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1.1.1.1.1Chapter Twenty-Two
Managing Interest Rate Risk and Insolvency Risk on
the Balance Sheet
1.1.1.2 I. Chapter Outline
1. Interest Rate and Insolvency Risk Management: Chapter Overview
2. Interest Rate Risk Measurement and Management
a. Repricing Model
b. Duration Model
3. Insolvency Risk Management
a. Capital and Insolvency Risk
II. Learning Goals
1. Define the repricing gap measure of interest rate risk.
2. Understand the weaknesses of the various interest rate risk models.
3. Define the duration gap measure of interest rate risk.
4. Discuss how capital protects against credit risk and interest rate risk.
5. Highlight the differences between the book value and market value of equity.
1.1.1.3 III. Chapter in Perspective
Providing maturity intermediation is a major function of FIs. Recall from Part I of the text that
institutions are intermediaries between ultimate borrowers and lenders. They serve as asset
transformers by providing claims designed to better meet the specific needs of borrowers and
lenders. The asset transformation function typically leaves the FI with longer term assets than
liabilities. Thus as interest rates change over time the spread between a FI’s asset earnings and
liability costs may increase or decrease, leading to major changes in a FI’s profitability.
Likewise, changes in the market value of the FI’s assets and liabilities will not be the same when
interest rates change. Changes in interest rates can cause the value of assets to change more or
less than the value of liabilities. The FI’s equity value can thus fluctuate sharply as interest rates
change because the market value of equity is equal to the market value of assets minus the
market value of liabilities. This chapter discusses the measurement of profitability and present
value risk and discusses methods of manipulating the balance sheets to manage these risks. The
chapter concludes by reemphasizing the role of equity capital in limiting insolvency risk and
discussing the benefits of market value accounting.
1.1.1.4 IV. Key Concepts and Definitions to Communicate to Students
Gap CGAP
Repricing model CGAP Effects
Funding gap model Spread Effects
Duration gap model Net effects of changing rates on NII
Net interest margin or net interest income Duration gap
Insolvency risk Duration gap and equity value
changes
Maturity buckets Market value of equity
Rate sensitive assets (RSAs) Book value of equity
Rate sensitive liabilities (RSLs) Immunization
Market to book ratio Maturity intermediation
Insolvency risk Mark to market accounting
Capital Purchase Program Refinancing risk
Reinvestment risk
1.1.1.5 V. Teaching Notes
1. Interest Rate And Insolvency Risk Management: Chapter Overview
The large interest rate movements of the 1980s illustrated the interest rate risk exposure of many
FIs as large numbers of lenders were bankrupted by the swings in interest rates in the early 1980s
coupled with regional problems in real estate loans.1
Changes in interest rates can impair a FI’s profitability and affect the market value of a FI’s
equity. The repricing model measures the effect of interest rate changes on profitability; the
duration model measures the predicted change in the market value of equity. Once the exposures
have been determined, it is possible to mitigate the effects of interest rate changes by
manipulating the balance sheet, or by using the off balance sheet tools discussed in Chapter 23.
Insolvency occurs if the value of liabilities exceeds the value of assets resulting in negative
equity. Insolvency normally occurs because of liquidity risk, credit risk and/or interest rate risk.
Maintaining sufficient equity capital and prudently managing the risks a FI faces provides the
surest protection against insolvency. The financial crisis led to large increases the number of
1Chapter 19 points out that interest rate risk and credit risk are interrelated. If rates begin to rise
precipitously once again, default rates will also rise, and an unhedged mortgage lender funding
the loans with short term liabilities will likely face some of the same problems that S&Ls faced
in the 1980s. Securitization allows DIs to largely avoid interest rate risk. Nonndiversified DIs
engaging in mortgage lending that do not securitize or otherwise hedge face potentially severe
insolvency risk from rising interest rates.
failures. From 2008 to 2012, 465 depository institutions (DIs) failed costing the FDIC $89
billion.
