x 1 2 3 4 5 6 7 8 9 10αx 0.172 0.055 -0.022 -0.344 -0.474 -0.327 -0.337 -0.067 0.384 0.959

βx 0.135 0.127 0.119 0.106 0.096 0.091 0.083 0.080 0.081 0.083


φ 0.977 0.005

Table 4: Female mortality rates. Parameter estimates are obtained from fittingmodels (4) and (5) based on the proposed least squares estimation.



x 1 2 3 4 5 6 7 8 9 10αx -2.068 -1.631 -0.789 -0.270 0.099 0.547 0.714 0.940 1.152 1.308

βx 0.088 0.094 0.106 0.109 0.108 0.108 0.103 0.099 0.096 0.090


φ 0.993 0.008

Table 5: Male mortality rates. Parameter estimates are obtained from fittingmodels (4) and (5) based on the proposed least squares estimation.



x 1 2 3 4 5 6 7 8 9 10αx -1.264 -0.949 -0.452 -0.272 -0.105 0.219 0.293 0.509 0.815 1.205

βx 0.105 0.108 0.112 0.108 0.103 0.101 0.094 0.091 0.089 0.088


φ 0.985 0.007

Table 6: Combined mortality rates. Parameter estimates are obtained from fittingmodels (4) and (5) based on the proposed least squares estimation.



Next we apply our proposed unit root test to the female, male andcombined mortality rates, where we use σ2e with L = 0.5

√T ,√T , 2√T

and σ2e (denoted by L = ∗) for independent errors. Note that (8)requires k0/T → 0 as T →∞. Since |Z1|/T is around 0.5 which is farlarger than zero, the limiting distribution of the proposed unit root testunder the unit root null hypothesis will be away from a chi-squareddistribution for the given T = 83. Therefore we apply the proposedunit root test to {Zt − Z1}Tt=1. The obtained variance estimates, teststatistics and Pvalues are reported in the following three tables for thefemale, male and combined mortality rates, respectively. As we see,these quantities are quite robust to the choice of L. Moreover, theproposed test rejects the unit root hypothesis for the female andcombined mortality rates, but fails to reject the unit root hypothesisfor the male mortality rates.



L σ2e Test statistic Pvalue

b12√T c 0.042 75.938 2.927e-18

b√T c 0.052 61.378 4.709e-15

b2√T c 0.038 85.207 2.687e-20

* 0.047 68.809 1.085e-16

Table 7: Female mortality rates. Variance estimates, test statistics and Pvalues arereported for L = b 1

2

√T c, b

√T c, b2

√T c, where ′L = ∗′ denotes σ2

e .




b12√T c 0.064 0.798 0.372

b√T c 0.073 0.692 0.405

b2√T c 0.064 0.793 0.373

* 0.073 0.699 0.403

Table 8: Male mortality rates. Variance estimates, test statistics and Pvalues arereported for L = b 1

2

√T c, b

√T c, b2


e .




b12√T c 0.051 12.229 4.704e-4

b√T c 0.061 10.289 1.338e-3

b2√T c 0.049 12.727 3.605e-4

* 0.058 10.789 1.021e-3

Table 9: Combined mortality rates. Variance estimates, test statistics and Pvaluesare reported for L = b 1

2

√T c, b

√T c, b2


e .



Next we compare the proposed bias corrected inference with theinference methods in Lee and Carter (1992) and Liu, Ling and Peng(2018) for the same dataset. The estimates for αx’s, βx’s, µ’s and φ’sare reported in Tables 10–12 for the U.S. female, male and combinedmortality rates, respectively. We observe that the proposed biascorrected estimate gives the smallest φ and largest |µ|, and a cleardifference in estimating αx for these three methods.



Table 10: US female mortality rates.

x 1 2 3 4 5 6 7 8 9 10

Lee and Carter (1992)αx -7.011 -6.736 -6.377 -5.984 -5.572 -5.159 -4.770 -4.348 -3.929 -3.465

βx 0.135 0.128 0.119 0.106 0.095 0.090 0.083 0.080 0.081 0.083

Liu, Ling and Peng (2018)αx 0.172 0.055 -0.022 -0.344 -0.474 -0.327 -0.337 -0.067 0.384 0.959

βx 0.135 0.127 0.119 0.106 0.096 0.091 0.083 0.080 0.081 0.083

Bias Corrected Inferenceαx -0.025 -0.070 -0.085 -0.318 -0.453 -0.276 -0.261 -0.023 0.463 1.047

βx 0.131 0.125 0.118 0.106 0.096 0.092 0.084 0.081 0.082 0.085

µ φLee and Carter (1992) -0.157 0.975

Liu, Ling and Peng (2018) -1.389 0.977Bias Corrected Inference -1.547 0.974



Table 11: US male mortality rates.

