Hedging and Risk Management of Variable Annuity Portfolios ......Variable Annuity (VA) A variable...

Hedging and Risk Management of Variable Annuity Portfolios

A Functional Data Approach

X. Sheldon Lin

Department of Statistical Sciences

University of Toronto

[email protected]

Joint work with Guojun Gan, University of Connecticut

Based on a paper that has been accepted by Insurance: Mathematics and Economics

WatRISQ Seminar, March 17, 2015

Outline

Introduction and Motivation

A Brief Introduction of Variable Annuity (VA)

Issues and Challenges when Risk Managing a VA Portfolio

Nested Simulation

Our Approach: Clustering and Universal Kriging

Numerical Illustration

Concluding Remarks and Selected References

Variable Annuity (VA)

◮ A variable annuity is a tax-deferred retirement saving plan (deferred

annuity).

◮ It has two phases: an accumulation phase and a payout phase:

◮ During the accumulation phase, the annuitant pays

premium(s) into the contract and accumulates assets;◮ During the payout phase, the annuitant has options to receive

a lump sum payment or to convert the account value into a

fixed immediate annuity.

◮ It is wrapped with guaranteed death benefits and/or guaranteed living

benefits.

Embedded Guarantees

A variable annuity is wrapped with one or more of two types of guarantees:

◮ The guaranteed minimum death benefit (GMDB);

◮ The guaranteed minimum living benefit (GMLB), which is further divided

into

◮ The guaranteed minimum accumulation benefit (GMAB)◮ The guaranteed minimum income benefit (GMIB)◮ The guaranteed minimum withdrawal benefit (GMWB)

Guaranteed Minimum Death Benefits (GMDB)

◮ A GMDB provides a level of protection against the loss in the

subaccount(s) in the event of death.

◮ Common types of GMDBs are ‘Return of Premium’, ‘Annual Rachet’ and

and ‘Annual Roll-up’

◮ ‘Return of Premium’ guarantees the beneficiary of the variable

annuity the greater of (a) the subaccount value or (b) the total

premiums paid in the past;◮ ‘Annual Rachet’ is an enhanced guarantee that pays the

greater of (a), (b) above or (c) the account value on a prior

contract anniversary date.◮ ‘Annual Roll-up’ is a rising floor GMDB, meaning that the

benefit level increases annually at a specified rate of interest.

‘Annual Roll-up’ is often combined with ‘Annual Rachet’.

Guaranteed Minimum Living Benefits (GMLB)

◮ A GMLB is typically offered as a rider.

◮ A GMAB usually is a ‘Return of Premium’ guarantee (or its enhanced

version).

The annuitant will receive a lump sum payment after a specified period

(8-10 years).

◮ A GMIB guarantees the greater of the subaccount value or a minimum

payout base (the total premiums paid with annual credited interest rate

(3%-5%)) and a guaranteed annuitization rate after a waiting period (7+

years).

◮ A GMWB guarantees a systematic withdrawal of a certain percentage

(5%-7%) of the total premiums paid annually until the total premiums

are exhausted, regardless of the market performance.

Essentially the VA guarantees are (path dependent) put options written on the

sub-account value.

As a result, single contracts can be valuated using option pricing theory.

Issues and Challenges when Risk Managing a VA Portfolio

◮ In-house hedging program: construction of an investment portfolio to

dynamically hedge the VA portfolio.

◮ Calculation of required capitals, Regulatory Capital (RC) and Economic

Capital (EC), to meet the regulatory requirements.

◮ The payoff function of the guarantee in most contracts is path dependent

that results in no closed-form formulas for the liability value of the

guarantees.

◮ The number of contracts is very large, often in the range of 100,000 to

1,000,000.

◮ The portfolio is highly non-homogeneous: each contract is different in

terms of age, gender, time to maturity, guarantee type, account value,

etc.

◮ In practice, insurance companies follow a market-to-model approach and

use the so-called nested simulation (to be described).

Nested Simulation

◮ Nested simulation is a two-level simulation procedure: outer loop

simulation and inner loop simulation.

◮ The outer loop involves projecting the VA liabilities along real world

scenarios. At each node of an outer loop scenario, quantities of interest

(the fair market value, greeks) are valuated at each node, using a large

number of simulated risk-neutral paths (the inner loop).

◮ Those are used to help construct a hedging portfolio and to determine

required capitals for the future mismatches between the VA portfolio and

the hedging portfolio.

