A stochastic asset-liability model is “a stochastic model of the main financial factors of an insurance company (and) a good model should simulate stochastically the asset elements, the liability elements and also the relationships between both types of random factors” (Kaufmann et al. (2001, p.214)).
Since the stochastic asset-liability (or stochastic solvency testing) model devel- oped in this thesis is intended for use in Australia, it is desirable that it be compatible with the existing Australian valuation philosophy wherever possible. Even though some elements of the existing valuation philosophy clearly must change, this is not the case with principles such as the realistic valuation of policy liabilities, which is one of the main objectives of the current Australian Life Insurance policy valuation standard, LPS1.04. Consequently, the model we outline in this section is based on
24 A Framework for Stochastic Solvency Testing many of the principles prescribed in LPS1.04, LPS2.04 and LPS3.04. The main principles that have been retained are as follows:
• Policy liabilities are calculated using Equations (2.10) and (2.11), that is: Policy Liability = Best Estimate Liability
+ EPV of future best est. bonuses (for participating business) + EPV of future best est. shareholder profit;
(3.1) and
Best Estimate Liability = EPV of future benefit payments + EPV of future expenses
−EPV of future receipts (ie. premiums); (3.2) but making no allowance for future shareholder profits (as is done in LPS2.04 and LPS3.04).
• Realistic valuation assumptions are used in the calculation of the policy lia- bilities and all significant cash-flows are allowed for in the calculation; and
• Projection techniques are used in calculating the policy liabilities1.
Assets are also assumed to be valued realistically (at market value) in accordance with standard Australian accounting practices.
Even though it is necessary to consider both policy and non-policy liabilities in solvency testing, non-policy liabilities (such as borrowings and accounts payable) are typically considered to be under the insurer’s control and assumed to be de- terministic in nature, and have generally been ignored in existing solvency testing models. Non-policy liabilities differ greatly in nature between insurers (so it is dif- ficult to construct a “typical” non-policy liability portfolio) and usually comprise a relatively small proportion of an insurer’s total liabilities. Data obtained from annual returns submitted by Australian Life Insurers to APRA2 show that, at the end of 2006, non-policy liabilities comprised only 10% of the total liabilities of all Australian Life Insurers. Consequently, non-policy liabilities are not as important as policy liabilities, from a solvency perspective, and they shall be ignored in this thesis. Note that this is the same as assuming that the assets backing the non-policy liabilities are perfectly matched with these liabilities and so cancel out at all times.
1
Note that LPS1.04 does not “prescribe a single methodology for the valuation of policy li- abilities.” However, it states that “the principles of the Standard will normally be achieved by adopting a projection methodology” (p.8).
A Framework for Stochastic Solvency Testing 25 Recall from Section 2.7 that LPS1.04 specifically lists seven factors that should be considered in calculating the policy liabilities:
• investment earnings;
• inflation;
• taxation;
• expenses;
• mortality and morbidity;
• policy discontinuance; and
• reinsurance.
These are the main factors to be considered in a Life Insurance policy valuation, and the parameters associated with these factors are usually assumed to be known and not subject to variability. In the case of the stochastic asset-liability model, some or all of these parameters are assumed to be stochastic.
Investment earnings, inflation, policy discontinuance and mortality and morbid- ity are generally considered to be stochastic processes, so stochastic interest rates (or investment yields) and inflation rates, and lapse and mortality table parameters will be used in our stochastic solvency testing model. Note that insurance classes for which morbidity is a valuation assumption (for example, disability income insur- ance) are not considered in this thesis, as data relating to these classes of business could not be obtained. Consequently, morbidity is not considered further.
Tax and real expenses (that is, inflation-adjusted expenses), on the other hand, will be assumed to be non-random in the stochastic solvency testing model. The tax rate paid by insurers is usually constant from year to year (it would only change following changes to the corporate tax system, which could be uncommon, although during economic hardships, tax incentives may be provided to stimulate the econ- omy); and real expenses are usually assumed to be under the control of the insurer and not random. Nominal expenses increase over time due to inflation and the stochastic nature of these expenses is modelled by allowing inflation to be stochas- tic. Reinsurance arrangements entered into by insurers are determined at the dis- cretion of the insurer itself. As is the case with non-policy liablities, reinsurance arrangements also differ greatly by policy type and by insurer, making it difficult to construct a “typical” set of reinsurance arrangements. Futhermore, for Australian Life Insurers, the expected present value (EPV) of future reinsurance benefits is rel- atively low compared with their gross policy liabilities. Data obtained from annual returns submitted to APRA shows that, in 2006, the total expected present value
26 A Framework for Stochastic Solvency Testing of reinsurance benefits for all Australian Life Insurers was less that 1% of the total gross policy liabilities for these insurers. For these reasons, although required under LPS1.04, reinsurance is not treated in this thesis. This approach is similar to that taken by Daykin et al. (1994). A major step in the development of our stochas- tic solvency testing model is, therefore, the construction of (i) stochastic mortality, (ii) lapsation and (iii) economic sub-models to describe the mortality experience, lapse experience and the economic environment (that is, the investment yields and inflation rates), respectively.
