Information and Accuracy in Interest-Rate-Risk Simulation

Mike Carhill 

Office of the Comptroller of the Currency

Economic & Policy Analysis

Working Paper 94 - 7

October 1994

The opinions expressed in this paper are those of the author and do not necessarily reflect those of

the Office of the Comptroller of the Currency or the Department of the Treasury. Any whole or

partial reproduction of material in this paper should include the following citation: Carhill,

Mike, “Information and Accuracy in Interest-Rate-Risk Simulation,” Office of the Comptroller

of the Currency, E&PA Working Paper 94-7, October 1994.

 

_______________________________________________________________________________

The views expressed herein are those of the author and do not necessarily reflect the views of the Office of the

Comptroller of the Currency. Christina Benson, Jeff Brown, Bob DeYoung, Ed Irmler, Kevin Jacques, Elizabeth

Mays, Jim O' Brien, and Mark Winer provided useful commentary and advice.

Questions regarding subscriptions to the OCC Economic and Policy Analysis Working Paper Series should be

addressed to Communications Division, Comptroller of the Currency, Washington, DC 20219.
 The OTS Final Rule on interest-rate risk is published as 12 CFR Parts 563, 567, 1

and 571 (Docket No. 93-100). The Banking Agencies' Notice of Proposed Rulemaking is

published as 12 CFR Part 3 (Docket No. 93-11). The Banking Agencies include the Federal

Reserve System, the Federal Deposit Insurance Corporation, and the Office of the Comptroller

of the Currency.

Any reproduction of material in this paper should include the citation that appears on the title page. 1

Introduction

Advances in computing technology are making interest-rate-risk simulations affordable for an

increasing number of intermediaries. Perhaps the major reason why simulations are superior to

"duration" and "repricing gap" approaches is that simulations can incorporate a much wider range

of information about firms' options exposures. This extra power requires the modeller to make

judgments about the costs and benefits of incorporating extra information into the model. To

explore those costs and benefits, we quantified the errors that result from the information

economies employed in the interest-rate-risk model proposed by the Bank Regulatory Agencies

(BA). We compared that model's results to results from more information-intensive models,

particularly the model used by the Office of Thrift Supervision (OTS). 

1

The OTS model requires some or all of the following information for about two dozen different

types of financial instruments (excluding off-balance-sheet positions):

(1) Weighted-average remaining term to maturity or repricing.

(2) Weighted-average coupons (WACs) for portfolios.

(3) Intra-portfolio coupon dispersions.

(4) Institution-specific coupon and balance history.

(5) Periodic and lifetime caps on adjustable-rate commercial and residential mortgage loans.

(6) Margins on ARMs.

(7) Indices for ARMs.

(8) Amortization class.

(9) The breakdown of loans of given types between adjustable- and fixed-rate loans.

(10) Original maturity.

In contrast, the only information input to the BA model is remaining maturity, which is collected

for only 13 separate classes of instruments.

To assess the value of the various information inputs, we calculate how each input changes the

instrument's estimated sensitivity to a 200 basis-point, parallel, upward shift in the yield curve

(interest-rate "shock"). For example, a typical adjustable-rate residential mortgage (ARM) with

no lifetime caps might decline in value by 100 basis points (one percent of its current or par value)

when interest rates increase by 200 basis points; an otherwise identical ARM with lifetime caps

might decline by 400 basis points in that interest-rate scenario. Therefore, the latter ARM's

lifetime caps cause a "model difference" in estimated sensitivity of 300 basis points.

There are two different types of model differences. The first type, exemplified by the maturityaggregation errors that we analyze in Chapter One, implies differences in the estimated "bottomline" sensitivity of portfolio equity. In our first two chapters, we describe the two main sources
Any reproduction of material in this paper should include the citation that appears on the title page. 2

of these bottom-line differences between the OTS and BA models: the superior maturity data

collected by the BA, and the superior coupon data collected by the OTS. 

The second type of model difference applies to specific assets or liabilities, such as the ARM with

lifetime caps discussed above. Depending on a particular institution's characteristics, such

instrument-specific differences may either offset or compound other differences when calculating

bottom-line sensitivity. Hence, there is not necessarily any systematic relation between

instrument-specific differences and bottom-line differences. However, in all cases instrumentspecific differences can create incentives for banks to structure their portfolios in ways that reduce

capital charges but do not necessarily reduce the "true" interest-rate exposures.

Those incentives become artificial incentives when model differences are in fact model "errors."

Where we discover significant model differences, we use published research and our own

modelling to discriminate between the two models' results. In most cases, we are able to establish

that one or both models is in error. When such discrimination is possible, we refer to model

differences as "model error."

With a few minor exceptions, we find that model errors are due to inadequate information rather

than the poor use of available information. There are several potential sources of instrumentspecific errors for every kind of financial instrument; we analyze only the most widely held

financial instruments. With regard to on-balance-sheet instruments, the two most important

information deficiencies in the BA model appear to be its lack of coupon data on fixed-rate

residential mortgage instruments (FRM), and the simplistic treatment of lifetime and periodic caps

for adjustable-rate residential loans. Our third and fourth chapters explain why either of those

information defects can lead to errors of several hundred basis points in estimated sensitivity.

The fifth chapter describes several miscellaneous information items. These miscellaneous items

produce model differences ranging from 0 to 100 basis points, and in many cases the model

differences appear to be errors in the BA model. However, the available data indicate that the

miscellaneous items would affect the on-balance-sheet estimates for relatively few banks. 

The final chapter discusses some of the omitted issues, the most important of which are core

deposits and off-balance-sheet instruments. We discovered that there is no market consensus

about core deposits, and so no given treatment is clearly in error. Off-balance-sheet and mortgage

derivatives are generally so idiosyncratic that small changes in the information inputs result in

large changes in the estimated interest-rate sensitivities, but we have been unable to independently

verify the existence of a consensus concerning the interest-rate sensitivity of any given instrument.

Those familiar with interest-rate-risk modelling might be surprised that, with the exception of core

deposits and derivatives, we have generally been able to characterize the errors in the models

with some precision. However, that precision is possible only after granting certain common

assumptions. Among other commonalities, both models use the Bloomberg model of the market

consensus for expected prepayments for residential mortgages. The models also share several

crucial assumptions: (1) that fixed-rate instruments other than residential mortgages carry

prepayment penalties that remove the relation between interest rates and prepayment incentives;

(2) that the covariance between interest-rate movements and changes in credit quality is zero; (3)

that rate changes take the form of instantaneous and parallel 200 basis-point shifts in the yield

curve; (4) that balance sheets are not altered in response to that rate shock, and (5) (where
Any reproduction of material in this paper should include the citation that appears on the title page. 3

relevant) the volatility of interest rates is constant. The apparent precision with which we quantify

errors results from those common assumptions. In the real world, violations of those assumptions

can be expected to produce divergences between model predictions and reality. That observation

implies that interest-rate-risk modelling is an inexact science, despite the apparent precision of the

model outputs.
Any reproduction of material in this paper should include the citation that appears on the title page. 4

Chapter One: The Two Models' Maturity-Aggregation Errors

Even if it were possible to perfectly model the interest-rate risk for a particular instrument, it is

generally too costly to model each of an institution's instruments separately. Instead, most

interest-rate-risk models aggregate--and treat identically--instruments that have similar but not

identical characteristics. That aggregation causes estimation error.

In this chapter, we analyze the error that results from aggregating the maturities of the instruments

of a given loan type, and calculate the errors that result from either the OTS' or the banking

agencies' approach to aggregating maturities. We employ both the available data and the Law of

Large Numbers to ensure that our estimated aggregation errors are representative for "real-world"

portfolios.

We evaluate the maturity-aggregation errors only for assets. In practice, maturity-aggregation

errors for liabilities will be negligible. About 50 percent of the median bank's liabilities are CDs,

and the remaining 50 percent are core deposits. For CDs, the average maturity is about one year,

so CDs have too little interest-rate sensitivity for aggregation errors to be large enough to

substantially affect the estimated interest-rate sensitivity of portfolio equity. For core deposits,

the contract maturity is zero, and the effective maturities are assigned by the modeller. 

Ignoring the liability side will misrepresent the estimation errors for the minority of banks that

have significant amounts of long-term CDs or nondeposit borrowing. For those banks, maturityaggregation errors on the liability side will sometimes ameliorate and in other cases aggravate the

maturity aggregation errors on the asset side.

I. The Two Regulatory Approaches to Maturity Aggregation

The two regulatory approaches to aggregating maturities represent alternative compromises

between model accuracy and reporting burden. The banking agencies employ a "bucket"

approach: For each portfolio of a given type of financial instrument, the individual instruments

are aggregated into several maturity ranges, called buckets (seven buckets in the 1993 Notice of

Proposed Rulemaking (NPR)). The maturity ranges are: 0 to 3 months, 3 to 12 months, 1 to 3

years, 3 to 5 years, 5 to 10 years, 10 to 20 years, and 20 to 30 years. The model assumes that

the maturities of all of the loans in each maturity range equal the midpoint of that maturity range.

The OTS' interest-rate-risk model uses a "weighted-average maturity" (WAM) approach instead

of buckets. For each type of financial instrument, each institution reports only the WAM for the

entire portfolio of that instrument type. For example, the remaining maturities for each thrift's

entire portfolio of Treasury securities are averaged into one WAM. The model then assumes that

the maturity of each individual security equals the WAM of the subportfolio.

Either the bucket or the WAM approach misrepresents the relation between the interest-rate

sensitivities and the maturities of an institution's individual financial instruments. The bucket

misrepresents average maturities because it is unlikely that the WAM for the instruments in any

bucket will equal the midpoint of that bucket. We label the estimation errors that result from that

problem the "maturity-estimation error." Furthermore, even when the maturity-estimation error

is negligible, the bucket approach produces what we label "within-bucket WAM-convexity error":

averaging the maturities of a group of instruments implicitly assumes that interest-rate sensitivities
Any reproduction of material in this paper should include the citation that appears on the title page. 5

are linear in maturities, but (as we will see) interest-rate sensitivities are highly convex in

maturities (interest-rate sensitivity increases at a declining rate as maturity increases). We label

the combined maturity-estimation error and within-bucket WAM-convexity error the "bucketaggregation error." 

The OTS model correctly assigns average maturities, so it has a zero maturity-estimation error.

However, it produces a much higher "WAM-convexity error" than does the bucket approach.

This is because the convexity error operates over the entire maturity range, rather than over the

short subsets of that range used by the bucket approach.

The OTS used a bucket approach from 1984-1992. Since then, the OTS has used the WAM

approach, which has two advantages. First, by using WAMs, the OTS requires thrifts to report

each relevant datum for each type of financial instrument only once. In contrast, when using

buckets, each relevant datum must be reported as many times as there are buckets. Therefore,

for any given length of the reporting form, the WAM approach allows the OTS to collect much

more information about each type of financial instrument.

The second advantage of the WAM approach is that it avoids the "within-bucket maturityestimation error." However, because the WAM approach produces a much higher convexity error

than does the bucket approach, there is a tradeoff that needs to be quantified.

II. The Bucket-Aggregation Errors.

In this section we analyze the within-bucket WAM-convexity errors and the within-bucket

maturity-estimation errors that result from the banking agencies' proposed model (BA model).

The 1984-1992 Thrift Financial Reports provide information on the distribution of instrument

maturities across buckets, but we are unaware of any data on actual within-bucket distributions

of maturities. Therefore, we rely on Monte Carlo simulations and what we believe to be

reasonable assumptions to characterize the errors statistically. We build a reality that, to the

extent that our assumptions characterize real-world portfolios, shows the extent to which the Law

of Large Numbers can be relied upon to eliminate the bucket-aggregation errors. However, even

if our assumptions correspond to reality, our approach has two major limitations.