x 1 2 3 4 5 6 7 8 9 10

Lee and Carter (1992)αx -6.262 -6.125 -5.844 -5.472 -5.048 -4.612 -4.204 -3.798 -3.415 -3.015

βx 0.088 0.094 0.106 0.109 0.108 0.108 0.103 0.099 0.096 0.090

Liu, Ling and Peng (2018)αx -2.068 -1.631 -0.789 -0.270 0.099 0.547 0.714 0.940 1.152 1.308

βx 0.088 0.094 0.106 0.109 0.108 0.108 0.103 0.099 0.096 0.090

Bias Corrected Inferenceαx -2.267 -1.838 -0.938 -0.305 0.105 0.623 0.856 1.063 1.300 1.400

βx 0.084 0.090 0.103 0.108 0.108 0.109 0.106 0.102 0.099 0.092

µ φLee and Carter (1992) -0.118 0.994




Table 12: US combined mortality rates.

x 1 2 3 4 5 6 7 8 9 10

Lee and Carter (1992)αx -6.562 -6.381 -6.075 -5.697 -5.279 -4.853 -4.454 -4.045 -3.656 -3.237

βx 0.106 0.109 0.112 0.108 0.103 0.101 0.094 0.090 0.089 0.088

Liu, Ling and Peng (2018)αx -1.264 -0.949 -0.452 -0.272 -0.105 0.219 0.293 0.509 0.815 1.205

βx 0.105 0.108 0.112 0.108 0.103 0.101 0.094 0.091 0.089 0.088

Bias Corrected Inferenceαx -1.495 -1.146 -0.576 -0.280 -0.083 0.304 0.430 0.613 0.939 1.294

βx 0.101 0.104 0.110 0.108 0.103 0.103 0.097 0.093 0.091 0.090

µ φLee and Carter (1992) -0.135 0.983




Next we apply these three methods to forecast future logarithms ofmortality rates for the next 50 years, and report the forecasts for thecombined mortality rates in Figure 1. The bias corrected inferenceforecasts larger mortality rates than the method in Liu, Ling and Peng(2018). As Liu, Ling and Peng (2018) rejected the unit root nullhypothesis for the combined mortality rates, only the forecasts basedon the proposed bias corrected inference in Figure 1 are theoreticallysuitable.



Figure 1: True data and next 50 years’ forecasts of logm(x, t) for US combined mortality rates. Thefirst five plots correspond to the age groups x = 2, 4, 6, 8, 10 and the last plot corresponds to the sum oflogm(x, t) over all age groups x = 1, 2, . . . , 10, where dashed line, dotted line and dashdotted line representthe Lee-Carter method, the method in Liu, Ling and Peng (2018) and the proposed bias corrected inference,respectively.

−7.0

−6.5

−6.0

−5.5

0 50 100

t

log

m(2

; t)

−6.5

−6.0

−5.5

−5.0

0 50 100

t

log

m(4

; t)



Figure 2: Continue

−5.6

−5.2

−4.8

−4.4

0 50 100

t

log

m(6

; t)

−4.50

−4.25

−4.00

−3.75

0 50 100

t

log

m(8

; t)



Figure 3: Continue

−3.6

−3.3

−3.0

−2.7

0 50 100

t

log

m(1

0; t)

−54

−50

−46

0 50 100

t

∑ x=1

x=10

log

m(x

; t)


Risk Allocation – Introduction

Risk Measures. In the banking regulatory frameworks of Basel II andIII, as well as the insurance regulatory regimes such as Solvency II andSwiss Solvency Test, an institution is required to hold a certain capitalaccording to a pre-specified regulatory risk measure. These regulatoryenvironments set the capital via two standard risk measures:Value-at-Risk (VaR) and Expected Shortfall (ES) at level p ∈ (0, 1),defined as

VaRp(Y ) := qp(Y ) := inf{x ∈ R : P (Y ≤ x) ≥ p

},

ESp(Y ) := cp(Y ) :=1

1− p

∫ 1

pVaRq(Y ) dq.



Risck or capital allocation. Risk measures such as VaR and ES are notonly used externally for calculating regulatory capital, but alsointernally for risk management and performance measurement. In theinternal use of risk measures, a crucial problem is the allocation oftotal capital to individual business lines in order to assess theperformance of each business line (performance analysis), for instancevia their Return on Risk-adjusted Capital (RORAC).



Suppose that a financial institution has d different lines of business,each with an individual risk Xj , j = 1, . . . , d. Let S be the total risk ofthis firm, that is, S := X1 + · · ·+Xd. The allocation of the totalregulatory capital, set via either ESp(S) or VaRp(S), is typicallycarried out in practice via the Euler capital allocation rule.If the total regulatory capital for the aggregate risk S is calculated viaESp(S), then the Euler allocation rule for the risk vector (X1, . . . , Xd)is given by

Cp(Xi|S) := E[Xi|S ≥ qp(S)], i = 1, . . . , d.

