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John G. Fulmer, Jr.
Vieth Professor and Head
Department of Accounting and Finance
The University of Tennessee at Chattanooga
615 McCallie Ave.
Chattanooga, TN 37403
John-Fulmer@utc.edu
Contents blurb: Constrained resources should not prevent a community bank from effectively analyzing and managing interest-rate risk. Smaller banks can efficiently implement strategies that protect their economic value.
Gregory A. Henry is a CPA and controller of First Volunteer Bank of Tennessee, Chattanooga.
Thomas I. Smythe, Jr., is assistant professor of economics and business administration, Furman
University, Greenville, South Carolina.
John G. Fulmer, Jr., is Vieth Professor and head, department of accounting and finance,
University of Tennessee at Chattanooga.
Gregory A. Henry; Thomas I. Smythe, Jr.; and John G. Fulmer, Jr.
Community banks constantly make decisions involving the trade-off between risk and return.
One of the most important risks faced is interest-rate risk. It is critical that all banks, including community banks, identify, measure, and manage interest-rate risk from both a net-interestincome and economic-value perspective.
The controller or chief financial officer for a community bank, who is frequently charged with interest-rate-risk management, may believe that the bank lacks the financial specialization and computer systems necessary to properly manage this risk. The purpose of this article is to clearly illustrate that this is not the case. As nebulous as interest-rate-risk-management concepts may appear, community banks can implement them with little effort. It is necessary first to identify the issues and then to develop strategies for managing and/or hedging interest-rate risk. This article will include examples that a community bank can use as tools.
Changing interest rates directly affect net-interest income. Mismatches in time to maturity, different dollar amounts maturing at different times, and the variable-rate features of many financial instruments create the potential for changes in interest rates to cause wide variation in net-interest income. Bank controllers need to understand the potential adverse changes caused by the variation and to implement strategies that will reduce this variability.
In addition to the impact on net-interest income, the controller knows that interest-rate variability has a significant impact on the economic value or capital position of the bank. A bank is a collection of interest-earning assets and interest-bearing liabilities. The economic value of the bank depends directly on the interest-rate characteristics of the individual assets and liabilities as well as on the current and projected interest-rate environment. Stated differently, interest-raterisk management has a direct and significant effect on the value of a community bank's equity.
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Interest-rate-risk management in community banks has historically focused on ensuring an adequate net-interest margin, or interest spread, between interest income from loans and securities and interest expenses from liabilities. A clear example of the need for active management in this area is the savings and loan crisis of the 1980s. Interest-rate risk played a critical role, as S&Ls made long-term fixed-rate mortgages and funded them with short-term deposits. When interest rates increased on their liabilities, net losses eroded capital to the point of rendering many S&Ls insolvent. This example, while extreme, illustrates the problem facing many community banks that make fixed-rate loans and fund them with shorter-term deposit accounts. The current interest rate environment offers a contrasting example to that just presented. With the Federal Reserve’s aggressive rate cutting during 2001 that has led to the lowest Fed Funds rate in 40 years, banks relying on floating rate assets have seen margins close rapidly, leading to severely reduced profits.
Gap analysis is the most common method used to measure the impact of interest-rate risk on netinterest income. To use gap analysis, bank managers and/or the asset and liability management committee must specify over what time period the analysis should cover. In doing so, time windows or buckets are specified.
1 The dollar amount of assets and liabilities that are either repricing or have maturing principal during a given time period are categorized in the appropriate time bucket. The repricing gap for each bucket is then computed as the difference between these rate-sensitive assets and rate-sensitive liabilities for that time bucket. The term "rate-sensitive" refers to the cash flows that receive a new interest rate during the time interval in question.
Usually the gap is computed on a cumulative basis as well, where the cumulative gap is the difference between all rate-sensitive assets and rate-sensitive liabilities from the current time bucket plus all prior time buckets. (See Exhibit 1 for an example.)
Using gap analysis in this manner is essentially a balance-sheet model, because it shows whether the bank is asset or liability sensitive. The analysis is more useful when it is converted into an income statement model. By assigning assumed rate changes to maturing and repricing asset and liability balances, the impact on net-interest income can be estimated, thus directly measuring the impact of changes in the balance sheet upon the income statement.
