Bank gap management and the useof financial futures

Elijah Brewer

Interest rate "gap" management has be-come an increasingly important part of bankfunds management over the past decade. Thismanagement technique matches liabilities toassets of similar maturity lengths and riskclasses.

As interest rates have become more vola-tile and have climbed to historically un-precedented high levels, the degree to whichvariable-rate assets are different from variable-rate liabilities (or, in other words, the amountof variable-rate assets supported by fixed-ratefunds) has caused concern. This "gap"—reallyan imbalance—measures the exposure of banknet interest margin, that is, interest income lessinterest expense, to unexpected changes inmarket interest rates.

Such changes can result in gains or lossesin a bank's portfolio. Losses result if the bankfinances its fixed-rate long-term loans with rel-atively short-term funds and market interestrates rise. Losses also occur if relatively fixed-rate longer-term funds are used and lendingrates fall. Gains can be made if interest ratesmove in the other direction. A bank, then, isexposed to interest rate risk whenever there isa quantitative imbalance between its fixed-rateliabilities and its fixed-rate assets of the samematurity.

Bankers have recognized the importanceof gap management in reducing interest raterisk and achieving acceptable bank perfor-mance.' Furthermore, bank regulators arepaying increased attention to a bank's gap po-sition. They are concerned that exposed assetand liability positions could threaten the prof-itability of some banks and, therefore, theircapital positions if interest rates should moveadversely. Controlling the size of the gap is animportant function of bank funds managementand managers are now using financial futurescontracts to hedge exposed asset and liabilitypositions.

To what extent can bank profits be sta-bilized by trading financial futures? To whatextent are bank futures trading decisions con-strained by the regulatory requirement that

futures positions represent bona fide hedges ofinterest rate exposure? This paper providessome insight into how financial futures can beused as vehicles for reducing interest rate ex-posure and managing the gap position, andmay aid regulators in their supervision of bankuse of these instruments.

The basic funds gap concept

In a typical gap management process,bank management is asked to dichotomize allitems, both assets and liabilities, on the balancesheet according to interest rate sensitivity. Anasset or liability with an interest rate subject tochange within a year is considered variable.One that cannot change for more than a yearis considered fixed. The imbalance betweenfixed-rate liabilities and fixed-rate assets is agap that can be expressed either as dollars oras a percentage of total earning assets. Iffixed-rate liabilities exceed fixed-rate assets, thebank has a positive gap. Under rising short-term interest rates, this positive gap would in-crease net interest margin. But decliningshort-term interest rates, with a positive gap,would exert downward pressure on net interestmargin.

If fixed-rate liabilities are less than fixed-rate assets, there would be a negative gap.With a negative gap, net interest margin woulddecline if short-term interest rates rose and in-crease if short-term interest rates fell.

Table 1 presents the gap position (thedifference between rate-sensitive assets andrate-sensitive liabilities divided by total assets)in the fourth quarter of 1983 and the first threequarters of 1984 for twenty large domesticbanks in the United States. During theDecember-September period, the 20-banksample became generally more liability-sensitive. The rate sensitivity gap as a percentof total assets in the third quarter of 1984ranged from — 12.1 percent at Bank of America

Elijah Brewer is an economist at the Federal Reserve Bankof Chicago.

