Modelling to reserve

A reserve (or provision) is the amount required to be set aside to meet the promised liabilities in future (allowing for the EPV of any future premiums or contributions).
higher degree of prudence usually required, so as to demonstrate ability to pay promised benefits in all reasonable future circumstances. also, basis often prescribed by legislation.
In general insurance, outstanding claims reserves are usually calculated using a statistical approach or an individual case estimate (where claims are large and infrequent (e.g. marine and aviation insurance). also, where each claim is very heterogeneous (and reserve held needs to reflect individual circumstances).).
the key assumptions underlying the basic chain ladder method

• claims development pattern is stable over time (i.e. with regard to speed of both reporting and settlement)
• all claims are fully run off after n years
• inflation pattern implicit in previous claims will continue in future

inflation-adjusted chain ladder method

• effects of claims inflation on past claims development removed, thereby allowing for separate projection of future claims inflation
• useful if claims inflation expected to be materially different in future (compared to past)

involve reserving on an individual basis
The main reasons for calculating reserves are to:

• determine a value for liabilities shown in published accounts
• demonstrate solvency on a statutory basis to regulators / (members in case of pension )
• determine a value for liabilities to assist with internal business management decisions
  ─ required when framing investment strategy and determining discretionary benefit increases (e.g. reversionary bonus for with-profit contracts or increases to pensions in payment)
• value business in event of merger or acquisition
• adjust contribution rate required by pension scheme sponsor in respect of any current surplus or deficit
• value the accrued benefits in the event of a pension scheme wind-up

Reserves calculated for the purposes outlined above generally involve aggregating the individual reserves required for each contract (or benefit promise) undertaken.

involve reserving on a global basis
However, it is often necessary to determine additional reserves on a “global” basis to allow for:

• risks associated with a mismatching of assets and liabilities
• the expected costs of any guarantees and options
• other risks faced by the provider (e.g. credit, operational)
• credit risk – failure of reinsurer (or other third party). operational risk – mis-selling, fraud.

Reasons for global risk evaluation

• reinsurance is usually purchased on a portfolio basis (rather than individually), so it is appropriate to allow for the risk collectively.
• whilst, potentially, credit and/or operational risks can give rise to very large losses, they should occur very infrequently if company has good internal risk management procedures in place. thus, additional reserves required for such risks should be small.
• A mismatching reserve is required to protect against the risk that a change in investment conditions results in a greater increase in liability cash flows than in asset cash flows, and thus that the solvency position is adversely affected.
• As investment strategies are usually determined on a global basis (i.e. for an entire portfolio or class of business), it is also appropriate for this reserve to be established on a global basis.
• For guarantees and options, allowing for a worst-case scenario on an individual basis is likely to result in unnecessarily large reserves being held.
• because it is highly unlikely that the guarantee will give rise to a cost for all in-force policies. in many cases, no additional cost will arise. thus, the total cost of guarantee over a portfolio of business is more relevant than the cost for an individual policy.