2
The formula to estimate the change in net-interest income for a projected interest-rate change follows:

NII = ($RSA x change in asset rates) - ($RSL x change in liability rates) where

NII is the change in net-interest income;
(1)
$RSA is the dollar amount of maturing and repricing assets;
$RSL is the dollar amount of maturing and repricing liabilities; and the change in asset (liability) rates represents an average change in rates across all assets
(liabilities). (See Exhibit 2 for an example.)
1 Typical buckets include one week, one month, one quarter, and/or one year.
2 "Maturing balances" should include any principal repaid in recurring installment loan and mortgage payments.
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Two points are worth noting. First, for greater precision, the formula can be subdivided for multiple categories of assets (liabilities). By doing so, we move away from the assumption that rates change equally for all categories of assets and liabilities. Second, the analysis assumes rates are annualized, just like the rates on individual asset and liability accounts. Alternatively, banks may be more interested in measuring the impact of interest-rate changes over a period that is not equal to exactly one year. For example, a bank may perform a quarterly evaluation of interest- income sensitivity until the end of the year. To do so, the formula is adjusted by multiplying the rates by n /365, where n is the number of days in the period being examined.
While intuitively appealing, traditional gap analysis suffers from several limitations. First, the necessary data may not be easy to collect and compile. Many bank computer systems include reports of loan and deposit maturities, but compiling the data into a useful format can be tedious.
Second, gap analysis measures the balances that mature or reprice but may not consider other cash flows, such as loan payments and bond interest, that are reinvested at the current rate when they are received. Third, traditional gap analysis (as in formula [1] above) does not consider that the magnitude of interest-rate changes may not be the same for assets and liabilities. A fed fundsfunds rate increase of 0.25% may affect loan rates by 0.25% but deposit rates by more or less than 0.25%. Finally, gap analysis does not consider that interest-rate changes do not occur simultaneously for every balance-sheet item within a time bucket. For example, if a bank were using a quarterly time bucket, repricing will occur at various points within the time bucket, but the analysis implicitly assumes they occur at the same time. This is problematic, because rates are likely to change between actual repricing points.
It is possible to overcome many of these limitations. As demonstrated in Exhibit 2, modifications to the change in net-interest-income formula can overcome the third limitation above. Similar formula changes can be made to account for the timing differences within a time bucket, as can having shorter time buckets. Finally, reasonably priced asset/liability software packages are available that take these factors into consideration.
3 These systems can import detailed data from main computer systems, minimizing the manual effort needed to manage the process.
The gap analysis provides an important starting point for managing the impact of interest-rate changes on net-interest income. As part of the bank's interest-rate-risk-management program, ranges of variation in net-interest income acceptable to the board of directors and/or asset/liability committee can be established. These ranges would normally be expressed in an interest-rate-risk policy statement. By doing sensitivity analysis such as interest-rate changes of plus or minus 0.50% and plus or minus 1.00%, potential violations of bank policy can be identified before they occur. Most of the software asset/liability management packages allow one to perform simulations over a variety of interest rate (yield curve) forecasts. Additionally,
Microsoft Excel also facilitates such analysis.
The most direct way to manage or hedge adverse changes in net-interest income is to use interestrate swaps, which are off-balance-sheet transactions. Because this hedging technique is designed to protect net-interest margins, a swap is normally employed during a volatile or uncertain interest-rate environment. Basically, the bank exchanges one cash flow stream for another with a counterparty at prespecified periods, such as quarterly, annually, or on a contractually agreedupon date (or dates). In practice, the exchange happens as a net cash settlement at the prespecified times. The most common type of swap is to exchange a fixed-rate cash flow stream
3 HNC Software and IPS-Sendero are two well-known PC-based asset/liability management packages in use by community banks. Both import detailed balance and repricing data as well as general ledger data.
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Page 4 of 14 for a variable-rate cash flow stream. Conceptually, this can lock in a net-interest spread over the life of the swap. (See Exhibit 3.)
There are two major limitations to interest-rate swaps. The first is that banks must find a counterparty of acceptable credit risk willing to enter into a swap in the amount and under the terms and maturity it desires. Many regional banks and the Federal Home Loan Bank have capital markets divisions, which can accommodate a variety of swap amounts and terms. The second limitation is basis risk, which is the risk that movements in two different indices may not be well correlated. For example, the bank's average CD or other short-term borrowing rate will not have perfect correlation with the prime rate or Libor rate on which the variable-rate portion of the swap might be based.
