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MANAGEMENT SCIENCE

Vol. 55, No. 12, December 2009, pp. 2019–2027

issn0025-1909?eissn1526-5501?09?5512?2019

informs®

doi10.1287/mnsc.1090.1090

©2009 INFORMS

Conditional Monte Carlo Estimation of

Quantile Sensitivities

Michael C. Fu

Robert H. Smith School of Business and Institute for Systems Research, University of Maryland,

College Park, Maryland 20742, mfu@rhsmith.umd.edu

L. Jeff Hong

Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology,

Clear Water Bay, Hong Kong, China, hongl@ust.hk

Jian-Qiang Hu

Department of Management Science, School of Management, Fudan University, 200433 Shanghai, China,

hujq@fudan.edu.cn

E

programming. Recently, Hong (Hong, L. J. 2009. Estimating quantile sensitivities. Oper. Res. 57 118–130) derived

a batched infinitesimal perturbation analysis estimator for quantile sensitivities, and Liu and Hong (Liu, G., L. J.

Hong. 2009. Kernel estimation of quantile sensitivities. Naval Res. Logist. 56 511–525) derived a kernel estimator.

Both of these estimators are consistent with convergence rates bounded by n−1/3and n−2/5, respectively. In this

paper, we use conditional Monte Carlo to derive a consistent quantile sensitivity estimator that improves upon

these convergence rates and requires no batching or binning. We illustrate the new estimator using a simple but

realistic portfolio credit risk example, for which the previous work is inapplicable.

stimating quantile sensitivities is important in many optimization applications, from hedging in financial

engineering to service-level constraints in inventory control to more general chance constraints in stochastic

Key words: quantiles; value at risk; credit risk; Monte Carlo simulation; gradient estimation

History: Received November 20, 2008; accepted August 4, 2009, by Wallace J. Hopp, stochastic models and

simulation. Published online in Articles in Advance October 30, 2009.

1.

There are many simulation settings in which the per-

formance measure of interest is a quantile rather than

an expected value. For example, in risk management,

value at risk (VaR) is a widely used measure by gov-

ernment regulators to specify minimal capital reserve

requirements in the financial industry, and in sup-

ply chain management, inventory control using safety

stocks are commonly specified. In the simulation set-

ting, derivative estimation of performance measures

is critical for optimization and sensitivity analysis.

However, there has been limited work on quantile

sensitivity estimation, until the recent work of Hong

(2009), who derived a batched infinitesimal pertur-

bation analysis (IPA) estimator, and Liu and Hong

(2009), who derived a kernel estimator. These estima-

tors are consistent with convergence rates bounded

by n−1/3and n−2/5, respectively. In this paper, we

apply conditional Monte Carlo, or smoothed pertur-

bation analysis, to derive a new estimator that has

the following advantages over the previously derived

estimators: (a) it doesn’t require batching or bin-

ning, and (b) it has a superior convergence rate. On

the other hand, applying conditional Monte Carlo

Introduction

requires choosing conditioning variables appropriate

for the problem setting. The convergence analysis of

this estimator uses a technique of handling order

statistics recently developed by Hong and Liu (2009),

who developed sensitivity estimators for conditional

value at risk (CVaR), an alternative performance mea-

sure in risk management related to VaR and possess-

ing some superior theoretical properties; however, the

results in that paper cannot be applied to VaR, which

is far more widely used in the financial services indus-

try. Thus, this paper fills an important gap in both

theory and practice.

A recent overview on estimating derivatives (or

sensitivities) in simulation is given in Fu (2008), and

a more in-depth technical discussion is contained

in Fu (2006), which includes numerous references.

Glasserman (1991) and Ho and Cao (1991) provide

the background on IPA, the technique that is used in

Hong (2009) and Hong and Liu (2009), and Fu and

Hu (1997) provide the background on smoothed per-

turbation analysis, or conditional Monte Carlo, which

is used in this paper. Literature on simulation estima-

tion of quantiles can be found in Hong (2009). In the

rest of this paper, we derive the general conditional

2019

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Fu, Hong, and Hu: Conditional Monte Carlo Estimation of Quantile Sensitivities

Management Science 55(12), pp. 2019–2027, ©2009 INFORMS

2020

Monte Carlo quantile sensitivity estimator, illustrating

it briefly using the single-server queue and then in

more detail on an application from finance in portfolio

credit risk, an example for which none of the results

of Hong (2009), Liu and Hong (2009), and Hong and

Liu (2009) are applicable.