• (1) Reserves for published accounts
  The key requirement here is to follow the relevant legislation and accounting principles.
  Accounts will usually be prepared on a going concern basis (i.e. assuming that the company will continue to trade as normal in future).
  Alternatively, it may be specified that the accounts are prepared on a break-up basis (i.e. assuming that no new business is written and cover on current policies is terminated immediately).
  discount rate should be based on current market conditions (rather than long-term return on assets held), as assets likely to be realised to meet obligations; future expenses may fall (e.g. no further sales and marketing); future withdrawal rates are not relevant
  Published accounts are often required to give a true and fair value of the assets and the liabilities. However, in general, there is no universally accepted decision as to what this means.
  Most practitioners have interpreted this to mean that, at the very least, a best estimate basis should be used to value the assets and liabilities (i.e. with no significant margins for prudence).
  However, some have taken this further and interpreted it as requiring the use of a market-consistent value for the liabilities.
  the price that would need to be paid to a counterparty to accept the liabilities in a competitive market (i.e. where there were a large number of potential buyers). however, in practice, such a market does not yet exist for insurance company and pension fund liabilities ... although, could we determine such a price if we assume such a market were to exist?
• (2) Reserves for demonstrating statutory solvency
  In practice, the method and assumptions used to demonstrate statutory solvency will often be prescribed by the regulator.
  prevents results from being manipulated by changing basis. also, allows objective comparison between different companies (and from year-to-year within the same company)
  The key feature of the method and assumptions laid down will be prudence, so that the company can be expected to meet future liabilities in full in all reasonable future circumstances.
  For demonstrating statutory solvency, reserves will often be calculated on a break-up basis (as, in the event of insolvency, the company would cease to write new business and existing liabilities will often be bought out by a third party).
  Similarly, it is common for assets to be taken at market value for the purposes of demonstrating statutory solvency (and there may also be restrictions of the types of assets included).
  Legislation may also prescribe for the calculation of additional reserves covering the other risks faced (e.g. mismatching reserve).
  Often, a further cushion will be included by specifying a statutory minimum solvency margin (SMSM) (amount by which value of assets must exceed value of statutory liabilities.)
  As well as increasing the security of the promised benefits, this can act as an early warning sign of possible problems ahead.
  What might the regulator require of a company that failed to meet the SMSM?
  immediate cash injection, stop writing new business, matched investment strategy, forced sale, immediate wind-up and payment of current liabilities.
  The size of the SMSM will usually depend on the strength of the reserving basis (i.e. stronger reserving basis → higher statutory reserve → lower SMSM, and vice versa).
• (3) Reserves for internal business management
  Reserves for internal business management will be required to assist with decision making in a number of areas, for example:
  • assessing the profitability of current business
  • choosing an appropriate investment strategy
  • setting bonus rates for with-profit contracts
  • determining the level of any discretionary benefits awarded
    ─ e.g. a pension scheme surplus may lead to pressure for discretionary increases to pensions in payment, and the effect of this on the security of accrued benefits must be examined
  • exploring the effectiveness of reinsurance arrangements
  Then, the reserves will usually be calculated on a best estimate basis (to give a realistic assessment of the effect of the proposed decision on the future financial position).
• (4) Reserves for valuing benefits in mergers/acquisitions
  For mergers and acquisitions, a best estimate basis is usually considered appropriate for valuing the liabilities. as this would usually be considered fair to both buyer and seller.
  However, in practice, the relative bargaining power of the two parties involved will also be important
• (5) Reserves for determining pension contributions
  The valuation of a defined-benefit pension fund has two key aims:
  • to assess the level of funding of the accrued benefits
    ─ defined as the ratio of the value of the current assets to the value of the current liabilities
  • to determine an appropriate contribution rate
    ─ based on expected cost of benefits accruing in future plus any adjustment for current surplus (or deficit)
  In this case, it is reasonable to assume that the scheme will continue as a going concern (unless advised otherwise by the sponsor), and the reserve would be calculated accordingly (i.e. traditionally using a discounted cash flow approach).
  However, there will often be a statutory minimum value of the liabilities based on a best estimate of the value of the accrued benefits were the scheme to wind up (or discontinue) immediately with no further benefits accruing.
  Why does this represent a sensible statutory minimum value?
  the employer always has the option to discontinue the scheme immediately, with no further benefit accrual. however, if the assets held are insufficient to cover the liabilities in this case, then the scheme is technically insolvent.
• (6) Reserves in the event of a pension scheme wind-up
  In this case, a break-up (or discontinuance) basis should be used to value the accrued liabilities, with assets taken at market value.
  Thus, current pensions in payment would usually be secured with an insurance company and a “transfer value” paid to current active members and deferred pensioners (reflecting the value of the accrued benefits on withdrawal at the valuation date).
  

There are two main approaches to carrying out a valuation of assets and liabilities, namely:

• a discounted cash flow approach
• a market-related approach
  • often based on the market value of a portfolio of assets that most closely replicates the duration and risk characteristics of the liabilities
  • known as a “replicating portfolio”

Regardless of which approach is used, when setting the discount rates used to value the assets and liabilities, it is crucial that a consistent rate is used in each case.
Discounted cash flow valuation
The key assumption in this case is the expected rate of return in future on the assets held to meet the liabilities, which is then used to discount the future cash flows to give a present value.
For consistency, assets must also be valued by discounting the expected future cash flows using similar long-term assumptions
The main criticism of this approach is that it is likely to place a different value on the assets from the market value. (can be difficult to justify to the scheme sponsor why the value placed on the assets by the actuary is different from that by the market ... particularly if it is a lower value (and, thus, requires additional contributions from the sponsor to make good any resulting deficit).)
This occurs because it is fundamentally incorrect to allow for risk (i.e. uncertainty in the future cash flows) by discounting the expected cash flows at a fixed rate of interest.
the fact remains that if a market were developed in future to trade the liability cash flows, the most sensible value to place on these cash flows would be this market value
Market-related valuation
the aim is to find a consistent value for the liabilities.

