Academic journal article Journal of Risk and Insurance

The Term Structure of Reserve Durations and the Duration of Aggregate Reserves

Academic journal article Journal of Risk and Insurance

The Term Structure of Reserve Durations and the Duration of Aggregate Reserves

Article excerpt


Estimating the duration gap of a life insurer demands the knowledge on the durations of liabilities and assets. The literature analyzed the durations of assets extensively but rendered limited analyses on the durations of insurance liabilities. This article calculated the reserve durations for individual policies and estimated the duration of the aggregate reserves. The results showed that the duration of the policy reserve might be negative and/or have a large figure. They further revealed an interesting pattern of the reserve duration with respect to the policy's time to maturity. A term structure with abnormal durations, however, does not result in an abnormal duration of the aggregate reserves.


Managing the company's interest rate risk is vital to a life insurer. Life insurance policies are long-term contracts. Small changes in interest rates can therefore cause large changes in the policy reserve liability, which usually constitutes more than 90 percent of a company's total liabilities. To offset the resulting fluctuations in the value of the reserve liability, the company must look for an asset portfolio that produces matched changes in values. If the match is not perfect, the high liability-to-surplus ratio prevalent in the life insurance industry will make the mismatch large relative to the insurer's surplus. Movements in interest rates, therefore, can have a significant adverse impact on the solvency of a life insurance company. (1)

A common measure of an institution's exposure to interest rate risk is the duration gap (DGAP). Samuelson (1945) was the first to introduce the concept of DGAP in analyzing how changes in interest rates may affect an institution. Redington (1952) invented the expression immunization and set up the fundamental equations for immunization strategies. One of the results of an immunization strategy is zero DGAP. Bierwag and Kaufman (1985) calculated four DGAPs to measure an institution's exposure to interest rate risk in accordance with alternative management goals. Further application and generalization of DGAP could be seen in Bierwag and Kaufman (1992, 1996). Fooladi and Roberts (2004) added the notion of convexity gap and default risk to DGAP to manage interest rate risk with better accuracy. The duration (gap) analysis has enjoyed widespread practitioner application in banks, insurance companies, and other financial institutions since the 1980s. Its usefulness as a measure of interest rate risk is undeniable and its use in finance markets today is extensive (Bierwag and Fooladi, 2006).

To calculate the DGAP of a life insurer, one usually has to calculate the durations of individual assets and liabilities. (2) The duration measure and the durations of the important assets held by life insurers such as bonds, mortgages, stocks, and real estate have been investigated extensively in the finance literature. As the review of Bierwag and Fooladi (2006) shows, the duration measure has been refined substantially with the considerations for stochastic interest rate processes, interest-rate-dependent cash flows, and default risk since the original work of Macaulay (1938). The durations of various asset classes, in addition to those of bonds, have also been explored substantially (e.g., Bierwag, Kaufman, and Toevs, 1983; Bierwag, 1987; Leibowitz et al., 1989; Bierwag, Corrado, and Kaufman, 1992; Babbel, Merrill, and Panning, 1997; Hayre and Chang, 1997; Hevert, McLaughlin, and Taggart, 1998; Cornell, 2000; Hamelink et al., 2002; Reilly, Wright, and Johnson, 2007) during the last quarter century.

In contrast, the durations of insurance liabilities have received limited attention. Babbel (1995) estimated the option-adjusted durations of the liabilities associated with a dozen life insurance products using a commercial software. (3) Santomero and Babbel (1997) listed the effective durations of the liabilities of several life insurance products based on their on-site investigation on the risk management practices of insurance companies. …

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