336 PA R T I V Duration Analysis The Management of Financial Institutions The gap analysis we have examined so far focuses only on the effect of interestrate changes on income Clearly, owners and managers of banks care not only about the effect of changes in interest rates on income but also about the effect of changes in interest rates on the market value of the net worth of the bank.4 An alternative method for measuring interest-rate risk, called duration analysis, examines the sensitivity of the market value of the bank s net worth to changes in interest rates Duration analysis is based on Macaulay s concept of duration, which measures the average lifetime of a security s stream of payments (described in the Web Appendix to Chapter 4) Recall that duration is a useful concept because it provides a good approximation, particularly when interest-rate changes are small, of the sensitivity of a security s market value to a change in its interest rate using the following formula: %*P DUR *i i (4) where % P DUR i (Pt Pt )/Pt duration interest rate percent change in market value of security After having determined the duration of all assets and liabilities on the bank s balance sheet, the bank manager could use this formula to calculate how the market value of each asset and liability changes when there is a change in interest rates and then calculate the effect on net worth There is, however, an easier way to go about doing this, derived from the basic fact about duration we learned in the Web Appendix to Chapter 4: Duration is additive; that is, the duration of a portfolio of securities is the weighted average of the durations of the individual securities, with the weights reflecting the proportion of the portfolio invested in each What this means is that the bank manager can figure out the effect that interest-rate changes will have on the market value of net worth by calculating the average duration for assets and for liabilities and then using those figures to estimate the effects of interest-rate changes To see how a bank manager would this, let s return to the balance sheet of the First Bank Suppose that the average duration of its assets is three years (that is, the average lifetime of the stream of payments is three years), while the average duration of its liabilities is two years In addition, the First Bank has $100 million of assets and, say, $90 million of liabilities, so its bank capital is 10% of assets With a 5-percentage-point increase in interest rates, the market value of the bank s assets falls by 15% ( 5% years), a decline of $15 million on the $100 million of assets However, the market value of the liabilities falls by 10% ( 5% years), a decline of $9 million on the $90 million of liabilities The net result is that the net worth (the market value of the assets minus the liabilities) has declined by $6 million, or 6% of the total original asset value Similarly, a 5-percentage-point decline in interest rates increases the net worth of the First Bank by 6% of the total asset value Note that accounting net worth is calculated on a historical-cost (book-value) basis, meaning that the value of assets and liabilities is based on their initial price However, book-value net worth does not give a complete picture of the true worth of a firm; the market value of net worth provides a more accurate measure This is why duration gap analysis focuses on what happens to the market value of net worth, and not on book value, when interest rates change