The first limitation is that we must omit managerial responses to the (stochastically determined)

interest-rate exposures. If managers have perfect knowledge of the outcome of that stochastic

process, they can change their exposures in either of two directions. First, risk-averse managers

that appear "match funded" using the bucket aggregations, but actually have significant

(stochastically determined) exposures, might take steps to reduce their exposures even though the

proposed BA model creates no regulatory incentives to do so. Conversely, risk-seeking managers

that have negligible (stochastically determined) exposures might seek to increase their risk when

they realize that such risk will elude detection by the regulatory model.

 

The second limitation is that the conclusions we reach, while otherwise robust to the assumptions

about the within-bucket distributions of maturities, are not valid for degenerate or badly skewed

distributions. To the extent that institutional factors produce such distributions, the errors are

much larger than we estimate here. Since the data offer little insight into the prevalence of "bad"

distributions, an important uncertainty remains that should be considered when evaluating the

tradeoff between information costs (reporting burden) and model accuracy.
 While no single distribution would characterize all portfolios, a uniform distribution 2

may be the most realistic globally as well as within buckets. An established institution that

pursues a stable buy-and-hold strategy will come to hold instruments spread evenly across the

maturity spectrum. As "old" instruments mature, they are replaced with new instruments at

the opposite end of the maturity spectrum. This implies that the average maturity will tend to

fall at the midpoint of the portfolio's maturity spectrum. 

The tendency toward uniform distributions is greatly weakened by a number of real-world

factors. For example, many instruments have amortizing payment structures, so that even for

uniform maturity distributions the weighted-average maturity exceeds the unweighted-average

maturity and hence the midpoint of the maturity spectrums. Obviously, there are many such

factors that influence the global maturity distributions.

Any reproduction of material in this paper should include the citation that appears on the title page. 6

Feid (1993) analyzes the maximum bucket-aggregation error that can result from the proposed BA

model. He demonstrates the theoretical possibility of "bottom-line" errors of several hundred

basis points in the estimated sensitivity of portfolio equity, even for relatively short-term

portfolios. In essence, we demonstrate below that such very large errors result only from

deliberate managerial gaming or unusual portfolios, and are highly unlikely to arise from

stochastic processes (see the Appendix to Chapter One for a brief discussion of Feid's techniques).

II.A. A Model of the Within-Bucket Maturity Distributions

We assume that the maturities of each institutions' instruments are uniformly but randomly

distributed within each bucket. There are many reasons why real-life portfolios might violate the

uniformity assumption, but whatever the true distribution of the maturities of a given type of

instrument for a given institution, by the Law of Large Numbers the observed distribution of

maturities within each bucket will tend toward uniformity as the number of buckets increases (as

an interval shrinks, the difference between the maxima and minima over that interval also

shrinks). Since the sum of uniform distributions tends to a normal distribution, this tendency 2

implies that the stochastically determined bucket-by-bucket WAMs are normally distributed.

Hence, our assumption that the instruments' maturities are uniformly distributed implies that the

WAMs within each bucket tend to a normal distribution.

For simplicity, we assume in this section that all instruments have nonamortizing payment

structures. That assumption overstates the errors for amortizing instruments and understates the

errors for zero coupon instruments. We assume that the instruments are distributed evenly across

maturities. 

We test both the seven-bucket model proposed for the BA model and the eight-bucket model used

by the OTS from 1989-1992 (the OTS gains that extra bucket by dividing the 3 to 12 month

bucket into a 3 to 6 month bucket and a 6 to 12 month bucket.) The latter produced only slightly

smaller errors. The numbers we report are for the eight-bucket model.

For each bucket, we employ the SAS random number generator to draw a separate random

number for the maturity of each instrument. The random number is drawn from the uniform

distribution U[(0,1),0.289] (0.289 represents the standard deviation for a uniform distribution on
 That standard deviation is implied by the Central Limit Theorem applied to results 3

from successive draws, with replacement, from the uniform distribution. To ensure that the

product of the SAS random number generator corresponds to theory, we inspected several

runs from the random number generator and found that the actual standard deviations varied

over a range of roughly 0.27/n to 0.3/n . For our purposes, relatively few draws (400) 1⁄2 1⁄2

produce sufficient accuracy, but in some modelling applications a much larger number of

draws is needed. 

Any reproduction of material in this paper should include the citation that appears on the title page. 7

that interval). The resulting random number P is then scaled up to occupy the interval

corresponding to each bucket's maturity. The scaling employs the formula ("+ $P), where "

represents the lower bound of the bucket in days and $ represents the bucket width in days

(365.25 days per year). The daily yields are adjusted to correspond to bond-equivalent yields

(BEYs) of 8 percent in the base case and 10 percent after the shock.

The resulting WAMs within each bucket are distributed N(["+ 0.5$],[{.289$}/n ]), where n 1⁄2

represents the number of financial instruments in each bucket. We ran each of our simulations 3

both with n= 10, to proxy a bank with a small number of instruments, and with n= 11,351 to

proxy a bank with a large number of instruments.

Varying the number of instruments in this way produces a rough proxy for differing bank sizes.

Suppose that the average instrument has a par value of $100,000. In an eight-bucket model, 10

instruments per bucket implies total financial assets of about $8 million, roughly representative

of the smallest existing banks. At the other extreme, 11,351 instruments per bucket implies a

bank with total financial assets of about $10 billion.

The number of instruments per bucket has different implications for the maturity-estimation error

than for the within-bucket WAM-convexity error. The maturity-estimation error has a stochastic

component, depending on the randomly determined within-bucket maturity dispersions. For

n= 11,351 the standard deviation of the WAMs is only .00271*$, and the expected value of hte

WAMs equals the bucket midpoint, so that within-bucket maturity-estimation error is negligible.

For n= 10, the standard deviation is 0.0914*$, large enough that the maturity-estimation error

remains important. However, the expected value of the WAM-convexity error is independent of

firm size, and so remains positive for the "large" bank. 

The maturity-estimation error could be captured by first aggregating and then valuing the loans

in each bucket, but capturing the within-bucket WAM-convexity error requires each loan to be

valued before aggregating the loans in each bucket. We assume each instrument is at par at a

level 8 percent BEY yield curve. We then calculate the value of each instrument after rates

increase to 10 percent BEY, summing the values to find the total value of the loans in each

bucket. 

To conserve computer resources, our simulations are run over 400 banks. Experimentation

showed that this sample size is large enough to ensure rough accuracy in our estimate of the

average maturity-aggregation error. However, with a small number of simulations, the

probability of producing outliers is reduced, so 400 simulations tends to understate the estimated

maximum possible errors. We assume that the maturities for each bank's instruments are evenly

distributed from zero through the portfolio's maximum maturity. We assume that banks choose
 We spread the maturities evenly across time by weighting the value of the 4

instruments in each bucket as follows: 1-3 months and 3-6 months, 1/120 each; 6-12 months,

1/60; 1-3 years and 3-5 years, 1/15 each; 5-10 years, 1/6; 10-20 years and 20-30 years, 1/3

each. The assumption that the instrument maturities are evenly distributed across time rather

than across the time buckets weights the long buckets more heavily, thus tending to overstate

the interest-rate sensitivity of the portfolios. This probably biases against our conclusion that

the bucket methodology produces small maturity-aggregation errors.

Any reproduction of material in this paper should include the citation that appears on the title page. 8

five different maximum maturities, corresponding to the five longest buckets in the BA model:

3 years, 5 years, 10 years, 20 years, and 30 years. 4

II.b. Error Estimates

We express the total maturity-aggregation error as 1-(true value/BA value). Positive errors mean

that the BA model has overstated sensitivity (understated the post-shock value of the instruments).

The expected error is positive, because the within-bucket WAM-convexity error is always positive

(i.e., the WAM convexity error represents an overstatement of the portfolios' true interest-rate

sensitivity), and the expected maturity-estimation error equals zero.

As we increase the maximum maturity, the potential error increases because the instruments that

are added to the portfolio become increasingly rate sensitive. For the "large" bank with 11,351

instruments per bucket, the errors are:

Distribution of large banks' estimated maturity-aggregation errors, as function of portfolio

maximum maturity:

Maximum

maturity

(years):

3 5 10 20 30

Smallest

error (basis

points):

0.2 0.7 4.3 11.6 10.4

Average

error:

2.0 2.1 6.4 13.9 12.2

Largest 

error:

3.7 3.5 8.4 16.0 14.0

Even in the worst case, with maximum maturities at 20 years, the bucket-aggregation error does

not exceed 16 basis points, and the mean error is only 14 basis points. The very tight range of

errors results because the large number of instruments in each bucket drives the stochastically

determined maturity-estimation error almost to zero. Evidently, provided that all large banks'

portfolios have maturity distributions that "cover" the buckets, the eight-bucket model produces

very small errors. However, the mean error remains positive even when the maturity-estimation

error goes to zero, because of the within-bucket WAM-convexity error.

For "small" banks, the same conclusion holds for the average error, but there are significant

outliers. For example, if the maximum maturity is 10 years, almost 25 percent of our
Any reproduction of material in this paper should include the citation that appears on the title page. 9

hypothetical small banks will have their portfolio sensitivity overstated by more than 25 basis

points, while over 10 percent of the banks will have their sensitivity underestimated by more than

25 basis points:

Distribution of small banks' estimated maturity-aggregation errors, as function of

portfolio maximum maturity:

Percentile

of errors

(basis

points)

Maximum maturity (years)/instruments per bank

(with 10 instruments per bucket).

3/40 5/50 10/60 20/70 30/80

Minimum -73.0 -54.7 -63.5 -66.8 -52.9

5th -33.4 -29.0 -39.8 -34.1 -22.3

10th -23.3 -18.1 -28.3 -24.1 -14.9

25th -13.3 -7.5 -10.7 -7.1 -2.3

50th 2.7 2.4 4.5 11.8 11.5

75th 14.6 12.0 23.2 32.1 24.2

90th 26.0 21.5 38.8 47.6 35.8

95th 32.2 27.3 45.7 57.1 44.2

Maximum 53.4 42.3 87.1 94.3 75.8

For most portfolios, the bucket-aggregation errors will lie between those experienced with 10

instruments per bucket and those experienced for 10,000 instruments per bucket. We repeated

the above simulation for a portfolio with 100 instruments per bucket. If each instrument has a par

value of $100,000, such a bank would have total financial assets approaching $80 million, roughly

representative of the median bank. 

Distribution of median banks' estimated maturity-aggregation errors, as function of

portfolio maximum maturity:

Percentile

of errors

(basis

points):

Maximum maturity (years)/instruments per bank

(with 100 instruments per bucket).

3/400 5/500 10/600 20/700 30/800

Minimum -19.0 -13.7 -15.7 -15.6 -8.6

25th -2.5 -1.7 -0.7 8.1 8.1

50th 1.9 2.0 6.2 14.2 12.3

75th 6.1 5.5 11.8 19.5 16.6

Maximum 19.4 17.7 29.5 38.5 31.5

For these median-sized portfolios, the Law of Large Numbers causes sufficient shrinkage of the

maturity-estimation error for the within-bucket WAM-convexity error to dominate, causing a

significant positive bias in the total maturity-aggregation error. Hence, the average error is higher

for the median-sized bank than for very small banks. However, despite the larger average error,

the range of estimation errors is much smaller for the median-sized banks than for the small

banks.
Any reproduction of material in this paper should include the citation that appears on the title page. 10

II.c. The Benefits of Alternative Bucketing Approaches

The results from our simulations suggest that the eight-bucket maturity-aggregation errors are

generally small. However, we have assumed that within-bucket maturities are uniformly

distributed. Our results hold for almost all distributional assumptions (since i.i.d. draws from

covering distributions are expected to sum to a normal by the Central Limit Theorem). However,

there will be cases where the within-bucket maturities tend to a degenerate or badly skewed

distribution. Feid has shown that for those distributions the within-bucket maturity-aggregation

errors can be extremely large.