12 Economic Perspectives

Table 1Rate sensitivity gap as a percentage of total assets*

twenty large banks

1983 1984Fourthquarter

Firstquarter

Secondquarter

Thirdquarter

Bank of America -11.4 -10.5 -13.8 -12.1

Bank of New York 6.6 7.2 5.8 0.8Bankers Trust Company 3.9 5.9 -1.1 1.7

Chase Manhattan Bank 9.0 -2.7 -3.5 -5.0

Chemical Bank 1.9 2.2 -2.8 -1.0

Citibank -1.8 -3.1 -2.9 -3.6

First Interstate Bank, California -1.8 -1.8 -0.3 -0.2

First National Bank of Boston -0.9 -2.5 -1.1 -0.4

First National Bank of Chicago -4.1 -9.0 -6.9 -8.6

Interfirst Bank, Dallas -4.2 -5.3 -2.5 -5.0

Irving Trust Company -4.0 2.5 4.7 2.0

Manufacturers Hanover Trust Company 5.9 5.4 4.6 2.7

Marine Midland Bank -1.9 -7.5 -5.8 -4.2

Mellon Bank -3.2 2.0 4.3 4.2

Morgan Guaranty Trust Company -2.4 -1.4 -0.7 -4.0

National Bank of Detroit 0.8 -0.2 0.7 -1.0

North Carolina National Bank -2.1 3.2 2.2 2.2RepublicBank, Dallas 2.4 3.6 1.2 2.1

Security Pacific National Bank -4.1 -4.7 -4.6 -6.8Wells Fargo Bank -1.9 -5.7 -5.9 -4.9

Average -0.3 -1.1 -1.4 -1.9

'One-year rate sensitivity gap.

Rate-sensitive assets include all assets repricing or maturing within one year and comprise loans and leases, debt security, andother interest-bearing assets.

Rate-sensitive liabilities are all those liabilities scheduled to reprice or mature within one year and include domestic time certif-icates of deposits of $100,000 or more, all other domestic time deposits, total deposits in foreign offices, money market depositaccounts, Super NOWs, and demand notes issued to the U.S. Treasury.

Source: Salomon Brothers, "Bank Analysts Rate Sensitivity Quarterly Handbook First Quarter 1984," July 27, 1984 and "BankAnalysts Quarterly Handbook Third Quarter 1984," January 29, 1985. The use of these figures does not constitute an endorse-ment of these estimates or the underlying methodology by the Federal Reserve System.

to 4.2 percent at Mellon Bank, compared witha range between -11.4 percent at Bank ofAmerica and 9.0 percent at Chase ManhattanBank in the fourth quarter of 1983.

Controlling the size of gaps such as thosein Table 1 is an important function of bankfunds management. To keep from relying toomuch on short-term funds, banks set a limit onthe use of variable-rate liabilities to financefixed-rate long-term assets. Thus, while federalfunds are a constant source of funds for some

banks, their use to finance fixed-rate long-termassets-with their potential for exposing banksto interest rate risk-is limited to a permissiblerange by, say, the ratio of variable-rate assetsto variable-rate liabilities.

The size of the gap has a major influenceon the volatility of earnings. If, for example,all variable interest rates changed 1 percent, a30 percent gap would have a $6 million effecton pretax earnings of a bank with $2 billion in

Federal Reserve Bank of Chicago

assets. The acceptable size of the gap, then,varies with a bank's interest rate expectations.

The tendency, of course, is for banks ex-pecting higher interest rates to accept largepositive gaps, with the plan being to reverse thegap before interest rates turn down. But be-cause demand for short-term loans is usuallyheaviest when interest rates are highest, mostbanks cannot close large gaps when they wantto. For banks expecting lower interest rates,the appropriate strategy would involve accept-ing negative gaps.

The gap, then, indicates the extent towhich banks have used fixed-rate liabilities tofund variable-rate assets. The larger this im-balance the more exposed the bank is tointerest-rate risk; the closer to zero this imbal-ance, the better off the bank is with regard tointerest rate risk. Such a gap, however, showsnothing of a bank's assets and liabilities thatare repriced within the gapping period.' Allthat matters with the "basic" gap approach isthat repricing occurs during the gapping pe-riod; it does not matter when during the periodthe repricing occurs. For example, suppose thegapping period is one year and all the rate-sensitive assets are repriced on day 1, while allthe rate-sensitive liabilities are repriced on thelast day of the year. If variable-rate assetsequal variable-rate liabilities, the gap mea-surement would show incorrectly that the bankportfolio is hedged against unexpected changesin market interest rates.

