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The relation between capital structure, interest rate sensitivity, and market value in the property-liability insurance industry: comment.


In an intriguing article appearing in this journal, Staking and Babbel (1995) empirically examine the relationship between interest rate risk and market value for firms operating in the insurance industry. Intuitively, firms with greater interest rate risk are expected to have lower market values when compared to firms with less risk exposure. And Staking and Babbel find that the expected relationship holds over low levels of risk. However, at higher levels of exposure, they find a positive relationship between interest rate risk and market value (as measured by Tobin's q). According to the authors, the direct relationship is a consequence of the ability of the firm to increase variability in asset value without fully suffering the consequence of reduced insurance premiums. Increased interest rate risk may increase shareholder value as long as the greater risk is not accurately priced in insurance contracts. Inaccurate pricing can be the result of poor information held by policyholders that might exist for a variety of reasons, including the presence of guarantee funds that make policyholders less interested in the financial condition of the primary insurer (assuming that the guarantee fund charges premiums that are not risk based).

This comment offers an additional explanation for the observed relationship between interest rate risk and market value. The authors' explanation may be part of the story, and is not quarreled with here. Rather, the purpose of this comment is to clarify the definition of interest rate risk and present another factor that is contributing to Staking and Babbel's findings.


When analyzing the interest rate sensitivity of a portfolio of assets, the most widely used measure, in part due to its simplicity, is the Macaulay duration of the portfolio. However, several alternative measures of duration exist. Bierwag (1978) demonstrates that the appropriate duration measure hinges upon the stochastic process underlying interest rate movements. Different measures are appropriate for alternative possible shifts in the yield curve. In addition, default and credit risk impact upon a portfolio's effective duration, as analyzed by Babbel, Merrill, and Panning (1997). Further issues of calculating the appropriate duration in the face of spreads between short-term and long-term yields are taken up by Khang (1979) and Bierwag, Kaufman, and Latta (1987).

In examining interest rate risk, Staking and Babbel rely on simple Macaulay duration to calculate the duration of assets and liabilities. The "duration of surplus" [D.sub.s] depends upon the duration of the firm's assets and liabilities and is given by

[D.sub.S] = [D.sub.A] - [D.sub.L]) A/S + [D.sub.L], (1)

where [D.sub.A] = a weighted-average of the Macaulay durations of the firm's assets,

[D.sub.L] = a weighted-average of the Macaulay durations of the firm's liabilities, A = the market value of assets, and

S = the market value of surplus, which is the difference between the market value of assets (A) and the market value of liabilities (L).

As shown below, using [D.sub.s] as a measure of interest rate risk is appropriate if the interest rate associated with the firm's assets ([r.sub.A]) is equal to the rate on liabilities ([r.sub.L]). However, if there is a positive spread between the two rates an alternative specification of interest rate risk is necessary.

Consider the market value of surplus for a firm with n assets and m liabilities:

S = [summation of] [A.sub.k] where k = 1 to n - [summation of] [L.sub.j] where j = 1 to m. (2)

Note that the market value of any asset - say, asset k - is a function of its interest rate ([r.sub.[A.sub.k]]). Similarly, the value of any liability - say, liability j - depends upon the associated yield ([r.sub.[L.sub.j]]). Then, totally differentiating equation (2) gives

dS = [summation of] d[A.sub.k]/[dr.sub.[A.sub.k]] [dr.sub.[A.sub.k]] where k = 1 to n - [summation of] d[L.sub.j]/[dr.sub.[L.sub.j]] [dr.sub.[L.sub.j]] where j = 1 to m. (3)

Suppose there is a general rise in interest rates, which uniformly changes rates on all assets and liabilities (dr = ([dr.sub.[A.sub.1]] = [dr.sub.[A.sub.2]] ... = [dr.sub.[A.sub.n]] = [dr.sub.[L.sub.1]] = [dr.sub.[L.sub.2]] ... = [dr.sub.[L.sub.m]]). Then, equation (3) can be rewritten as

Ds = [[summation of] d[A.sub.k]/dr where k = 1 to n - [summation of] d[L.sub.j]/dr where j = 1 to m] dr. (4)