2. Conditional Monte Carlo

Form of q?

Suppose that L??? is a continuous random variable

that depends on a parameter ? ∈?, where ? ⊂? is an

open set. Let F?·??? denote the distribution function

of L???, and q???? denote the ?-quantile of L??? for

any 0<?<1. Then, q???? satisfies

????

F?q???????=??

(1)

In this paper, we are interested in estimating q?

dq????/d? for any ? ∈?.

If F?t??? and q???? are both differentiable, then by

differentiating with respect to ? on both sides of Equa-

tion (1), we have (where ? denotes the partial deriva-

tive with respect to the subscripted argument)

???? =

?tF?t????t=q????·q?

Therefore,

????+??F?t????t=q????=0?

q?

????=−??F?t???

?tF?t???

????

t=q????

?

(2)

Because F?t???=Pr?L???≤t?=E?1?L???≤t??, and the

indicator function 1?L???≤t? is a discontinuous func-

tion with respect to either ? or t, we cannot inter-

change the order of differentiation and expectation to

compute ??F?t??? and ?tF?t???, so we apply condi-

tional Monte Carlo, and summarize the results in the

following theorem.

Theorem 1. Suppose that there exist random variables

X1??? and X2??? such that

F?t??? = E?Pr?L????Xi?????

= E?Gi?t?Xi????????

G1?t?X1?????? is differentiable with probability 1 (w.p.1)

with respect to ? and G2?t?X2?????? is differentiable w.p.1

with respect to t, and

i =1?2?

?G1?t?X1??+?????+???−G1?t?X1???????≤K1????? (3)

?G2?t+?t?X2??????−G2?t?X2???????≤K2??t??

for some random variables K1and K2with E?K1?<? and

E?K2?<?. If F?t??? and q???? are both differentiable, then

????=−E???G1?t?X1???????

E??tG2?t?X2???????

(4)

q?

????

t=q????

?

Proof. By the dominated convergence theorem

(Rudin 1987) and Equations (3) and (4), we have

??E?G1?t?X1??????? = E???G1?t?X1????????

?tE?G2?t?X2??????? = E??tG2?t?X2????????

Then the conclusion of the theorem follows directly

fromEquation(2)and

and G2.

?

Remark 1. It is important to note that L is not

required to be Lipschitz continuous with respect to ?,

which leads to wider applicability than the results in

Hong (2009), Liu and Hong (2009), and Hong and

Liu (2009), as the portfolio credit risk example in §4

demonstrates.

Remark 2. Although we condition on two random

variables X1??? and X2??? in Theorem 1, the two ran-

dom variables may be the same, as shown in the

single-server queue example and the example of §4.1.

However, allowing the two random variables to be

different provides more flexibility as shown in the

example in §4.2.

As an example, we consider a first-come, first-

served G/G/1 queue. Let Ai, Si, Ti, and Widenote

the interarrival time, service time, system time, and

waiting time of customer i, respectively. Suppose that

Sihas a distribution function G?t??? and a density

function g?t???, and we are interested in estimating

the quantile sensitivity of Tiwith respect to ?. By

Lindley’s equation for the G/G/1 queue, Wi= ?Ti−1−

Ai?+. From Ti= Si+ Wi, and because Siand Wiare

independent,

thedefinitions of

G1

Pr?Ti≤t?=Pr?Si≤t −Wi? = E?Pr?Si≤t −Wi?Wi??

= E?G?t −Wi?????

Note that

??E?G?t −Wi????

=E

?tE?G?t −Wi????=E?g?t −Wi?????

Then, by Theorem 1,

?

−g?t −Wi???dWi

d?

+??G?t −Wi???

?

?

q?

????=E?g?t−Wi???dWi/?d??−??G?t−Wi????

E?g?t−Wi????

????

t=q????

?

(5)

Computation of dWi/d? in Equation (5) can be carried

out using infinitesimal perturbation analysis (Ho and

Cao 1991) as follows. Because Ti= Si+ Wiand Wi=

?Ti−1−Ai?+, we have

dTi−1

d?

d?

dWi

d?d?

