“Duration Ratio” as a New Risk Measure in Bank Risk Management
Yoon K. Choi
Department of Finance
University of Central Florida
Orlando, FL 32816-1400
(407) 823-5023; FAX: (407) 823-6676
E-mail: Yoon.Choi@bus.ucf.edu
ABSTRACT
I propose a duration gap measure, “duration ratio”, which measures a relative duration
gap between assets and liabilities. The new measure may be a true risk measure in the
sense that it captures not only the effect of interest rate changes, but also other exogenous
shocks, such as market risk and exchange rate risk.
This new approach provides us with a good proxy for the duration ratio based on interest
income and expenses, and U.S. large banks are examined for their risk management
based on the duration ratios.
June 14, 2001
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“Duration Ratio” as a New Risk Measure in Bank Risk Management
Risk management has been a crucial element of success in the bank industry. However,
with so many hedging instruments and strategies, it becomes so difficult to measure risk
and assess the quality of these risk management techniques. This note proposes a new
risk measure that would help investors and bank monitors and regulators in assessing risk
in the bank industry. I also empirically examine risk management in the large banks in
U.S. using this new approach.
1. Duration Gap and Immunization (Saunders, 2000)
The overall interest rate risk faced by bank owners whose equity is the difference
between assets and liabilities is normally defined as sensitivity of the value of equity with
respect to interest rate changes. In notation,
E / I = A / I - L / I,
(1)
where E = equity, I = interest rate, A = assets, and L = liabilities.
The well-known duration model shows:
A / A = - DA * I / (1 + I),
(2)
L / L = - DL * I / (1 + I),
(3)
where DA and DL are the durations of the bank’s asset and liability portfolio, respectively.
From equations (2) and (3), we obtain
A = - DA * A * I / (1 + I),
(4)
L = - DL * L * I / (1 + I).
(5)
Following the “Macauley” duration model which assumes that the interest rate and its
shock are the same for both assets and liabilities,
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E = - [DA * A - DL * L] * I / (1 + I).
(6)
Rearranging terms, we get
E = - [DA - DL * k] * A * I / (1 + I),
(7)
where k = L/A is a measure of the bank’s leverage.
According to the traditional immunization strategy based on equation (7), managers
should structure the balance sheet so that DA = DL * k such that E = 0.
2. “Duration Ratio” as Elasticity between Assets and Liabilities
Instead of calculating the difference, dividing (2) by (3) leads to the following:
(A / A) / (L / L) = DA / DL
(8)
Now the duration ratio in equation (8) can be interpreted as the elasticity of the asset
value with respect to liability values. I attempt to provide an economic meaning to the
duration ratio in terms of the interaction between assets and liabilities. In general, bank
risk management involves a way to offset asset risk by liabilities risk (e.g., maturity or
duration matching), such that equity risk will be immunized from the source of risk.
Suppose that both assets and liabilities are stochastic, and bank managers monitor assets
and liabilities so as to generate riskless equities. Equity can be seen a portfolio of assets
and liabilities. In a mathematical term, E = A – L.
And
VAR(E) = VAR(A) + VAR(L) – 2 * COV(A, L).
(11)
VAR and COV indicate variance and covariance, respectively.
For simplicity, assume VAR(A) = VAR(L). Then,
VAR(E) = 2 * VAR(A) * (1 - CORR(A, L)).
(12)
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Thus, equation (12) suggests that managers can hedge the total portfolio against any risk
by achieving a perfect correlation (e.g., CORR = 1) between assets and liabilities. Just
like elasticity, correlation is a statistical measure of a linear relationship between asset
and liabilities. Therefore, the duration ratio is interpreted to measure overall risks rather
than interest rate risk faced by banks.