Nested Simulation: Graphical Illustration

Figure: Nested simulation has two loops: the outer loop and the inner

loop.

Nested Simulation: Challenges

◮ The computation of nested simulation of a large VA portfolio is highly

intensive and often prohibitive.

◮ For example, if we use 1000 real world scenarios in the outer loop and

1000 risk-neutral paths in the inner loop, and project the liabilities or a

quantity of interest at yearly steps for 30 years, then the total number of

projections for each contract is

1000× 1000× 30× 31/2 = 4.65× 108,

which is a very big number. For a portfolio of 100,000 contracts, the

number of projections would be 4.65× 1013.

Our Approach

◮ Select a small number of representative VA policies from the portfolio.

◮ Evaluate the representative VA policies using a nested simulation model.

◮ Build a universal kriging model to evaluate all the VA policies in the

portfolio without simulation.

Selection of Representative VA Policies by Clustering

◮ Let X = {x1, x2, · · · , xn} denote the portfolio of VA contracts, where n is

the number of VA contracts and xi represents the i-th VA contract.

◮ Without loss of generality, we assume that every VA contract is

characterized by d attributes (e.g., gender, age, time to maturity,

guarantee type, etc.) and that the first d1 attributes are numeric and the

remaining d2 = d − d1 attributes are categorical.

◮ Define the distance/metrics between two VA policies x and y as

D(x, y) =

√

√

√

√

d1∑

h=1

wh(xh − yh)2 +

d∑

h=d1+1

whδ(xh, yh),

where xh and yh are the hth component of x and y respectively, wh > 0 is

a weight assigned to the hth component, and δ(·, ·) is the simple

matching distance defined as

δ(xh, yh) =

{

0, if xh = yh,

1, if xh 6= yh.

Choice of Weights wh

◮ For the numeric attributes, assume the form

wh =1

R2h

,

where R2h measures the degree of variability of the data of the h-th

attribute.

◮ The most sensible choice for R2h is the corresponding sample

variance:

R2h =

1

n − 1

n∑

i=1

(xih − x̄h)2,

where x̄h is the sample mean.◮ Another choice is to use the range of the data set:

Rh = max1≤i≤n

xih − min1≤i≤n

xih.

However, this choice would give a small weight to a numeric

variable with a large extreme value.

◮ For the categorical attributes, a common weight, say α, is assigned (e.g.

α = 0.5, 1, or 2).

The K-prototypes (Modified K-Mean) Clustering Algorithm

◮ Initialize cluster centers (representative policies)

selecting k distinct contracts from portfolio X randomly: Suppose

µ(0)1 ,µ

(0)2 , · · · , µ

(0)k .

◮ Update cluster memberships

Updated the cluster memberships γ1, γ2, · · · , γn as follows:

γi = argmin1≤j≤k

D(xi ,µ(0)j ).

◮ Update cluster centers

µ(1)jh =

1

|Cj |

∑

x∈Cj

xh, h = 1, 2, · · · , d1,

µ(1)jh = modeh(Cj), h = d1 + 1, · · · , d .

◮ Repeat the above steps until the memberships of each cluster is stabilized.

◮ The k distinct representative VA contracts Z = {z1, z2, . . ., zk} are the

VA contracts that are the closest to each of the cluster centers.

The Universal Kriging Model

◮ Let Z = {z1, z2, . . ., zk} now be the k distinct representative VA

contracts.

◮ Let Xzj (t) be a quantity of interest, say the dollar Delta, at time t of the

representative contract zj along an outer loop scenario obtained by a

nested simulation model.

◮ The dollar Delta at time t of an arbitrary contract xi in the VA portfolio

is expressed linearly as

X̂xi (t) = λtri XZ (t) =

k∑

j=1

λijXzj (t),

where λi = (λi1, λi2, . . . , λik)

◮ The key is how to obtain the optimal value of λij ! This is an optimal

projection (in math) or process regression (in statistics) problem.

Universal Kriging◮ It is a method in spatial functional data analysis

◮ Let {Xx(t)}, x ∈ P, where x represents a ‘location’ or a VA contract of

the VA portfolio P in our case.

◮ All the stochastic processes {Xx(t), x ∈ P} are assumed to have the

following stationary properties:

E [Xx(t)] = m(t)

Var[Xx(t)] = σ2(t)

Cov(Xxi (t),Xxj (t)) = C(h; t),

where h is the distance between xi and xj .