Because of these and other concerns, many advocate doubling the number of buckets employed

by the BA model. Each concern is at least partially mitigated if more buckets are employed. To

see if using more buckets would reduce the estimation error, we redid the above simulations after

doubling the number of buckets by splitting each of the eight OTS buckets in two. This also

required that we cut the number of loans by one-half, so that there were 5 loans in each bucket

for the smallest banks and 50 loans in each bucket for the median-sized bank. We found that

doubling the number of buckets cut the errors by approximately one-half; and is helpful for small

banks that may suffer a large measurement error under the eight-bucket approach. For example,

the estimation errors for the small banks ranged from -34 basis points to 52 basis points with 16

buckets (the case with maturities ranging from zero to 20 years), compared to -67 to 94 basis

points using 8 buckets.

While 16 buckets are more accurate than 8 buckets, employing WAMs within 8 buckets is more

accurate than using bucket midpoints within 16 buckets. For the 10-instrument-per-bucket case

with 8 buckets, but using WAMs in place of the bucket midpoint, the estimation errors range

between 5 and 22 basis points. 

The advantage of using within-bucket WAMs relative to doubling the number of buckets becomes

even greater when our simulations are run with variable loan sizes. Varying the loan sizes

increases the maturity-estimation errors inherent in the bucket approach, because even if the

within-bucket distribution of maturities is uniform, the loans might vary in size. In our stochastic

simulations, variable loan sizes increase the error that results when the average maturity differs

from the bucket midpoint. For the small bank with 10 loans per bucket, randomly varying the

loan sizes from $1,000 to $300,000 causes the estimation errors to increase by about 50 percent

for those banks for which the bucket approach overstates sensitivity. (Varying loan size actually

slightly reduces the error for those banks for which sensitivity is understated. In that case, the

negative maturity-estimation error partially offsets the positive within-bucket WAM-convexity

error.) However, if within-bucket WAMs are used, variable loan sizes have no effect on the

estimation errors.

A disadvantage of the within-8-bucket WAM approach relative to the 16-bucket approach is that

the former has a larger mean bucket-aggregation error than the latter. This is because the WAM

approach always overstates sensitivity, while the replacement of actual maturities with the bucket

midpoints can produce either positive or negative errors. Because of that bias, employing WAMs

addresses only the maturity-estimation errors, not the WAM-convexity errors. Doubling the

number of buckets addresses both errors.
Any reproduction of material in this paper should include the citation that appears on the title page. 11

On net, however, a positive but known bias with a small range of errors seems preferable to

smaller positive bias with a much larger range of errors. Perhaps more importantly, withinbucket WAMs address the problems of degenerate/skewed distributions and possible management

gaming much more effectively than does doubling the number of buckets. Of course, if the

reporting costs are deemed negligible, within-16-bucket WAMs would be the preferred solution.

III. Estimation Errors Resulting from Replacing "Buckets" with Weighted-Average

Remaining Maturities (WAMs).

The OTS model aggregates all maturities for each type of instrument by requiring thrifts to report

their portfolio "weighted-average maturities" (WAMs). For example, each institution aggregates

all of its fixed-rate 30-year residential together and calculates that portfolio's WAM. In this

section, we show that for many instruments, this procedure results in a large estimation error,

both in theory and in practice. 

WAM aggregation was introduced in the 1993 (CMR) version of the OTS model; the 1984-1992

OTS models had employed the "bucket" approach proposed in the Banking Agencies' NPR. The

OTS made this change because the 1984-1992 models omitted several pieces of information that

the OTS believed essential to measuring the interest-rate risk in residential mortgages. The OTS

replaced the buckets, which required eight line items for every datum, with weighted-average

maturities (WAMs), which require only one line item per datum, in order to collect that additional

information without greatly expanding the reporting form.

III.a. Why the WAM Error Occurs

The basic problem with the WAM approach is convexity. The interest-rate sensitivity of an

instrument or portfolio increases in its maturity, but the rate of increase in that sensitivity declines

as maturities increase. An 8 percent coupon Treasury bond with 7.5 years to maturity will decline

in value by 10.38 percent if market rates go from 8 percent to 10 percent BEYs. If we double the

maturity to 15 years, the decline in value increases by 5 percent, to 15.37 percent. However, if

we increase the maturity from 15 years to 22 years, the decline in value increases by only 2.4

percent, to 17.74 percent. 

The convexity error can be illustrated by considering several $300 million portfolios of financial

instruments, each distributed differently across six five-year buckets, but each having a WAM of

15 years: 
 The CPR when the BEY is at 8 percent is irrelevant for calculating its value, because the 5

mortgage is at par. However, in some prepayment models a high initial prepayment rate may

remain relatively high even after a 200 basis point increase in market rates. Therefore, we

also used a 12 percent CPR for the 8-percent fixed-rate mortgages at 10 percent BEYs, as all

the models we have seen agree that the CPR will be less than 12 percent for such discount

mortgages. If a 12 percent CPR is assumed instead of a 6 percent CPR, the mortgages decline

in value by 6.82 percent instead of 8.39 percent when rates jump from 8 percent to 10 percent

BEY.

Any reproduction of material in this paper should include the citation that appears on the title page. 12

Bucket

(years):

0-5 5-10 10-15 15-20 20-25 25-30 WAM

Years to

maturity:

2.5 7.5 12.5 17.5 22.5 27.5

Portfolio A $50 $50 $50 $50 $50 $50 15

Portfolio B 95 4 11 90 46 54 15

Portfolio C 150 0 0 0 0 150 15

Portfolio

D

0 0 150 150 0 0 15

Portfolio E 0 150 0 0 150 0 15

Portfolio F 59 24 42 76 81 18 15

Portfolio G 0 180 10 0 0 110 15

Portfolio

H

0 40 110 110 40 0 15

For now, suppose that all of the instruments within each bucket have maturities equal to the

midpoint of that bucket, so that no convexity or maturity estimation error is introduced by

grouping subportfolios into only six buckets (as we have seen, that error is likely to be relatively

small unless the bank has very few instruments in each bucket). For rate increases from 8 percent

to 10 percent, the OTS estimates the loss in value for each portfolio at 15.37 percent (for

nonamortizing Treasury Bonds), and at 8.39 percent (for a 1-4-family mortgage at a 6 percent

CPR). However, the actual decline in values for the portfolios are: 5

Portfolio: As T-Bond (%) As FRM (%)

A 13.60 7.50

B 13.02 7.22

C 11.48 6.44

D 15.23 8.32

E 14.08 7.73

F 13.72 7.57

G 13.52 7.43

H 14.93 8.16

OTS estimate: 15.37 8.39
 The linearization needed for a general solution can be achieved with a Taylor Series 6

expansion to the nth order derivatives. However, the solution is computationally intractable,

so we employed a polynomial expansion to characterize the WAM convexity error. The

expansion fits a curve where (a) the dependent variable is the difference between the sensitivity

at each bucket's maturity midpoint and the sensitivity at the portfolio-wide WAM and (b) the

independent variables are the difference between each bucket's maturity and the portfolio

WAM, raised to powers implied by the degree of the polynomial expansion. We found that a

9th-order polynomial, applied to the 8-maturity case implied by the 1990-1992 OTS Section

MR, produced an R of 1 and estimation of the WAM error that was accurate to the fourth 2

decimal place.

Any reproduction of material in this paper should include the citation that appears on the title page. 13

For the Treasury Bond portfolio D, which has maturities that "bunch" relatively tightly around

its WAM, the convexity error from replacing buckets with WAMs is only 14 bp (for FRMs, the

corresponding error is only 7 bp.) However, for A, which is dispersed evenly across the buckets,

the error is 177 bp for the bond and 89 bp for the FRM; for D, which is evenly divided between

the two most extreme buckets (a "barbell portfolio"), the estimation error is 389 bp for the bond

and 195 bp for the FRM. To the extent that real-world portfolios look like either A or D,

aggregating maturities across buckets produces a very large error.

As a percentage of sensitivity, convexity error declines as the WAM increases. Expressing

sensitivities as a percentage of par, as we do in this paper, the WAM convexity is greatest in the

middle maturities; at short WAMs, the low interest-rate sensitivity dominates the high convexity

error; at long WAMs, the low convexity error--combined with the low dispersion needed to attain

a long WAM--dominates the high interest-rate sensitivity. At the extreme, as maturities increase

past 40 years, interest-rate sensitivities (asymptotically) become constant, so that the convexity

becomes irrelevant (e.g., a negligible WAM-convexity error results from using a 50-year WAM

to represent a portfolio that has equal proportions of 40-year and 60-year instruments).

III.b. Characterizing the Data

 

Unfortunately, banks' call reports do not include maturity distributions. However, the Thrift

Financial Reports (TFRs) included data on thrifts' intraportfolio maturity distributions buckets

from 1984 through 1992. We assume that portfolios with given WAMs have the same bucket

dispersions whether held by banks or by thrifts, so that we can use the thrift data to characterize

the WAM-convexity error that would result if the current OTS model were applied to banks. To

calculate the portfolio WAM, we assume that within-bucket WAMs equal the bucket midpoints.

For an arbitrary number of buckets, arbitrary WAMs, or arbitrary interest-rate scenarios,

characterizing the WAM-convexity error requires a sophisticated and tedious mathematical

treatment. However, for the specific case of eight buckets, where we have a parallel shift in a 6

level yield curve from 8 percent to 10 percent BEY, a much more simple and direct solution is

available. We compare the errors from the WAM and bucket approaches by (1) valuing each

bucket separately and then summing the resulting values, and (2) calculating the WAM for each

loan type and then valuing the portfolio as if the WAM represented the actual maturity of each

instrument. 
Any reproduction of material in this paper should include the citation that appears on the title page. 14

We applied these calculations to the TFR data for the first quarter of 1990 and the fourth quarter

of 1992. The conclusion is essentially identical no matter which period is used. However, the

sample is about 50 percent larger for 1990 than for 1992 (2900 v. 1900 thrifts), so we report the

numbers for the earlier period. We calculate the WAM-convexity errors for three different types

of instruments: 1-4 family fixed-rate residential mortgage loans and MBS; Treasury and agency

bonds ("investment securities type 1" on the TFR); and 5+ family and nonresidential real estate

loans. Together, those three types of instruments are about 40 percent of the average bank's total

assets, in roughly equal shares. We assume that the 1-4 family loans had 30-year original

maturities, and amortize with a constant prepayment rate (CPR) that falls to either 6 percent or

12 percent per year after the 200 bp shock to interest rates. We assume that the other real estate

loans and bonds are nonamortizing.

 

The average thrift reported that its fixed-rate 1-4 family residential loan portfolio had a WAM of

16.2 years in the first quarter of 1990 and 14.3 years in the fourth quarter of 1992. For the other

real estate loans, the average maturity was nine years. Few thrifts hold bond portfolios with long

maturities, with the average WAM at only about two years.

We calculate the global WAM-convexity "error" as [1-(OTS value/BA value)]. This term is a

pure error term only under our assumption that the BA value contains no within-bucket maturity

aggregation error, whereas we discovered above that even under the best of conditions that error

ranges from 3 to 13 basis points. Because convexity overstates sensitivity, the OTS value is

always less than or equal to the BA value.