Maturity bucket approach

The maturity bucket approach attemptsto solve the intraperiod problem by measuringthe gap for each of several subintervals of thegapping period. Balance sheet items aregrouped in a number of maturity "buckets"; forexample, one day, one to three months, threeto twelve months, one to five years and so on.Balance, or maturity, gaps, are computed foreach bucket. These separate dollar gap valuesare called incremental gaps and they algebra-ically sum to the total that is measured by thebasic funds gap model.

Asset and liability positions can behedged by setting each incremental gap equalto zero. If rates are expected to rise, positivegaps should be put into place; the oppositeholds for expected rate declines. The use ofincremental gap rather than the basic funds

gap model increases the probability that netearnings will turn out to be as expected.

One of the drawbacks of this technique,as well as of the basic funds gap concept, is thatit assumes interest rate changes for assets andliabilities of all maturities are of the samemagnitude. There is overwhelming evidencethat interest rate changes occur in varyingmagnitude. 2 The gap literature has handledthis issue of different interest rate change mag-nitudes by assuming that the volatility of theinterest rates in question is in constant propor-tion to the volatility of some standard interestrate.

The standardized gap

The standardized gap is a concept thatadjusts for the relative volatilities of variousinstruments. A more volatile interest rate fi-nancial instrument has a greater impact on in-come when it is reset, so it should contributemore to the standardized gap than other, lessvolatile, interest rate financial instruments. Inthe gap literature, historical interest ratechange data on various rate-sensitive assets andliabilities are used to estimate interest ratechange proportionalities. These proportionalfactors measure the rate volatility of rate-sensitive assets and liabilities relative to astandard of account. Consider for example thebank depicted in Figure 1. If the rate-sensitiveliabilities are $500 and the rate-sensitive assetsare $200, there is a naive gap of — $300. Butsuppose the rate-sensitive liabilities are treatedas $500 in 90-day large certificates of deposit(CDs) and the rate-sensitive assets as $200 in30-day commercial paper and the 90-day largeCD rate is 105 percent as volatile as the yieldof 90-day Treasury bill futures while the 30-daycommercial paper rate is 31 percent asvolatile. 3 Then the standardized gap is — $463.(The $500 in 90-day large CDs is equivalent tothe volatility of 1.05 x $500 = $525 in 90-dayTreasury bill futures. The $200 in 30-daycommercial paper is equivalent to the volatilityof 0.31 x $200 = $62 in 90-day Treasury billfutures. The standardized gap position is$200 x 0.31 — ($500 x 1.05 = — $463).) 4

Now let the rate-sensitive liabilities be6-month money market certificates of deposits(MMCs). Dew has estimated that the yield of6-month MMCs was 185 percent as volatile asthe yield of 90-day Treasury bill futures con-

14

N ono mic Per.spec t

Figure 1Rate sensitivity gape

Assets

Liabilities

Money marketdeposit accounts

Variable rate CDs

Federal fundspurchased

Other nondepositfunds

Floating rate notesand debentures

Fixed-ratebusiness loans

Long-term U Sgovernments

Long-term stateand local

securitiesFixed rate

other loans

Other Fixed-rateassets

•There can be some trade-off between maturity and fix versusvariable rate instruments on bank balance sheets.

tracts. The standardized gap of the abovebank whose rate-sensitive liabilities are6-month MMCs is — $863. The naive gap, inboth cases, remains — $300.

Therefore, a bank that has morevariable-rate liabilities than variable-rate assetsand whose variable rate liabilities are, say,90-day large CDs, has a different exposure torising rates than one whose variable-rate li-abilities are 6-month MMCs. This is becausevarious assets and liabilities of different matu-rity have different sensitivities to changes ininterest rates. By taking into account relativeinterest rate volatilities, the standardized gapincreases the probability that net earnings willturn out to be as expected.