Another important implication of equation (8) is that we can obtain a simple
proxy for the overall duration ratio: The ratio of the percentage change in the market
value of assets to that of liabilities. Since these variables are in market value, I use two
alternative variables in the balance sheet and income statement: book values of asset and
liabilities and interest revenues and expenses. I use these variables of the U.S. large banks
in order to determine its validity as a proxy for the duration ratio.
3. Other Issues: Convexity and Capital Ratio Requirement
3.1. Convexity and Duration
It is well known that the duration model assumes a linear relationship between
rate shocks and bond price changes. However, the actual bond price-yield relationship
exhibits a property called convexity rather than linearity. Specifically, the increase in
price due to a rate fall tends to be greater than as predicted by the duration model, while
the price falls more for rate increases. We can incorporate the convexity factor in the
duration model by adding the curvature parameter as follows:
A / A = - DA * I / (1 + I) + ½ * C * (I )2,
(9)
The second term on the right hand side of the equation indicates the curvature term.
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Given the convexity property, we need to adjust the duration ratio accordingly
because we do not obtain the simple duration ratio as in equation (8), due to the
convexity factor. Therefore, we suggest a modified duration ratio as follows:
(A / A) / (L / L) = DA / DL + CV
(10)
where CV is a convexity factor with a property being that greater curvature is associated
with higher CV. Of course the duration model implies CV = 0.
3.2. Capital Ratios and Immunization
The result in equation (8) has a very interesting implication for risk management
under capital ratio restriction. Regulators set minimum target ratios for a bank’s net
worth, E. The simplest one is the ratio of bank capital to its assets, E/A. Suppose that a
bank wants to immunize against falling below this target ratio; that is (E/A) = 0.
Kaufman (1984) shows that in order to immunize the net worth to set (E/A) = 0, the
manager needs to set DA = DL, instead of setting DA = k * DL such that E = 0. This
implies, according to equation (8), that the percentage change in assets should be the
same as the percentage change in liability. Furthermore, the risk management approach
would be different depending on the bank’s objective of risk management, including the
bank’s capital ratio requirement.
4. Data and Empirical Results
4.1. Data
I collected the data on assets and liabilities for the top 100 largest bank holding
companies as of December 31, 1997. The data were from the National Information
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Center of the Federal Financial Institutions Examination Council. Only 89 of the top 100
bank holding companies had the variables we use in the empirical investigation. Table 1
shows the summary statistics of our sample. The average asset size (AA) over the sample
period ranges from 6.19 billion to 281 billion dollars, with a mean of about 33 billion
dollars. On average, the interest revenue doubled interest expenses, indicating a profitable
performance in 1997. Tier I capital ratio is about 7.5% on average, which is well above
the 5% benchmark rate for “well-capitalization” according to the category defined by the
1991 FDIC Improvement Act. In fact, the minimum Tier I leverage ratio in our sample
was 5.03%.
4.2. Duration ratio in changes in assets and liabilities in book value
Although we use the book value in calculating duration ratios using assets and liabilities
(DR1), the ratio may capture the current market value changes since we use the recent
percentage changes in assets and liabilities instead of the levels. The average duration
ratio based on assets and liabilities was 0.919 close to 1, which implies successful
hedging strategies used in our sample banks. However, if we examine the results
carefully, the duration ratio may not be reliable for some individual banks. I find a few
negative sign, sometimes with a very high number (e.g., -6.2), and this is not acceptable
because the duration ratio is supposed to be positive.
4.3. Duration ratio in interest revenues and expenses (DR2)
Since the price changes in assets and liabilities in banks are accurately reflected in
interest revenues and expenses, we may obtain the duration ratio using interest revenues
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and expense changes instead. Table 2 shows that the duration ratio is again very close to
1.0. DR2 seems to measure the duration ratio more accurately, since the standard
deviation (0.06) of the measure is much smaller than DR1. Furthermore, all duration
ratios calculated in this sample are positive, which is another indication of its reliability
for the proxy for the duration ratio.