◮ C(h; t) is often expressed in terms of the so-called semivariogram function

γ(h; t) =1

2Var[Xxi (t)−Xxj (t)] = σ2(t)− C(h; t).

◮ The accumulative semivariogram function

γ(h) =

∫ T

0

γ(h; t)dt

plays an important role in obtaining optimal values.

Determining the Semivariogram Function γ(h)

◮ Choose the spherical function of the form:

γ(h; a, b, c) =

0, if h = 0;

a + b[

3h2c

− 12

(

hc

)3]

, if 0 < h ≤ c;

a + b, if h > c.

◮ Estimate parameters a, b and c by minimizing the L2 distance between

the spherical function and the empirical semivariogram function:

γ̂(h) =1

2|N(h)|

∑

(x,y)∈N(h)

S∑

s=1

(Xx(ts)−Xy(ts))2 .

Here S is the number of discretization steps, and

N(h) = {(x, y) : D(x, y) = h, x ∈ Z , y ∈ Z},

where |N(h)| is the number of elements in N(h).

The Optimal Solution

The optimal values λi = (λi1, λi2, . . . , λik) is the solution of the system of

linear equations:(

A(Z ) B(Z )

Btr (Z ) 0

)

·

(

λi

vi

)

=

(

A(Z , xi )

B(xi )tr

)

.

Here, A(Z ) is a k × k matrix defined as

A(Z ) =

γ(D(z1, z1)) γ(D(z1, z2)) · · · γ(D(z1, zk))

γ(D(z2, z1)) γ(D(z2, z2)) · · · γ(D(z2, zk))...

.... . .

...

γ(D(zk , z1)) γ(D(zk , z2)) · · · γ(D(zk , zk))

.

The Optimal Solution

A(Z , xi ) =

γ(D(z1, xi ))

γ(D(z2, xi ))...

γ(D(zk , xi ))

.

B(Z ) =

f1(z1) f2(z1) · · · fq(z1)

f1(z2) f2(z2) · · · fq(z2)...

.... . .

...

f1(zk) f2(zk) · · · fq(zk)

,

where f1(·), f2(·), . . ., fq(·) are q basis functions linking to key features of a VA

contract.

B(xi ) =(

f1(xi ) f2(xi ) · · · fq(xi ))

.

Numerical Illustration: A Synthetic VA Portfolio

100,000 VA policies with a total account value of 25.552 billion dollars are

randomly generated.

Attribute Values Distribution

Guarantee type {GMDB, GMDB + GMWB} 50%, 50%

Gender {Male, Female} 50%, 50%

Age {20, 21, 22, . . ., 60} Uniform

Account value [10000, 500000] Uniform

GMWB withdrawal rate {0.04, 0.05, 0.06, 0.07, 0.08} Uniform

Maturity {10, 11, 12, . . ., 25} Uniform

Model Specifications

◮ The underlying equity model: a regime switching lognormal model with

two regimes.

◮ 30 year projection period with annual time step.

◮ Projecting dollar Deltas of the portfolio along each of 1,000 outer loop

scenarios.

◮ The dollar Deltas of each of representative VA policies are calculated

based on 1,000 inner loop risk neutral paths.

Model Specifications

◮ We use 6 basis functions (i.e., q = 6) defined as follows:

f1(x) = 1, f2(x) =

{

0, if x is male;

1, if x is female,

f3(x) =

{

0, if x contains GMDB only;

1, if x contains GMDB and GMWB,

f4(x) = normalized age of x,

f5(x) = normalized guaranteed withdrawal rate of x,

f6(x) = normalized maturity of x.

◮ All contracts are scaled to have the same account value to ensure the

weak stationary conditions. We then scale the dollar Deltas estimated by

the UK method back to reflect the account value of the original contracts.

Testing

◮ We test the UK method with the target number of representative

contracts being 550, 1,100 and 2,200, respectively. These numbers

represents roughly 0.5%, 1%, and 2% of the total number of contracts in

the VA portfolio.

◮ For illustration purposes, we randomly select 5 outer loop real world

scenarios and calculate the annual dollar Deltas along the scenarios for

each of the representative contracts using the nested simulation model.

◮ We then estimate the annual dollar Deltas of every scaled contract in the

portfolio using the UK method.

◮ For comparison, we also calculate the annual dollar Deltas of every scaled

contract using nested simulation (extremely time consuming).

