More on Gap Models

The last post briefly discussed the two models for the Earnings at Risk (EAR) measure of interest rate risk - the gap model and the income simulation model. We then went on to discuss the three types of accounts - fixed, variable and NIS. And then we discussed how each type of account gets modelled in the gap model. (We will cover the income simulation model later.) We concluded that there are many complications to consider, and that you would have a very poor measure of interest rate risk if these complications are not taken into account.

Let's start with a discussion with a simplistic look at interest rate risk. There was a time (early 1980's?) when interest rate risk was not measured. It was also a time when financial institutions made extra margin by extending loans along the yield curve - basically they would borrow short (say via a savings account) and then invest long (say via 5-year mortgages.) This was very profitable at first because long rates were much higher than short savings account rates, and a much of that extra spread made its way to the bottom line. Then we hit the eighties and interest rates soared. The rate on savings accounts also soared -sometimes to rates that were higher than those booked 5-year mortgages. Now that very profitable spread was actually negative, and the financial institutions lost money - some even went bankrupt.

So from that experience financial institutions recognized and started to measure interest rate risk. It became conventional wisdom that you shouldn't finance long assets (like mortgages) with short liabilities (like savings accounts). It was much more sensible to match fixed mortgages with fixed deposits.

And here is where it starts to get complicated. One reason for this complication is that interest rate risk models give better results when cashflows are used instead of principal values. When entering data into the interest rate model, cashflows provide a much more accurate interest rate risk measure than principal values. What does that mean? Here's an example:

Let's say a credit union has $5 million in 60 month mortgages and $5 million in 60-month deposits. In our little interest rate risk story, this position would be considered matched and no credit union would lose money on that position simply because interest rates changed. Given our gap model so far, that would have zero interest rate risk. All of which is wrong, wrong, wrong. The error is that this example uses principal values, not cashflows. The 60-month mortgage pool really has a cashflow every month - the members make a payment each month and each payment consists of some interest and some principal. Each principal and interest payment is a cashflow. Plus, mortgages allow prepayments and any pool of mortgages will have some prepayments, and each prepayment is also a cashflow. That means that cashflows will be spread from 1 month to 60 months, unlike the principal amount – where there is just one term – all at 60 months. Depending on the average amortization period of the mortgage pool and the rate of prepayments, the principal amount left in the 60th month might only be $2.5 million (one-half the total principal) or even less.

And what about the 60-month deposits? What if the 5 year fixed deposit pool accrued interest and paid interest at maturity. If that was the case there would only be 1 cashflow on the maturity date and that cashflow amount would be larger than the $5 million principal amount because of the accrued interest paid on maturity.

So the position does have interest rate risk. All those monthly mortgage payments have no matching liability. And the maturing deposit is likley only half covered. And because asset cashflows tend to happen sooner than liability cashflows, this interest rate risk exposure is to falling rates. In other words, the credit union with a 5-year mortgage portfolio and a matching 5-year deposit portfolio is considerable more exposed to falling rates than you might think.

Also recall that EAR measures (both gap and income simulation) completely ignore maturities beyond 1 year. Once again the mortgage portfolio will have cashflows in the first year - the payments and prepayments from 1 month through to 12 months. But the deposit portfolio will have no payments less than 1 year - an exposure to falling rates. And, incidentally, EVR is even more exposed to falling rates because the long items are badly mismatched in the 60-month term.

In fact, because assets (loans) tend to pay more frequently than liabilities (deposits), using principal values generally tends to underestimate falling rate exposures. (You could also express this as overestimating rising rate exposures, but this is really just the flipside of the first statement - an underestimate of a falling rate exposure also means an overestimate of rising rate exposures and vice-versa. To save the confusion, we will just discuss the effect on falling rate exposures, but remember, there is also an opposite effect on rising rate exposures.)

We also need to adjust slightly our earlier statements on how to calculate the interest rate effects of the various types of accounts. For fixed accounts (specifically, for a five-month maturity) we said, "...take the gap between all the 5 month assets and all the 5 month liabilities and multiply that difference by 1% further multiplied by 7/12." This needs to be adjusted to - take the gap between all the 5 month asset cashflows and all the 5 month liability cashflows and multiply that difference by 1% further multiplied by 7/12. Principal values are only an estimate of the cashflow amount. Sometimes the principal amount is a good estimate (they are pretty good for variable accounts), but sometimes it is terrible estimate as shown in our example above.

