Financial Risk Management of Insurance Enterprises

Finding the Immunizing Investment for Insurance Liabilities: The Case

of the SPDA

The SPDA Product

• SPDA stands for Single Premium Deferred Annuity (the name is very descriptive)

• Purchaser makes one deposit into an interest-bearing account that will be converted into a life annuity at some future date

• A life annuity converts a lump sum of money into a lifetime stream of payments– Annuities are insurance because they protect annuitants

from “living too long”

The SPDA Product (p.2)

• If purchaser dies before converting to an annuity, usually the deposit is returned with interest

• Typically, SPDAs are tax-deferred so that interest income is not taxed as long as the money remains with the insurer

• Unlike a 401(k) invested in stocks or bonds, SPDAs only receive an annual crediting rate on their deposit and do not directly own assets

Policy Features• The following features are typical for SPDAs• Initial crediting rate and guarantee period

– Insurers guarantee an initial interest rate usually through the first year

• Minimum credited rate– Lower bound on amount earned by deposits

• Policyholder can withdraw at any time– Surrender charges apply, but decrease with the age of the

policy

Insurer Crediting Strategy• Defining the crediting strategy is an important

part of SPDA product for an insurer

• If market rates are not competitive with similar products, policyholder will lapse– Crediting strategy and lapse rate are inextricably

linked

• Must balance the cost of providing higher returns vs. the ability to sell more business

Approach of Noris and Epstein*

• First case considers a fixed liability• Look at various investment strategies• Determine the best investment strategy that will meet the liabilities “perfectly”• Immunization aims to have no deficiency

nor surplus after the liability is paid off

* Peter D. Noris and Sheldon Epstein, 1988, Finding the Immunizing Investment for Insurance Liabilities: The Case of the SPDA, Strategy Notes, Morgan Stanley & Co., Inc.

Methodology of Paper

• Generate 100 future interest rate paths• Consider each investment strategy in turn and

continue the strategy until the liability is paid off• For each path, determine the surplus or deficiency• Better investment strategies have more compact

distributions– The ideal strategy would have zero surplus in all 100

paths

Immunization

Good Close Poor Bad

Asset Strategy

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1st Quintile 2nd Quintile3rd Quintile 4th Quintile5th Quintile

Fixed Liabilities - Results and Comments

• The “best” single bond investment strategy uses comparable average duration over the entire liability term

• With exact duration match, rebalancing of the investment portfolio is required. This is because asset and liability durations may change at different rates as:– Time passes (duration drift)– Interest rates change (convexity)

Goal of Paper

• Now consider SPDAs which are not fixed cash flows

• Use the same type of methodology as in the fixed cash flow example– For SPDAs, cash flows are path dependent

• Which investment strategies would immunize the SPDA?

Options in SPDAs• Policyholder lapse

– If interest rates increase, competition may provide the higher “new money” rates

– Option to lapse is a put option on bonds since policyholder has right to “sell” at account value without a market value adjustment

– Surrender charge may help offset (if applicable)

• Minimum guarantee– Insurer has sold an interest rate floor to policyholder

Modeling SPDA Market Values

• Project cash flow along each interest rate path– Existence of options make flows path dependent– Amount and timing of cash flow based on level of

interest rates

• Discount the cash flows along each path

• Average all paths to develop market value

• Note: If approach is applied to Treasuries, resulting price should be current market values (interest model must be arbitrage-free)

Critical Assumptions

• Lapse behavior– As interest rates change, how does this affect

policyholder lapse?

• Competitor crediting rates

• Interest rate volatility– Affects options cost

• Interest rate process

Immunization Goals

• The perfect immunization strategy would have zero surplus in every interest rate path

• Even if duration matched, the paper finds that perfect immunization is not realistic

• Another immunization goal could be that the probability of deficiency should be less than some percentage– e.g., Only a 10% chance of a deficiency

Conclusion

• Evaluating SPDAs requires Monte Carlo simulation due to path dependent options

• Duration matching is not a static technique– The strategy rebalances the portfolio every 3

months

• Using duration matching, we reduce the probability of a deficiency in insurer surplus– The resulting immunization is still not “perfect”