=dSi−1

=dTi−1

+dWi−1

d?

and

·1?Wi>0?

w.p.1?

(6)

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Management Science 55(12), pp. 2019–2027, ©2009 INFORMS

2021

where dSi/?d?? = −??G?Si???/g?Si???. Suppose that

the system starts empty. Then, dW1/d? =0, and Equa-

tion (6) can be used to iteratively calculate dWi/d? for

any i =2?3?????

3.Conditional Monte Carlo Estimator

of q?

Y?t? = ??G1?t?X1??????

· ?t?X2?????? for any t ∈ ?. Suppose that, for any

t ∈?, we can observe n independent and identi-

cally distributed (i.i.d.) samples ?L1?Y1?t??Z1?t???

?????Ln?Yn?t??Zn?t??, e.g., from running simulation

experiments. Let Lk?ndenote the kth-order statistic

of L1?L2?????Ln, and let ˆ q?= L?n???n. Note that ˆ q?is

a commonly used estimator of q???? (Serfling 1980).

Let? Y?t? = ?1/n??n

ˆ q?

where? Y?·?,?Z?·?, and ˆ q?are all computed using the

same set of samples. In the rest of this section, we

show that ˆ q?

n−1/2under some regularity conditions. The major dif-

ficulty in the convergence analysis is the dependence

between ?L1?Y1?t??Z1?t?? and ˆ q?. We use a technique

developed by Hong and Liu (2009) to circumvent

this difficulty. A detailed analysis is provided in the

appendix.

Let

u1?t? = E?Y1?t?1?L1< t???u2?t? = E?Y1?t? ·

1?L1> t??, and u3?t? = E?Y1?L1?1?L1< t??, and let

v1?t? = E?Z1?t?1?L1< t??, v2?t? = E?Z1?t?1?L1> t??, and

v3?t? = E?Z1?L1?1?L1< t??. Then, the following theo-

rem shows that ˆ q?

as n→?. The proof of the theorem is provided in the

appendix.

????

Letand

Z?t? = ?tG2

i=1Yi?t? and?Z?t? = ?1/n??n

????=−? Y?ˆ q??/?Z?ˆ q???

i=1Zi?t?.

Then, we can estimate q?

???? by

(7)

???? is consistent with convergence rate

???? is a consistent estimator of q?

????

Theorem 2. Under the assumptions in Theorem 1, if

E?K2+?

j

? < ? for some ? > 0 for both j = 1?2, where K1

and K2are the random variables defined in the assumptions

of Theorem 1, and if ui?t? and vi?t?, i = 1?2?3, are all

continuous at t =q????, then ˆ q?

Remark 3. The conditions on ui?t? and vi?t? in The-

orem 2 are typically easy to verify in practice. By

Equation (3), we have ?Y1?t?1?L1<t??≤?Y1?t??≤K1. If

Y1?t? is continuous at t = q???? w.p.1 and if Pr?L1=

q????? = 0, then Y1?t?1?L1< t? is continuous at t =

q???? w.p.1. Because E?K1?<?, by the dominated con-

vergence theorem (Rudin 1987), u1?t? = E?Y1?t?1?L1<

t?? is continuous at t =q????. Similarly, ui?t? and vi?t?,

i = 1?2?3, are all continuous at t =q???? if Y1?t? is

at t =q???? w.p.1 and if Pr?L1= q????? = 0. For both

the G/G/1 queue example of §2 and the credit risk

example of §4, these conditions can be verified easily.

????

P

−→q?

???? as n→?.

We now analyze the rate of convergence of ˆ q?

The big O notation is often used to denote the

rate of convergence of a sequence of determinis-

tic numbers, where an= O?bn? as n → ? denotes

limsupn→??an/bn?<?? For the rate of convergence of

a sequence of random variables, we use the concept

of bounded in probability. A sequence of random vari-

ables ?Xn? is said to be bounded in probability if for

every ? >0 there exists M?>0 such that

????.

limsup

n→?