Finally, although each participant has protected its net-interest margin, the net cash settlement of the swap at the end of each period can only benefit one party. By locking in a net-interest spread, each bank has given up the opportunity to maximize net-interest income if rates move in its favor. In other words, if rates favor Short Bank, with its variable-rate assets, it still incurs the expense of paying the net swap settlement, thereby lowering an otherwise higher net-interest margin. On the other side, Long Bank will have the additional income of receiving the net swap settlement, which cushions the decrease in net-interest income caused by rising rates with shortterm deposits and fixed-rate loans.
To calculate the dollar or notional amount of the swap needed to hedge the risk, use the formula:
$Swap =
[(Dur
Assets
- ( k x Dur
Liab
)] x Assets
Dur
Fixed
- Dur
Var
(2) where $Swap = the face (notional) value of the swap needed to hedge; k = the ratio of total liabilities to total assets;
Dur
Assets
= Macaulay Duration of assets on balance sheet; 4
Dur
Liab
= Macaulay Duration of liabilities on the balance sheet;
Dur
Fixed
= Macaulay Duration of the fixed-rate swap cash flows;
Dur
Var
= Macaulay Duration of the variable-rate swap cash flows; and
Assets = total assets on bank balance sheet.
The bank may not wish to hedge the entire amount of net-interest income. For example, a partial hedge may be desirable if interest-rate forecasts are unclear or if only a partial hedge is necessary to keep net-interest income variability within policy guidelines for the relevant interest- rate shocks. In these cases, substitute a lower number for total assets in the equation above.
Because community banks traditionally place a priority on managing net-interest income, the potential impact of changes in interest rates on the economic or market value of the institution has been overlooked to some extent. Returning to the example of the S&L crisis in the late
1980s, as interest rates rose dramatically, the value of fixed-rate mortgage loans declined. At some point, the market value of assets declines to the point where they are less than the market value of liabilities, making the value of the bank's equity capital less than zero, thereby rendering
[?] the bank economically insolvent. As this example shows, this could happen long before the net-interest margin becomes negative or cumulative net losses erode the bank's capital.
4 Macaulay Duration is defined and discussed below.
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The best way to measure economic value at risk is with duration gap analysis. Duration is a property of all fixed-income securities. Conceptually, it is the weighted-average time to maturity of the security, where the weights are derived from the timing and size of the instrument’s individual cash flows. More important from a practical perspective, duration is an objective measure of the relative level of interest-rate risk among different securities. Manually calculating duration is a tedious process, but software packages are available to perform the calculations easily. Also, spreadsheet software generally has a function available to compute duration by entering several variables.
5
What makes duration a valuable component of interest-rate-risk management is that when duration is known, asset and liability price or value changes can be reasonably estimated for predicted changes in interest rates. More specifically, the longer the duration, the greater the impact a given change in interest rates has on the price or market value of the securities in question. In addition, a weighted-average portfolio duration can be calculated by grouping assets and liabilities so that a bank can measure the average duration for categories of assets or liabilities or for assets and liabilities as a whole.
Once these portfolio durations are calculated, the duration gap, or duration of the bank's capital, can be calculated as the difference between asset duration and liability duration. However, unlike the static gap income model, a bank's duration gap requires an adjustment for leverage, which is measured as the proportion of assets funded with liabilities. The formula to compute the duration of the bank's capital is related to the basic accounting identity that Assets = Liabilities + Capital and also related to the method for calculating the bank's static gap:
Dur
Gap
= Dur
Capital
= Dur
Assets
- (( k ) x Dur
Liab
) (3) where k = (liabilities / assets).
(See panel A of Exhibit 4 for an example.)
After determining the duration gap, the impact of interest-rate changes on the economic value of the bank's capital can be estimated for various interest-rate shocks. The impact can be measured in both percentage and dollar terms. If we initially assume that any changes in interest rates affect both sides of the balance sheet identically, the estimated percentage change in the bank's equity value can be calculated using the following formula, which assumes some change in interest rates from their current level:
% Change
Capital
= - Dur
Capital
x Change i where i
(1 + i )
= the current market interest rate
Change i
= the expected change in i .