4.4. Two Alternative Risk Management: DA = DL vs. DA = k * DL
The previous section discussed two different risk management objectives: DA =
DL vs. DA = k * DL . If management wants to hedge the level of equity against risks
(e.g., E = 0), then the manager needs to set DA = k * DL (Strategy I). In this case, the
duration ratio is close to the leverage ratio. If the manager wants to hedge the capital ratio
instead (e.g., (E/A) = 0), he needs to set DA = DL (Strategy II). Here, the duration ratio
is close to one. Table II shows summary statistics for the two different sets of banks,
based on these strategies. There are 17 banks whose immunization strategy is to DA = k
* DL so as to keep the level of equity stable. These banks seem to be larger in size and
produce less interest revenues than those banks using Strategy II. Finally, the capital
leverage ratio for the banks using Strategy I seems to be lower than the banks using
Strategy II.
5. Summary and Conclusion
We examine two proxies for the “duration ratio”. One is based on assets and
liabilities; the other is based on interest revenues and expenses. Given the calculated
duration ratios of the well-capitalized large U.S. banks, the second measure based on
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interest revenues and expenses seems to be a more reliable proxy. The underlying
assumption is that these U.S. banks in our sample are engaged in successful risk
management. We find that an interesting pattern exists among our sample banks in terms
of immunization strategies. Those banks using Strategy I seem to be larger in size, have
lower capital leverage ratios, and produce less interest revenues than those banks using
Strategy II.
Several extensions are in order.
1. We can extend our analysis with much longer time-series data. We may observe how
duration ratio evolves over time.
2. We may examine smaller banks of different capital ratios to see whether different
duration ratios are associated with these banks in a systematic way.
References
Kaufman, G. “Measuring and Managing Interest Rate Risk: A Primer.” Federal Reserve
of Chicago, Economic Perspectives, 1984, p. 16-29.
Saunders, A. Financial Institution Management: A Modern Perspective, McGraw-Hill
Co. 2000.
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Table I
Summary Statistics of the Sample
This table contains the summary statistics of the major variables considered for 1997.
INTREV is the interest income per average assets, INTEXP is the interest expense per
average assets, AA is the average asset size in millions of dollars. CAP is the ratio of Tier
I capital ratio. DURAT_1 ((A / A) / (L / L)) is the first proxy for duration ratio.
DURAT_2 is the second proxy, defined as (int rev / int rev) / (int exp / int exp).
____________________________________________________________________
Variables
Mean
Minimum
Maximum
Std Dev.
____________________________________________________________________
INTREV (%)
7.5%
3.55%
14.15%
1.74%
INTEXP (%)
3.55%
1.83%
5.32%
0.66%
AA (Million)
32,907
6,190
281,678
52,346
CAP (%)
7.53%
5.03%
15.92%
1.9%
2.215
0.8
DURAT_1
0.9196
-6.169
DURAT_2
1.0011
0.8846
1.291
0.06
____________________________________________________________________
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Table II
Immunization Strategies
This table contains the summary statistics of the major variables considered for 1997 for
the banks with different immunization strategies. INTREV is the interest income per
average assets, INTEXP is the interest expense per average assets, AA is the average
asset size in millions of dollars. CAP is the ratio of Tier I capital ratio. L/A is the leverage
ratio. DA / DL is the duration ratio. Strategy I is to set DA = k * DL, while Strategy II is to
set DA = DL .
____________________________________________________________________
Strategy I
Strategy II
Variables
Mean
Median
Mean
Median
____________________________________________________________________
INTREV (%)
6.89%
6.97%
7.64%
7.43%
INTEXP (%)
3.34%
3.35%
3.59%
3.51%
AA (Million)
46,302
15,532
29,743
13,340
CAP (%)
6.5%
6.36%
7.77%
7.24%
DA / DL
93.4%
94.2%
101.7%
99.9%
L/A
93.5%
93.6%
92.2%
92.7%
____________________________________________________________________