The distributions of WAM convexity errors for the three types of instrument are:
Any reproduction of material in this paper should include the citation that appears on the title page. 15

Estimated errors from replacing eight buckets with WAMs, for thrifts in 1990 quarter four:

Percentile of

errors

(basis points)

Instrument\number of thrifts with positive balances (1990 Quarter one)

Fixed-rate

1-4 family

loans and MBS (2871

observations)

Other real estate

(2724 observations)

Investment 

securities

(2793 observations)

12 % postshock CPR6 % postshock CPR

Minimum 0 0 0 0

10th 17.7 22.7 24.6 0

25th 24.8 31.3 65.7 0.5

50th 35.2 43.0 113.5 7.5

75th 54.8 63.4 173.7 49.0

90th 86.8 93.7 232.7 248.7

Maximum 230.9 236.1 522.9 564.0

This table probably overstates the errors in the current OTS model for FRM, because the 1993

OTS model segregates those instruments between 15-year and 30-year original maturity. With the

1993 reporting form, the average WAM for 30-year original maturity FRM is about 22 years,

compared to the 14.3 years reported when the 15-years and 30-year original maturities were

combined. Conversely, the table understates the errors for investments securities that would result

if the OTS model were applied to commercial banks, because few thrifts but most banks hold

intermediate-term investment securities. The average maturity of banks' investment portfolios is

about twice that of thrifts' portfolios (four years compared to two years). To correct for those

biases, we divide the sample into ranges of WAMs, ranging from 0 to 5 years to 20 to 25 years.

The OTS model error is generally very high for Treasury portfolios but is generally relatively

small for FRM:
Any reproduction of material in this paper should include the citation that appears on the title page. 16

Estimated errors from replacing eight buckets with WAMs, 1990 quarter four:

Range for

WAM in

years

Median convexity error in basis points

(1990 quarter one)

Fixed-rate 1-to-4-

family mortgages

and MBS

(number of

observations)

Other real estate

(number of

observations)

Investment

securities 

(number of

observations)

0-5 45.5

(39)

61.1

(498)

6.2

(2522)

5-10 92.3

(111)

136.4

(1137)

359.6

(163)

10-15 43.2

(925)

108.8

(884)

476.1

(62)

15-20 46.4

(1292)

135.9

(166)

369.5

(28)

20-25 30.9

(504)

56.2

(39)

70.9

(18)

The errors are highest in the intermediate WAMs, both because the convexity error is highest at

intermediate maturities and because portfolio with intermediate WAMs tend to have the most

widely dispersed instrument maturities. The errors are extremely large for investment securities,

due to the frequency of barbell portfolios for thrifts. Barbell distributions may be more rare

among banks; the call reports suggest that banks generally concentrate on intermediate-maturity

Treasuries. 

IV. Conclusion

From the OTS perspective as a thrift regulator, portfolios of fixed-rate amortizing instruments

with more than 20 years to maturity, together with ARMs with less than five years remaining to

repricing, are much larger proportions of balance sheets than are portfolios of nonamortizing

instruments with intermediate WAMs. The information gained in the 1993 reporting change may

be more valuable than the buckets that were sacrificed. However, even for amortizing

instruments, the WAM-convexity error in the OTS model is significant in proportion to the

banking agencies' proposed threshold sensitivity of 1 percent of assets.

For the average bank, long-term Treasuries and commercial real estate loans are more than 20

percent of assets, while 1-4 family fixed-rate mortgage instruments are less than 15 percent of

instruments. For bonds and commercial real estate, the global WAM approach can produce large

measurement errors. For example, if the WAM-convexity is 100 basis points for Treasuries and

commercial real estate loans, and those loans are 20 percent of assets, the errors in estimated networth sensitivity will be about 20 basis points (less any WAM-convexity error relating to fixedrate CDs and nondeposit borrowings plus any WAM-convexity error for the bank's other assets).

And it appears that the WAM-convexity errors may approach 300 basis points for the commercial

real estate loans and Treasuries held by a significant minority of the industry.
Any reproduction of material in this paper should include the citation that appears on the title page. 17

The bucket approach to maturity aggregation, in contrast, produces errors approaching 100 basis

points only very rarely. Furthermore, those rare events can be prevented by using more buckets

or weighted-average maturities within the current seven- or eight-bucket approach. Using withinbuckets WAM would reduce the errors much more effectively than would doubling the number

of buckets. 
Any reproduction of material in this paper should include the citation that appears on the title page. 18

Appendix to Chapter One: Stochastic Processes and Feid's Error Estimates

Feid (1993) uses the seven buckets proposed for the banking agencies' IRR model, constructs

hypothetical portfolios for which the interest-rate sensitivity is known with certainty, and

calculates the maximum error in the estimation of the sensitivity of portfolio net worth that can

result from assuming that the weighted-average maturity within each bucket equals the midpoint

of that bucket. He does this by assuming that all of the assets within each bucket have the

maximum maturity possible within that bucket, while all of the liabilities within each bucket have

the minimum maturity possible for that bucket.

Feid finds that, even where the institution has no assets or liabilities of more than 3 years, the

maximum error from bucket aggregation totals 1.2 percent of assets (given a 5 percent capital-toasset ratio). Where the institution has assets and liabilities spread out evenly across the buckets,

but still has no assets or liabilities of more than 20 years, the maximum error is 2.39 percent of

assets.

Because such large errors call into question the usefulness of the model's results, we replicated

Feid's paper (including the simultaneous valuation of both assets and liabilities), but replaced his

deterministic maturities with stochastic maturities. In particular, we were interested in determining

if random variation of within-bucket maturities could produce the same large errors that Feid

proves can result from management exploitation. For all loan maturities, stochastic errors

approach the size of the deterministic errors only when there are six or fewer financial instruments

per bucket (42 instruments per institution). For the case with 10,000 instruments per bucket,

stochastic processes can generate measurement errors of at most 20 basis points.
Any reproduction of material in this paper should include the citation that appears on the title page. 19

Chapter Two: Portfolio-Wide Coupon Levels

The OTS collects the weighted-average coupon for the portfolio of each type of instrument, while

the banking agencies' (BA) model forgoes coupon information entirely. Coupon data have both

instrument-specific and general importance. In this chapter, we analyze the general importance

of coupon data for bottom-line accuracy, deferring the instrument-specific issues to the next

chapter. To analyze the bottom-line importance, we simultaneously assess the errors on both the

asset and the liability sides of the balance sheet. 

I. Why the BA Approach Can Cause Errors

The quasi-duration approach used by the BA model actually uses coupons and discount rates as

inputs for evaluating every instrument. However, rather than requiring banks to report their

portfolio weighted-average coupons, the proposed model assumes coupons instead. The approach

is to observe the banking system's "typical" net interest margin (NIM), taken to be 375 basis

points. To choose a discount rate, flat term structures are assumed for both sides of the balance

sheet, with all liabilities yielding 4.75 percent and all assets yielding 8.50 percent (except for

FRM, where rates reflect current market rates of 7.5 percent for 20-to-30-year mortgages and 7.0

percent for shorter-maturity mortgages). Because the assumed discount rate equals the assumed

coupons, all instruments are always at par.

If the BA model's explanation for a bank's NIM is inappropriate, or if a bank's instruments are

not at par (coupons are not equal to discount rates), model errors result. While we can not

determine how the errors from those two sources will interact, we can separately quantify the

errors.

II. Errors That Result When the BA Theory of the NIM Is Wrong

When the BA model assumes flat yield curves, with all coupons on assets 375 basis points higher

than all coupons on liabilities, it implies that maturity mismatching does not contribute to the

typical bank's 375 basis point NIM. It is true that banks' practice of using asset yields to

recapture administrative and default costs explains a significant part of the NIM, but for many

institutions maturity mismatching also plays a role. In fact, for the thrift industry the matchfunded NIM appears to be as close to zero as to the 375 basis points assumed in the BA model

(the call reports do not include coupon-by-maturity information, and so are not adequate for

estimating match-funded NIMs). Coupons for most of thrifts' short-term assets are in the 4 to 5

percent range, compared to 8 percent for long-term assets (coupons are substantially higher for

credit-risky assets). Similarly, CD yields increase from 3 percent to 5 percent as CD maturities

increase from 3 months to 36 months, with long-term liabilities yielding in the 7 percent range.

 

We quantified the error that might result from an inappropriate specification of the match-funded

NIM by constructing a hypothetical bank with a 5 percent capital-to-asset ratio. The bank has

$100 in assets and $95 in liabilities in each of the seven time "buckets" used by the BA model.

This implies that the bank's maximum asset maturity is 30 years (we also construct banks with

maximum maturities of both 20 years and 10 years), and that asset maturities equal the liability

maturities. All instruments are initially at par. We assumed that the bank's asset coupons at each

maturity are 250 basis points higher than the liability coupons (i.e., the match-funded NIM is 250
Any reproduction of material in this paper should include the citation that appears on the title page. 20

basis points instead of the 375 assumed by the BA model). We specified a yield curve that

roughly corresponds to the present yield curve, rising from 3 percent for maturities of less than

one year to 8 percent at the longest maturities:

Hypothetical banks' yields as function of maturities 

Maturity

Bucket

1-3

months

3-12

months

1-3

years

3-5 

years

5-10

years

10-20

years

20-30

years

Asset 

yields

(%)

5.5 5.5 6.5 7.5 8.5 9.5 10.5

Liability

yields

(%)

3.0 3.0 4.0 5.0 6.0 7.0 8.0

Finally, we assume that all instruments are nonamortizing and options-insensitive.

If the average coupon differs between assets and liabilities, match-funding does not completely

eliminate interest-rate risk. This is because the interest-rate sensitivity of an instrument's market

values declines as the discount rate increases. Therefore, when interest rates rise, the value of

our hypothetical bank's assets declines by a smaller amount than does the value of its liabilities,

so its equity ratio increases. Conversely, when interest rates decline, the value of assets increases

less than does the value of liabilities, so equity value declines. We report the results for the

adverse declining scenario here. 

For a bank that has assets and liabilities with maturities extending to 30 years, the 200 basis-point

interest-rate decline causes equity to fall from 5.0 to 2.6 percent if the match-funded NIM is 375

basis points, but only to 3.8 percent if the match-funded NIM is 250 basis points. This means that

assuming that the match-funded NIM is 375 basis points when the "true" match-funded NIM is

250 basis points produces a bottom-line estimation error of 120 basis points (3.8 percent minus

2.6 percent). Fortunately, the error is much smaller for banks with assets and liabilities of shorter

maturities:
 Most of the coupon data is from the thrift industry's Thrift Financial Reports. For banks 7

portfolios of Treasury securities, we were able to use call report data to infer the

interfirm coupon dispersions. To limit the sample to only one type of securities, we used only

those banks that have more than 85 percent of their portfolios in Treasuries. From the set of

Any reproduction of material in this paper should include the citation that appears on the title page. 21

Post-shock equity ratio for hypothetical bank following a 200 basis point

rate decrease (upsloping yield curve, 5% initial equity-to-asset ratio, all

instruments initially at par):

Maximum

instrument

maturity: 

Post-shock equity ratios:

Match-funded

NIM @ 

375 bp (BA

assumption)

Match-funded

NIM @ 

250 bp

("Truth") 

Match-funded

NIM @ zero

30 years 2.60% 3.77% 5.00%

20 years 3.82 4.30 5.00

10 years 4.55 4.70 5.00

For most banks, the maximum maturity of 10 or 20 years is probably most representative. First,

for the bank with maximum maturities of 30 years, 3/7 or 42 percent of the assets have maturities

of greater than 5 years (the three longest of the seven buckets are 5-10 years, 10-20 years, and

20-30 years). In contrast, the call reports show that the average bank has only 15 percent of

assets at maturities exceeding 5 years. Hence, given our assumption of options-insensitive

instruments, it appears that the BA coupon assumptions will probably only rarely result in errors

approaching 100 basis points. However, our analysis to this point grants the BA assumption that

all instruments are always at par.

We also repeated this exercise by comparing the bank's sensitivity under various assumed matchfunded NIMs and flat yield curves. The size of the error that results from assuming the wrong

match-funded NIM is not materially affected by the assumed slope of the yield curve. Similarly,

we found that error to be no higher for banks that are highly mismatched than for banks that have

the same maturities for both assets and liabilities.

III. Instruments that Differ from Par

The BA assumption that all instruments are at par may be an excellent device given that the model

forgoes coupon data, but will cause errors for instruments that are at either premium or discount.