The best benchmark

There are several factors to consider inchoosing the benchmark to use in estimatingthe effective contribution of money market in-struments to the standardized gap. First, therelationship between the benchmark rate andother interest rates affecting the net interestmargin of the institution should not vary sub-stantially with the passage of time, since thecontribution of other instruments to the rateexposure of the firm has been based on thehistorical relationship between the benchmarkrate and those other interest rates.

One property that would make abenchmark rate desirable from this point ofview is that it should have a maturity as closeas possible to the average maturity of all otherinstruments affecting the incremental gap posi-tion. This will minimize the impact , of shifts inthe slope of the yield curve on the accuracy ofthe estimated relationship between thebenchmark rate and the other interest rates af-fecting the standardized gap.

A second way to assure the reliability ofestimates of the standardized gap is to choosea benchmark rate that is market-determined.Administered rates may change their relation-ships to predominantly market-determined in-terest rates found on the balance sheets offinancial institutions. Therefore, it seems rea-sonable to avoid the prime rate, the FederalReserve discount rate, and perhaps the federalfunds rate, in choosing a benchmark. Thecurrent gap literature recommends that finan-cial futures contracts be used as benchmark in-struments because futures rates aremarket-determined and the contracts them-selves are useful in adjusting the gap positionin the direction desired by the bank. If thecalculation of the standardized gap yields apositive number for a given month, the firm isasset-sensitive 5 and therefore should go long in,say, 90-day Treasury bill futures for delivery inthat month or the month nearest it. If thecalculation yields a negative number, the firmis liability-sensitive and therefore should goshort in 90-day Treasury bill futures.

In interpreting the standardized gap con-cept, as well as the other gap concepts, it itimportant to remember that it does not meas-ure the interest rate risk resulting from the ef-fect of changes in interest rates on presentvalues of cash flows and periodic principalpayments of assets and liabilities.

Duration gap model

Duration, a concept first introduced byFrederick R. Macaulay, has recently been usedin the gap literature to measure interest raterisk resulting from the effect of changes in in-terest rates on present values of cash flows andperiodic principal payments of assets and li-abilities. Duration is defined as the period oftime that elapses before a stream of paymentsgenerates one-half of its present value.

Floating ratebusiness loans

Federal fundssold

Variable rateconsumer loans

Short-termU.S. governments

Short-term stateand localsecurities

Variable-rate20%

Variable-rate50%

Gap30%

Demand deposits

Other savingsdeposits

Fixed-rateconsumer-typedeposits

Fixed-rate notesand debentures

Equity capital

Federal Reserve Bank of Chicago

Conceptually, duration is computed byweighting the present value of each future cashflow by the number of periods until receipt ofpayment and then dividing by the current priceof the security, or

E t PV(Ft)t=1

ZPV(Ft)t=1

where D is duration, t is length of time (numberof months, years, etc.) to the date of payment,PV(Ft) represents the present value of payment(F) made at t, or Ft • i represents the appro-0 +, NE .priate discount rate and is the summationfrom the first to the last pL.Vment (N).

Duration is an important measure of theaverage life of a security because it recognizesthat not all the cash flow from a typical securityoccurs at its maturity. Duration of a stream ofpositive payments is always less than the timeuntil the last payment or maturity, unless thesecurity is a zero-coupon issue, in which caseduration is equal to maturity. Duration ex-presses also the elasticity of a security's pricerelative to changes in the interest rate andmeasures a security's responsiveness to changesin market conditions.

Consider the extreme case of two banks,each holding loans with ten-year terms to ma-turity and with identical yields to maturity.Bank A loans make no interest payments dur-ing the term of the contract and return theirface value at the end of the ten-year period.Bank B loans make 6 percent interest paymentsper year for each of the ten years. Further,assume that the two banks purchased the loansduring a period when the yield curve was risingand the loans are funded with 7-1/2 year zero-coupon deposits. Thus, the interest rate on thefinancing is lower than the interest rate on theloans. A summary of these conditions and ananalysis of the banks' exposure to interest raterisk are presented in Table 2.