Pr??Xn?≥M??≤??

which we denote by Xn= Op?1? (see, for instance,

Bhat 1985, Lehmann 1999). Furthermore, for two

sequences of random variables ?Un? and ?Vn?, the

notation Un=Op?Vn? denotes that Un/Vn=Op?1?.

Let f?t? denote the density of L, and fur-

ther let u4?t? = E?Y1?L1?Y2?L1?1?L1> L2?1?L1< t??,

u5?t? = E?Y1?L1?Y2?L1?1?L1< L2?1?L1< t??, v4?t? =

E?Z1?L1?Z2?L1?1?L1> L2?1?L1< t??,

E?Z1?L1?Z2?L1?1?L1<L2?1?L1<t??. Then, the following

theorem shows that the rate of convergence of ˆ q?

is n−1/2. The proof of the theorem is provided in the

appendix.

and

v5?t? =

????

Theorem 3. Under the assumptions in Theorem 1, if

E?K2

dom variables defined in the assumptions of Theorem 1,

and if f?t? is continuously differentiable at t = q????,

f?q????? > 0, and if ui?t? and vi?t?, i = 1?????5, are all

twice differentiable at t = q????, ?u?

?v?

all i =1?????5, then ˆ q?

Remark 4. Although the conditions on ui?t? and

vi?t? in Theorem 3 are typically difficult to verify, we

expect them to hold in practice. Similar conditions

have also been used in Hong (2009) and Liu and Hong

(2009) to analyze other types of estimators of quan-

tile sensitivities. Furthermore, the numerical results in

§4.3 also support the conclusion of the theorem.

By Serfling (1980, p. 52), Xn= Op?1? if Xn⇒ X

as n → ?, where ⇒ denotes “convergence in dis-

tribution.” Hong (2009) proposes a batching estima-

tor? Dmkof q?

N?0??2? as n → ? with some ?2> 0, where k satis-

fies that k → ? and k3/n2→ 0 as n→?. Therefore,

? Dmk=Op?k−1/2?=Op?n−1/3?. Liu and Hong (2009) pro-

pose a kernel estimator? Vnof q?

?n?n?? Vn− q?

some c ≥ 0 as n → ?, and supn→??n?3

fore,? Vn=Op??n?n?−1/2?=Op?n−2/5?. By Theorem 3, we

see that the rate of convergence of ˆ q?

It is clear that ˆ q?

than the other two estimators.

j?<? for both j =1?2, where K1and K2are the ran-

i?t?? ≤ M, ?u??

i?t?? ≤ M,

i?t??≤M, and ?v??

i?t??≤M for some constant M >0 for

????−q?

????=Op?n−1/2?.

????. He shows that

√k?? Dmk− q?

????? ⇒

????. They show that

2? as n → ? with some ?

????? ⇒ N????2

and ?2>0, where ?nsatisfies ?n→0 and n?5

n→c with

n?−1< ?. There-

?is of Op?n−1/2?.

???? is asymptotically more efficient

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Fu, Hong, and Hu: Conditional Monte Carlo Estimation of Quantile Sensitivities

Management Science 55(12), pp. 2019–2027, ©2009 INFORMS

2022

4. Sensitivity Analysis for Portfolio

Credit Risk

Portfolio credit risk refers to the losses due to defaults

of obligors in a portfolio. Let L denote the loss, which

we model as

L=

m?

i=1

li·1?Xi<xi??

where m is the number of obligors in the portfolio.

In this model, we use a latent random variable Xito

determine the default of obligator i. If Xi< xi, where

xiis a threshold, then obligator i defaults and it will

cause a (random) loss of li. In this section, we assume

that liis a continuous random variable independent

of Xi. Under this assumption, L is a continuous ran-

dom variable, and thus our estimator can be applied.

With respect to parameters that affect Xi, however,

the loss function L is not Lipschitz continuous, so the

estimators of Hong (2009), Liu and Hong (2009), and

Hong and Liu (2009) cannot be applied.

To capture the dependence between the defaults of

obligors, ?X1?X2?????Xm? are often assumed depen-

dent. The normal (Gaussian) copula model is widely

used in the industry (see, for instance, Li 2000,

Glasserman 2004). It is the basis of the CreditMetrics

models developed by J. P. Morgan and the Moody’s

KMV system. However, the model fails to capture

extreme credit risk scenarios where many oblig-

ors default simultaneously, which is often observed

empirically (Mashal and Zeevi 2003). Bassamboo et al.

