Optimal Multi-Period Asset Allocation for Life Insurance

Policies

Hong-Chih Huang

Associate Professor, Department of Risk Management and Insurance, National

Chengchi University, Taiwan

Yung-Tsung Lee

Ph.D. Student, Department of Risk Management and Insurance, National Chengchi University, Taiwan

Purpose

employ a multi-assets model and investigate the multi-period optimal asset allocation on life insurance reserves. for a general portfolio of life insurance policy

Literature Review

Marceau and Gaillardetz (1999)

Huang and Cairns (2006).

Contributions provide a good contribution on

solving multi-period asset allocation problems of the application of life insurance policies.

find that the optimal investment strategy will be very different under different durations of policy portfolios.

The liability model

the prospective loss random variable can be expressed as the present value of the stochastic cash flows for the whole portfolio

0

nPTF

j

L CF j v j

where valuation date is at 0j and 0CF considers the premium income only.

The mortality process and the asset returns are assumed to be independent.

Thus, the expectation of PTFL is given by

0

nPTF

j

E L E CF j E v j

The liability model Moreover, by conditional on assets returns’ information, it

follows that

* *PTF PTF PTFVar L E Var L i Var E L i

where * , 1, 2,...,i i k k n represents the information set of

assets returns until time n, and

*

1 1

,n n

PTF

j k

E Var L i E v j v k Cov CF j CF k

*

1 1

,n n

PTF

j k

Var E L i E CF j E CF k Cov v j v k

Multi-asset return model

Three assets model: a one-year bond (cash), a long-dates

bond and an equity asset.

The log-return on cash between 1t and t is 1y t ,

which is assumed to follow the AR(1) process

( ) ( ( 1) ) ( )y yy t y y t y Z t

From 1t to t , the log-return rate on the bond is b t

and the log-return rate on the equity is e t

2-2. Multi-asset model

These two log-return rates on risky assets are assumed to be

positive connected with 1y t and there processes are

following

1 1

1 1

b b b by y b b

e e e ey y eb b e e

t y t t y t Z t Z t

t y t t y t Z t Z t Z t

where yZ t , bZ t and eZ t are 0,1N random variables

that are independent of one another and identical distributed

through time t.

Moments of Loss Functions Derive the formulae of 0[ ( ) | ]nE v k and

0[ ( ) ( ) | ]E v k v k m .

Obtain the analytic expression of 0PTFE L and

0PTFVar L , which are functions of the insurer’s asset

allocation.

Explore the optimal investment strategy through these two analytic moments.

Three Cases of Policy Portfolios

All cases have 10 endowments policies, with the same term 10 years and the same sum assumed 1.

Case A: 10 new policies at the valuation date Case B: 10 policies with different uniform

maturity dates Case C: The maturity date is selected

randomly.

Three Cases of Policy Portfolios

The maturity dates of case C are as follows:

Maturity dates

1 2 3 4 6 8 10

Policy amounts

1 2 1 1 3 1 1

Optimal asset allocation

Single-Period RebalanceMulti-Period Rebalance

Single-Period RebalanceMean–variance plot: case B

proportion of stocks

prop

ortio

n of

long

bon

ds

contour plot of mean(L) and var(L)

3.6

3.65

3.73.75

3.83.85

3.65

3.7

3.75

0.25

0.5

0.5

0.75

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

mean

varianceefficienct frontier

Mean–variance plot: case B

An efficient frontier can be found at the left part of the plot.

Insurance company can minimize variance of loss under a contour line of mean; or minimize mean under a contour line of variance.

Objective Function

0.5min , 1

2PTF PTFk

E L Var L k

Optimal Asset AllocationSingle-period rebalance

Case cash long bond stock

A 0.3106 0.4715 0.2178

B 0.4056 0.4072 0.1872

C 0.4748 0.3605 0.1646

Optimal Asset AllocationMulti-period rebalance

case A and case B

1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

term

wei

ght

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

term

wei

ght

cash

long bondsstocks

Optimal Asset AllocationMulti-period rebalance

case A and case B(with short constraints)

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

term

wei

ght

cash

long bondsstocks

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

term

wei

ght

4-2. Multi-period rebalance - case A and case B

The holding pattern of riskless/risky asset are totally different between case A and case B, regardless of a short constrain exist or not.

Under case A, the proportion of cash is increasing and the proportion of risky assets is decreasing; whereas an opposite pattern arise under case B.

Optimal Asset AllocationMulti-period rebalance-case C

1 2 3 4 5 6 7 8 9 10-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

term

wei

ght

cash

long bonds

stocks

Multi-period rebalance- case C Due to the randomness of the maturity

dates of the policies, the optimal investment strategy appears a saw-toothed variation, whereas the pattern is similar with case B (the uniform case).

The optimal asset allocation with short constrain under case C is almost the same as the without constrain one, so we display the result of without constrain only.

Sensitivity Analysis of k

1 2 3 4 5 6 7 8 9 10-1

0

1

term

wei

ght

of c

ash

case B, k=0.5

case B, k=1case B, k=2

1 2 3 4 5 6 7 8 9 100

0.5

1

term

wei

ght

of lo

ng b

onds

1 2 3 4 5 6 7 8 9 100

0.5

1

term

wei

ght

of s

tock

sThe optimal asset allocations of multi-period rebalance, k=0.5, 1 and2

Sensitivity Analysis of asset model

High excess mean: The optimal asset allocations under case B

1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

term

wei

ght

of c

ash

1 2 3 4 5 6 7 8 9 100

0.5

1

term

wei

ght

of lo

ng b

onds

1 2 3 4 5 6 7 8 9 100

0.5

1

term

wei

ght

of s

tock

s case B

case B, high excess mean

Sensitivity Analysis of asset model

High variance: The optimal asset allocations under case B

1 2 3 4 5 6 7 8 9 100

0.5

1

term

wei

ght

of c

ash

1 2 3 4 5 6 7 8 9 100

0.5

1

term

wei

ght

of lo

ng b

onds

1 2 3 4 5 6 7 8 9 100

0.2

0.4

term

wei

ght

of s

tock

s

case B

case B, high variance

The Case of large sample We examine the optimal asset allocations

under 4 policy portfolios. These 4 portfolios have a same statistic

property: the maturity dates of a same portfolio has p-value 0.9513 under chi-square goodness of fit test.

The null hypothesis is that the maturity dates are selected form a discrete uniform distribution.

Thus, these 4 portfolios are unlike uniformly distributed in a statistical sense.

The Case of large sample The optimal asset allocation of the 4 special portfolios

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

term

wei

ght

long bonds

stocks

The Case of large sample The optimal asset allocation of a specific portfolio

2 4 6 8 100

5

10

15

20

2 4 6 8 100

0.2

0.4

0.6

term

wei

ght

cash

2 4 6 8 100

0.2

0.4

0.6

term

wei

ght

long bonds

2 4 6 8 100

0.2

0.4

0.6

term

wei

ght

stocks

Uniform case

Special Case

6. Conclusion This paper successfully derives the

formulae of the first and second moments of loss functions based on a multi-assets return model.

With these formulae, we can analyze the portfolio problems and obtain optimal investment strategies.

Under single-period rule, we found an efficient frontier in the mean-variance plot. This efficient frontier can be found under an arbitrary policies portfolio.

6. Conclusion In multi-period case, we found that

the optimal asset allocation can vary enormously under different policy portfolios. A. “Top-Down” strategy for a single

policy B. “Down-Top” strategy for a portfolio

with numbers of policies