One more point - remember the horribly mismatched portfolio that caused financial institutions to go bankrupt – the 5-year mortgages funded by savings accounts? Well today, that might actually be a better match than the 5-year deposit. Say the savings account pays a very small rate (0.10%), that it has paid that rate for a decade and is expected to pay that rate for the foreseeable future. Obviously this is not a variable account. In fact, it is what is called a 'Core Deposit'. Core deposits are a big topic worthy of a complete post all by itself (the next one), but for now here is how these accounts are modelled. They are treated like fixed deposits and spread from one month out to your credit union's maximum maturity. If you look at the first year (where EAR is affected), the mortgage cashflows offset by the savings account cashflows is at least a partial match - certainly there is a better match than the fixed deposits which have no cashflows in the first 12 months.

There are two lessons here. One, cashflows are what should be modelled, not principal values. And two, how an account reacts to changes to interest rates is key. The savings account in this example does not change much with interest rate changes, and so it would not have caused nearly the problems that savings account did in the 1980's. Ironically, the mortgages funded by such savings accounts have a much lower EAR than the case where mortgages are funded by similar term deposits.

Core deposits are another complication. Since they are covered more extensively in the next post, I'll just say this: Core deposits are really important to model correctly, especially for EVR. The effect of treating core deposits as variable deposits tends to underestimate falling rate exposures - both EAR and EVR. Its a big mistake, and it is a common mistake.

The next complication is interest rate floors. As a minimum, your interest rate risk model should not allow negative interest rates. That means you should not let rates fall lower than 0.00%. So, if your savings account has a rate of 0.25% and your rate shock is 1%, then this account’s rate would move to 0%, not negative 0.75%. As this mainly effects deposit accounts (because they tend to have lower rates), an interest rate risk model allowing negative rates will tend to underestimate falling rate exposures.

More recently, we have seen where interest rate floors can also affect assets accounts. The current level of prime is at a minimum - it cannot go lower. So asset accounts with a rate based on prime (i.e. variable accounts) cannot drop further. That means there are currently no falling rate exposures for variable loan accounts. Rates can't drop any more - in effect they are 'floored out'. That is an adjustment to your model - now when the rate shock is applied to your variable assets, the rate is unchanged. This can have a huge effect on interest rate risk. Not adjusting for floors on assets tends to result in an overestimate of falling rate exposures. (This is the first complication that overestimates the falling rate exposures.) And again, it is a common mistake to not apply these floors.

And before we leave floors, some credit unions stopped dropping their prime rates before prime bottomed out at the Bank of Canada - say at 3.50% prime. Those credit unions have higher floor rates on their variable assets, which also needs to be modelled. For a credit union that ceased dropping primes, the model should not allow rates to go any lower. And, to be even more complicated, the model must now adjust for what the credit union will do when interest rates start rising again. Will a variable loan account that has had its rates frozen for the last one percent of Bank of Canada bank rate drops, start raising their rates when the Bank starts raising them, or will they leave their variable loan rates constant until their prime is the same as the major banks, or will they do something in between? Whatever the answer, it is important that your interest rate risk model incorporates that answer, or the results could be substantially incorrect. There is a recommended further discussion in my May and June 2008 posts on Low Rate EAR.

Another complication has to do with accuracy. There are a number of unavoidable approximations made when estimating interest rate risk. Core deposit modelling is an excellent example of unavoidable approximations. So, because there are already inaccuracies in the model, it is important that it interest rate risk model reduces estimation error wherever possible.

