For any given scenario, there is a unique hedging solution immunizing the N11 against changes in interest rates. This hedging solution is the one that offsets the variable interest rate gap.

A first implication is that, with several scenarios, it is not possible to lock in the N11 for all scenarios by hedging, because gaps differ from one business scenario to the other. We can consider as hedging solutions those who close the variable interest rate gaps under business scenarios A and B. The issue is to find the best hedging solution among these two. By modifying the hedge, we change the entire matrix of values. With two hedging scenarios, there are eight combinations with equal probabilities resulting from two matrices and each hedging solution is summarized by a couple of values, the average N11 and the N11 volatility.

The first hedging scenario locks in the interest rate for a debt of 4. The remaining debt, 12 -4 = 8, required to bridge the liquidity gap, is a floating rate debt. The second solution locks in the interest rate for an amount of 8, the remaining 4 being floating rate debt. Those are the HI and H2 hedging solutions, summarized in Table 24.8. The hedges simply change the mix of variable rate/fixed rate financing of the same liquidity gap of 12.

Since the first hedge makes the N11 constant under business scenario A, the variable rate gap is zero with A. With an initial variable rate gap of-4, the hedge offsets this gap with a payer swap (paying the fixed rate) of notional 4, making the financing fixed rate for an amount of 4. The variable rate gap after hedge is increased by 4 for both scenarios A and B. The variable rate gap for A is zero. The variable rate gap with B is initially -8 and becomes -4. The same rationale applies when fully hedging the N11 under scenario B. For B, the swap closes the variable rate gap and, for A, it results in a positive variable rate gap of+4.

TABLE 24.8 Funding scenarios

"The hedging solution is the fraction of total funding of which rate is locked in. "fr" and 'Vr" designate respectively the fractions of debt with fixed rate and variable rate.

The interest rate gap is that of the banking portfolio less the floating rate debt.

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