As is well known, the relationship between the Macaulay duration of an asset ([D.sub.[A.sub.k]]) and the derivative d[A.sub.k]/dr is given by

d[A.sub.k]/dr = -[D.sub.[A.sub.k]] [A.sub.k]/(1 + [r.sub.[A.sub.k]]). (5)

Likewise, for liabilities, the following relationship holds:

d[L.sub.j]/dr = -[D.sub.[L.sub.j]] [L.sub.j]/(1 + [r.sub.[L.sub.j]]). (6)

Substituting equations (5) and (6) into equation (4) gives

dS = - [([summation of] [D.sub.[A.sub.k]] [A.sub.k]/(1 + [r.sub.[A.sub.k]) where k = 1 to n) - ([summation of] [D.sub.[L.sub.j]] [L.sub.j]/(1 + [r.sub.[L.sub.j]]) where j = 1 to m)] dr. (7)

For simplicity, assume that all assets have the same rate ([r.sub.A]), so [r.sub.A] = [r.sub.[A.sub.1]] = [r.sub.[A.sub.2]] = ... [r.sub.[A.sub.n]]. Likewise, let all liabilities possess interest rate [r.sub.L] ([r.sub.L] = [r.sub.[L.sub.1]] = [r.sub.[L.sub.2]] = ... [r.sub.[L.sub.m]])..

Then, factoring gives

dS = -[A/(1 + [r.sub.A]) ([summation of] [D.sub.[A.sub.k] [W.sub.[A.sub.k]] where k = 1 to n) - L/(1 + [r.sub.L]) ([summation of] [D.sub.[L.sub.j] [W.sub.[L.sub.j]] where j = 1 to m)] dr, (8)

where [W.sub.[A.sub.k]] = [A.sub.k]/A, and [W.sub.[L.sub.j]] = [L.sub.j]/L.

Let x represent the spread between the interest rate on assets and liabilities, so

[r.sub.A] = [r.sub.L] + x. Then, substituting [D.sub.A] (= [summation of] [D.sub.[A.sub.k]] [W.sub.[A.sub.k]] where k = 1 to n), [D.sub.L] (= [summation of] [D.sub.[L.sub.j]] [W.sub.[L.sub.j]] where j = 1 to m), and

[r.sub.L] + x into equation (8) gives

dS = -[A [D.sub.A]/(1 + [r.sub.L] + x) - L [D.sub.L]/(1 + [r.sub.L])] dr, (9)

which can be rewritten as

dS = -[[(1 + [r.sub.L] + x) (1 + [r.sub.L])].sup.-1] [A [D.sub.A] (1 + [r.sub.L]) - [D.sub.L](A - S) (1 + [r.sub.L] + x)] dr. (10)

Dividing through by S and rearranging yields

dS/S = -[[(1 + [r.sub.L] + x) (1 + [r.sub.L])].sup.-1] [[D.sub.S](1 + [r.sub.L]) - [D.sub.L] (A/S) x + [D.sub.L]x] dr, (11)

where [D.sub.S] = ([D.sub.A] - [D.sub.L]) A/S + [D.sub.L]).

Staking and Babbel refer to [D.sub.S] as the "duration of surplus," and use it as a measure of interest rate risk throughout their article. However, referring to equation (11), it is evident that [D.sub.S] is not generally correct as a measure of interest rate risk. The percentage change in equity (dS/S) resulting from a change in interest rates depends on the interest rate on liabilities, the spread between rates on assets and liabilities, the average duration of liabilities, and Staking and Babbel's duration of surplus ([D.sub.S]). Using [D.sub.S] as a measure of interest rate risk assumes there is no spread (x = 0), resulting in the following simplification of equation (11):

dS/S = -[D.sub.S]/(1 + [r.sub.L]) dr. (12)

A careful examination of expressions (11) and (12) provides insight into the findings of Staking and Babbel regarding interest rate risk and firm value. Perfect immunization implies that a change in interest rates has no effect on firm value (dS/S = 0). Consider two firms, called firms A and B, which are perfectly immunized. However, suppose firms A and B differ in the spreads they are earning: Firm A is earning no spread, while Firm B earns a positive spread. Referring to equation (12), it is evident that the firm with no spread will perfectly immunize by setting [D.sub.S] = 0. For Firm B, the spread is positive (x [greater than] 0), and expression (11) is relevant. Referring to equation (11), perfect immunization implies that

