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.
2.
3.
4.
5.
6.
7.
Introduction
General theoretical framework
Modelling the assets and the short-rate
Data and parameter estimation
Results
Conclusions
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:
Assets
Liabilities
A0
L0= αA0
E0= (1-α)A0
t=T:
alpha=0.5, delta=0.4, policy payment=100
180
160
140
120
100
80
60
40
20
0
0
50
100
150
liabilities
200
250
equity
300
350
General theoretical framework
0<=t<=T:
L*T
Lt  L*T P(t , T )  Put ( At , L*T )  Call ( At , )

*
LT
*
Et  Call ( At , LT )  Call ( At , )

t=0: no cross-subsidizing
L*T P(0, T )  Put ( A0 , L*T )  Call ( A0 ,
L*T
Note: prices under risk-neutral measure

)  A0
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
alpha=0.95, delta=0.91, roffered=0.04, T=10
8000
Ser ies : POLIC YR ETU R N
Sample 1 50000
Obs erv ations 50000
6000
Mean
Median
Max imum
Minimum
Std. D ev .
Sk ew nes s
Kur tos is
4000
2000
J ar que-Ber a
Probability
7.340863
7.252698
23.45385
-8.229739
3.329559
0.100822
3.092011
102.3469
0.000000
0
-5
20000
0
5
10
15
20
Ser ies : EQU ITYR ETU R N
Sample 1 50000
Obs erv ations 50000
15000
Mean
Median
Max imum
Minimum
Std. D ev .
Sk ew nes s
Kur tos is
10000
5000
J ar que-Ber a
Probability
0
**
-80
-60
-40
-20
0
20
-2.948042
11.25932
32.70101
-100.0000
38.46484
-2.070338
5.435791
48079.74
0.000000
Results
alpha=0.8, delta=0.72, roffered=0.04, T=10
12000
Ser ies : POLIC YR ETU R N
Sample 1 50000
Obs erv ations 50000
10000
Mean
Median
Max imum
Minimum
Std. D ev .
Sk ew nes s
Kur tos is
8000
6000
4000
2000
J ar que-Ber a
Probability
6.856239
6.610520
21.32206
-6.511236
2.620875
0.432164
3.288499
1729.783
0.000000
0
-5
20000
0
5
10
15
20
Ser ies : EQU ITYR ETU R N
Sample 1 50000
Obs erv ations 50000
15000
Mean
Median
Max imum
Minimum
Std. D ev .
Sk ew nes s
Kur tos is
10000
5000
J ar que-Ber a
Probability
0
**
-80
-60
-40
-20
0
20
3.718068
10.41540
31.10035
-100.0000
26.28205
-3.434603
13.69329
336526.0
0.000000
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 debtequity ratios, conflicting practice.
Further research
 Further investigate causes of unsatisfactory riskreturn 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.