# Interest Rate Risk

Duration is a key indicator for the measurement of the direct interest rate risk. The principal usefulness of duration stems from the fact that it approximates the elasticity of the market values of assets and liabilities to the respective rates of return,15

&MrA) _-РлАгл AA(r, ) _ - D, Ar, A(rA) (! + /•„)’ A(rL) (1 + rJ

where A(rA) and L(rL) are market values of assets and liabilities of a banking system, and where rA and rL are annual interest rates on assets and liabilities. This feature of duration can be used to summarize the effect of changes in interest rates on banks’ capital. In particular, we can define capital as A(rA) - L(rL), and can express it as a ratio to risk weighted assets.16 Differentiating capital with respect to the interest rate on assets, and substituting from equation 6, the sensitivity of the C/ARW ratio to interest rate changes can be expressed as

Assuming that the risk-weighted assets move proportionately to total assets, that is, AArw/Arw=AA/A, equation 7 can be simplified to

where GAPd is the duration gap, defined as

OAP„-DA-D,'-^-j^. (9)

1 + Ъ Агд

The duration gap and the direct interest rate stress test are two analytical tools that can often be viewed as substitutes for each other. Equations 8 and 9 illustrate the relationship between the two duration FSIs and the capital adequacy FSI.17 In particular, equation 8 characterizes the relationship between the “interest rate exposure FSI” and the corresponding stress test in a similar way as equation 2 for the exchange rate risk. The interest rate exposure FSI is the duration gap, which is a function of the two duration FSIs. In the special case when the interest rates for assets and liabilities move simultaneously, the duration gap can be approximated as a difference of the two durations: DA - DL. Similar to the exchange rate risk, the effect on capital adequacy can generally be expressed as a product of the shock and the “exposure FSI.” In both cases, however, this shortcut formula is subject to simplifying assumptions, such as the one on the relationship between total and risk-weighted assets.

The duration gap is a reliable estimator of the effect of interest rate changes only for small shocks. Durations can change with changes in interest rates. Because stress tests typically involve large changes in interest rates, it is advisable to include second derivative terms to account for convexity. However, given the complexities involved in such calculations, FSAP stress tests so far have not been able to satisfactorily reflect possible changes in duration. In fact, most FSAP missions used much simpler approaches than

those based on duration.18 A related issue is the calculation of a combined interest rate and exchange rate shock, when the combination of the aggregate duration and the aggregate net open position may give only an approximate indication of the overall effect. A currency breakdown of duration would help to identify maturity mismatches by currencies. Again, this analysis was typically not done in FSAP missions, mostly because of the lack of data.

The calculation of duration of total assets and total liabilities of a financial system can be a difficult computational task; however, alternative approaches are possible. In practice, alternative and less-costly approaches to measuring the interest rate risk are often used. Assets and liabilities can be lumped into groups that are based on common features, such as coupon rates (or comparable contractual rates), maturities, and credit risk. Within such cells, one can estimate the implied cash flow stream and the relevant market yields and can compute duration, which can then be aggregated across the cells.

A simplified measure of interest rate sensitivity that is often used in place of duration is based on the traditional “maturity gap analysis.” Under this approach, expected payments on assets and liabilities are sorted into “buckets” according to the time to repricing or when payments are due (e. g., period until financial instruments are redeemed or the interest rates on them are reset or reindexed).19 Similar to duration, the net difference (gap) in each time bucket can be multiplied by an assumed change in interest rates to gain an indication of the sensitivity of banks’ income to changes in interest rates.

Maturity gap data are useful, but they are inferior to duration measures and could conceal actual risks in the system. Ahmed, Beatty, and Bettinghaus (1999), using empirical data on U. S. banks, 1991-99, found that maturity gaps reported by the banks were useful in assessing the loss potential of banks’ interest rate risk positions, because there was a significant statistical relationship between the maturity gap and future changes in net interest income. However, it is possible that the maturities of financial assets and liabilities match, but the timing of the cash flows on assets and liabilities is not matched (i. e., their durations differ) and banks are, thereby, open to interest rate gains or losses. Bierwag (1987) showed practical examples of banks that have zero maturity gaps but that, in fact, have extremely risky positions (measured by duration).

Similar to the net open position in foreign exchange, duration gaps capture only the direct effect of an interest rate change on the bank. They do not reflect indirect effects, in particular the effect that an increase in lending interest rates is likely to have on the credit risk of banks’ borrowers. This risk could be approximated by using the encouraged FSI of corporate earnings to interest and principal expenses. In practice, however, this indicator has so far been reported relatively infrequently, even though it has been used more frequently in the recent FSAP mission. Those FSAP missions that attempted to assess this type of risk typically estimated a regression model for the share of NPLs to TLs, with interest rates among the explanatory variables. The panel data estimate presented by IMF (2003) did not find a significant relationship between interest rates and the NPL/TL ratio, although this lack of relationship may reflect the limitation of the data set. However, for individual countries using time series data, the slope coefficient was often significantly negative.20 Similar to the exchange rate risk, the integration of the direct and indirect interest rate risk is easier to implement with the help of stress tests.

In some stress-testing exercises, the values of a set of correlated risk factors (e. g., a set of prices, macrovariables, financial ratios, yield curve shifts) are simulated assuming a joint probability distribution of those factors, typically a joint normal distribution that is based on empirically determined parameters. The values drawn from the distribution— through Monte Carlo simulations—are used to stress the portfolio so that probability of specified extreme outcomes or the size of potential losses at specified probability level can be calculated.