[D.sub.S](1 + [r.sub.L]) = [D.sub.L]x(A/S - 1). (13)

As the right-hand side of equation (13) is positive, it is clear that Firm B will immunize its portfolio by setting [D.sub.S] [greater than] 0. Then, when comparing the two perfectly immunized companies, the firm earning the greater return on its assets when compared to its liabilities will also have a greater [D.sub.S], the measure of duration of surplus used by Staking and Babbel. The implication is that Staking and Babbel's finding of a positive relationship between market values and interest rate risk is due to the inadequacy of [D.sub.s] as a risk measure. Perfectly immunized firms will have different values of Ds depending upon the extent to which assets are earning a higher rate than is being paid on liabilities. Firms with higher yielding assets will have larger values of [D.sub.s] and will likely have higher market values of equity as well.

Similar arguments apply when examining companies that are interest rate sensitive. When comparing two firms possessing equal amounts of interest rate risk but having different spreads, the firm with a higher spread will also have a larger [D.sub.S]. To see this, consider the usual case where an insurance company is exposed to interest rate changes such that an increase in rates (dr [greater than] 0) reduces the market value of its equity (dS/S [less than] 0). From equation (11), it is evident that the percentage decline in equity caused by rising rates depends on K ([less than] 0), where K is defined as follows:

K = -[[(1 + [r.sub.L] + x) (1 + [r.sub.L])].sup.-1] [[D.sub.S](1 + [r.sub.L]) - [D.sub.L]x(L/S)]. (14)

Thus, dS/S = Kdr and K is the appropriate measure of interest rate risk. Holding K constant while differentiating equation (14) with respect to x gives

[[(1 + [r.sub.L] + x) (1 + [r.sub.L])].sup.-2] (l + [r.sub.L]) [[D.sub.S] (1 + [r.sub.L]) - [D.sub.L]x(L/S)] -[[(1 + [r.sup.L] + x) (1 + [r.sub.L]).sup.-1] [d[D.sub.S]/dx(1 + [r.sub.L]) - [D.sub.L](L/S)] = 0. (15)

Solving equation (15) for d[D.sub.S]/dx results in

d[D.sub.S] / dx = -K + [D.sub.L](L/S) [(1 + [r.sub.L]).sub.-1] [greater than] 0. (16)

Thus, when comparing firms with like interest rate risk, higher spreads are associated with longer surplus durations as measured by [D.sub.S].


Staking and Babbel find what appears to be a surprising relationship between interest rate risk and firm value: At high levels of exposure, greater interest rate risk is associated with higher market values. This article demonstrates that their result is at least partially attributable to their choice of duration surplus as a measure of interest rate risk. When the interest rate associated with assets exceeds the rate on liabilities, Staking and Babbel's duration surplus is not a complete measure. Further, when comparing firms with identical interest rate risk exposure, those firms earning higher spreads will have larger surplus durations. Thus, the positive relationship between the duration of surplus and market value found by Staking and Babbel (for a portion of their sample) might be explained as the natural consequence of the market placing higher values on the equity of the firms earning higher spreads. Further examination of the data would be required to determine whether there is in fact a positive relationship between firm values and interest rate risk when a more complete measure of interest rate risk is utilized.


Babbel, D. F., C. B. Merrill, and W. Panning, 1997, Default Risk and the Effective Duration of Bonds, Financial Analysts Journal, 53: 35-44.

Bierwag, G. O., 1978, Measure of Duration, Economic Inquiry, 16: 497-507.

Bierwag, G. O., G. Kaufman, and C. Latta, 1987, Bond Portfolio Immunization: Tests of Maturity, One- and Two-Factor Duration Matching Strategies, Financial Review, 22: 203-219.

Khang, C., 1979, Bond Immunization When Short Term Rates Fluctuate More Than Long Term Rates, Journal of Financial and Quantitative Analysis, 14: 1085-1090.

Staking, Kim B. and David F. Babbel, 1995, The Relation Between Capital Structure, Interest Rate Sensitivity, and Market Value in the Property-Liability Insurance Industry, Journal o fRisk and Insurance, 62:690-718.

L. Dwayne Barney is Professor of Finance at Boise State University.
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Author:Barney, L. Dwayne
Publication:Journal of Risk and Insurance
Date:Dec 1, 1997
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