(2008) suggest using the following model:

Xi=?Z +?1−?2?i

W

where Z denotes the common factor that affects all

obligors, ?idenotes obligor i’s idiosyncratic risk, W is

a nonnegative random variable that captures a com-

mon shock to all obligors, and Z, W, and ?iare mutu-

ally independent. When Z and ?iare all independent

standard normal random variables and W = 1, the

model becomes the one-factor normal copula model.

When W is a random variable, a small W value will

create a common shock to all obligors and cause many

of them to default simultaneously. Bassamboo et al.

(2008) show that the model can explain extreme credit

risk when W or W2follows a gamma distribution.

Let q?denote the ?-VaR of the portfolio loss L.

When ? is close to 1, e.g., ?=0?95, q?is an important

risk measure of the portfolio credit risk. Because the

loss L depends on many parameters, we are interested

in finding the sensitivities of the ?-VaR with respect

to these parameters.

?i =1?2?????m?

4.1.

We first consider the situation where ? affects only

a single obligor and the goal is to find q?

Parameter of an Individual Obligor

????. For

instance, we consider ? as a parameter of the distri-

bution function of ?1, and we are interested in esti-

mating q?

We let Hi?·? and Fi?·? denote the distribution func-

tions of liand ?i, respectively, let hi?·? denote the

densities of li, and let FW?·? and FZ?·? denote the

distribution functions of W and Z, respectively. Let

?i=?xiW −?Z?/?1−?2for all i =1?????m. Then ?iis

mined by (measurable with respect to) W and Z. Let

FL?·? denote the distribution function of L. Then

?

Note that for any t >0,

?m?

=Pr??1<?1?W?Z?·Pr

????.

a random variable whose value is completely deter-

FL?t?=Pr?L≤t?=E Pr

?m?

i=1

li·1??i<?i?≤t

????W?Z

??

? (8)

Pr

i=1

li·1??i<?i?≤t

????W?Z

?

?

l1≤t−

?m?

li·1??i<?i?

m?

li·1??i<?i?≤t

?????W?Z

i=2

li·1??i<?i?

????W?Z

?

+ Pr??1≥?1?W?Z?·Pr

?

?m?

where?Fi?t? = 1 − Fi?t?. We may analyze Pr??m

after m iterations, we have

?m?

=

i=1

i=2

????W?Z

?

=F1??1?·E

H1

?

t−

m?

li·1??i<?i?≤t

i=2

?

+?F1??1?·Pr

i=2

????W?Z

?

?

i=2li·

1??i< ?i? ≤ t?W?Z? using the same approach. Then,

Pr

i=1

m?

li·1??i<?i?≤t

?

????W?Z

t −

?

Fi??i?·E

?

Hi

?

m?

j=i+1

lj·1??j<?j?

?????W?Z

+

i=1

?

·

i−1

?

j=1

?Fj??j?

?

m?

?Fi??i??

so by Equation (8) we have

?

?m?

Because ? is a parameter of F1?·?, differentiating Equa-

tion (9) with respect to ? yields

FL?t? =

m?

+E

i=1

E

Fi??i?·Hi

?

t−

m?

j=i+1

lj·1??j<?j?

?

·

i−1

?

j=1

?Fj??j?

?

i=1

?Fi??i?

?

?

(9)

??FL?t?

=E

?

??F1??1?·H1

?

t−

m?

i=2

li·1??i<?i?

??

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Fu, Hong, and Hu: Conditional Monte Carlo Estimation of Quantile Sensitivities

Management Science 55(12), pp. 2019–2027, ©2009 INFORMS

2023

−

m?

i=2

E

?

??F1??1?·Fi??i?·Hi

?

t−

m?

j=i+1

lj·1??j<?j?

?

?

·

i−1

?

j=2

?Fj??j?

−E

?

??F1??1?·

m?

i=2

?Fi??i?

?

?

(10)

Furthermore, for any t > 0, differentiating Equa-

tion (9) with respect to t yields

?

?tFL?t?=

m?

i=1

E

Fi??i?·hi

?

t−

m?

j=i+1

lj·1??j<?j?

?