This allocation satisfies that ESp(S) =∑d

i=1Cp(Xi|S), which is anatural and required condition for any capital allocation rules. On theother hand, if the total regulatory capital is calculated via VaRp(S),then the Euler allocation rule for (X1, . . . , Xd) is given by

Qp(Xi|S) := E[Xi|S = qp(S)], i = 1, . . . , d.

Once again, we have VaRp(S) =∑d

i=1Qp(Xi|S).Liang Peng (RMI at GSU) Statistical Inference for Mortality Model and Risk AllocationMarch 14, 2019 63 / 103


Question. A crucial observation is that, for a classic nonparametricestimator of Qp(Xi|S), the convergence rate is slower than the standardconvergence rate n−1/2 of a nonparametric estimator for Cp(Xi|S),where n is the sample size. Therefore, a practical question is whetherone could find an allocation (C1, · · · , Cd) such that

∑di=1Ci = VaRp(S)

and each of C1, . . . , Cd could be estimated nonparametrically at thestandard rate of convergence n−1/2. Ideally, this new allocation shouldalso be close to Qp(Xi|S), so that it roughly gives the Euler allocation.Sensitivity analysis. Another major interpretation of Cp(Xi|S) andQp(Xi|S) is in the context of sensitivity analysis, which refers to theevaluation of the impact of some model inputs over the model outputs.



Parametric inference. If (X1, . . . , Xd) is modeled by a parametricdistribution family and independent observations are available,Glasserman (2005) and Glasserman and Li (2005) employedimportance sampling technique to estimate Cp(Xi|S) and Qp(Xi|S)with applications to portfolio credit risk.



Nonparametric Inference. Nonparametric estimation for Cp(Xi|S) hasbeen studied in Scaillet (2005). For estimating Qp(Xi|S)nonparametrically, we refer to Gourieroux et al. (2000), Liu and Hong(2009), Hong (2009), where, as mentioned previously, these methodssuffer from a slower convergence rate than the standard convergencerate n−1/2. Under some special cases, Qp(Xi|S) could be written as aratio of two quantities and each quantity is estimatednonparametrically at the standard rate of convergence, i.e. Qp(Xi|S) isestimated at the standard rate of convergence; see Fu et al. (2009) andJiang and Fu (2015).



Inference for sensitivity. Quantile sensitivity has appeared in theliterature in various forms. For example, the linear risk portfolio isdiscussed in Gourieroux et al. (2000); kernel estimation andimportance sampling techniques are combined by Tasche (2009) for aportfolio credit risk model; other nonparametric estimations areemployed via infinitesimal perturbation analysis (see Hong (2009));importance sampling techniques for a parametric portfolio credit riskmodel are detailed in Glasserman (2005) and Glasserman and Li (2005)for both quantiles and average tail risk. Finally, sensitivity analysis forthe VaR-based and ES-based measures of performance are investigatedby using the special features of Fourier Transform Monte Carlo in Siller(2013) within the portfolio credit risk setting.



Aim. Find a quantile-based allocation which can be estimated at thestandard rate of convergence nonparametrically. The allocation rulestudied here is briefly mentioned in Kalkbrener (2005) without eitherformal theoretical development or statistical analysis. It is ourintention to give a theoretical treatment of this idea and to provide anefficient inference procedure with the standard rate of convergence.


Risk Allocation – Methodology

New allocation. Let p∗ ∈ (0, 1) be given by the following equation

p∗ := inf{t ∈ [0, 1] : ct(Y ) ≥ qp(Y )}. (15)

Note that in the above notation, we omit the dependence of p∗ on Yand p, which is clear from the context. First, let us verify the existenceof p∗.

Theorem

For Y ∈ L1 and p ∈ (0, 1), assume E[Y ] ≤ qp(Y ). Then p∗ ∈ [0, p].Moreover, if Y is continuously distributed, then cp∗(Y ) = qp(Y ).

Note that in insurance and financial applications, p is typically close to1 (e.g. p = 0.99, 0.995, 0.999, . . .), which makes the condition,E[Y ] ≤ qp(Y ), a realistic assumption.



For a portfolio of risks X := (X1, . . . , Xd), we take X = Xi andY = X1 + · · ·+Xd. In the sequel, we shall refer to Cp∗(X|Y ) as theES-based allocation for the p-quantile. If Y is continuously distributed,an immediate feature of the allocation (Cp∗(X1|Y ), . . . , Cp∗(Xd|Y )) is

d∑i=1

Cp∗(Xi|Y ) = Cp∗

(d∑i=1

Xi

∣∣∣Y) = cp∗(Y ) = qp(Y ). (16)

Therefore, (Cp∗(X1|Y ), . . . , Cp∗(Xd|Y )) indeed gives an allocation ofthe VaR-based total risk capital qp(Y ).



