In practice, a number of different

In practice, a number of different approaches are used to calculate interest rate deltas. One approach is to define delta as the dollar duration. This is the sensitivity of the portfolio to a parallel shift in the zero-coupon yield curve. A measure related to this definition of delta is DV01. This is the impact of a one-basis-point increase in all rates. It is the dollar duration multiplied by 0.0001. Alternatively, it is the duration of the portfolio multiplied by the value of the portfolio multiplied by 0.0001. Analysts like to calculate several deltas to reflect their exposures to all the differ-ent ways in which the yield curve can move. There are a number of different ways this can be done. One approach corresponds to the partial duration approach that we outlined in the previous section. It involves computing the impact of a one-basis-point change similar to the one illustrated in Figure 9.5 for points on the zero-coupon yield curve. This delta is the partial duration calculated in Table 9.5 multiplied by the value of the portfolio multiplied by 0.0001. The sum of the deltas for all the points on the yield curve equals the DV01. Suppose that the portfolio in Table 9.5 is worth $1 million. The deltas are shown in Table 9.6.
TABLE 9.0 Deltas for Portfolio in Table 9.5
Maturity (years) Delta
1 -20 2 -60 3 -90 4 -160 5-200 7 210 10 300 Total -20
Value of portfolio is $1 million. The dollar impact of a one-basis-point increase in points on the zero curve is shown.
182 MARKET RISK
6 5 4 3 2 1 0
Zero rate (%)
Maturity (years)
O 2 4 6 8 10 12
FIGURE 8.7 Change Considered to Yield Curve When Bucketing Approach Is Used
A variation on this approach is to divide the yield curve into a number of seg-ments or buckets and calculate for each bucket the impact of changing all the zero rates corresponding to the bucket by one basis point while keeping all other zero rates unchanged. This approach is often used in asset-liability management (see Sec-tion 9.1) and is referred to as GAP management. Figure 9.7 shows the type of change that would be considered for the segment of the zero curve between 2.0 and 3.0 years in Figure 9.4. As with the partial duration approach, the sum of the deltas for all the segments equals the DV01.
Calculatleg Deltas to Facilitate HedeIcl One of the problems with the delta measures that we have considered so far is that they are not designed to make hedging easy. Consider the deltas in Table 9.6. If we plan to hedge our portfolio with zero-coupon bonds, we can calculate the position in a one-year zero coupon bond to zero out the $200 per basis point exposure to the one-year rate, the position in a two-year zero-coupon bond to zero out the exposure to the two-year rate, and no on. But, if other instruments are used, a much more complicated analysis is necessary. In practice, traders tend to use positions in the instruments that have been used to construct the zero curve to hedge their exposure. For example, a government bond trader is likely to take positions in the actively traded government bonds that were used to construct the Treasury zero curve when hedging. A trader of instruments dependent on the LIBOR/swap yield curve is likely take a position in LIBOR deposits, Eurodollar futures, and swaps when hedging. To facilitate hedging, traders therefore often calculate the impact of small changes in the quotes for each of the instruments used to construct the zero curve. The quote for the instrument is changed by a small amount, the zero-coupon yield curve is recomputed, and the portfolio revalued. Consider a trader responsible for interest rate caps and swap options. Suppose that, when there is a one-basis-point
Interest Rate Risk 183
change in a Eurodollar futures quote, the portfolio value increases by $500. Each Eurodollar futures contract changes in value by $25 for a one-basis-point change in the Eurodollar futures quote. It follows that the trader's exposure can be hedged with 20 contracts. Suppose that the exposure to a one-basis-point change in the five-year swap rate is $4,000 and that a five-year swap with a notional principal of $1 million changes in value by $400 for a one-basis-point change in the five-year swap rate. The exposure can be hedged by trading swaps with a notional principal of $10 million.