To see how large such errors might be in practice, we used the available data for banks and thrifts

to indicate what the distributions of bank portfolio's weighted-average coupons might look like.

There appears to be substantial variation in those portfolio coupons. The range from the 10th

percentile to the 90th percentile is as high as 500 basis points for such heterogeneous instruments

as consumer loans, and about 150 to 200 basis points for homogeneous instruments such as 30-

year original maturity FRM or Treasury securities. We use a range of plus-or-minus 100 basis

points to represent the range across real-world portfolios.

7
banks that met that criterion, we chose only those banks that reported specific weightedaverage maturities for their portfolios (we chose banks with portfolios with weighted-average

maturities falling within specific ranges, e.g. 2.5 to 3 years.) For the banks that met both of

those criteria, we observed the range for the banks' reported portfolio premiums. We then 

solved back for the range of coupons necessary to produce that range of premiums. This

technique yields an estimated range of from 150 to 200 basis points between the 10th and 90th

percentile of the banking' industry's coupons on Treasury portfolios.

Any reproduction of material in this paper should include the citation that appears on the title page. 22

In principle, incorrect coupon assumptions produce only moderate sensitivity-estimation errors.

Discounting a 25-year bond at an 8 percent BEY, varying the coupons from 7 percent to 9 percent

causes the sensitivity to a 200 basis point shock to vary between 17.9 percent and 18.6 percent.

That range is even more tight at shorter maturities. 

Percentage Decline in value of Treasury Bonds at various coupons, for a

discount-rate increase from 8 percent to 10 percent (BEYs):

Maturity 7% Coupon 8 % Coupon 9% Coupon

5 Years 7.8 7.7 7.6

10 Years 12.8 12.5 12.2

25 Years 18.6 18.3 17.9

These sensitivity ranges characterize the error that results from violating the BA assumption that

all instruments are at par. For a portfolio that is characterized by the 10-year instruments, the

range of sensitivities is 60 basis points (12.8 percent minus 12.2 percent). Hence, provided that

the BA model assumes a coupon that falls within the 10th-to-90th-percentile range of the

industry's actual coupons, the error that results from incorrectly assuming that all instruments are

at par appears unlikely to ever exceed 60 basis points as a percentage of total assets. For a

portfolio that is characterized by 5-year instruments, that error appears unlikely to exceed 20 basis

points. 

It is difficult to interpret the significance of that apparently small error for individual banks. The

BA error for not-at-par instruments may either compound or offset the error that results if the BA

model inaccurately characterizes the match-funded NIM. When those two errors compound, the

bottom-line estimation error approaches 40 basis points for banks with only short-term assets (15

basis points for the NIM-estimation error plus 20 basis points for the premium/discount error),

and can approach 200 basis points for banks with long-term assets and liabilities. Unfortunately,

we have been unable to devise a way to determine how often those errors will compound in

practice.
Any reproduction of material in this paper should include the citation that appears on the title page. 23

IV. Conclusions

Deferring the relation between coupons and options sensitivity to the next chapter, it appears that

foregoing coupon data will generally result in bottom-line errors that average less than 20 basis

points. However, under particular circumstances, there may be bottom-line errors of more than

100 basis points.

We would prefer to have better data for assessing the frequency of those large errors. In

particular, we would like to be able to better estimate the extent to which the potentially large

errors on one side of the balance sheet (a) are prevalent, and (b) systematically cancel out

potentially large errors on the other side of the balance sheet. Absent that data, we can not ensure

that large errors (approaching 100 basis points) will not result from foregoing coupon data.

However, it does appear that such large errors will seldom occur, except at banks with significant

amounts of greater-than-10-year-maturity assets. (The call report data are inadequate for

estimating how many banks hold such long-term assets.)

The lack of coupon data has provoked a great deal of criticism of the BA model. For example,

Hugh Cohen of the Federal Reserve Bank of Atlanta (1994) estimates that a bottom-line error of

(plus or minus) 335 basis points results from the BA approach to coupons and maturity (he does

not decompose the source of that error into coupons versus maturity errors). Furthermore, he

believes that those large errors will commonly result in practice, even for "conservative" banks.

We have some analytical differences with Cohen, and do not endorse his conclusion. However,

given the lack of data, it may be that the truth is somewhere between our optimistic assessment

and his pessimistic conclusion.
Any reproduction of material in this paper should include the citation that appears on the title page. 24

Chapter Three: Coupons and Options-Sensitive Instruments

For assessing the importance of maturity and coupon data, we expressed errors as a "bottom-line"

percentage of total assets, because those types of errors apply simultaneously to both assets and

liabilities. For the remainder of this report, we analyze errors that apply only to specific

instruments. These errors do not necessarily increase bottom-line errors; in some cases

instrument-specific errors cancel other errors, leaving the bottom-line errors unchanged, while

in other cases those errors compound each other, causing the bottom-line errors to balloon.

However, regardless of the effect on bottom-line accuracy, instrument-specific errors will

inappropriately penalize or reward portfolio choices that have different implications for "true"

interest-rate risk than for the regulators' measure of interest-rate risk.

The two instruments for which coupons have special importance are fixed-rate residential

mortgages (FRMs) and core deposits. For these instruments, the coupons help determine the

strength of the incentives to prepay or withdraw early. By motivating prepayments, coupons help

determine the instruments' effective maturities, which determine interest-rate sensitivity. This

phenomenon is potentially important for any long-term fixed-rate instrument, but except for FRMs

and core deposits the financial markets generally impose "prepayment penalties" that generally

remove the relation between coupons and interest-rate sensitivity. 

For FRMs, we analyze the importance of coupon data at two distinct levels of detail. First, we

analyze the importance of knowing the weighted-average coupon for a given portfolio. We then

consider the importance of knowing the dispersions of coupons within that portfolio, because any

given WAC can result from many different combinations of coupons.

I. Portfolio Weighted-Average Coupon Levels (WACs). 

I.a. Coupons and FRMs

To see how coupons can determine the interest-rate sensitivity of FRMs, consider the way that

the OTS evaluated prepayments during the low interest-rate environment of mid-1993. Almost all

FRM borrowers had a strong incentive to prepay, because interest rates were at the lowest levels

in two decades. If a FRM had a coupon that was relatively high, the borrower's incentive to

prepay was expected to weaken only slightly if rates rose by 200 basis points, because those

relatively high coupons would still remain above market rates. However, for FRM with relatively

low coupons, a rate increase of 200 basis points would cause prepayments to fall quite sharply,

because market rates would then be higher than the mortgages' coupons. Hence, FRM portfolios'

effective maturities were expected to increase when rates increased, and that effect was expected

to be much more pronounced for portfolios with low coupons than for portfolios with high

coupons.

This effect causes portfolios with low coupons to have longer effective maturities than do

portfolios with high coupons. Because instruments with long effective maturities also have highly

rate-sensitive market values, FRM portfolios with relatively low WACs have much more

sensitivity than portfolios with high WACs. For a 30-year original-maturity whole-loan portfolio

with an 8 percent WAC--roughly the thrift industry's 10th percentile of FRM WACs as of mid-
 Aggregation generally produces estimation error, except for linear functions. Interest- 8

rate sensitivities are linear in the coupon if and only if the payment structure is independent of

the discount rate:

PV = COUP((1-(1+ i) )/i); *COUP/*i = 0 ] * (*PV/*i)/(*COUP) / 0. -t 2 2

Any reproduction of material in this paper should include the citation that appears on the title page. 25

1993--the OTS estimates that a 7.6 percent decline in value would result for a 200 basis-point

interest-rate increase. For the 90th percentile of the thrift industry's WACs (about 9.25 percent),

the estimated decline in value is only 5.2 percent. Hence, interportfolio differences in the WACs

cause a difference in FRM sensitivity of over 200 basis points in the OTS model. 

Because the BA model assumes that the coupons on all FRMs equal the current market rate, it

estimates the same sensitivity for all FRM portfolios of given maturity distributions. In the NPR,

the assumed coupon for the portfolios described above is 7.5 percent, and the estimated decline

in value for the 200 basis-point interest-rate increase is 9.4 percent.

We used a "constant prepayment rate" (CPR) model that we developed using Federal Home Loan

Mortgage Corporation prepayment data to independently test for the importance of coupon

information (Benston, Carhill and Olasov, 1992). The results from that model correspond to

those of the OTS model concerning the importance of the portfolio WACs.

I.b. Coupons and Core Deposits

Portfolio coupons are also important in the OTS assessment of core deposits. As the coupon on

the various types of core deposits is varied from the thrift industry's 10th percentile to its 90th

percentile, the estimated sensitivities to the 200 basis point shock change by about 100 basis

points. 

We cannot independently replicate the OTS result for core deposits. There is as yet no consensus

about the proper technique for evaluating the interest-rate sensitivity of core deposits.

II. Intraportfolio Coupon Dispersions.

For FRMs and fixed-rate nondeposit borrowings, the OTS augments the instruments' WAC with

detailed information on the balances within each of several coupon ranges, which increase from

"Less Than 8%" to "11.00% & Above" in increments of 100 basis points. The OTS requires this

detail because, for options-sensitive instruments, aggregating coupons can result in estimation

error. 

8

To see why coupon dispersions can be important, consider an example of two portfolios. Both

have WACs of 8 percent, but the first consists only of mortgages with 8 percent coupons while

the second is comprised of equal shares of 7 percent and 9 percent coupons. The market might

expect that prepayments on mortgages with coupons below 8.5 percent would fall nearly to zero

if interest rates rose by 200 basis points, while prepayments on 9 percent mortgages would fall

only to 20 percent per year. Hence, even though the two portfolios have the same WACs, a rate

increase of 200 basis points drives the expected prepayments on the first portfolio to zero, while

the expected prepayments on the second fall only 10 percent (the 20 percent prepayment rate

multiplied by the 50 percent of the portfolio with WACs at 9.0 percent, plus 0 percent prepayment

multiplied by the 50 percent of the portfolio with WACs at 7.0 percent.) 
Any reproduction of material in this paper should include the citation that appears on the title page. 26

That example is illustrative but is also highly simplified. In practice, the effect of coupon

dispersions on expected prepayments is highly complex. For some portfolios, coupon dispersions

have a major effect on the estimated prepayments, while for other portfolios the effect is very

small. This is reflected in the OTS' estimated market-value sensitivity to the 200 basis point

changes in interest rates. For perhaps 50 percent of portfolios, the OTS model produces roughly

the same results either with or without taking account of coupon dispersions. For most of the

remainder, taking account of the dispersions changes the sensitivity by around 40 basis points.

For a small minority, acknowledging coupon dispersions can change the estimated sensitivity by

over 100 basis points. 

When we attempt to replicate the OTS results for coupon dispersions, we obtain the somewhat

stronger result that recognizing the effect of coupon dispersions usually changes the sensitivity

estimates by 100 to 200 basis points. We discussed this issue with the OTS modelers. Their view

is that the relation between prepayments and coupons varies systematically over time. In contrast,

the Benston/Carhill/Olasov model treats that temporal variance as random. The OTS believes

that, in the current environment, ignoring FRM coupon dispersions will produce errors of 100

basis points or more only for a minority of institutions. However, the OTS expects that, in other

interest-rate environments, ignoring dispersions could produce those large errors for a majority

of institutions. 

For the nondeposit borrowings, the coupon dispersions had no effect on the estimated sensitivity

in the OTS model. We were able to replicate that finding on our own models. Most thrift

borrowings are from the Federal Home Loan Bank System, and most have prepayment penalties

that remove the institution's incentives to prepay when market rates of interest decline. The

coupon dispersions on nondeposit borrowings would become important only if a bank had

substantial borrowings that are either callable or puttable. 