Bank A is more exposed to interest raterisk than Bank B. The average term-to-maturity per dollar of payment stream for theinterest-payment loan is approximately equalto that for the deposit. That is, the durationor the "true" term to maturity of Bank B'sinterest-payment loan is less than 10 years be-cause the bank is getting its funds back fasterwith the interest-payment loan. Fisher and

Weil estimate the duration of the 6 percentloan to be 7.45 years. 6 Since the durationequals maturity for zero-coupon instruments,the duration of 7.50 years for Bank B's depositis approximately equal to the duration of theloan so that the bank is hedged against unex-pected changes in interest rates. On the otherhand, the term to maturity and duration areten years for Bank A's noninterest-paymentloan. As a result, Bank A is exposed to gainsor losses from unexpected changes in interestrates because the duration of its assets is greaterthan the duration of its liabilities.

Banks, then, can hedge against uncertainfluctuations in the prices and yields of financialinstruments by managing their loans and in-vestments so that the duration composition oftheir asset portfolio matches the duration com-position of their liabilities. Because of the typ-ically short duration of banks' liabilities andthe traditional emphasis on liquidity, they oftenprefer to hold short-duration to medium-duration assets.

If a bank accepts a liability, say, a depositof short duration, it can offset that liability bylending for the same duration. In theory, cashflows from the asset can be used to pay off thedebt coming due at the same time. The bankis, presumably, content to make its profit on thespread between the interest rate paid on the li-ability and the rate charged on the loan.

To the extent, however, that banks try tomatch the duration of an asset with the dura-tion of a liability, they may give up opportu-nities for profits because asset duration does notfit into the duration structure of the existingportfolio. Although the duration of the loanmay initially be equal to that of the liability,it may not remain so over the life of the loan.As the loan ages, its duration may change at adifferent rate than that of the liability fundingit. So the bank will be exposed to interest raterisk.

Furthermore, duration will be accurateonly if the yield curve is presumed to shift in aparallel fashion—i.e., where the slope of thecurve remains flat. The assumption of a stableyield and slope is unrealistic since normal in-terest rate movements involve greater fluctu-ations in short-term than in longer-terminterest rates. Despite these shortcomings, theapplication of duration analysis to gap man-agement helps improve bank understanding ofinterest rate risk.

(1)

Economic Perspectives

Table 2Analysis of bank exposure to interest rate risk

Bank A

Assets

Duration of the ten-year maturityloan is ten years since itis a zero-coupon instrument.

Liabilities

Duration of the 7.50-yearzero-coupon deposit is 7.50 years.

Because the duration of its assets is greater than that of its liabilities, Bank A is exposed to gains or losses fromunexpected changes in interest rates. That is, when interest rates move, capital value of the loan will move morethan that of the deposit.

Bank BAssets

Duration of the ten-year 6 percentcoupon loan is 7.45 years.

Liabilities

Duration of the 7.50-yearzero-coupon deposit is 7.50 years.

Since the duration of the loan is approximately equal to the duration of the deposit, Bank B is protected againstunexpected movements in interest rates.

In a typical gap management process, thebank attempts to protect net interest incomeagainst unexpected changes in interest ratesover some gapping period. One year is usuallychosen for this gapping period. Expected netinterest income over the gapping period can beexpressed as

NII = RSA[(1 + 2,;,a)Trsa(1 Kna)(1—Trsa) —1] (2)— RSL[(1 + r„1)Trs1(1 Kr,i)(1—Tr51) —1]

where T„„(T„,) is the length of time (fractionof a year) to the date of payment of the rate-sensitive asset (liability); RSA (RSL) is therate-sensitive asset (liability) book value at thebeginning of the year of a single cash inflow(outflow) that will occur at timeTrsa(Trs); rrsa(rrs1) is the rate-sensitive asset (li-ability) contractual interest rate; and Krso (K,I)is the rate-sensitive asset (liability) expectedinterest rate upon any repricing during thegapping period.