Before approaching the estimation for Cp∗(Xi|Y ), we first discuss someuseful properties. We would like the new allocation Cp∗(Xi|Y ) to beclose to Qp(Xi|Y ). Indeed, for elliptically distributed (X,Y ), we havethe equality Cp∗(X|Y ) = Qp(X|Y ), as illustrated by the followingexample. The elliptical family includes the multivariate normal and t-distributions. Due to its ease of implementation, this family ofdistributions is commonly used in risk management (see McNeil et al.(2015)) and credit risk analysis, see Glasserman (2005) and Glassermanand Li (2005). Further results on the relation between Cp∗(X|Y ) andQp(X|Y ) are presented later.



Example (Elliptical distributions)

For a vector µ = (µ1, µ2) and a positive semi-definite matrix Σ = (σij) ∈ R2×2, let (X, Y ) follow from anelliptical distribution E2(µ,Σ, φ), where φ is the generator of E2(µ,Σ, φ) (see Cambanis et al. (1981)).Assume E[|Y |] <∞. By Corollary 5 of Cambanis et al. (1981), for p ∈ (0, 1),

Qp(X|Y ) = E[X|Y = qp(Y )] = µ1 +σ12

σ22(qp(Y )− µ2).

Analogously, (e.g. Theorems 2-3 of Landsman and Valdez (2003)), for p ∈ (0, 1),

Cp(X|Y ) = E[X|Y > qp(Y )] = µ1 +σ12

σ22(cp(Y )− µ2).

If p ∈ [1/2, 1), then p∗ ∈ [0, 1). Therefore, we have Cp∗ (X|Y ) = Qp(X|Y ) for p ≥ 1/2. In the capital

allocation setting, if (X1, . . . , Xd) ∼ Ed(µ,Σ, φ) and Y = X1 + · · ·+Xd is non-degenerate and integrable,then (Xi, Y ) is elliptically distributed for all i = 1, . . . , d, and in turn, for p ≥ 1/2 we have that

(Cp∗ (X1|Y ), . . . , Cp∗ (Xd|Y )) = (Qp(X1|Y ), . . . , Qp(Xd|Y )),

which can also be derived from Corollary 8.43 of McNeil et al. (2015).



Next, we build up a simple example to show that Qp(X|Y ) 6= Cp∗(X|Y )is possible, although the two values are typically quite close.

Example (Pareto risks)

For α > 1, take X ∼ Pareto(2α), i.e. P (X > x) = x−2α, x ≥ 1. Let Y = X2 and therefore,Y ∼ Pareto(α). One may calculate, for t > 1,

y(t) = E[Y |Y > t] =

∫∞t αy−αy.

t−α= t

α

α− 1,

and hence, E[X|Y = y(t)] =√t αα−1

.

On the other hand,

E[X|Y > t] = E[X|X >√t] =

√t

2α

2α− 1.

Therefore,

E[X|Y = y(t)]

E[X|Y > t]=

√αα−1

2α2α−1

=2α− 1

2√α(α− 1)

6= 1. (17)

For some p with qp(Y ) > E[Y ], we can replace t by qp∗ (Y ). Then y(t) = cp∗ (Y ) = qp(Y ), and (17) gives

that Qp(X|Y ) 6= Cp∗ (X|Y ). Moreover we note that the value in (17) is quite close to 1 if α ≥ 2 (e.g., it

is 1.06066 for α = 2 and 1.00416 for α = 6).



Quantity of interest. Let p ∈ (0, 1) and (X1, . . . , Xd) be a randomvector such that Y := X1 + · · ·+Xd is continuously distributed. Forsimplicity, write Cj = Cp∗(Xj |Y ) for all j = 1, . . . , d, which is ourtarget to estimate.Observations. Let {Xt = (X1,t, · · · , Xd,t)}nt=1 be a stationary sequencefrom the distribution of (X1, . . . , Xd) and write Yt := X1,t + · · ·+Xd,t

for all t = 1, . . . , n.



Estimation. A nonparametric estimator p∗ for p∗ in (15) is obtained bysolving the following equation for t ∈ [0, 1):

1

1− t

∫ 1

tG−n (s) ds = G−n (p),

where Gn(s) := 1n

∑nt=1 I(Yt ≤ s) and G−n denotes the generalized

inverse function of Gn. It is not difficult to check that the aboveequation has a unique solution when

∫ 10 G

−n (s) ds < G−n (p). Therefore,

a nonparametric estimator for Cj = Cp∗(Xj |Y ) is given by

Cj :=1

1− p∗1

n

n∑t=1

Xj,tI(Yt > G−n (p∗)). (18)



To derive the asymptotic limit of the above estimator,{Xt := (X1,t, · · · , Xd,t)

}∞t=−∞ is assumed to be a strictly stationary

α-mixing sequence, i.e.

