Fair Valuation of Guaranteed Contracts:

the Interaction Between Assets and Liabilities

Erwin Charlier

Tilburg University and ABN AMRO Bank

Joint work with Ruud Kleynen

Maastricht University and Kleynen Consultants

Overview

1. Introduction

2. General theoretical framework

3. Modelling the assets and the short-rate

4. Data and parameter estimation

5. Results

6. Conclusions

7. Further research

Introduction

Balance sheet: book value accounting fair or market value of assets and liabilities

Market value of assets:

Market prices for publicly traded assets (stocks, bonds)

Valuation models for less liquid assets like real estate

Market value of liabilities:

Very little traded liabilities

Optionalities

Introduction

In this presentation:

Simple insurer:

Assets: investments in stocks and bonds

Liabilities and equity:

Single guaranteed return contract (policy)

Equity

Policy characteristics:

Guaranteed return, roffered

Bonus: if the return on equity exceeds roffered then fraction of surplus to policyholder

General theoretical framework

t=0:

t=T:

Assets Liabilities

A0 L0= αA0

E0= (1-α)A0

alpha=0.5, delta=0.4, policy payment=100

020406080

100120140160180

0 50 100 150 200 250 300 350

liabilities equity

General theoretical framework

0<=t<=T:

t=0: no cross-subsidizing

Note: prices under risk-neutral measure

),(),(),(*

**

T

tTtTtL

ACallLAPutTtPLL

),(),(*

*

T

tTttL

ACallLACallE

0

*

0*

0* ),(),(),0( A

LACallLAPutTPL T

TT

Modelling the assets and the instantaneous short-rate

Instantaneous short-rate: stochastic, Vasicek

LN gross asset returns: normal

Geometric Brownian motions correlated

Under risk-neutral measure: analytic formulae for price of put and call

Real-world measure used to describe economy at time t, also input for prices

Data and parameter estimation

Parameters in process for instantaneous short-rate:

Cross-section of FR bond prices (Feb 28, 2002)

Time-series of 1-month FIBOR rates

Also used to derive instantaneous short-rate series

Parameters in process for assets:

Assume two investment categories: stocks and bonds (monthly, Nov 1990-Feb 2002)

Use weights to construct time-series of portfolio returns

But: high mean used Dimson(2002)

Correlation: use imputed instantaneous short-rate and portfolio returns

Results

0

2000

4000

6000

8000

-5 0 5 10 15 20

Series: POLICYRETURNSample 1 50000Observations 50000

Mean 7.340863Median 7.252698Maximum 23.45385Minimum -8.229739Std. Dev. 3.329559Skewness 0.100822Kurtosis 3.092011

Jarque-Bera 102.3469Probability 0.000000

0

5000

10000

15000

20000

** -80 -60 -40 -20 0 20

Series: EQUITYRETURNSample 1 50000Observations 50000

Mean -2.948042Median 11.25932Maximum 32.70101Minimum -100.0000Std. Dev. 38.46484Skewness -2.070338Kurtosis 5.435791

Jarque-Bera 48079.74Probability 0.000000

alpha=0.95, delta=0.91, roffered=0.04, T=10

Results

0

2000

4000

6000

8000

10000

12000

-5 0 5 10 15 20

Series: POLICYRETURNSample 1 50000Observations 50000

Mean 6.856239Median 6.610520Maximum 21.32206Minimum -6.511236Std. Dev. 2.620875Skewness 0.432164Kurtosis 3.288499

Jarque-Bera 1729.783Probability 0.000000

0

5000

10000

15000

20000

** -80 -60 -40 -20 0 20

Series: EQUITYRETURNSample 1 50000Observations 50000

Mean 3.718068Median 10.41540Maximum 31.10035Minimum -100.0000Std. Dev. 26.28205Skewness -3.434603Kurtosis 13.69329

Jarque-Bera 336526.0Probability 0.000000

alpha=0.8, delta=0.72, roffered=0.04, T=10

Conclusions

Model allows for stochastic interest rates that can be correlated with process for assets.

Parameters in the model estimated from data instead of choosing some value.

Using both risk-neutral and real-world measure we can derive risk-return profiles for both policyholders and equityholders.

Different specifications of the debt-equity ratio and the contract did not lead to satisfying return profiles for both policyholders and equityholders.

Best results for equityholder occur with low debt-equity ratios, conflicting practice.

Further research

Further investigate causes of unsatisfactory risk-return profiles.

Extend to more complicated balance sheet (more than one product, different maturities for the policies, etc.).

Consider balance sheet at intermediate times with rule for regulator to interfere.

Use more advanced models to describe the instantaneous short-rate and the assets, while keeping closed-form solutions for the options.

Drop the requirement of no cross-subsidizing.