It can be inferred from equation (2) thatnet interest income will be protected againstunexpected changes in interest rates providedthat the weighted market value of the rate-sensitive asset equals the weighted market valueof the rate-sensitive liability where the weightsare equal to the fraction of the year from re-pricing to the end of the a year. 7 Since both theasset and liability are single-payment instru-ments, duration is equal to maturity expressed

as fractions of a year. The duration of therate-sensitive asset, Drsa , is 7,,„ and that of therate-sensitive liability, D„„ is Trsl.

The duration gap (DG) that measures theexposure of net interest income to unexpectedchanges in interest rates can be defined mostsimply as the difference between the presentvalue of rate-sensitive assets times one minustheir duration and the present value of rate-sensitive liabilities times one minus their dura-tion', or

DG = MVRSA (1 — Drsa)— MVRSL(1 — D„,)

where MVRSA and MVRSL are the presentvalues of rate-sensitive assets and liabilities, re-spectively.

The sign of DG indicates the type of raterisk to which the bank is currently exposed.The larger DG is in absolute value, the greateris the risk. If the calculation of DG yields apositive number, then the bank is asset-sensitiveand exposed to falling interest rates. If thecalculation yields a negative number, then thebank is liability-sensitive and exposed to risinginterest rates. The duration gap thus definedyields a single-valued risk index that is not onlyconvenient but at least as accurate an indicatorof risk as the risk level derived from the matu-rity bucket approach.

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Federal Reserve Bank of Chicago -17

Financial futures reduce bank exposure

Financial futures markets give banks achance to hedge exposed asset and liability po-sitions. The primary function of futures mar-kets is to transfer the risk of commodity pricechanges to speculators who are willing to takethe risks. Financial futures provide protectionagainst losses from unexpected adverse pricechanges by enabling participants to lock into afuture price, currently quoted in the futuresmarket.

A futures contract is a standardized con-tract which establishes, in advance, the pur-chase (and sale) price of a commodity fordelivery and settlement at a fixed future time.The futures price embodies the market's ex-pectations of the spot price of the item that willprevail at the time of delivery.'

Hedging involves taking a position in thefutures market that is equal and opposite to acurrent or a planned future position in the spotor cash market. Therefore, regardless of themovement in prices, losses in one market willbe offset by gains in the other. A successfulhedge requires that cash market prices and fu-tures market prices move in the same direction.The difference between the prices in the twomarkets is called the basis.

The hedge would be perfect if the basisdid not change—that is, if the futures and cashprices moved in the same direction by the sameamount. In real life, the basis rarely remainsconstant. 10 Hedgers watch for changes in basisrisk, that is, in the relationship between futuresand cash prices that could expose them to a lossor gain.

A bank can hedge the interest rate riskcaused by the mismatch in the duration of thefirm's assets and liabilities. When a negativeduration gap exists, a normal bank responsewould be to extend the duration of liabilitiesor reduce the duration of the assets. But alter-natively, financial futures could be sold tohedge this exposure. When a positive durationgap exists, a normal bank response would be toextend the duration of the assets or reduce theduration of the liabilities. But a banker alsocan hedge this asset-sensitivity by buying fi-nancial futures.