Estimated Dollar Delta vs Actual Dollar Delta (550

clusters)

0 5 10 15 20 25−8

−7

−6

−5

−4

−3

−2

−1

0x 10

9

Year

Dol

lar

Del

ta

NS1UKFD1NS2UKFD2NS3UKFD3NS4UKFD4NS5UKFD5

Figure: The annual dollar Delta along five outer loop scenarios estimated

by the UK method with 512 representative contracts.


clusters)

0 5 10 15 20 250

0.5

1

1.5

2

2.5x 10

4

Year

MS

E

RW1RW2RW3RW4RW5

(a)

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

(b)

Figure: Figure (a) shows the mean squared errors (MSE) at each point in

time along the five outer loop scenarios. Figure (b) shows the histogram

of the differences of dollar Deltas at time 0.


clusters)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 105

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

4

Account Value

Diff

eren

ces

of D

olla

r D

elta

s

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 105

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Account Value

Nor

mal

ized

Diff

eren

ces

of D

olla

r D

elta

s

(b)

Figure: Figure (a) is the scatter plot between the account values and the

differences of the dollar Deltas calculated by the simulation model and

those estimated by the UK model.

Figure (b) is the scatter plot between the account values and the

normalized differences of the dollar Deltas.


clusters)

0 5 10 15 20 25−8

−7

−6

−5

−4

−3

−2

−1

0x 10

9

Year

Dol

lar

Del

ta





clusters)

0 5 10 15 20 250

2000

4000

6000

8000

10000

12000

14000

16000

18000

Year

MS

E

RW1RW2RW3RW4RW5

(a)

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

(b)





clusters)

0 5 10 15 20 25−8

−7

−6

−5

−4

−3

−2

−1

0x 10

9

Year

Dol

lar

Del

ta





clusters)

0 5 10 15 20 250

2000

4000

6000

8000

10000

12000

14000

Year

MS

E

RW1RW2RW3RW4RW5

(a)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 104

0

1

2

3

4

5

6

7x 10

4

(b)




Mean Absolute Percentage Error

Number of ContractsMean Absolute Percentage Error

RW1 RW2 RW3 RW4 RW5

512 4.49% 8.75% 10.17% 5.42% 4.18%

987 2.82% 5.75% 6.67% 3.52% 2.62%

1,845 1.72% 3.31% 3.90% 2.07% 1.59%

Table: The mean absolute percentage errors of annual dollar Deltas

estimated by the UK method and those calculated by the simulation

model. RW1 to RW5 denote the five outer loop real world scenarios.

Computing Times

Number of Contracts Entire Portfolio

512 987 1845 100,000

k-prototypes 4.52 3.83 4.22 NA

Nested simulation 166.50 317.75 580.18 32520.68

UKFD 48.16 100.71 199.05 NA

Total 219.18 422.29 783.45 32520.68

Table: Computing times used by the clustering algorithm, the nested

simulation model, and the UKFD method. The numbers are in seconds.

Concluding Remarks and Ongoing/Future Work

◮ The UK method performs well in terms of accuracy and speed.

◮ Although we used dollar Deltas to illustrate the effectiveness of the

proposed approach, it can be applied to other quantities of interest such

as dollar Rho, reserves and risk measures.

◮ In our test cases, we used annual time steps in the nested simulation

model. The method can be also applied to cases when monthly steps or

variable-time steps are used. When the method is applied to monthly

data, the gain in computation time will be even more significant because

the UK method’s performance does not depend on the number of time

steps.

Ongoing/Future Work

◮ Better clustering/sampling methods.

◮ Utilize time-dependent variogram function for more accurate projection.

◮ Application to dynamic hedging via meta-modelling.

◮ Risk capital calculation: VaR and CVaR.

Selected References

Bauer, D., A. Reuss and D. Singer (2012). “On the calculation of the

solvency capital requirement based on nested simulations”, ASTIN

Bulletin, 42, 453–499.

Caballero, W., R. Giraldo and J. Mateu (2013). “A universal kriging

approach for spatial functional data”, Stochastic Environmental Research

and Risk Assessment, 27, 1553–1563.

Gan, G. (2013). “Application of data clustering and machine learning in

variable annuity valuation”, Insurance: Mathematics and Economics, 5,

795–801.

Gan, G. and X.S. Lin (2015). “Valuation of large variable annuity

portfolios under nested simulation: a functional data approach”,

Insurance: Mathematics and Economics, accepted.

Ramsay, J.O. and B.W. Silverman (2005). Functional Data Analysis, 2nd

Edition, Springer Series in Statistics, New York, NY.

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