αX(k) = sup{∣∣P (A ∩B)− P (A)P (B)

∣∣ : A ∈ F i−∞,B ∈ F∞i+k,−∞ < i <∞

}→ 0

as k →∞, where Fba denotes the σ-field generated by {Xt}bt=a.



Let Fj and G denote the distribution function of Xj,t and Yt,respectively. Further, define

C(x, y; j) := P(Fj(Xj,t) ≤ x, 1−G(Yt) ≤ y

),

Fjn(x) :=1

n

n∑t=1

I(Xj,t ≤ x),

Cn(x, y; j) :=1

n

n∑t=1

I(Fj(Xj,t) ≤ x, 1−G(Yt) ≤ y

).

A key technique in deriving our theoretical results is the followingweighted approximation of empirical copula process.



Under the regularity condition C1) given below, it follows fromProposition 4.4 of Berghaus et al. (2017) that

sup1/n≤u1,u2≤1−1/n

∣∣√n(Cn(u1, u2; j)− C(u1, u2; j))−W (u1, u2; j)

∣∣{min(u1, u2, 1− u1, 1− u2)}δ

= op(1)

(19)for any δ ∈ (0, 1/2), where W (u1, u2; j) is a bivariate centered Gaussianprocess with covariance

Cov(W (u1, u2; j),W (v1, v2; j))

=

∞∑i=−∞

Cov(I(Fj(Xj1) ≤ u1, G(Y1) ≤ u2

),

I(Fj(Xj,1+i) ≤ v1, G(Y1+i) ≤ v2

)).



Regularity conditions:

C1) αX(k) = O(ak) for some a ∈ (0, 1);

C2) E[Y ] < qp(Y );

C3) G is differentiable, G′(x) is continuous at G−(p) andC(2)(x, y; j) = ∂

∂yC(x, y; j) is continuous at 1− p∗ for all x ∈ [0, 1];

C4) There exist η0 > 0 and δ0 > 0 such thatE[|Yt|2+δ0I

(Yt > G−(p∗)− η0

)]<∞,

E[|Xj,t|2+δ0I

(St > G−(p∗)− η0

)]<∞,

sup|s−p∗|≤η0

∫ 1

0C(2)(x, 1− s; j) dF−j (x) <∞.



TheoremUnder conditions C1)–C4), for a given p ∈ (0, 1), we have

√n(p∗−p∗) = −

1

G−(p)−G−(p∗)

∫ 1

p∗

W (1, 1−s; j)G′(G−(s))

ds

+(1− p∗)W (1, 1− p; j)

G′(G−(p∗)

)(G−(p)−G−(p∗)

) + op(1)

√n(Cj−Cj) =

(−

1

G−(p)−G−(p∗)

∫ 1

p∗

W (1, 1− s; j)G′(G−(s))

ds +(1− p∗)W (1, 1− p; j)

G′(G−(p∗))(G−(p)−G−(p∗))

)

×(

1

(1−p∗)2

∫ 1

0F−j (x) dC(x, 1− p∗; j) +

1

1− p∗

∫ 1

0C

(2)(x, 1− p∗; j) dF−j (x)

)

+1

1−p∗

(−∫ 1

0W (x, 1−p∗; j) dF−j (x) +W (1, 1−p∗; j)

∫ 1

0C

(2)(x, 1−p∗; j) dF−j (x)

)+op(1).



Interval Estimation. Obviously the above theorem shows that Cj canbe estimated nonparametrically at the standard rate of convergencen−1/2 and the asymptotic distribution of the proposed nonparametricestimator Cj is normal, but with a complicated asymptotic variance.To construct a confidence interval for Cj , one may simply employ theblockwise bootstrap method for a mixing sequence. However, since weare dealing with quantiles, nonparametric inferences become veryinefficient due to a small number of blocks. A simple remedy is tomodel the dependence of each {Xi,t}nt=1 by a time series model andthen to employ a bootstrap method based on residuals to constructconfident intervals.



Specifically, we assume that each {Xi,t}nt=1 follows an AR-GARCHmodel (for details, see Chen and Fan (2006)), i.e.,{

Xj,t = µj +∑pj

i=1 aj,iXj,t−i + ej,t, ej,t = h1/2j,t ηj,t,

hj,t = wj +∑qj

i=1 αj,ie2j,t−i +

∑pjk=1 βj,khj,t−k

(20)

for all j = 1, . . . , d, where{ηt := (η1,t, . . . , ηd,t)

}is a sequence of

independent and identically distributed random vectors with zeromeans and variances of one.



In order to estimate the asymptotic variance of Cj via a bootstrapmethod, one has to estimate ej,t, which requires to estimate theunknown parameters in (20). An obvious estimator is the so-calledquasi maximum likelihood estimator; its asymptotic normality isavailable in Francq and Zakoıan (2004), which requires finite fourthmoments of both ej,t and ηj,t. However, it is quite often in practicethat

∑qji=1 αj,i +

∑pjk=1 βj,k are close to one, and thus, assuming

Ee4j,t <∞ may be questionable.