(4)
Alternatively, to estimate the dollar change in equity, modify equation (4) to be:
$ Change
Capital
= - Dur
Capital
x A x Change i
,
(1 + i )
(5)
5 In Microsoft Excel, the function is =Duration, and the input variables are settlement date, maturity date, coupon, yield, and cash flow frequency. In Lotus 1-2-3, the formula is @Duration, and the input variables are settlement date, maturity date, coupon, yield, cash flow frequency, and day-count basis (30/360, actual/365, etc).
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Page 6 of 14 where A is the total assets on the bank's balance sheet. See panel B of Exhibit 4 for a numerical example of equations (4) and (5). From the example in panel B, the conclusion is that the economic value of the example bank's capital is expected to decline by approximately 10.05% with a 1% increase in the relevant market interest rate. A more valuable way to use this analysis at the community bank level is to determine the percentage change in economic value that the bank is willing to risk in a worst-case interest-rate scenario and then work back to the interest change that leads to the worst-case outcome. (See panel C of Exhibit 4 for such an example.)
In panel C of Exhibit 4, the bank in this example could endure an increase of up to 1.58% in the relevant market interest rate before the decline in economic value exceeds the bank's worst-case risk tolerance. Since the benchmark interest rate was 6%, the benchmark rate would have to increase to only 7.6% before policy limits are exceeded. While increases of this size are rare, they do occur as evidenced between February 1999 and January 2000, when the rate on the 10year constant maturity Treasury rose from 5.00% to 6.66%. Based on this analysis, the bank's board of directors and management attains a certain degree of knowledge about the bank's exposure to changes in economic value that result from interest-rate risk.
In the practical realm, changes in interest rates will not usually affect both sides of the balance sheet identically. As such, we can expand equation (5) to account for this likelihood:
Change
Capital
= A x [(Dur
Assets
)(Change i
A
) - ( k ) (Dur
Liab
)(Change i
L
)] (6)
(1 + i
A
) (1 + i
L
) where A = total assets, i
A
= average rate on earning assets, i
L
= average rate on interest-bearing liabilities, and k = liabilities / assets.
The example in Exhibit 4 is continued in panel D by incorporating equation (6). The conclusion is that the market rate increase of 1% causes an estimated decrease of $5.181 million in bank equity value, or 5.181% of its economic value. As is evident under this more realistic scenario, a
1% rate shock leads to a much larger change in equity value.
Since the impact of interest-rate changes upon the bank's economic value is dependent upon the duration of bank capital, it follows that adjusting the duration of capital to within a target range will help insulate the bank from changes in economic value caused by changes in interest rates.
Most bank assets have longer durations than their liabilities. The longer the net duration of capital, the greater the impact of interest-rate changes upon economic value.
In the example in Exhibit 4, Dur
Capital
= 10.65, but assume the bank's risk tolerance indicates that
Dur
Capital
should not be greater than 1.5. Operationally, the bank should either shorten the duration of the assets or lengthen the duration of the liabilities, or both. Conceivably, this could be done in the normal course of bank business by holding more fed funds and less investment securities, making more balloon loans and fewer fully amortizing loans, accepting longer-term time deposits, or borrowing longer-term nondeposit funding. However, the adverse impact of these balance-sheet changes on net-interest margin are immediately evident, as this strategy would normally drive down asset yields and drive up the cost of funding. While conceptually possible, this strategy may not be practical or desirable. Below, we discuss financial hedging alternatives.
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The bank can also shorten the duration of capital by transacting in the futures market. Futures contracts are obligations to deliver or receive securities at a definite point in the future known as the expiration date. In our example, we will use highly liquid Treasury futures, which are available with quarterly expirations extending out one year. Treasury bond contracts trade in blocks of $100,000 on the Chicago Board of Trade. Since the contracts are traded on a national exchange, there is an active market and virtually no counterparty default risk. Since an actual
Treasury bond underlies the futures contract, the contract's price is determined by the interest rate/price of the underlying security and current market interest rates. Stated differently, the price behavior of a futures contract is similar to that of the underlying Treasury security. Price quotations for the contracts are available daily in the Wall Street Journal . Prices are expressed as a percentage of face value, similar to bonds. Futures contracts can be bought or sold at any time prior to expiration, but positions are typically closed out before expiration by taking the opposite position. By doing so, no delivery of the underlying Treasury security occurs.
