“Each year, it seems, larger and more daunting mountains of text rise from the lush lowlands of visual reproduction.... You are likely to find yourself scaling craggy massifs of prose.... hacking a path through thickets of Nietzsche, Kierkegaard, Baudelaire and Marx.” Calvin Tomkins, in Simpson's Contemporary Quotations, 1988.

Calvin Tomkins' lament about the changing topography of coffee-table art books applies equally to the challenge of mastering the concepts and terminology involved in financial risk management. An insurance practitioner needs to scale the prose of such words and phrases as general equilibrium, arbitrage-free, lattice, stochastic, spot rate and path-dependence, and then hack through the thickets of the interest rate models of Vasicek, Cox-Ingersoll-Ross, Hull-White, and Heath-Jarrow-Morton. The purpose of this article is to serve as a hitchhiker's guide for a journey along this path.

In a previous article (“A sensitive subject,” Global Reinsurance, Volume 8, Issue 7), we discussed the nature of property/casualty insurer liabilities and their sensitivity to changes in interest rates. Since we live in a world characterized by interest rate volatility, properly measuring changes in liability values resulting from potential movements in interest rates should be, we suggested, a critical part of an insurer's efforts to manage financial risks via asset-liability management. In this article, we explore these issues further, and describe an approach for determining the impact of stochastic interest rate movements on an insurance company's net worth.

An insurance company's operations are exposed to a variety of different types of risks. Insurers and reinsurers are most accustomed to considering and addressing traditional insurance risks, such as catastrophic risks. Indeed, the potential impact of natural catastrophes on property/casualty insurers is given extensive attention and recognition by the industry; sophisticated models exist to permit companies to quantify this exposure and securitized insurance products are being designed to permit the trading of such risks with the capital markets. Other underwriting-related risks have also traditionally received attention from the industry.

However, as with other companies within the financial services industry, insurers are also significantly exposed to a variety of financial risks. For example, with the rising level of globalization in the insurance industry, the risk of fluctuations in foreign exchange rates is becoming an increasing concern for insurers. Another critical area of risk faced by insurers involves fluctuations in value due to interest rate movements.

The values of an insurer's assets and liabilities can be affected by changes in future interest rates. The reason for this is that the economic value of a financial asset or liability is the discounted value of its future cash flows. Thus, if interest rates increase, the economic value of future cash flows will decrease; if interest rates decrease, economic value will increase. The direction of the movement in values of both the assets and the liabilities, according to this principle, will be the same for a given change in interest rates. The problem, however, is that asset and liability values will generally not move by the same amount (unless specifically and accurately set up to do so). If they do not move similarly, the net worth of an insurer will change over time due to the volatility of interest rates.

To understand why assets and liabilities may not move by the same amount, look at Figure 1. This chart shows the future cash flows for two financial items - an asset and a liability - superimposed upon one another. The asset is a five-year, 7.25%-annual-coupon bond; the liability is a possible payment stream associated with general liability loss reserves. For each item, the economic value is determined by discounting each of the item's expected cash flows and summing across all the flows. From this figure, it can be seen that the nature and timing of the cash flows are completely different, and it is thus far from certain that the discounted values of the asset and the liability will exhibit a similar response to changes in interest rates.

As discussed in our previous article, “duration” is an important measure of the sensitivity of the value of a financial item to changes in interest rates. A traditional measure of duration, called “modified duration”, represents the percentage change in the economic value of a financial item resulting from a one percentage point change in interest rates. In theory, and somewhat overly simplistically, if the durations of assets and liabilities can be matched, then a movement in interest rates will cause similar changes in asset and liability values and thus the net value of the company will be unaffected. In fact, the asset in Figure 1 was selected so that it would have the same modified duration as the loss reserve liability. However, if the duration values of assets and liabilities are not the same, the net economic value will change when interest rates change. It is therefore critical to an insurer's financial risk management process that the durations of both assets and liabilities be accurately calculated.

Traditional modified duration makes three crucial assumptions about the potential movements of interest rates:

1. The term structure of interest rates is flat. Thus, short-term and long-term interest rates - and all rates in between - are equal to each other at any given time.

2. Movements in the term structure occur in parallel fashion - the entire structure shifts upward and downward uniformly. In other words, if short-term rates increase by one percentage point, intermediate and long-term rates also increase by one percentage point.

3. Changes in interest rates do not affect the magnitude or timing of the future cash flows of the assets and liabilities being valued.

All three of these assumptions are unrealistic. For example, with regard to the first assumption, the term structure of interest rates is typically not flat. More often than not, the term structure is upward-sloping, meaning that short-term rates are lower than long-term rates. Occasionally, one also encounters a downward-sloping term structure, or even a humped structure.

The second assumption is also unrealistic. Historically, short-term interest rates have tended to be more volatile than longer term interest rates. More generally, as interest rates change, the term structure can move in at least three different ways simultaneously: shifting upward or downward, becoming more or less steeply sloped, and changing shape (eg, developing a hump where none existed before).