Here, we propose to employ the self-weighted estimator from Ling(2007) with the following weights

δj,t =

{max

(1,

1

dj

t−1∑k=1

|Xt−k|I(|Xt−k| > dj

)k9

)}−4to estimate

θj = (µj , aj,1, . . . , aj,Pj , αj,1, . . . , αj,qj , βj,1, . . . , βj,pj ),

say θj , where the asymptotic normality only requires E[|ej,t|] < 1 andE[η2j,t] <∞. Here, dj is chosen as the 90% sample quantile of{Xj,t}nt=1, as suggested in Zhu and Ling (2011).



After obtaining θ1, . . . , θd, we get our estimators for ηt, t = 1, . . . , n,say ηt. Therefore, we resample from {ηt}nt=1 with sample size n andthen refit models (20) to obtain bootstrap samples X∗j,t for j = 1, . . . , dand t = 1, . . . , n. Based on this bootstrap sample, one may calculatethe bootstrapped estimator of Cj . By repeating this procedure, abootstrap confidence interval for Cj via Cj could be constructed.


Risk Allocation – Data Analysis

The proposed nonparametric estimator is now investigated for financialdata. The bootstrap method is applied to the 100 times log-returns ofIBM stock price (X1,t) and S&P 500 index (X2,t) from December 1,2005 to December 31, 2015. Hence, Yt := X1,t +X2,t.First, we compute the proposed nonparametric estimates Cp∗(X1,1|Y1)and Cp∗(X2,1|Y1) for levels p = 0.95 and 0.99. To examine the closenessof these new allocations to the VaR-based allocations Qp(X1,1|Y1) andQp(X2,1|Y1), we estimate them nonparametrically by

Qp(X1,1|Y1) =

∑nt=1X1,tk(Yt−θh )∑nt=1 k(Yt−θh )

and Qp(X2,1|Y1) =

∑nt=1X2,tk(Yt−θh )∑nt=1 k(Yt−θh )

,

where θ is the p-th sample quantile of Yt’s. We use k(x) = 34(1− x2) for

x ∈ (−1, 1) and h = 0.5n−1/5, n−1/5, 1.5n−1/5, which have the samerate of convergence as the optimal bandwidth in terms of minimizingthe mean squared error.



We also estimate the percentagesCp∗ (X1,1|Y1)

Cp∗ (X1,1|Y1)+Cp∗ (X2,1|Y1)and

Cp∗ (X2,1|Y1)

Cp∗ (X1,1|Y1)+Cp∗ (X2,1|Y1)by

Cp∗ (X1,1|Y1)

Cp∗ (X1,1|Y1)+Cp∗ (X2,1|Y1)and

Cp∗ (X2,1|Y1)

Cp∗ (X1,1|Y1)+Cp∗ (X2,1|Y1), respectively. Since the sum of these two

estimators is one, it is easy to check that these two estimators have thesame standard deviation. Hence we only report results for the firstestimator.



To compute the standard deviations of the proposed estimators by thebootstrap method given after Theorem 7, which uses the self-weightedquasi maximum likelihood estimation in Ling (2007) to fitAR(1)-GARCH(1,1) models to our data. The estimates are

{µ1 = 0.0389, a1,1 = −0.0264, w1 = 0.1201, α1,1 = 0.1187, β1,1 = 0.8161,µ2 = 0.0770, a2,1 = −0.063, w2 = 0.0366, α2,1 = 0.1386, β2,1 = 0.8393.

(21)

Based on the above estimates and drawing 1, 000 bootstrap samples,we compute the bootstrapped standard deviations for Cp∗(X1,1|Y1),Cp∗(X2,1|Y1), Qp(X1,1|Y1) and Qp(X2,1|Y1). We also compute the

bootstrapped correlation coefficients between Cp∗(X1,1|Y1) and

Cp∗(X2,1|Y1), and between Qp(X1,1|Y1) and Qp(X2,1|Y1). These resultsare reported in Table 13. We remark the bootstrapped standarddeviations of Qp(X1,1|Y1) and Qp(X2,1|Y1) are incorrect as Hall (1990)showed that bootstrap method can not catch the asymptotic bias inkernel density estimation.



