THE ECONOMICS OF MONEY,BANKING, AND FINANCIAL MARKETS 368
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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
1
i
(4)
where
% P
DUR
i
(Pt 1 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 do 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% 3 years), a decline of $15 million on the $100 million of assets. However, the market value of the liabilities falls by 10% (
5%
2 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.
4
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.