If a hedge is properly executed, the value of the futures contract rises at the same time and by approximately the same amount that the economic value of the bank falls. The typical community bank has a positive duration gap with assets having longer maturities than liabilities, and its economic value decreases when rates rise. To hedge against this possibility, the bank should take a position in the futures market, which increases in value when rates rise.
Typically, bank transactions involve buying a security before selling the same security, known in trading vernacular as being long. This strategy is appropriate when prices are expected to rise
(rates are expected to fall). However, it is also possible to sell a security that is not yet owned and then purchase it later and deliver what was already sold.
6 This process, known as being short a security, illustrates the process of selling a futures contract. By selling a futures contract, a bank receives cash today in exchange for the obligation to deliver a security at the expiration date of the contract in the future. This strategy is appropriate when security prices are expected to fall
(rates are expected to rise). The typical community bank wishing to hedge the bank's capital position requires a futures position that increases in value when rates rise (prices fall). As such, they would sell one or more futures contracts. The contract will be closed out financially before expiration by entering the futures market and buying the same contract that was originally sold.
In this way, no underlying security is delivered.
It is important to recognize that the best futures hedging vehicle is a contract for a security with duration similar to that of the bank's capital. For any movement in interest rates, the price behavior of the futures contract will be similar to that of the change in economic value of the bank's capital. A closely matched duration helps to achieve the objective of offsetting decreases in equity value with increases in the value of the futures contract.
The question now turns to calculating the total dollar amount of the hedge and how many contracts should be bought or sold. In Exhibit 4, the example bank is estimated to lose $5.18 million in economic value for a benchmark interest-rate increase of 1%. The hedge should be constructed so that the increase in value of the futures contract will be $5.18 million. Since the bank's equity value falls when rates rise, the hedge position's value should rise when rates rise.
6 It may be helpful to think in terms of purchasing a stock. If the price were expected to rise, one would purchase it first and sell later. On the other hand, if a stock price were expected to fall, one could "borrow" the stock from a broker, sell it first, and then later buy it back at a lower price and deliver it, a process known as "short-selling."
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To achieve this, Treasury bond contracts must be sold. To determine the number of contracts to sell, use the following formula:
(7) N f
= (Dur
Assets
- ( k x Dur
Lib
)) x Assets
Dur f
x P f where Dur f
= the duration of the futures contract and
P f
= the price of the futures contract and see Exhibit 5 for a numerical example.
From Exhibit 5, the bank should sell 2,080 Treasury contracts at the current price of $95.14. If the expected interest-rate increase occurs, the value of the underlying Treasury bond will decrease, as will the price of the futures contract. The bank will then enter the futures market again prior to expiration and buy 2,080 contracts at a price below $95.14. This closes out the futures position, and the gain from the transaction approximately offsets the decrease in equity value resulting from the interest-rate increase.
Two things should be noted with regard to hedging with futures contracts. First, the change in hedge value will not equal, but approximate, the change in equity value. This is because the price of the underlying security of the futures contract will not move exactly in tandem with changes in equity value. This error is known as basis risk. Second, like the case of hedging net-interest income with swaps, hedging with futures contracts is a zero sum game. While the bank is protected against downside risk, it cannot participate in gains associated with favorable interest- rate changes, interest-rate decreases in our example.
An alternative way to manage economic value at risk is by hedging with options on Treasury securities. As the name implies, an option gives its owner the option, but not the obligation, to buy or sell a specified Treasury security at a stated price (the exercise or strike price) at any time up to and including the expiration of the option. Options trade on a recognized national exchange, and price quotes are available daily. The option price is known as the option premium.
An option transaction may be thought of as an insurance policy. For a relatively small price, the option premium, the option owner (long position) is protected from downside risk while maintaining unlimited upside potential, unlike hedging with futures contracts. The concept of hedging with options is similar to that of hedging with futures. The objective is that gains on the option transaction offset losses of economic value caused by interest-rate changes. However, in contrast with futures, if interest-rate changes cause an increase in economic value, the loss on the option transaction will not be realized because the owner will allow the option to expire unexercised, limiting losses to the option premium.