Table 13: Estimates, Bootstrapped Standard Deviations and BootstrappedCorrelation Coefficient. Estimates and bootstrapped standard deviations arereported for Cp∗(X1,1|Y1), Cp∗(X2,1|Y1), Qp(X1,1|Y1) and Qp(X2,1|Y1) for levelsp = 0.95 and 0.99. The bootstrapped correlation coefficients between Cp∗(X1,1|Y1)and Cp∗(X2,1|Y1) are −0.0891 for p = 0.95 and −0.2432 for p = 0.99. Thebootstrapped correlation coefficients between Qp(X1,1|Y1) and Qp(X2,1|Y1) withh = 0.5n−1/5, n−1/5, 1.5n−1/5 are, respectively, −0.4493,−0.2008, 0.0427 for p = 0.95and −0.5996,−0.6117,−0.4389 for p = 0.99.

p = 0.95 p = 0.99Estimate Bootstrapped SD Estimate Bootstrapped SD

p∗ 0.8484 0.0084 0.9687 0.0035Cp∗ (X1,1|Y1) 1.9441 0.1318 3.3539 0.3921

Cp∗ (X2,1|Y1) 1.7000 0.1817 3.3379 0.5869

Qp(X1,1|Y1), h = 0.5n−1/5 1.8889 0.2000 3.4250 0.7938

Qp(X2,1|Y1), h = 0.5n−1/5 1.7562 0.2097 3.2555 0.8989

Qp(X1,1|Y1), h = n−1/5 1.9545 0.1632 3.3627 0.6044

Qp(X2,1|Y1), h = n−1/5 1.6771 0.1789 3.3111 0.7781

Qp(X1,1|Y1), h = 1.5n−1/5 2.0329 0.1395 3.3840 0.4862

Qp(X2,1|Y1), h = 1.5n−1/5 1.5974 0.1643 3.2869 0.6773Cp∗ (X1,1|Y1)

Cp∗ (X1,1|Y1)+Cp∗ (X2,1|Y1)0.5335 0.0344 0.5012 0.0645



From Table 13, we observe that i) the new allocations are close to theircorresponding VaR-based allocations; ii) the capital allocation estimatefor S&P 500 index has a larger variance than that for the IBM stockprice, which may be due to the fact that β2,1 + α2,1 is closer to onethan β1,1 + α1,1; iii) the variance for each estimate increases as pbecomes larger; iv) the variances for the VaR-based allocationestimates are quite sensitive to the choice of bandwidth, which may beexplained by the inconsistency of the employed bootstrap method asargued in Hall (1990); v) the variances of the VaR-based allocationestimates are larger than those for the new allocation estimates whenp = 0.99, which may be explained by the faster rate of convergence ofthe new allocation estimates.



To distinguish different time periods, we use 10-day return data and amoving window of 250 or 500 observations to compute p∗ and thepercentage of capital allocated to the S&P 500 index with respect tothe level p = 0.99. These estimates are reported in Figures 4-5. Duringa majority of time spots reported in Figures 4, the value of p∗ isslightly less than 0.975, indicating two conclusions for this particulardata set. First, the Basel Committee on Banking Supervision recentlyreplaced VaR0.99 by ES0.975 as the risk measure for 10-day market risk.Figure 4 shows that the transition from VaR0.99 to ES0.975 will slightlyincrease the capital requirement for market risk. Second, sincep∗ ≈ 0.974 for normal risks, the plots in Figure 4 show that the 10-dayreturn data have a heavier tail than normal distribution. In Figure 5,we observe that the percentage of the capital allocated to S&P takesthe smallest value during the financial crisis. This may be explained bythe fact that the IBM stock volatility were high during the period of2007-2009, and at the same time the value of S&P decreased drastically(hence, a less portion of the portfolio is invested in S&P).



Figure 4: p∗ based on real data. p∗ is calculated for p = 0.99 based on ten days’returns with a moving window of observations 250 or 500.

2008 2010 2012 2014 2016

0.9

40

.95

0.9

60

.97

0.9

8

ph

at*

moving window with 250 observations

2008 2010 2012 2014 2016

0.9

40

.95

0.9

60

.97

0.9

8

ph

at*




Figure 5: Percentage estimate based on real data.Cp∗ (X1,1|Y1)

Cp∗ (X1,1|Y1)+Cp∗ (X2,1|Y1)is

calculated for p = 0.99 based on ten days’ returns with a moving window ofobservations 250 or 500.

2008 2010 2012 2014 2016

0.5

0.6

0.7

0.8

pe

rce

nta

ge


2008 2010 2012 2014 2016

0.4

50

.50

0.5

50

.60

0.6

50

.70

0.7

5

pe

rce

nta

ge




Finally, in order to simulate data close to these two real data sets, weneed to fit a parametric family to ηt. Although we could use the Rpackage ’sn’ to fit a skewed bivariate t-distribution, the mean andvariance of ηj,t may not be zero and one, respectively, which arerequired by the models (20). Instead, we simply use the R function’stdFit’ to fit a t distribution to each marginal and use the R package’copula’ to fit a t copula to the copula of ηt. This gives marginaldistributions t(4.9504) and t(5.3462), and the t copula with ρ = 0.6976and ν = 5.3355. These parameters will be employed to generatesamples to examine the finite sample performance of the proposednonparametric estimator and its bootstrap method in the nextsimulaion study.


