Riassunto esame Risk management, Prof. Capizzi Vincenzo, libro consigliato Risk Management, Mario Valletta

Net Interest Income Formula

A Lcourse, consistent with t). So, we can write the formula comprising the components of the Net Interest Income: therefore, * (SA + NSA) … vedi “risultato” della prima formula. The NII is equal to r A. Actually, if we pass from the absolute measure of NII to the formula describing the change (!) in Net Interest Income and we consider the changes in the average levels of interest rates on assets and on liabilities, we understand that the change in NII is not (!) affected by the level of non-sensitive (!) assets and non-sensitive liabilities because, if they are not sensitive, by definition, they are not impacted by changes in the level of interest rates in the period of time we are observing. So, at the end, the change in NII is the difference between 2 products: change in average * … vedi ultima equazione. interest rate on sensitive assets. Now, we can even simplify our approach by assuming that the change in the average interest rate on assets is perfectly equivalent to

the change in interest rate on liabilities: therefore, Δr Δr Δr. rather than writing "and", we can simply write A L. But, at this point, the change in Net Interest Income will be the product of the change in the Δr, level of market interest rates, * the difference between sensitive assets and sensitive liabilities. Where this difference is exactly the gap, G (!). So, … 315

Now, let's consider the following table, in which the gap and the changes in the level of market interest rates can be positive or negative. So, in other words, we can imagine that interest rates increase or decrease. Homogeneously for Δr Δr … both assets and liabilities because we are assuming that equals A L. The impact of higher rates on the Net Interest Income, if the gap is positive, this means that sensitive assets exceed sensitive liabilities. 316 So, there's a higher amount of assets maturing in the period than the amount of liabilities: therefore, there are

more assets to be repriced with the new level of interest rates (which is higher) than liabilities to be repriced.

So, if we assume an increase in the level of interest rates and a positive gap, the impact on the Net Interest Income is positive: the NII will increase.

But, as we have seen in the 2 examples, in case interest rates increase and the gap now is negative, then, the impact on the NII would be negative.

And so on for the other 2 combinations.

What should a bank do to be perfectly immunized against interest rate changes, that is to say a bank that wants to protect the level of the Net Interest Income?

Now, the bank should constantly maintain a gap equal to 0 because, if the change in Net Δr * gap, Interest Income is if the gap is 0, then the change in NII will be 0, regardless of market interest rate increases or decreases.

But these conditions should be maintained for all (!) possible gapping periods, that is to say continuously.

Is this realistic? No: indeed, we have to remember

One of the reasons why banks exist is that they perform the maturity transformation: depositors want short maturities, while borrowers (firms) borrowing from banks try to secure funding for a long period of time and prefer long maturities. So, a certain degree of mismatching is unavoidable for a bank. And, if the yield curve or the term structure of interest rates is positively sloped, the maturity mismatching is a source of profit for a bank because the liabilities have rather short maturities and assets have, on average, longer maturities. Therefore, the interest rate paid on liabilities is lower than the interest rate earned on assets... 317

So, let's see some guidelines that may be inferred from this framework... Some indicators in interest rate risk management can be derived from the gap concept... 318 [ultimo ratio]... And, if sensitive assets are greater than sensitive liabilities, the gap ratio will be greater than 1; and vice versa. Let's

punto 1

“market Thisrates”… is the formula we worked out: change in net interest income equals change in interest rate * the gap.

Let's consider the first 2 examples we made at the beginning of the lecture: we had the CD maturing in 6 months whose fixed interest rate is 2% and the loan maturing in 12 months whose interest rate is 6%.

What would be the impact of an increase in the level of interest rates of 100 basis points? No impact on the NII, but the impact would be felt on interest expenses. 10€ = 20€.st

What about the NII in the 1 semester? The NII is 30€ - 320nd

What about the NII in the 2 semester? The interest rate on the loan is constant at 6%, therefore, interest income is 30€, but the interest expenses increase from 10€ to 15€: the NII is 15€.worth 35€.Quindi, il NII su

base annua (!) sarebbe pari aA quanto ammonta la diminuzione rispetto all’expected NII valutato all’inizio dell’anno in40€, per effetto dell’aumento di 1%ipotesi di tassi costanti? Il NII sarebbe stato ma, dei tassil’impatto sul NIIdi interesse di mercato, è di -5€…Δr * ΔrVediamo se funziona applicando la formula gap: dato che il è pari a 1% e il gap,–siccome è dato da sensitive assets (0) sensitive liabilities (1.000, il principal del CD), abbiamoΔNIIun pari a -10…Therefore, it seems that the model does not work because the actual decrease in the NII is -5,while applying the formula that we worked out the result would be -10. 321How can we explain this discrepancy?è l’estrapolazione su base annua dell’impatto dell’aumento dei tassi di interessePerché -10€nell’esempio: cioè, mi chiedo: se i tassi dovessero aumentare istantaneamente dell’1%,

rate-sensitive liabilities, 60 = 140. The cumulative gap from 0 to 3 months is 230 - 260 = -30. – Then, the repricing gap making reference to a gapping period of 6 months would be 350 34010€. = – The repricing gap for a gapping period of 12 months (230 + 350 + 430 + 70 =) 500 (260 + 340=) 500 = 0: therefore, if we make reference to a gapping period of 12 months, it seems that the bank is perfectly immunized against changes in market interest rates… But there are 3 observations that deserve our attention:

1. firstly, it is true that the bank is immunized making reference to a 12-months gapping period, but what happens afterwards (!) could be that the gap for a gapping period longer than 12 months is not 0;
2. secondly, it is true that, if we look at the entire period of 12 months, the bank appears to be perfectly immunized, but it is also true that the gap corresponding to certain sub-periods (!) are not all identically equal to 0.

This means that, when we consider

The exposure of the bank to the interest rate risk on the banking book and we look at a certain period of time in the future, not only it is important to specify what could be the change in market interest rates, but it is also important to specify when (!) this increase will take place because the change in market interest rates will produce effects only from that point (!) (3 or 12 months, for instance) onwards…

Another point that has to be underlined is that, if we consider the initial and the final levels of interest rates, they could be identical, but, notwithstanding with the fact that the starting and the finishing points are equals, there could be changes in the sub-periods(!). So, the situation could be: the following blue line are the interest rates: the initial and the ending levels are the same, but, during this period, market interest rates may have fluctuated……

Therefore, if we have a CD maturing at time T, the interest rate at which we will have 1 to

remunerate a newly issued CD is higher than 5%. So, we have to consider that changes in interest rates impact the NII only from the moment when they materialize and, then, we have to take into account what happens within the several sub-periods that are comprised in a given gapping period. 324 So, let's imagine that today the gapping period is 12 months and that we have interbank deposits with 1 month maturity: they are fixed rate for 1 month. The new rate would become effective only after 1 month, that is to say when we renew the interbank deposit. Even if the increase in market interest rates occurs 7 days from today, that increase will produce an impact on the NII only starting from the maturity of these instruments we are considering. Therefore, the increase is 7 days after today, but th