Fed policy strongly affects interest rates. In the summer of 2004 the Fed began a series of
interest rate increases to curb inflationary pressures in the economy. The Fed increased interest
rates 17 consecutive times, each time by 25 basis points. In 2007 and 2008 the Fed reversed the
increases, rapidly bringing rates down as the subprime crisis worsened. Rates were eventually
lowered to near zero. As of June 2014 the Fed continued to keep interest rates very low.
However, the FDIC has recently issued warnings to institutions to prepare for upcoming interest
rate increases that may occur as early as 2015.
2. Interest Rate Risk Measurement and Management
The repricing model (sometimes called the funding gap model) has historically been the
accepted method of measuring a FI’s interest rate risk and is used by the majority of institutions.
With the recent advances in computer technology and the ability to easily generate more
complex calculations, the duration gap model is now becoming a supplemental measure of
interest rate risk.
Teaching Tip: Both are still used. The repricing gap model is easier for bankers (and students) to
understand conceptually and is used at many smaller banks. Understanding the duration gap
model presented here requires an understanding of Chapter 3, understanding the repricing model
does not. Institutions that concentrate on long term lending funded by short term deposits face
greater interest rate risk. All DIs are now required to measure and report interest rate risk. In
addition the BIS proposed that all DIs report the level of capital at risk from interest rate
changes. Although this chapter presents these models as means for DIs to limit risk, banks and
others can (and do) choose to take positions on interest rates in order to bolster profitability. A
high level committee usually called the “Asset and Liability Committee” or something similar
manages the institution’s interest rate risk. Members of the committee will normally include the
bank president and senior VPs.
a. The Repricing Model
The repricing model attempts to measure how a FI’s interest income will change relative to
interest expense over a given time period if interest rates change. The time periods (called
maturity buckets) typically include one day, 3 months, 6 months, 1 year, 5 years, and greater
than 5 years.2 The model classifies assets and liabilities as “fixed rate” or “rate sensitive” based
on whether the earnings or costs will change on these accounts during the planning period if
interest rates change. Rate sensitive accounts are those where the cash inflows on an asset or
outflows on a liability will change at some point within the planning period if interest rates
change. Accounts are classified as fixed rate if the cash inflows on an asset or outflows on a
liability will not change within the planning period even if interest rates change. Conceptually
one can then compare the rate sensitivities of the two sides of the balance sheet and estimate how
Net Interest Income (NII) is likely to change if interest rates change.
2Cumulative repricing gaps are then calculated across the maturity buckets. The text calls these
CGAP.
For example, a simple balance sheet has been classified for a 6 month maturity bucket below:
Assets Liabilities
Rate Sensitive Assets (RSAs) $10
0
Rate Sensitive Liabilities
(RSLs)
$
50
Fixed Rate Assets (FRAs) $20
6
Fixed Rate Liabilities (FRLs) $25
6
Nonearning Assets (NEAs) $
34
Equity $
34
Total $34
0
Total $34
0
Because we can think of every asset as financed by a liability or equity account we can think of
the individual asset categories as financed by a given liability or equity account. Students can
readily grasp that there is very little profit risk from an interest rate change on the $34 of NEA
financed by equity. Likewise there is little profit risk from the $206 FRAs financed by FRLs
because the cash inflows and outflows on these accounts do not change over the given maturity
bucket. Notice that this pairing leaves $50 in FRLs not yet accounted for. There should not be
an excessive amount of risk for the amount of RSAs financed by RSLs, because both are rates
sensitive. For instance, if interest rates rise, the earnings on RSAs and the costs on RSLs should
both rise and the spread should be roughly unchanged. If the spread changes this is termed a
‘spread effect’ (as described below). There are $50 (out of the total $100) RSAs financed by
RSLs. This leaves a final category, the remaining $50 in RSAs that are financed by the
remaining $50 in FRA. This category is a major source of interest rate risk because one side (the
assets) is rate sensitive and the other side is not. This category is called the repricing gap. The
repricing model measures this ‘GAP’ or the difference in sensitivity of interest income and
interest expense in the given maturity bucket
If R = the general level of interest rates then we can predict the ΔNII resulting from a given ΔR
as follows:
R)RSLRSA(RGAPNII
where RSA and RSL are equal to the balance sheet
quantity of rate sensitive assets and liabilities respectively. The change in NII over a given time
period is a function of the size and sign of the gap and the size and sign of the interest rate
change. A negative repricing gap means the FI is exposed to refinancing risk which means the
institution will be hurt if interest rates increase because funding costs will upward more quickly
than asset returns, thus reducing the net interest margin. A positive repricing gap implies the FI
faces reinvestment risk, which is the risk that interest rates fall and funds will have to be
reinvested at lower rates while more liabilities will retain the same interest rate cost.