Rsik Allocation – Simulation Study

We examine the finite sample performance of the proposednonparametric estimation by considering the models described in (20)with d = 2, θ1 and θ2 as given in (21), while η′ts are independentrandom vectors with t-copula and t-distributed marginals with allparameters obtained as in the above real data analysis.Note that our new capital allocation Cp∗(Xj |Y ) could be different fromthe studied one Qp(Xj |Y ) in the literature, and our proposednonparametric estimator for Cp∗(Xj |Y ) has a faster rate of convergencethan the one for Qp(Xj |Y ).



Firstly, in order to obtain the true values of C1 and C2 at p = 0.95 and0.99, we draw 100, 000 random samples of size n = 100, 000 from ourmodels. The next step is to approximate the true values by theaverages of these C1 and C2, which are 2.7294 for p = 0.95 and 5.6655for p = 0.99, as reported above.



Secondly, we draw 400 random samples of size n = 2000 and 3000 fromthe above models and then compute C1 and C2, whose mean andstandard deviation based on these 400 estimates are reported in Table1 as well. The reason why we do not repeat a larger number of times isthat bootstrap method for a time series model is computationallyintensive, since one has to refit the time series model to the resamplingfrom the residuals. Due to the computational constraint, the bootstrapestimates of the standard deviations of the proposed estimates, asdescribed after Theorem 7, are based on 400 repetitions, which arereported in Table 14 below.



Table 14: Nonparametric Estimate: the true values and proposed nonparametricestimates are given when p = 0.95 and 0.99; the numbers in brackets are thestandard deviations of C1 and C2, respectively.

(p, n) True C1 C1 Bootstrap SD True C2 C2 Bootstrap SD(0.95, 2000) 1.9062 1.9137 (0.1451) 0.1545 1.6872 1.6778 (0.2678) 0.2648(0.99, 2000) 3.2956 3.3177 (0.4242) 0.5265 3.1799 3.1391 (1.0224) 1.1787(0.95, 3000) 1.9062 1.9055 (0.1216) 0.1203 1.6872 1.6546 (0.2106) 0.2018(0.99, 3000) 3.2956 3.3138 (0.3835) 0.3741 3.1797 3.0588 (0.6767) 0.9099



From Table 14, we observe that the proposed nonparametric estimatorbecomes more accurate in terms of mean squared errors when thesample size increases. Moreover, the asymptotic variance increases as pbecomes larger, while C1 has a smaller variance than C2 under theconsidered setting, which is consistent with the observation in the realdata analysis.


Risk Allocation – Some Properties

We would like Cp∗(X|Y ) is close to Qp(X|Y ) for a larger p. That is, wewould like to investigate when

Qp(X|Y )

Cp∗(X|Y )→ 1 as p ↑ 1. (22)

Again recall that p∗ depends on p.Assuming that Y is continuously distributed with X and Y beingunbounded from above, i.e. qp(X)→∞ and qp(Y )→∞ as p ↑ 1, it isobvious that (22) is equivalent to

E[X|Y = y(t)]

E[X|Y > t]→ 1 as t→∞, (23)

where y(t) = E[Y |Y > t].



Theorem

Let X,Y ∈ L1, unbounded from above, and Y is continuouslydistributed.

i) If limt→∞

E[X|Y = t]/t exists, is finite and non-zero, then (23)

holds.

ii) Assume that X ≥ 0, there exists t0 > 0 such thatE[X|Y = t] <∞ for all t > t0, and further,

limt→∞

P (X > tx|Y = t) := g1(x) (24)

holds uniformly on <+ := [0,∞) with g1 being integrable on <+.Then, E[X|Y = t]/t→

∫∞0 g1(x) dx as t→∞ and in turn, (23)

holds as long as∫∞0 g1(x) dx > 0.



Theorem (Continue)iii) Assume that x0 := sup{x ∈ R : F (x) = 0} > −∞ and there exists t0 > 0 such thatE[X|Y = t] <∞ for all t > t0. If

limt→∞

P (X > x|Y = t) := g2(x) (25)

holds uniformly on [x0,∞), then (23) holds whenever x0 +∫∞x0

g2(x) dx is finite and non-zero.

iv) Assume that (X, Y ) has a density function f such that

limt→∞

f(tx, ty)

v(t):= q(x, y) for (x, y) ∈ R2

, (26)

where v(t) is a regularly varying function at infinity with index −α for some α > 2 (that is,

limt→∞ v(tx)/v(t) = x−α for all x > 0). Then, (23) holds as long as∫∞−∞ zq(z, 1) dz and∫∞

−∞ q(z, 1) dz are finite and nonzero.

v) If X and Y are jointly log-normally distributed, i.e. (logX, log Y ) is bivariate normal, then (23)holds.


Thank You!


Documents

Statistical Inference for Mortality Model and Risk Allocation