III. FRM Errors and Bottom-Line Errors

Given that the BA approach to FRMs can produce sensitivity-estimation errors exceeding 200

basis points, it is useful to know how many banks have extensive FRM holdings. Including both

MBS and portfolio loans in FRM, the median bank holds about 12 percent of its assets in FRM

portfolios. About 2600 banks, or 25 percent of the banking system, have as much as 20 percent

of their assets in FRMs, and 500 banks have as much as 30 percent of their assets in FRMs.

(These numbers all increase if collateralized mortgage obligations (CMOs) are included among

FRMs.) Given the BA model's large errors in measuring the sensitivities of FRMs, and the large

number of banks with significant FRM holdings, it appears that forgoing coupons on FRMs will

cause the BA model to improperly assess many banks' interest-rate sensitivity.

The importance of the problem for the bottom-line error depends of the share of assets in FRM.

If a bank holds 12 percent of its assets in FRMs, and the sensitivity-estimation error is 200 basis

points, then that error will equal 25 basis points as a percentage of total assets (200 bp x 12%).

For a bank with 40 percent of its assets in FRMs, that error would be 80 basis points as a

percentage of total assets. In some cases, those errors will offset other errors at the bottom line,

but in some cases those errors would compound with other errors.
Any reproduction of material in this paper should include the citation that appears on the title page. 27

IV. Conclusions

Both interest-rate sensitivities and market values fluctuate considerably over the range of realworld FRM portfolio WACs, so that data on WAC levels appear indispensable for modelling the

interest-rate sensitivity of FRM portfolios. The evidence is less clear with respect to FRM coupon

dispersions. In the current environment, we know that errors of over 100 basis points will

occasionally result if those dispersions are ignored, and that errors of over 40 basis points will

be common. In other environments, errors of 100-200 basis points could be common if coupon

dispersions are ignored.

For core deposits, the importance of coupon data is more difficult to rigorously document. Our

judgement is that a better understanding of how to model the interest-rate sensitivities of core

deposits is now evolving. Those new models rely on both current coupons and coupon histories.
 The "lagging-index" ARMs are sometimes called "COFI" ARMs, because most are 9

indexed to the thrifts' Cost-of-Funds Index developed by the Federal Home Loan Bank of San

Francisco. Changes in that index usually lag changes in other interest rates.

Any reproduction of material in this paper should include the citation that appears on the title page. 28

Chapter Four: Evaluating Caps on Adjustable-Rate Instruments

In Chapter Three, we saw that the interest-rate sensitivity of FRM portfolios varies greatly with

relatively small changes in their coupon distributions. In this chapter, we look at several

information items that the OTS model uses in evaluating adjustable-rate residential mortgages

(ARMs).

Estimating interest-rate sensitivity is much more complex for adjustable-rate instruments than for

fixed-rate instruments. This greater complexity can cause larger estimation errors for adjustablerate loans than for fixed-rate loans, even though the latter generally have greater interest-rate

sensitivity. Hence, for any given error tolerance, modelling adjustable-rate loans requires the

largest amount of information.

I. Information Differences Between the OTS and BA Models.

I.a. The OTS Approach

Because neither adjustable-rate commercial loans or consumer loans are a large part of thrift

portfolios, the OTS focusses on adjustable-rate residential mortgages (ARMs). Thrifts segregate

their ARM portfolios both by the index and by period used for the loans' "reset" (aka "repricing,"

the length of time between coupon adjustments). There are three categories of "current-index"

(Treasury) ARMs: one-to-six-months reset period, seven-months-to-two-years reset period, and

two-to-five-years reset period. There are two categories of "lagging-index" ARMs: One-month

reset, and 2-month-to-five-year reset. For each category the following items are collected: 9

Balances at "teaser" (below-market) coupons; weighted-average margin above the index;

weighted-average coupon, remaining maturity, and time remaining to "reset;" and the proportion

of the portfolio that has coupons falling within various distances to the loans' lifetime and periodic

interest-rate caps and floors. For adjustable-rate commercial and consumer loans, only a subset

of that information is collected.

The original (1984-1989) OTS model collected only time to repricing and coupon information,

essentially the same treatment as in the proposed BA model. At that time, it was widely believed

that ARMs were a solution to the interest-rate risk inherent in FRMs.

The OTS' current understanding is much different. It appears that the 1984-1989 OTS model

significantly understated the interest-rate risk of ARMs. This created incentives for thrifts to

underprice ARMs relative to FRMs in their residential mortgage portfolios, and probably

contributed to the widespread substitution of ARMs for FRMs in thrifts portfolios: ARMs were

essentially zero percent of assets in the early 1980s, but increased to over 40 percent of assets

by 1993, while FRMs were falling from about 70 percent of assets to only 30 percent. That
 In the BA model, ARMs within 100 basis points of lifetime caps are treated as fixed-rate 10

loans. In our analysis, all of the ARMs are more than 100 basis points from the lifetime caps,

and so would be treated as ARMs without caps by the BA model.

Any reproduction of material in this paper should include the citation that appears on the title page. 29

substitution, in turn, is associated with a moderate increase in the thrift industry's rate of failure

and with reduced thrift profitability (Benston, Carhill and Olasov, 1991; Getman, 1989). 

To correct that situation, the OTS ARM model has been expanded twice, first in June 1989 to

incorporate information about lifetime caps and ARM indices, and then in 1993 to its present

form. 

I.b. The BA Approach

In contrast to the detailed information collected for the OTS model, the BA model uses only the

time remaining to coupon reset and reset margins. This is potentially an important limitation 10

of the BA model, because adjustable-rate loans are as important in the banking industry as ARMs

are in the thrift industry. Adjustable-rate loans constitute over 50 percent of all bank loans, and

about 28 percent of total assets. Adjustable-rate investment securities constitute about 12 percent

of bank securities, and about 2 percent of assets. Thus, almost one-third of bank assets are

adjustable-rate commercial, consumer, and residential instruments.

Over 70 percent of banks' adjustable-rate instruments have coupons that adjust instantly to

changes in market rates of interest. In the jargon, such loans have zero "reset periods." The call

report does not explicitly separate the adjustable-rate loans by loan types, but we believe that most

commercial and industrial loans have zero reset periods. If that is true, then circumstantial

evidence suggests that the remaining 30 percent that have nonzero reset periods are roughly evenly

divided between ARMs, consumer loans (such as adjustable-rate second mortgages), and

commercial real estate loans. 

A disproportionate share of adjustable-rate instruments are held by the largest banks, as the

median share of assets in adjustable-rate instruments is about 20 percent rather than the aggregate

30 percent. About 5 percent of banks hold more than 50 percent of their assets in adjustable-rate

instruments.

We lack data on the share of banks' adjustable-rate loans that have either periodic or lifetime caps.

However, interviews with various commercial lenders indicate that caps are not uncommon on

adjustable-rate commercial loans. For thrifts, about 90 percent of ARMs have lifetime caps, and

about 90 percent have periodic caps. The data on commercial real estate loans are incomplete,

but indicate that a similar proportion of thrifts' commercial real estate loans have lifetime caps.

Thrifts do not report caps for other commercial or consumer adjustable-rate loans. 

II. The Relation Between Model Inputs and Estimated Sensitivities

The method we use is to begin with a very simple ARM, with no caps or floor, sequentially vary

the information input, and observe the change in the OTS' estimated sensitivities. Because most

of the information items used by the OTS are not collected at all by the BA, any changes in the
Any reproduction of material in this paper should include the citation that appears on the title page. 30

OTS sensitivity that result from changes in the information inputs represent model differences

between the BA and the OTS.

Analyzing caps requires an explicit treatment of prospective interest-rate volatility, and most

models employ a Monte Carlo simulation. We have not developed and published a Monte Carlo

model. For an independent analysis of the OTS model for ARMs, we rely on the work of

Schwartz and Tourus, who develop a Monte Carlo model that uses the same information inputs

as the OTS (Schwartz and Tourus, 1991). The OTS and Schwartz and Tourus models roughly

agree about quantitative marginal effects of the information inputs that we describe here. 

II.a. Coupons and Interest-Rate Sensitivity

We varied the weighted-average coupons (WACs) on the portfolios evaluated by the OTS and

found that, within the current range of interindustry variance in WACs, the portfolio WACs do

not affect the estimated sensitivity to a 200 basis-point shock. Hence, coupon information does

not appear to be important for estimating the market-value sensitivity of ARMs.

II.b. Lifetime Caps

Lifetime caps are crucial to evaluating ARMs. However, the effect of lifetime caps on interestrate sensitivity varies with several other factors: the ARM's periodic caps and floors, its lifetime

floors, its maturity, its coupon and reset margin, and the current interest rate environment. To

focus on lifetime caps, we specified an ARM portfolio that had lifetime caps but not periodic caps,

and was typical in other respects. This portfolio is useful for illustrating the effect of the interestrate environment in determining the importance of lifetime caps.

In the mid-1993 interest-rate environment, the OTS estimated that an ARM portfolio with no

lifetime or periodic caps would decline in value by 1.5 percent of par (150 basis points) in the up200 interest-rate shock. The BA estimated that the same portfolio would decline in value by about

180 basis points. None of the information items that we discuss below would change that BA

estimate, but all change the OTS estimates substantially. (In the BA model, ARMs within 100

basis points of lifetime caps are treated as fixed-rate loans. In the following analysis, all of our

ARMs are more than 100 basis points from the lifetime caps, and so would be treated as ARMs

without caps by the BA model.)

About 90 percent of all ARMs have lifetime caps. When the loans are initially issued, these caps

are typically 500 basis points above the loan's contract rate. If rates subsequently decline, as has

occurred over the last few years, the lifetime caps become progressively less important.

Conversely, if market interest rates and the portfolio coupons begin to move upward, the caps

become more important. The case of a given one-year ARM with lifetime caps but no floors or

periodic caps, a 28-year remaining maturity, and otherwise typical characteristics illustrates that

point:
 Prices for 5-year interest-rate caps on 3-month LIBOR are reported regularly in Swaps 11

Monitor. In the first six months of 1994, 3-month LIBOR increased from 3.35 percent to

4.80. Over the same period, the price of a newly issued 5 year cap with a strike price of 7.0

percent increased from 1.66 percent of notional to 4.14 percent of notional. Similarly, the

Bloomberg reports showed that ARMs MBS fell from 3 to 7 percent in the first half of 1994.

Though changing volatility estimates and the steep yield curve also contributed, the magnitude

of the 1994 increase in caps prices is consistent with the OTS and Schwartz and Tourus

predictions about interest-rate sensitivity. 

Any reproduction of material in this paper should include the citation that appears on the title page. 31

OTS' estimated sensitivity of typical ARM with 28 years to maturity:

Distance of

current 

coupon 

from

lifetime 

cap

(basis

points):

- 4 -800 -600 -400 -200

(OTS') basis

point decline

in value for 

200 bp rate

increase:

170 180 220 400 780

As market rates rise above the cap (so that the ARM's coupon reaches the cap), the estimated

sensitivity asymptotically approaches that of a 30 year FRM (about 10 percent for a 28-year

remaining maturity, depending on expected prepayments). 

For real-world portfolios with typical characteristics (as opposed to the single ARM presented in

the table), the difference between the OTS and BA sensitivity estimates are about 50 basis points

in the current environment (230 basis points in the OTS model minus 180 basis points in the BA

model). Following an increase in interest rates, a lender might find that one-half of the same

portfolio was within 300 basis points of its lifetime cap, while the other one-half was within 150

basis points of its lifetime caps. At that point, the OTS estimated market-value sensitivity to the

200-basis-point increase would be about 500 basis points. Since the BA estimate would remain

at 180 basis points, the model difference at that point in the rate cycle would be 320 basis points.