Teaching Tip: When comparing the interest sensitivities of two or more institutions of different
size, or when comparing one institution to peer averages the percentage gap (= dollar gap /
Assets) is more useful than the dollar gap. Sometimes one also calculates the Gap Ratio (=
RSA / RSL) (see the text). This measure can lead to incorrect comparisons about interest
sensitivity if used to compare the interest sensitivity of a bank with a positive dollar gap to a
bank with a negative dollar gap. The bank with the ratio furthest from 1 may not be the most
interest sensitive. For this reason the percentage gap is a better comparison tool than the gap
ratio.
Teaching Tip: The following paradigm can be used to measure the repricing gap for a particular
maturity bucket and simultaneously analyze the profitability of the two sensitivity classes:3
1. Classify each asset on the balance sheet as either:
RSA FRA NEA
2. Classify each liability/equity account:
RSL FRL Equity
3. Group assets and liabilities into the following groups:
RSAs financed by RSLs
FRAs financed by FRLs
NEA financed by Equity
Gap: Positive dollar RS Gap: Indicates that excess RSAs financed by remaining FRLs
Negative dollar RS Gap: Excess FRAs financed by remaining RSLs
The leftovers:
Whatever is leftover financed by equity OR Equity financing whatever is leftover
This analysis highlights the idea that the quantity of interest rate risk depends upon the size
of the gap.
4. Calculate the average annual % rate of return on each asset category and the average annual
% cost rate on each liability category and then calculate the spreads. Spreads are the
difference between the income rate and the cost rate per dollar invested in the category.
5. Calculate the dollar contribution to profit from each category as the product of the amount
times the spread.
6. Add up the profits. The banker is now in a position to both understand the major sources of
profitability and compare pricing with other institutions. One can also easily forecast
changes in profitability for various projected changes in interest rates.
This method is illustrated in the example that follows.
The dollar gap for each maturity bucket is measured as the dollar quantity of rate sensitive assets
(RSAs) minus the dollar quantity of rate sensitive liabilities (RSLs). The cumulative gap
(CGAP) is calculated by adding the gaps for subsequent time periods.
For a positive CGAP, rising interest rates over the maturity period will normally increase
profitability, all else equal, and falling interest rates will decrease profitability. In other
3See Gardiner, Mills and Cooperman, Managing Financial Institutions: An Asset/Liability
Approach, Dryden Press, 2000.
words, interest rates and profitability move in the same direction if CGAP is positive.
For a negative CGAP, rising interest rates will decrease profitability, all else equal, and
falling interest rates will increase profitability. Interest rates and profitability move in
opposite directions if the CGAP is negative.
These effects are termed CGAP effects.
Unequal changes in rates on RSAs and RSLs
The repricing gap analysis is more complicated than previously indicated because although the
rates of return on RSAs and RSLs will generally move in the same direction as interest rates
change, they will only rarely move identically. Thus, the spread between the interest income
earned on the RSAs and the interest cost on the RSLs will normally change as interest rates
change.4 The change in income from this category as interest rates change is called the Spread
Effect.