• Method 1: fair value (defination)
  A “market‐consistent” value for the cash flows can be thought of as the amount that another investor in the market would be required to be paid to accept responsibility for meeting the liabilities , assuming that such a fully competitive market were to exist and that both parties to the transaction were fully informed).
• Method 2: Marketing to market
  This is usually done by constructing a replicating portfolio of assets that most closely matches the liabilities by both duration and risk characteristics.
  Then, the discount rate for the liabilities is based on the return on the assets making up this replicating portfolio (rather than the return on the actual assets held to meet the liabilities).
  Typically, this is done by a process known as marking‐to‐market.
  In this case, a portfolio of fixed‐interest bonds can be designed such that the cash flows match those expected from the portfolio of non‐profit whole life assurance contracts.
  Then, the market value of the replicating portfolio can be taken as the “market” value of the corresponding liabilities.
  However, it should be noted that, due to uncertainties in future mortality experience, the timing of liability cash flows will not be known in advance.
  This is particularly true when the number of policies in the portfolio is small.
  Thus, the “market” value obtained will not reflect the full risk characteristics of the liability cash flows
  However, it should be noted that, due to uncertainties in future mortality experience, the timing of liability cash flows will not be known in advance. This is particularly true when the number of policies in the portfolio is small.
  Thus, the “market” value obtained will not reflect the full risk characteristics of the liability cash flows, and as such, cannot be thought of as representing a true and fair value (at which two fully informed counterparties would willingly trade the uncertain future cash flows).
  Thus, in practice, whilst it may be possible to construct a replicating portfolio reflecting many of the financial risks inherent in the liability cash flows, it is very difficult to construct a portfolio that fully reflects all the risks involved.
  However, as more sophisticated investment instruments are developed (e.g. mortality‐linked securities where the payments are linked to actual mortality experience in a given population), it is likely that this approach will become more common.
  But, as the mortality experience driving the payments on mortality‐linked securities will not fully reflect the experience of a company’s own portfolio of business, the full risk characteristics of a particular set of liabilities will not readily be captured until a market for trading the cash flows actually exists.
• Method 3: discount the cash flow using market rate of interest
  When such a replicating portfolio does not readily exist, the aim is to discount the liability cashflows at the current yield on an investment that best match these cash flows.
  In practice, this often means that current market yields on long-term government bonds are used to discount the future expected liability cash flows. It may be considered appropriate to use discount rates that vary with term to reflect the current shape of the yield curve.
  However, it is much more difficult to appropriately allow for many of the risk characteristics (e.g. uncertainty resulting from factors affecting future claims experience, such as mortality) MTM value for uncertain cash flows is higher than true “market” value.
  Then, whilst MTM will often give a prudent value for the liability cash flows, it cannot be thought of as representing a true and fair value (at which two fully informed counterparties would willingly trade the uncertain future cash flows)
  
• Method 4: State price deflators
  we can use a stochastic asset model to generate a range of possible future scenarios and then discount the resulting cash flows using a state-price deflator (or stochastic discount factor) to arrive at a market-consistent value for the liabilities (that takes appropriate account of the uncertainties attached to the individual cash flows).
  However, it can be difficult to derive an appropriate structure for the state-price deflator for a chosen stochastic asset model (unless it was constructed with this aim in mind) and it can also be difficult to ensure that the model is correctly parameterised to reflect current market conditions.

Allowing for risk in cash flows in reality
In practice, actuaries have taken a somewhat simpler approach to allowing for the risk (or uncertainty) in the future cash flows:

• best estimate plus margin ─ i.e. a margin reflecting the risk involved is included explicitly in each assumption made
• contingency loading ─ i.e. a single contingency loading is applied to the best estimate of the value of the future liability cash flows
• discounting cash flows using a risk premium ─ i.e. best estimate cash flows are discounted at a rate of return that reflects the overall risk of the contract

Methods of calculating reserves in non-life insurance

• (1) Statistical analysis
  If the population exposed to a particular risk is large enough, then by the Law of Large Numbers, a mathematical approach to establishing a reserve for the risk will give a valid estimate (e.g. based on the chain ladder techniques outlined above).
  • Suitable when the claim settlement pattern is stable.
  • suitable for claims with high frequency and low severity
  • particularly poor for bodily injuries
  • Suitable when there are a large number of policies and it appears to be unrealistic to estimate the reserve on a case-by-case basis
• (2) Case-by-case estimates
  However, if the insured risks are heterogeneous or claim events are rare (with large variability in the claim amount), then such statistical techniques can break down.
  because chain ladder techniques assume that the claim development process is stable (with regard to reporting and settlement). unlikely to be true with very heterogeneous claim events that often need to be assessed individually.
  Disadvantages:
  • time-consuming and, thus, expensive
  • subjective (as different claims assessors with place a different value on the liabilities).
  • information required to estimate reserve may not be readily available (e.g. if it is dependent on a future court ruling).
• (3) Equalisation reserves
  For low probability risks with large and volatile claim amounts, it may also be appropriate to set up a claims equalisation reserve in years when few claims arise (and profits would otherwise be high). This can then be used to smooth results in years when more claims arise.