The Schwartz and Tourus analysis is consistent with that of the OTS, and options-pricing theory

in general suggests that lifetime caps will have a large effect on the interest-rate sensitivity of an

ARM. We conclude that the 50-to-320-basis-point difference between the OTS and BA models 11

represent a BA model error for the case of ARMs with lifetime caps but no floors or periodic

caps.
Any reproduction of material in this paper should include the citation that appears on the title page. 32

Lifetime caps can be even more important for adjustable-rate commercial real estate loans (the

only other adjustable-rate loan for which the OTS collects information on lifetime caps) than for

ARMs. For commercial mortgages, adding a cap at a distance of 150 basis points from current

market rates causes the sensitivity to rise from 0.3 percent to 12.1 percent. This extreme decline

results because in that rate environment, lifetime caps essentially convert the adjustable-rate

commercial real estate loans into fixed-rate loans. Since those loans often have long maturities

and nonamortizing payment structures, their market values can be even more rate sensitive than

the market values of FRMs.

II.c. Periodic Caps

We evaluated the importance of periodic caps both in a portfolio without lifetime caps and then

in a portfolio with nearly binding lifetime caps. In an otherwise typical portfolio without lifetime

caps, the OTS model projects that a 200 basis point rate increase would cause a decline of about

150 basis points in the value of a portfolio with no periodic caps, a decline of about 260 basis

points in a portfolio with 200 basis point periodic caps, and a decline of about 500 basis points

in a portfolio with periodic caps of 100 basis points. If the same portfolio has the nearly binding

lifetime caps discussed above, the periodic caps increase the sensitivity by only about 50 basis

points, to about 550 basis points. As before, the BA model projects a 180 basis point sensitivity

for this ARM portfolio, so ignoring periodic caps creates model differences of between 30 and

370 basis points, depending on the lifetime caps.

Schwartz and Tourus generally focus on lifetime rather than periodic caps, but their limited

analysis of periodic caps is consistent with the results of the OTS model, insofar as confirming

that periodic caps have a significant effect on the sensitivity of ARMs. However, we have not

independently replicated the OTS' specific quantitative assessment.

II.d. Reset Margin

"Reset margin" refers to the gap between an adjustable-rate loans' coupon and its index rate. For

example, a floating-rate loan might have a coupon that equals the prime rate plus 100 basis points;

the index is the prime rate and the margin is 100 basis points. The OTS collects the weightedaverage margin for all types of adjustable-rate loans. 

We varied the reset margin for several kinds of the adjustable-rate commercial-loan portfolios that

the OTS evaluated. Varying the margin causes the estimated interest-rate sensitivity to vary by

only 0-30 basis points. Subsequent investigation suggests that, in general, the reason for that

change in sensitivity was that varying the reset margin permanently varies the instruments'

coupons, which changes interest-rate sensitivity for the reasons described in chapter 2 above.

While unimportant in the absence of caps, the reset margin can interact with lifetime and periodic

caps to have an important effect on the estimated sensitivity. For example, given a one-year

Treasury ARM with both periodic and lifetime caps, increasing the reset margin from 100 basis

points to 200 basis points can increase the OTS' estimated market-value sensitivity by as much

as 350 basis points, depending on the ARM's other characteristics. 
Any reproduction of material in this paper should include the citation that appears on the title page. 33

II.e. Remaining Maturity

For both commercial and residential adjustable-rate loans, the OTS collects both weighted-average

remaining maturity and time remaining to reset, while the BA model collects only time remaining

to reset. Remaining maturity can have significant effects on sensitivity under either of two

conditions. First, if the instrument has a long reset period, the remaining maturity becomes a

prime determinant of sensitivity. For example, if rates rise by 200 basis points, a 4-year-to-reset,

adjustable-rate, options-insensitive, amortizing loan with a remaining maturity of 30 years will

decline in value by 600 basis points. If the same loan has a remaining maturity of only 5 years,

it will decline in value by only 400 basis points.

While that example suggests that ignoring remaining maturity can lead to large estimation errors,

the effect of remaining maturity declines rapidly as the reset period shortens, becoming negligible

for reset periods of one year or less. Both call report and TFR data indicate that only about 10

percent of adjustable-rate loans have reset periods in excess of one year.

The second condition under which remaining maturity becomes important is when the lifetime

caps become binding or nearly binding. As that happens, an adjustable-rate loan begins to take

on the characteristics of a fixed-rate loan, so remaining maturity becomes a key determinant of

sensitivity. In the OTS model, if the current coupon on an ARM is 400 basis points below the

lifetime cap, increasing the maturity from 20 to 30 years increases the sensitivity by about 20

basis points. If the ARM is 200 basis points below the lifetime cap, that increase in maturity

increases the sensitivity by about 80 to 100 basis points. This result holds regardless of the

ARM's time to reset. 

This analysis suggests that in most interest-rate environments, the remaining maturity of

adjustable-rate loans has little value as a datum, unless a bank makes adjustable-rate loans that

have long reset periods. However, in interest-rate environments where lifetime caps are

beginning to come into play, the remaining maturity becomes an important determinant of

sensitivity for even short-term-to-reset adjustable-rate loans. In other words, as the probability

that lifetime caps will become binding increases from zero to one, remaining maturity

progressively replaces term to reset in determining interest-rate sensitivity. 

III. Conclusions

The values of adjustable-rate instruments with caps are very sensitive to interest-rate changes.

While conclusive data are not available, the available data suggest that banks issue caps on the

majority of their adjustable-rate residential loans, and perhaps on their adjustable-rate commercial

loans as well.

Unfortunately, information that is irrelevant in the absence of caps becomes crucial in the

presence of caps. As caps come into play, every item that the OTS collects can have major effects

on the interest-rate sensitivity of the values of adjustable-rate loans.

If the regulatory model fails to account for the importance of caps on adjustable-rate loans, there

will be three major consequences. First, a strong artificial regulatory incentive will be created

for risk-seeking banks to issue caps on their adjustable-rate loans, because such loans create

considerable interest-rate risk but will not explicitly require capital funding. Second, even at
Any reproduction of material in this paper should include the citation that appears on the title page. 34

banks that are risk averse, the regulators may appear to endorse the myth that adjustable-rate loans

have no interest-rate risk. Third, because adjustable-rate loans are a large part of banks' assets,

a substantial bottom-line error will exist in the regulatory interest-rate-risk assessment. At the

median, adjustable-rate instruments are about 20 percent of bank assets; unless offset by other

errors, an error of 300 basis points as a percentage of par would translate into a bottom-line error

of 60 basis points as a percentage of total assets.
Any reproduction of material in this paper should include the citation that appears on the title page. 35

Chapter Five: The Effects of Miscellaneous Information Items

The preceding chapters analyzed four specific controversies about the information parsimony of

the BA model. However, as we described in the introduction, there are several other pieces of

information that are used by the OTS model but not by the BA model. In this chapter, we

measure the model differences that result from those information items, by varying the portfolio

characteristics within the range of the inter-firm variance reported in the Thrift Financial Reports.

I. Amortization Structure

Financial instruments can be fully amortizing, partially amortizing, nonamortizing, or (more

rarely) negatively amortizing. (Partially amortizing or nonamortizing loans are sometimes referred

to as "balloons.") Roughly speaking, the BA model assumes the consumer loans and residential

mortgages are fully amortizing, while the various types of commercial loans are nonamortizing.

In contrast, the OTS requires thrifts to segregate commercial real estate (CRE) loans between

amortizing versus nonamortizing loans, though it assumes that all construction and all commercial

and industrial (C&I) loans are nonamortizing.

It is usually necessary to know whether a loan is amortizing or nonamortizing in order to

determine its interest-rate sensitivity. As the table below shows, a 200-basis-point upward shift

in the yield curve causes declines in value that vary greatly, depending on whether the loan is

amortizing or nonamortizing (we assume options-insensitive, zero-prepayment instruments, with

semiannual payments):

Decline in value as percent of par, for rate increase from 8 percent to 10 percent BEYs:

Remaining

maturity or

time to

repricing:

1 year 2 years 4 years 7.5 years 15 years 25 years

Amortizing 1.4 2.3 4.0 6.4 11.1 15.0

Nonamortizing1.9 3.5 6.5 10.4 15.4 18.3

The current call report does not distinguish between amortizing and nonamortizing loans. It does

report that about 20 percent of banks' assets are in commercial lending, split roughly equally

between relatively long-maturity CRE and relatively short-maturity C&I loans. 

I.a. Commercial and Multi-Family Real Estate Loans

The Thrift Financial Reports indicate that, for thrifts, about 60 percent of CRE loans are fully

amortizing, so if we assume that banks' CRE portfolios are similar to thrifts' , we can comment

on the significance of the BA model's assumption that all CRE loans are nonamortizing. Of the

thrifts' amortizing CRE loans, about 70 percent have maturities or times to reset of 2 years or
Any reproduction of material in this paper should include the citation that appears on the title page. 36

less, so we estimate that an average of about 4 percent of bank assets are in short-term amortizing

CRE loans (70 percent of amortizing CRE, which are 60 percent of total CRE, which are 10

percent of total assets). For those loans, the BA model would overstate sensitivity by 0 to 120

basis points, an estimation error that would contribute only 0 to 5 basis points to the bottom-line

estimation error as a percentage of total assets (4 percent of assets multiplied by the maximum 1.2

percent estimation error). If we suppose a four-year maturity/time-to-reset for the remaining 30

percent of amortizing CRE, for which the BA model would overstate sensitivity by 250 basis

points, an additional 5 basis points of bottom-line estimation error would result.

These results suggest that, for the average bank, ignoring the payment structure of the CRE

portfolio would make only a small contribution to the bottom-line estimation error. However, for

banks that concentrate in amortizing CRE (or amortizing C&I loans, if there are such animals),

the BA model would create a moderate disincentive for extending amortizing loans, since such

loans have lower interest-rate risk but receive the same regulatory treatment as do nonamortizing

loans.

I.b. FHA-VA FRM

One reason the OTS decided to forego detailed maturity information was to expand the detail

devoted to loan types without lengthening the reporting form (see Chapter 1). One specific place

that the OTS chose to increase the detail on loan type was to distinguish 30-year FRM between

those guaranteed by the FHA-VA and so-called "conventional" guarantors.

The reason for distinguishing between FHA-VA and conventional FRMs is that the former are

assumable, and so have different prepayment patterns than the latter. (As we saw in chapter 2,

both the interest-rate sensitivity and base-case market values of FRMs are very dependent on their

expected prepayments.) When interest rates rise, borrowers would prefer not to prepay, while

lenders would prefer that their borrowers do prepay. With most conventional loans, borrowers

who sell their homes must prepay regardless of the interest-rate environment; this baseline

prepayment rate is about 5 percent per year. However, with assumable FRMs, that 5 percent

need not prepay. Hence, in rising interest-rate environments, the effective maturity of assumable

loans increases more rapidly than does the effective maturity of conventional loans, and this

causes FHA-VA FRMs to decline in value by a greater amount than do conventional FRMs.

To test for the importance of this datum, we changed the percentage of FHA/VA mortgage-backed

securities (MBS) in one of our hypothetical portfolios from 30 percent --the industry average-- to

100 percent. The OTS estimates that a portfolio with 30 percent share of FHA/VA, and otherwise

typical characteristics, would decline in value by about 700 basis points if rates increased by 200

basis points. A portfolio composed entirely of FHA/VA MBS would decline by about 800 basis

points. Hence, this information datum creates a model difference of about 100 basis points.

While the economic argument for the extra sensitivity of FHA/VA makes sense, we lack the data

to independently verify that the effect is as large as the OTS model suggests. The OTS derives

its prepayment projections from the Bloomberg system, which reports a purported consensus of

the prepayment projections of a small number of Wall Street firms. Independent verification

would require prepayment history for both conventional and assumable mortgages.
Any reproduction of material in this paper should include the citation that appears on the title page. 37

MBS comprise about 10 percent of the average bank's assets, so if banks' portfolios look like

thrift portfolios, FHA/VA MBS average about 3 percent of bank assets. Thus, the one-percent-ofpar-value model difference would translate into 3 basis points of "bottom-line" interest-rate

sensitivity. For the thrift industry, about 10 percent of MBS holders specialize entirely in

FHA/VA, and the bottom-line errors would be correspondingly higher to the extent that those

specialists hold significant amounts of FHA/VA MBS. Only about one percent of portfolio

lenders specialize in FHA/VA loans, apparently because it is difficult to focus one's marketing

efforts entirely on FHA/VA-qualified borrowers.