If the Spread Effect is positive, when interest rates either rise or fall the spread of interest
income earned on RSAs less the interest cost on RSLs tends to increase, thereby contributing
to higher NII.
If the Spread Effect is negative, when interest rates rise or fall, the spread of interest income
earned on RSAs less the interest cost on RSLs tends to fall, thereby contributing to lower
NII.
Conclusions about CGAP and Spread Effects:
Dollar GAP Spread Effect R Direction of NII
Positive 1.1.1.6 Posi
tive
1.1.1.7 Incr
ease
1.1.1.8 Incr
ease
Negative Increase Ambiguous
Positive Decrease Ambiguous
Negative Decrease Decrease
Negative Positive Increase Ambiguous
Negative Increase Decrease
Positive Decrease Increase
Negative Decrease Ambiguous
The following tables contain a more detailed example of a calculation of the repricing model for
a 1 year maturity bucket.5
Assets ($ Mill) Liabilities & Equity
4Their correlation is less than +1 for reasons indicated below.
5Detailed examples of this type (from which this example is drawn) can be found in the
aforementioned Gardiner, Mills and Cooperman, Managing Financial Institutions: An Asset
Liability Approach, Dryden Press. 2000. More realistic applications of both the repricing and
duration gap models can be found in Saunders, Financial Institutions Management: A Modern
Perspective, Irwin, 1994.
Investments under 1 year @ 5% $ 100 Deposits < 1 year @ 4% $ 900
Loans < 1 year @ 7% $ 350 All Long Term Liabilities @ 7% $ 500
Variable rate loans
(rate reset in 6 months) @ 6.5% $ 300 Equity $ 200
Fixed Rate Assets > 1 year
maturity @ 8% $ 850
Total $1,600
Total $1,600
The percentages are the average interest rate earned or paid on the given account. All assets and
liabilities that mature in less than one year or have an interest rate reset within one year are
potentially rate sensitive because their income could change if interest rates change.
Rearranging the assets and liabilities into the appropriate sensitivity categories based on maturity
and payment pattern results gives the following results:
Rate Sensitive Assets Rate Sensitive Liabilities
Amnt Income Amnt Cost
Investments under 1 year @ 5% $ 100 $ 5.00 Deposits < 1 year
@ 4% $ 900 $ 36.00
Loans < 1 year @ 7% $ 350 $24.50
Variable rate loans
(rate reset in 6 months) @ 6.5% $ 300 $19.50
Total RSAs $ 750 Total $ 900
Total Income $49.00 Total Cost $ 36.00
NII from this category $13.00
Average rate of return 6.533% Average cost rate 4.000%
Spread on RSAs financed by RSLs 2.533% (6.533% - 4%)
The spread indicates the contribution to profit from this category per dollar invested (ignoring
noninterest income and costs.) Note that some RSLs are used to finance something other than
RSAs since there are only $750 RSAs but $900 RSLs.
Dollar Gap = RSAs – RSLs = -$ 150
Percentage Gap = -$150 / $1,600 = -9.375%
Gap ratio = $750 / $900 = 0.833
The negative dollar gap indicates that some
fixed rate assets are financed by rate sensitive
liabilities.
The ‘gap’ indicates the imbalance in
sensitivities of the liabilities that are funding
the assets.
Fixed rate assets and liabilities
Fixed Rate Assets Fixed Rate Liabilities
Amnt Income Amnt Cost
Fixed Rate Assets > 1 year
maturity @ 8% $ 850 $68.00
All Long Term
Liabilities @ 7% $ 500 $ 35.00
Total FRAs $ 850 Total $ 500
Total Income $68.00 Total Cost $ 35.00
NII from this category $33.00
Average rate of return 8.000% Average cost rate 7.000%
Spread on FRAs financed by FRLs = 1.000% (8% - 7%)
The spread indicates the contribution to profit from this category per dollar invested (ignoring
noninterest income and costs.) Note that only $500 of FRAs are actually financed by FRLs.