While hedging with options would appear to be a fruitful alternative to using futures contracts, regulatory and tax scrutiny make them less appealing. Options fall within deferred hedge accounting rules and are often frowned upon by regulators as a hedging instrument, primarily when they are sold. As such, a detailed methodology and example are not provided.
All banks, including community banks, must deal with interest-rate risk and its impact on the bank. The controller or chief financial officer of a community bank may believe that the bank does not have the financial or computer capabilities necessary to cope with this risk. Further, he or she may wonder if the benefits derived from managing such risk are worth the time required.
However, if one ignores interest-rate risk and its effects, the results can be devastating. For
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Page 9 of 14 example, in Exhibit 4, a market-rate increase of 1% causes a 5% decrease in the bank’s economic value.
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Exhibit 1
Repricing Gap Buckets
Assume a bank's simplified balance sheet below:
Assets
$5 cash and reserves
$3 in fed fundsfunds
$5 in 3-month T-bills
$5 in 1-year T-bills
$5 in 10-year T-notes
$15 in fixed-rate loans
$15 in ARMs [spell out?]
Liabilities + Equity
$6 demand deposits
$10 in 6-month retail CDs
$5 in 3-month Jumbo CDs
$7 in fed fundsfunds
$15 in 1-year FHLB loans
$2 in 7-year subordinated debt
$2 other assets
$55
$10 in equity
$55
If we use the one-week, one-month, one-quarter, and one-year time frames, the corresponding repricing gaps are:
One Week
$3 fed funds (A) - $7 fed funds (L) = -$2
One Month
$3 fed funds (A) - $7 fed funds (L) = -$2
One Quarter
[$3 fed funds + $5 3-month T-bills] - [$7 fed funds + $5 3-month CDs] = -$4
One Year
[$3 fed funds + $5 3-month T-bills + $5 1-year T-bills + $15 ARMs]
[$7 fed funds + $5 3-month CDs + $10 6-month retail CDs] = $6
In the example as it is presented, we assume that all principal from the ARMs reprices after one quarter. In reality, any assets that have intermittent principal payments, such as ARMs and fixed-rate installment credit, would be included in the repricing assets/liabilities.
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Exhibit 2
Example of Estimating Change in Net-Interest Income
Using the assets and liabilities from Exhibit 1 and assuming the one-year gap, assume the Fed is expected to announce an increase of 0.25% in their target for the fed funds rate at the beginning of the time bucket.
Under traditional gap analysis, the impact of the interest-rate increase is assumed to be the same for all assets and liabilities.
($RSA x change in asset rates) - ($RSL x change in liability rates)
$3 in fed funds x 0.25% = 0.0075
$5 in 3-month T-bills x 0.25% = 0.0125
$5 in 1-year T-bills x 0.25%
$15 in ARMs x 0.25% = 0.0375
0.0700
= 0.0125
$10 in 6-month retail CDs x 0.25% = 0.0250
$5 in 3-month Jumbo CDs x 0.25% = 0.0125
$7 in fed funds x 0.25% = 0.0175
0.0550
The estimated change in net-interest income = 0.07 - 0.055 = $0.015
Moving away from the assumption that the interest-rate impact is identical for both sides of the balance sheet or even different asset (liability) classes, assume the bank expects ARM rates to increase by 0.20% and retail CDs by 0.10%.
($RSA x change in asset rates) - ($RSL x change in liability rates)
$3 in fed funds x 0.25% = 0.0075 $10 in 6-month retail CDs x 0.10% = 0.0100
$5 in 3-month T-bills x 0.25% = 0.0125
$5 in 1-year T-bills x 0.25% = 0.0125
$5 in 3-month Jumbo CDs x 0.25%
$7 in fed funds x 0.25%
$15 in ARMs x 0.20% = 0.0300 0.0400
= 0.0125
= 0.0175
0.0625
The estimated change in net-interest income = 0.0625 - 0.04 = 0.0225
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Exhibit 3
Swap Cash Flows
To illustrate a swap, consider the following two banks:
Short Bank
Loans: Variable/Short Deposits: Long Loans: Fixed/Long
Long Bank
Deposits: Short
Short Bank experiences a decrease in net-interest income when rates fall since liabilities have fixed rates while asset rates vary. On the other hand, Long Bank experiences a decrease in net-interest income when rates rise. To hedge against this possibility, the banks enter into a swap arrangement whereby Short Bank pays a variable-rate cash flow to Long Bank in exchange for a fixed-rate cash flow. Instead of one inflow from earning assets and one outflow for deposits, both banks have two cash outflows, one for deposits and one for the swap, and two cash inflows, one from earning assets and one from the swap.