It appears that ignoring the distinction between FHA/VA and conventional mortgages will create

moderate artificial incentives for risk-seeking lenders to extend those loans. However, such loans

will cause at worst moderate errors in estimating banks' bottom-line sensitivity, and those errors

only rarely. 

II. Basis Risk

Different interest rates change by different amounts at different times. For example, T-Bill rates

can decline while rates on core deposits and CDs remain unchanged. Similarly, rates on longterm Treasury Bonds can increase while rates on Treasury Bills are unchanged. When either or

both events happen, banks that own adjustable-rate loans or off-balance-sheet instruments indexed

to the Treasury rate suffer economic losses. That risk is called basis risk; the first type of basis

risk depends on the instrument used for the basis, while the second depends on the point of the

term structure used. 

To measure such risk, the OTS requires thrifts to report the index most commonly used for the

various types of adjustable-rate loans. In contrast the BA model ignores the index, except that

banks are required to assign all of their so-called "lagging-index" ARMs to the 3-5 year time

bucket, regardless of the lagging-ARM's actual time to repricing. 

We tested for the importance of this variable in the OTS model by varying the indices for a

number of adjustable-rate loans. For most kinds of loans, varying the index made almost no

difference in estimated sensitivity. The largest effect was on ARMs; lagging-index ARMs

declined by about 20 basis points more than current-index ARMs, except when (nearly) binding

lifetime caps came into play. In that case, lagging-index ARMs decline by 100 basis points less

than do current-index ARMs. In contrast, the BA model assigns considerable significance to the

index for ARMs. In their model, the values of all lagging-index ARMs decline by 300 basis

points when rates increase. In the OTS model, lagging-index ARMs decline by about 150 basis

points absent lifetime caps and by about 450 basis points with (nearly) binding lifetime caps.

We have not independently verified either model's results about the importance of basis risk. The

BA treatment of lagging-index ARMs is admittedly ad hoc. The OTS has yet to publish a

comprehensive description of its approach to basis risk. 

Given that caveat about our inability to verify either model's results, we tentatively conclude that

basis risk can be ignored for on-balance-sheet instruments without creating significant errors in

estimating interest-rate sensitivity. However, it would be inadvisable to extend that conclusion

to off-balance-sheet instruments, where the basis often determines the entire cash flow.
Any reproduction of material in this paper should include the citation that appears on the title page. 38

III. Original Maturity

The OTS collects data on original maturity for all FRMs and for CDs. Original maturity is

supposed to affect the retention rate on CDs and the prepayments on FRMs. However, we found

that varying the original maturity had negligible effects on the estimated sensitivity of CDs. 

For FRM with given remaining maturity and coupons, the OTS expects higher baseline

prepayments for higher original maturities. As a result, the OTS projects that a 7.5 percent

coupon, 30-year original-maturity FRM with a 180-month remaining maturity would decline in

value by 8.7 percent in the up-200 scenario. An otherwise identical FRM but with 15-year

original maturity would decline by 9.4 percent in the same scenario.

It is not clear how to specify the relation between coupons on 15-year mortgages, coupons on 30-

year mortgages, and current market rates. The OTS relies on the Bloomberg system, which in

turn relies on a survey of market participants, who do not reveal their prepayment methodologies.

Hence, the greater sensitivity that the OTS attaches to 15-year original-maturity FRM does not

appear to be independently verifiable. 

IV. Conclusions

All of the miscellaneous items appear to be relatively unimportant with regard to estimating the

bottom-line sensitivity of most banks' on-balance-sheet instruments. However, either the payment

structure of commercial loans or the assumability of mortgages might be important for a small

minority of banks. Given a replicable source of data, we would anticipate verifying the OTS

contention that the original maturity of FRM is an important datum.
 Letter from Jill M. Considine, President of the New York Clearinghouse, to Karen 12

Carter, Office of the Comptroller of the Currency, dated October 28, 1993. The eight

members of the clearinghouse are: The Bank of New York; the Chase Manhattan Bank,

N.A.; CitiBank, N.A.; Chemical Bank; Morgan Guaranty Trust Company of New York;

Bankers Trust Company; Marine Midland Bank, N.A.; United States Trust Company of

New York; National Westminster Bank USA; European American Bank; and Republic

National Bank of New York. The eight banks that participated in the survey were not

identified.

Any reproduction of material in this paper should include the citation that appears on the title page. 39

Chapter Six: Caveats and Conclusions

For most issues, the analysis to this point has been able to use either third-party judgements or

original modelling to discriminate between the two models. To conclude, we discuss two major

areas in which such discrimination was not possible, and discuss what our work suggests about

interest-rate-risk modelling.

I. Core Deposits

For financial intermediaries, determining the interest-rate sensitivity of "core deposits" is essential

for modelling interest-rate risk. Core deposits fund an average of about one-half of all bank

assets. The deposits have a contract maturity of zero, but it is universally recognized that the all-in

cost of core deposits exhibit at least some stickiness with respect to market rates of interest, so

that core deposits have positive effective maturity.

While all agree that core deposits have positive effective maturity, there exists no consensus on

the actual maturity that should be assigned to any given type of core deposit. For example, to

help lobby the regulators concerning the proposed BA model, the New York Clearing House

surveyed its eight members concerning the interest-rate sensitivity of their demand deposits. The 12

estimates of the market-value decline that would result from a 200 basis-point rate increase were

essentially uniformly distributed from 1.6 percent to 9.5 percent. We observed similarly wide

ranges of estimates for other types of core deposits in the proprietary disclosures that banks make

to the regulators as part of the supervision process.

The current lack of consensus on the rate sensitivity of core deposits may be reduced as banks

devote more attention to interest-rate-risk modelling. One reason why present estimates show

such a broad range is that most banks' core-deposit models focus on acquisitions rather than rate

sensitivity. However, some banks attempt to directly model the interest-rate elasticity of their

deposits. Most such efforts are highly proprietary, but Tempo Savings Bank has published its

analysis, and the most sophisticated proprietary model we have seen produces conclusions that are

very similar to those published by Tempo (Stroh 1993). This suggests that one reason for the lack

of consensus on core deposits might be that different banks apply incompatible statistical

approaches when measuring their deposits' interest-rate risk.

Recent academic work shows promise of rigorously documenting the rate sensitivity of deposits

(Hutchison and Pennachi, 1993; Hutchison, 1993; O'Brien, Orphanides, and Small, 1994). That

work suggests that sensitivities may be highly variable across institutions, though the evidence at

this point is not sufficient to distinguish estimation error from true institution specificity. If core
Any reproduction of material in this paper should include the citation that appears on the title page. 40

deposits are shown to have highly institution-specific characteristics, the regulators will find

themselves in a dilemma between imposing inappropriately generic core-deposit assumptions or

encouraging endless appeals from banks concerning their measured interest-rate risk.

II. Mortgage and Off-Balance-Sheet Derivatives

Despite the extensive options-pricing literature, we have found few pieces that directly address

the interest-rate sensitivity of derivative products. The BA model essentially ignores the

idiosyncratic characteristics of given derivatives. The OTS does value derivatives, and our work

with their model suggests (not surprisingly) that the interest-rate sensitivities of derivatives are

extremely dependent on those idiosyncracies. Accordingly, for complex derivatives, the OTS

requires institutions to report their own sensitivity estimates (for simple positions, the institution

can report their own positions, but have the option of relying on the OTS model). Since

derivatives can overwhelm the balance sheet in determining an institution's interest-rate-risk

exposure, it appears that off-site measurement systems are inapplicable to institutions that hold

complex instruments, and that the regulation or characterization of such banks will necessarily

depend on the banks' own models. A similar conclusion probably holds for banks with significant

foreign-exchange exposures.

III. Conclusions

Not surprising, there are significant differences between the estimates of the BA and OTS models.

For basic on-balance-sheet instruments, we have been able to categorize those differences into

three broad orders of seriousness. Almost all of the model differences we identified results from

differences in information inputs. The most serious class, or "first-order differences," exceed 100

basis points as a percent of the instrument's par value and result from different information about:

caps and floors on adjustable-rate loans; maturity distributions for long-term instruments;

coupons, original maturity, and assumability for fixed-rate residential mortgages; amortization

structures on commercial real estate loans, and the put option on core deposits. (Neither model

deals with call options on securities, whether liabilities or assets, but ignoring call options and

perhaps other miscellany could also create first-order errors.) "Second-order" differences,

between 30-100 basis points, occur from dispensing with general coupon data and from the bucket

approach to maturity distributions. Finally, third-order differences of 0 to 30 basis points can

result from different treatments of almost any instrument characteristic.

While the existence of model differences is not surprising, it is surprising that we were able to

discriminate between differences and errors down to the third order, for every (on-balance-sheet)

item except core deposits. This is because the estimates of independent models repeatedly fell

within 30 basis points of the estimates of the more information-intensive model. Though our

analysis has hardly exhausted all possible information parsimonies, it seems safe to conclude that

information parsimony in interest-rate-risk modelling generally produces measurable estimation

error. 

The fact that parsimony produces error does not imply that completeness precludes error. The

apparent precision in our model estimates, and our discrimination between estimates, is possible

only because we grant the model's underlying assumptions, as discussed in the introduction. Even

where there is apparent consensus about the underlying assumptions, those assumptions can

mischaracterize reality in important and unpredictable ways, causing false predictions. 
Any reproduction of material in this paper should include the citation that appears on the title page. 41

An example is the prepayment functions for fixed-rate mortgages, one of the most important

assumptions for modelling the risk of firms with significant involvement in that asset (portfolio

holdings, originations, or mortgage servicing). The experience of the 1980s had indicated that

the prepayment function was highly stable. For example, Benston et. al. used aggregated 1984-

1988 prepayment-by-coupon data to achieve an R of .7 in their prepayment function (Benston, 2

Carhill, and Olasov, 1992). Wall Street firms, which use much more detailed data, presumably

achieved an even greater explanatory power in their proprietary models. 

That apparent stability of the prepayment function created great confidence among market

participants that prepayments were highly predictable, but that predictability vanished in the early

1990s. For example, data provided the authors by the Federal National Mortgage Association

reports that the annualized quarterly prepayment rate for mortgages with coupons 200 basis points

above market has ranged from below 20 percent to above 60 percent since December 1990,

instability that is clearly inconsistent with the .7 R reported by Benston, even though the average 2

40 percent CPR at that coupon-to-market spread is consistent with the prior parameter estimates.

The breakdown of the historical relation between interest rates and prepayments caused the failure

of several firms. A classic example is Coastal States Life Insurance Company, a $128 million

company that had invested $120 million in a CMO portfolio that was supposedly immune to

interest-rate risk. However, that immunity held only so long as actual prepayments were equal

to those predicted in each given interest-rate scenario. When prepayments did not perform as

expected, the company's losses produced deep insolvency (Jereski 1993). 

Similar caveats hold for the relation between interest rates and credit losses, and any number of

other underlying assumptions that interest-rate-risk models are forced to adopt. Yet, as a

mathematical exercise, interest-rate simulations require such assumptions in order to fix the

variables. 

For these reasons, we conclude that complete information is necessary but not sufficient for

accurate interest-rate-risk simulation. Even at the level of detail in the OTS model, and even

granting its underlying assumptions, important inaccuracies remain (e.g., its maturity aggregation

error described in Chapter One). And we know that the underlying assumptions should be

questioned. Despite being an almost entirely mathematical exercise, interest-rate-risk modelling

is demonstrably an art form rather than a science.
Any reproduction of material in this paper should include the citation that appears on the title page. 42

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