$200 FRAs are financed by equity and the remaining $150 FRAs are financed by RSLs.
Notice the GAP FRAs – FRLs
The profit calculations per category can be found as the product of the amount and the spread:
Category Amount Spread $ Profit
RSAs financed by RSLs $ 750 2.533% $19.00
FRAs financed by FRLs $ 500 1.000% $ 5.00
FRAs financed by equity6$ 200 8.000% $16.00
The Gap: FRAs financed by RSLs $ 150 4.000% $ 6.00
NII $46.00
Average rate of return per dollar invested 2.875%
The two categories that are subject to interest rate risk are the categories in bold type: RSAs
financed by RSLs and the Gap, which in this case is FRAs financed by RSLs.7 In each case the
spreads are calculated as the average rate of return for the given asset category less the average
cost rate for the liability category used to finance those assets. Note that equity has a contractual
cost rate of zero so the spread on that category is simply the given asset rate of return. The
spread on the gap is the rate of return on the FRAs minus the cost rate on the RSLs. If the gap
had been positive this spread would be calculated differently (the rate of return on the RSAs less
the cost of the FRLs). The return on equity can be calculated by dividing NII by equity (ignoring
noninterest income and expenses).
Using the calculations: Suppose interest rates increase 100 basis points and the spread effect is a
negative 30 basis points:
Category Amount Spread $ Profit
RSAs financed by RSLs $ 750 2.233% $16.75
FRAs financed by FRLs $ 500 1.000% $ 5.00
FRAs financed by equity8$ 200 8.000% $16.00
The Gap: FRAs financed by RSLs $ 150 3.000% $ 4.50
NII $42.25
Average rate of return per dollar invested 2.641%
The change in ROA is 2.641% - 2.875% = - 23 basis points.
If the spread effect had been positive the profit drop would have been smaller.
The bank could reduce the amount of RSLs and increase the amount of FRLs to minimize the
effect of the rising interest rates. The bank may also wish to focus on RSL accounts that are not
6FRAs financed by equity are not a part of the gap since the assets and liabilities in this category
are both fixed rate. Instructors please be aware that the profit table has to be constructed based
on the size of the given categories. For instance, one will not always include a line where FRA is
financed by equity. If the gap had been positive the third row would have been RSA financed by
equity.
7Had the gap been positive the gap would have been represented by RSAs financed by FRLs.
8FRAs financed by equity are not a part of the gap because the assets and liabilities in this
category are both fixed rate. Instructors please be aware that the profit table has to be
constructed based on the size of the given categories, for instance, it will not always include a
line where FRA is financed by equity. If the gap had been positive the third row would have
been RSA financed by equity.
as interest sensitive and attempt to increase the interest sensitivity of the RSAs to minimize the
negative spread effect. A problem with balance sheet manipulations of this type is that the
customer will normally desire the opposite of what the bank wishes to offer them. That is, in a
period of rising rates customers will desire long term, fixed rate loans (bank FRAs) and short
term or variable rate deposits (bank RSLs) while the bank will desire to offer them short term,
floating rate loans (bank RSAs) and long term, fixed rate deposits (bank FRLs) to maximize the
bank’s Net Interest Margin (NIM) or equivalently, NII.
Problems with the repricing model include:
It is not always clear which category an account belongs in. Demand deposits can now
pay interest, but most banks don’t pay interest on them. This would make them a FRL.
NOW accounts do pay interest but, along with demand deposits, may act like core
deposits which are long term sources of funds. Some would argue that demand deposits
should be included with RSLs because as interest rates rise some holders will switch to
higher paying accounts. Managers must determine customer behavior on these accounts
and categorize them accordingly.