Remember this? "...take the gap between all the 5 month assets and all the 5 month liabilities and multiply that difference by 1% further multiplied by 7/12." The truth is that there will be many maturities during the month of July, but relatively few will fall exactly on July 31, which the 7/12 implicitly assumes. A more accurate number would be 6.5/12, which assumes that items maturing in July are spread evenly over the month. Even better, the 7/12 should be replaced by the number of days between Jan 1 and July 16th divided by the number of days in the year (yes, incorporating leap years). For 2009 the 7/12 should be replaced with 0.539726027397 (compared to the .5833333 that 7/12 equates to) taken to as many decimals as possible. Further, it is even better if the exact maturity date is used (rather than July 31, or any one single day in the month) for each loan and deposits and gaps are calculated for every day of the year instead of only monthly.

Yes, these are only small errors, but the whole process has many assumptions and estimations. In my opinion it is best to avoid estimation errors where possible in order to get the very best calculation possible. Modern computers make most avoidable approximation errors inexcusable. Many models simply use the 7/12. In my opinion, that is inexcusable. Equating variable accounts to fixed accounts with a 0 to 3 month maturity - also inexcusably inaccurate.

Many models accept aggregate data as opposed to transaction level data. Do you input 60 entries for your mortgages or do you enter thousands representing every single mortgage outstanding at the credit union? Clearly, its more accurate to use every single transaction if possible. Any aggregation will likely reduce accuracy.

The nature of accounts is another complication. All accounts must be classified as NIS, variable rate, or fixed rate, but the truth is that sometimes it is hard to tell. Which account type is a standard on-demand savings account - NIS, variable or fixed? The truth is that such accounts are sometimes a bit of all three account types and sometimes they are just one, and sometimes a combination of two types – there are an infinite number of possibilities. For instance, if the rate never changes, and the rate is never going to change no matter what happens, then it is 100% NIS. If the rate is tied directly to prime and moves in lock step with the prime rate then it is variable. And if it is determined that it is a core deposit, then it is really fixed. And, more typically, it is a little bit fixed, and a little bit variable, and a little bit NIS. For instance, say the savings account rate moves slightly with prime – say when prime moves 1.00% it’s rate moves 0.10% - then it is 10% variable. And if it is also determined that the savings account is 85% core, then it is 85% fixed and it must also be 5% NIS. The important thing is that each account needs to be properly modelled, because small differences can have a big effect. This takes effort and work, but in my opinion it is really well worth it.

One final complication needs to be discussed and that is interest rate sensitivity. The interest rate on some variable accounts is only partially based on prime. Like in the example above, where the account is determined to be 10% variable. The best way to model such an account is to give it an interest rate sensitivity of 10%. But that account is also 85% core. That means that 85% of the balance is going to be removed and spread out over various fixed terms. The remaining 15% of the savings account is 67% variable (10% divided by 15%) and that is how the interest rate sensitivity of that account should be modelled - 67% interest sensitive. Another example is the case where the credit union stopped lowering its prime at 3.50%. Going forward, they have decided that they will not keep their prime steady when the Bank of Canada starts to raise rates again – rather they will increase their prime at 50% of the Bank of Canada increase until their prime is equal again to the major banks. How to model that? Answer, give the account a 50% interest rate sensitivity.

Also note that it is quite possible that the interest rate sensitivity is different for downward movements in prime than it is for upward moves, and should be modelled accordingly. That means that at least two interest rate sensitivities should be tracked for each variable account.

Some complications are just not worth modelling. You always have to weigh the benefit of obtaining the extra information against the cost of getting it. Sometimes it’s just not worth the time and effort - core deposits are an excellent example of this. There are outfits in the states that will model your core deposits and give you very precise ways that they should be treated in your interest rate risk model, but their models require years of information, which can be hard to come by, and their analyses are not inexpensive. Even if you gather your data and pay for such an analysis, you are still left with the fact that the analysis is based on historical information, which might not be relevant going forward. I’m of the view that a straight forward, rational, consistent approach can give comparable results for much less time and expense.

In summary

• Accuracy is important and should be a key objective of the model. There also must be a reasonable balance between the cost of obtaining more accuracy and the benefit of having it.

• A gap model that does not account for the many complications of interest rate modelling is likely worthless.

• Not modelling many of complications tends to result in an underestimate of falling rate exposures. This is surely one of the main reasons that credit unions tend to have an exposure to falling rates.

• A gap model that incorporates these many complications will give results that are not radically different from for the income simulation model.