The third assumption may or may not be unrealistic, depending upon the specific type of asset or liability being considered - but when this assumption does fail, it can have a large impact on the accuracy of the duration measure. For certain types of assets and liabilities, including many held by insurers and reinsurers, a change in interest rates will affect future cash flows. On the asset side, for example, the timing of future cash flows on collateralized mortgage obligations depends upon interest rates, since mortgage prepayment rates will increase as interest rates fall, due to more frequent refinancings. Similarly, corporate bonds may be “called” (a form of debt refinancing) if interest rates fall sufficiently. Insurers frequently hold a variety of such assets with interest-sensitive cash flows. On the liability side, the magnitude of property/casualty loss payments will be affected by future inflationary pressures. Since interest rates and inflation are correlated, property/casualty liability loss reserves are also interest-sensitive.

The third assumption can be addressed by using a measure called “effective duration”, which explicitly considers the impact of interest rate changes on the magnitude and timing of future cash flows when determining the interest-sensitivity of current asset and liability values. The first and second assumptions can be addressed by using a stochastic model of the term structure of interest rates; such a model recognizes the random element inherent in future interest rates.

There are a variety of term structure models for interest rates which have been proposed, including the formidable names cited in the first paragraph of this article; each model has its particular strengths and weaknesses. Term structure models provide a mathematical representation of interest rate movements. For example, historical interest rate patterns suggest that short-term interest rates might be “mean-reverting” (rates may be volatile, but over time the level of interest rates tends to move back toward a long-run mean value), and that the volatility of interest rate movements is greater when rates are at higher levels. Term structure models can mathematically represent these and other patterns found in the historical and current market data.

With this mathematical foundation, a simulation approach can be used to determine the impact of future interest rate movements on current financial values. The steps of this process are:

1. Choose a term structure model.

2. Select the parameters for the model, based upon statistical analysis of historical interest rate experience and/or judgment regarding market conditions.

3. Generate an interest rate path over an appropriate time period, at least as long as the anticipated cash flows stemming from the assets and liabilities - say 30 years. This path represents one possible “scenario” as to the future pattern of interest rates.

4. Based on the correlations of interest rates with inflation, mortgage prepayment patterns, and other economic variables, calculate the cash flows of assets and liabilities that will emanate from the interest rate path generated in step 3. For example, if the interest rate path contains very high interest rates, the associated inflation rates are likely to be much higher than originally envisioned, causing inflation-sensitive liability payments to be larger than anticipated. The authors have developed a model that reflects this interest sensitivity of loss reserve payments by simulating inflationary pressures and developing the pattern of future loss payments in light of those pressures.

5. Discount the re-stated cash flows from step 4, based on the interest rates generated in step 3. The discounted figures represent the present (or economic) values of the future asset and liability cash flows anticipated under the simulated interest rate environment.

6. Repeat steps 3 through 5 many times, which will generate a variety of interest rate scenarios, and observe the resulting economic values of assets and liabilities (and thus of net worth). By analyzing the distribution of these economic values, one can determine the range of likely outcomes, the likelihood of a particular unfavorable outcome and the expected value.

An example of interest rate scenario generation based on the Cox-Ingersoll-Ross term structure model is provided in Figure 2. This model reflects the mean reverting nature of interest rates (low rates tend to increase and high rates tend to decrease). The graph shows several illustrative ten-year paths for the short-term interest rate. Each line shows the stochastically generated path of future interest rates resulting from one simulated scenario. The dispersion of these lines after the hypothetical beginning value of 5% reflects the stochastic nature of interest rate emergence.

Consider again the general liability situation in Figure 1. The cash flows of the loss reserve payout pattern and the bond were set up to have the same modified duration of 4.19. Thus, based on modified duration, the economic value of each set of cash flows will change by the same amount - 4.19% - in response to a one percentage point movement in interest rates. Traditional asset-liability management techniques might consider this loss reserve liability to be “immunized” by this asset. However, modified duration is subject to the three generally unrealistic assumptions noted earlier. Using the stochastic interest rate model approach outlined above, we find that the effective duration of the asset is 2.61 and the effective duration of the liability is 1.67. Thus, there is actually an asset-liability mismatch: a company that thought it had immunized its net worth against changes in interest rates by equating the asset and liability modified durations would, in reality, be exposed to interest rate risk, as evidenced by the different effective durations between the asset and the liability.

Effective durations of assets and liabilities are often much lower than traditional duration measures would indicate. Since the effective duration framework described above provides a more realistic view of how interest rates behave than traditional modified duration, and because it reflects the sensitivities of future cash flows to inflationary and other pressures, it is critical that insurers, when they employ financial risk management techniques such as asset-liability management, use the proper duration measure. If durations are mismatched, an insurer or reinsurer could unknowingly be subject to adverse consequences in the event of interest rate movements.

Despite the apparent complexity of interest rate modeling, the process is relatively straightforward once the terminology and general approach are understood. The mathematical models for the term structure of interest rates can be quite involved, but the overall approach used to quantify interest rate risk is no more complicated than many other calculations frequently used in insurance and reinsurance. Hopefully, this guide can help the reader to understand this important concept.

Stephen P. D'Arcy and Richard W. Gorvett teach at the University of Illinois at Urbana-Champaign, Mr D'Arcy in the finance department and Mr Gorvett in the actuarial science program in the mathematics department.