As was mentioned in Section 2.5, with the exception of the interrelationship be- tween interest rates and inflation, dependency relationships between variables have generally been ignored in previously proposed stochastic valuation models. How- ever, lapse rates are commonly believed to be influenced by the economy; mortality rates are often believed to depend on past lapse rates; and a number of studies have suggested that fluctuations in the economy can cause fluctuation in mortality rates (these dependency relationships are discussed in detail in Chapter 6). For a desired realistic solvency testing model, the possibility of interconnected sub-models should be considered, as we do.
Once the three abovementioned stochastic sub-models and the relevant distribu- tions have been specified, and the model parameters have been estimated, observa- tions can be simulated and the insurer’s capital at some future point in time can be estimated. Repeating this procedure a large number of times produces many esti- mates of the capital values which can be used to calculate an empirical distribution of the capital amount. From this distribution, solvency capital requirements can be found. This simulation-based approach was also used in Daykin et al. (1994), Lee (2000) and Tsai et al. (2001).
Figure 3.1 shows a simplified view of the solvency testing model framework used in this thesis. This framework is assumed to be applicable to a specified block of business whose basic characteristics (such as the number of policies, policy type, sums insured and the sex and age of each policyholder in the block) and the basic characteristics of the assets backing this business (that is, the proportion allocated to each class of assets) are included as model inputs. The arrows in the diagram in- dicate flows of information. For example, random observations of economic variables simulated using the economic sub-model are inputs to the mortality sub-model, the asset model and the payment model.
The economic, mortality and lapsation stochastic sub-models are all potentially interconnected, but we will assume that the mortality and lapsation experiences of a relatively small group of lives (that is, the group of policyholders that comprise the block of business under consideration) do not have a significant effect on the economy as a whole. The simulated observations act as inputs to the four spreadsheet models,
A Framework for Stochastic Solvency Testing 27 that is, the asset model, the policy model, the capital model and the payment model. In policy liability valuation, projection techniques are usually implemented using spreadsheet models comprised of a policy model and a payment model, and we shall follow this practice here. Lapsation and mortality are inputs to our policy model, while the economic variables, interest and inflation, are inputs to a payment model. Deterministic tax and expense assumptions are also inputs to the payment model, though not explicitly shown in Figure 3.1. Two more spreadsheet models are required for solvency testing: an asset model, for determining the insurer’s asset position at each point in time, and a capital model, for determining the insurer’s capital position at each point in time. The economic investment yield variables and the payment model outputs are inputs to the asset model, while the outputs of the asset and payment models are inputs to the capital model.
The capital distribution, once estimated, is used to calculate target solvency capital amounts which will be compared with those calculated deterministically under the current Australian standards. As discussed in Section 2.6, a number of different methods can be used to determine solvency capital and there is no general consensus as to which of these methods is the most suitable. Consequently, we will consider two versions of the model, in which solvency capital is calculated using either the VaR or the TVaR method, as described in Section 2.6.
The capital distribution can also be used to calculate the minimum value of assets that an insurer must hold at a particular point in time so that it satisfies its solvency capital requirement, holds sufficient funds so that its liabilities can be met, and could provide adequate risk compensation to a hypothetical insurer who may take over the portfolio in the future. This amount, which is referred to as the stochastic minimum asset requirement (SMAR), is calculated as:
SMAR = Best estimate liability + Cost of capital risk margin
+ Solvency capital requirement. (3.3)
This quantity is similar to the minimum asset requirement under the Swiss Solvency Test, although in this case, a credit margin is not included.
Solvency capital in the model is calculated using a 99.5% confidence level over a one year time horizon, in keeping with the recommendations of the International Actuarial Association in IAA (2004) and with precedents throughout the world3, and using a 95% confidence level over a three year time horizon, as this is the level of sufficiency that the current Australian Solvency standard is calibrated to (according to Karp (2002)).
28 A Framework for Stochastic Solvency Testing
Figure 3.1: A Simplified Pictorial View of the Solvency Testing Model Framework
Stochastic Sub-Models: Economic Sub-Model Mortality Sub-Model Lapsation Sub-Model Spreadsheet Models: Asset Model Policy Model Capital Model Payment Model Outputs: Capital Distribution Stochastic Solvency Capital Stochastic Min. Asset Req.
A Framework for Stochastic Solvency Testing 29