·

i−1

?

j=1

?Fj??j?

?

? (11)

Therefore, we may apply the conditional Monte Carlo

estimator of §3 to estimate q?

and (11).

???? using Equations (10)

4.2.

Now we consider ? as a parameter of the distribu-

tion function of W, and we are interested in esti-

mating q?

Equation (9) may not be used to obtain ??FL?t? when

? is a parameter of W, because ? would occur in the

indicator function. Therefore, we need another repre-

sentation of FL?t? to compute ??FL?t?.

Without loss of generality, we assume that xi< 0

for all i = 1?????m. Let ?i= ??Z +?1−?2?i?/xi. Sort

the ith smallest one. We also define ??0?= −? and

??m+1?=+?. Furthermore, we let l?i?denote the obser-

vation of l1?????lmthat corresponds to ??i?for all i =

1?????m (e.g., if ?1=??i?, then l?i?=l1). Then,

?

?m+1

Parameter of Common Shock

????. Note that the representation of FL?t? of

?ifrom the smallest to the largest, and let ??i?denote

FL?t? = EPr

?m?

1

i=1

?m?

li·1?W <?i?≤t

?????i?li?i=1?????m

??

= E

?

·Pr???i−1?≤W <??i?

?

i=1

j=i

l?j?≤t

?

?????1??????m?

·?FW???i??−FW???i−1???

?

=

m+1

?

i=1

E1

?m?

j=i

l?j?≤t

??

? (12)

Because ? is a parameter of FW?·?, then

m+1

?

??FL?t? =

i=1

E

?

1

?m?

·???FW???i??−??FW???i−1???

j=i

l?j?≤t

?

?

? (13)

where we define ??FW???0??=??FW???m+1??=0. Combin-

ing Equations (11) and (13), we can apply the condi-

tional Monte Carlo estimator of §3 to estimate q?

????.

4.3.

We consider a simple example to illustrate the perfor-

mance of the conditional Monte Carlo quantile sen-

sitivity estimators. We suppose that there are two

obligors (i.e., m=2), ?=0?6, Z follows a standard nor-

mal distribution, ?ifollows a normal distribution with

mean ?iand variance 1, i = 1?2, and W follows an

exponential distribution with rate ?. Furthermore, we

suppose that x1= x2= −2 and liare independently

uniformly distributed on ?0?1?. We let ?1=?2=0, ? =

1/0?3, and ?=0?95. We are interested in estimating the

sensitivity of the ?-quantile of the loss L with respect

to ?1(a parameter of an individual obligor) and ?

(a parameter of the common shock).

By using Equation (12), and because we only have

two obligors, we can compute the quantile sensi-

tivities by combining numerical integration with the

finite-difference method. With a very large sample

size (109), we estimate the true values of ??1q??L?

and ??q??L? to be approximately −0?2521 and 0?0628,

respectively. We use these values as the benchmarks

to evaluate the performance of the conditional Monte

Carlo estimators. Note that we cannot compare our

estimators to the batched IPA estimator of Hong

(2009) and the kernel estimator of Liu and Hong

(2009) for this example, because neither of them can

be applied to the portfolio credit risk example.

In Tables 1 and 2, we report the estimated mean,

standard deviation (SD), and root mean square error

(RMSE) of our estimators under different sample

sizes. All values reported in the tables are based

on 1?000 independent replications. From the tables,

we see that our estimators have good performance.

For both ??1q??L? and ??q??L?, the RMSEs are less

than 1% of the true values when the sample size is

105. Furthermore, the RMSEs appear to be decreasing

according to the n−1/2rate (i.e., an additional decimal

place of accuracy is obtained when the sample size

Numerical Example

Table 1Performance of ??1q??L? (True Value=−0?2521)

103

104

Sample size

105

106

Mean

SD

RMSE

−0?2533

0?022

0?022

−0?2525

0?0067

0?0067

−0?2520

0?0020

0?0020

−0?2521

0?00065

0?00065

Table 2Performance of ??q??L? (True Value=0?0628)

103

Sample size104

105

106

Mean

SD

RMSE

0?0631

0?0057

0?0057

0?0629

0?0019

0?0019

0?0628

0?00060

0?00060

0?0628

0?00019

0?00019