The objective is to receive cash inflows from the swap that mirror the cash outflows on actual deposits and borrowed funding (fixed in / fixed out, variable in / variable out). The bank then pays cash outflows for the swap, which mirror the cash inflows received from actual earning assets. The result is that cash inflows received from earning assets are on a similar basis as the net cash outflows, thereby locking in a net-interest spread. The following table illustrates this point.
Short Bank
Outflow from deposits - Fixed
Inflow from swap + Fixed
Outflow from swap
Net cash outflow
Inflow from assets
- Variable
- Variable
+ Variable
Long Bank
- Variable
+ Variable
- Fixed
- Fixed
+ Fixed
Because of the swap, Short Bank has locked in variable-rate funding, which matches its variable-rate assets, protecting net-interest income if rates fall. On the other hand, Long Bank has locked in fixed-rate funding, which matches its fixed-rate assets, protecting net-interest income if rates rise.
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Exhibit 4
Duration Gap Analysis Example
Panel A: Calculating Duration of Capital (Duration Gap)
Assume a bank has a duration of assets equal to 12.0 and a duration of liabilities equal to 1.5, and a 10% capital to assets ratio.
7 Then:
Dur
Capital
= Dur
Assets
- (k) x Dur
Liab
Dur
Capital
= 12.0 - (0.90) x 1.5
Dur
Capital
= 10.65
Panel B: Calculating Percentage and Dollar Changes in Bank Equity Value
Assuming the bank's current interest rate is 6% and the relevant rate shock is an increase of 1%, we get the following result using equation (4):
% Change

Capital
= - Dur
Capital
x Change i
1 + i
% Change

Capital
= - 10.65 x 0.01 / (1.06)
% Change

Capital
= - 0.1005 = -10.05%
OR
If the bank has $100 million in assets,
$ Change

Capital
= - A x Dur
Capital
x Change i
(1 + i )
$ Change

Capital
= -100mill x 10.65 x 0.01 / (1.06)
$ Change

Capital
= -$10.05
Panel C: Maximum Percentage Capital Loss
Assume the maximum reduction in economic value that the bank's management is willing to accept under any circumstances is 15%. Rearranging the above equation:
Change i
= % Change

Capital
x (1 + i )
- Dur
Capital
Change i
= - 0.15 x (1.06)
- 10.05
Change i
= 0.0158 = 1.58%
Panel D: Capital Loss with Unequal Changes in Asset and Liability Rates
Assume the bank has $100 million in total assets, the weighted-average yield on earning assets is 9.0%, the weighted-average cost of funds is 4.25%, the bank funds assets with 90% liabilities, and that the rate shock of 1% is expected to increase asset yields by 0.50% and funding costs by 0.25%. Then:
$Change
Capital
= A x [(Dur
Assets
)(Change i
A
) - (k) (Dur
Liab
)(Change i
L
)]
(1 + i
A
) (1 + i
L
)
$Change
Capital
= -$100 mill x [(12) (0.005)] - [(0.90) (1.5) (0.0025)]
(1.09) (1.0425)
$Change
Capital
= - $5.181 mill
7 Note that this is the basic balance-sheet capital/assets ratio and not risk-based capital as calculated for regulatory reporting.
Bank Accounting & Finance Filename: 726852349 4/11/2020 8:27 AM
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Page 14 of 14
Exhibit 5
Calculation of Number of Futures Contracts
T-bond contracts trade in blocks of $100,000. Assume a price quote on this contract is 95:05 = 95 5/32 =
95.1429, or 95.1429% of face value. Then, P f
= 0.951429 x 100,000 = $95,142.9. Also assume that D f
=
5.3 years. Then the number of contracts to sell is:
N f
=
=
[12 - ((.9)(1.5)] x $100 mill
5.38 x $95,142.9
2,080.60 contracts (always round down).
Bank Accounting & Finance Filename: 726852349 4/11/2020 8:27 AM
5494 words