The repricing model (RPM) measures only short term profit changes, not shareholder
wealth changes. As such it suffers from the same problems as the goal of maximizing
profits. In particular the RPM ignores cash flows changes that occur outside the maturity
bucket and ignores the change in current value of future cash flows as interest rates
change.
The maturity buckets are arbitrarily chosen and can be difficult to manage. It is possible
to have a positive 3 month RS gap, a negative 6 month RS gap and a positive 1 year RS
gap. Managing this requires detailed forecasts of interest rate changes over the various
arbitrarily chosen time periods.
All assets and liabilities that mature within the maturity bucket are considered equally
rate sensitive. This is defacto not true if a spread effect exists.
The RPM ignores runoffs. Runoffs are receipts of cash on FRA or payments due on
FRLs that occur during the maturity bucket period.9 This cash must be reinvested by the
intermediary and it is rate sensitive. Runoffs are not calculated in the basic version of the
RPM presented here.
The RPM ignores prepayments. Prepayment patterns are affected by changing interest
rates and are difficult to predict. Prepayments increase with declining rates so assets that
were considered fixed rate may become rate sensitive by being prepaid within the
maturity bucket.
The RPM ignores cash flows generated from off balance sheet activities. These cash
flows are also often sensitive to the level of interest rates, so the RPM underestimates the
interest rate sensitivity of the institution.
Teaching Tip: Many accounts do not have fixed maturities and the classification of RS or FR
must be based on historical turnover patterns and management’s subjective evaluation.
Investors’ desire for liquidity may change as interest rates change, and accounts that were
9Payments on FRLs that require additional borrowing would result in a change in interest
expense on a given account, making it rate sensitive. Similarly, if the bank had to liquidate part
of a fixed rate asset to pay the liability, this would change the income on fixed rate assets.
previously fixed rate may become rate sensitive or vice versa.
b. The Duration Model
Even if a bank could set the repricing gap for all maturity buckets to zero (and the model had no
deficiencies) so that the bank could ensure that a given level of profits would occur no matter
how interest rates changed, the bank could not ensure that the present value of the given profits
would be the same if interest rates changed. If rates increased the present value of the given
hedged future profit stream would decline. Equity value is theoretically equal to the present
value of future profits so in this case the market value of equity would decline if rates rose and
rise if interest rates fell. The market value of equity is also equal to the market value of assets
minus the market value of liabilities. Banks can thus do a better job of managing stockholder
risk and rate of return by estimating how much the value of assets will change relative to how
much the value of liabilities will change when interest rates move. These values can be
estimated by measuring and comparing the duration of the asset portfolio (DA) and the duration
of the liability portfolio (DL). DA is the weighted average of the durations of each asset:
)DX(D
iiA
where Xi is the percentage of total assets invested in asset i and Di is the
duration of the ith asset. DL is calculated similarly.
The accounting identity states that A = L + E or E = A - L where E = Equity, A = total assets
and L = total liabilities.
The percentage change in A and L for a given change in rates is given by (from Chapter 3):
)R1(
R
D
A
A
A
)R1(
R
D
L
L
L
respectively.
Dollar changes in A and L are given by:
)R1(
R
DAA A
)R1(
R
DLL
L
so that
)R1(
R
DL
)R1(
R
DAE
LA
If R and (1+R) are the same for assets and liabilities then E can be simplified as:
)R1(
R
DLDAE
LA
and by multiplying by A / A:
)R1(
R
AkDDE
LA
where k = L / A or the total debt ratio.
{DA – kDL} is termed the duration gap.
Example calculation: Suppose a bank with $500 million in assets has an average asset duration
of 3 years, and an average liability duration of 1 year. The bank also has a total debt ratio of
90%. R may be thought of as the required return on equity (see Gardner and Mills) or perhaps as
the average interest rate level. If R is 12% and the bank is expecting a 50 basis point increase in
interest rates, by how much will the equity value change?