Consider the case of a bank whose netasset and liability positions are shown in Table3. It has initially extended loans with facevalues of $500, $600, $1000, and $1400 to be

Table 3Interest-sensitive assets and liabilities

Days Assets Liabilities

90 $ 500 $3,299.18

180 600

270 1,000

360 1,400

repaid in a single payment at the end of 90days, 180 days, 270 days, and 360 days, re-spectively. For simplicity, loans that matureat the end of 90 days, 180 days, 270 days, and360 days are assumed to be rolled over for 360days, 270 days, 180 days, and 90 days, respec-tively. The interest rate for any loan accountis 12 percent. 11 The present value of theseloans, and, therefore the total value of the loanportfolio, is $3,221.50 (= $500(1.12)' 25 +$600 / (1.12) 5° + $1,000 / (1.12) ./5 + $1400 /(1.12) ). To finance the loan portfolio, thebank borrows $3,221.50 in 90 day large certif-icates of deposits (CDs) at 10 percent interest.The two percentage-point spread is the returnearned by the bank for employing its special-ized capital in intermediating between bor-rowers and lenders. This will be the spreadbank funds management is content to makeover the planning period.

The amount that the bank will owe in 90days is $3299.18 ( = $3,221.50(1.10) .25), whichit plans to pay by borrowing this amount foranother 90 days. The bank anticipates beingable to roll the large CDs over every 90 daysat the same interest rate.

A summary of the present value of theasset and liability positions and the corre-sponding net interest income in each of the90-day periods is presented in Table 4. As thattable reveals, the net interest income on theinitial investment of $3,221.50 yields a returnof 2 percent ($64.63 / $3221.50).

In this example, the bank is subject toconsiderable interest rate risk because its fixed-rate loans mature at various times during theyear while all of its deposit liabilities must berolled over every 90 days. The duration of thelarge CDs is .25 years—the duration of a singlepayment is always the time to the paymentdate. The duration of the loan portfolio is .73years (.25($486.03 / $3,221.50) + .50($566.95)/$3,221.50) + .75($918.52 / $3,221.50) +

18 Economic Perspectives

($1,250 / $3,221.50)). The duration gap (DG)is negative and equals —$1,514 ($3,221.50 (1— .73) — $3,221.50 (1 — .25)). 12 As a practicalmatter, the assets' longer duration implies thata given change in interest rates will change thepresent value of the assets more than it will af-fect the present value of the liabilities. Thedifference, of course, will change the value ofthe bank's equity. By appropriately structuringa hedge, the bank can effectively insure thatnet interest income will turn out to be asexpected—yielding the 2 percent return.

The financial futures market can be usedin at least two ways to remove this durationimbalance: 1) to shorten the duration of theassets to .25 years, or 2) to lengthen the dura-tion of the liabilities to .73 years. Since thisbank is net long, i.e., the duration of its assetsis longer than the duration of its liabilities, theappropriate futures positions for a hedge willalways be short; i.e., it will involve the sale offutures contracts. Suppose, to hedge its expo-sure to interest rate risk, the bank decides toform a "loan-with-futures" portfolio consistingof both cash loans and futures contracts. Theduration of a portfolio containing cash loansand futures contracts is given most simply by la

NfFPDp = D„0 + Df u (4)

V rsa

where Dp is the duration of the entire portfolio;D„, is the duration of the cash loan portfolio;Dd is the duration of the deliverable securitiesinvolved in the hypothetical futures contractfrom the delivery date; Vrsa is the market valueof the cash loan portfolio; Nf is the number offutures contracts, and FP is the futures price.

Table 4Current value of assets and liabilities

Days Assets Liability

Netinterestincome

0 $3,221.50 $3,221.50 $ 0.00

90 3,314.08 3,299.18 14.90

180 3,409.31 3,378.74 30.57

270 3,507.28 3,460.21 47.07

360 3,608.08 3,543.65 64.43

Return on assets = $64.43/$3,221.50 = 2.0 percent

Table 5Portfolio characteristics for the

duration analysis

.25 years

Drsa .73 years

D1 .25 years

FP $97.21

Visa $3,221.50

Since the goal is to shorten the asset du-ration to .25 years, it must be that Dp = .25years. Table 5 summarizes the relevant data.The price of each future contract is $100 /(1.12)25 = 97.21. These (hypothetical) con-tracts call for delivery of $100 face value ofTreasury bills having 90 days remaining untilmaturity. Since Treasury bills are pure dis-count instruments, their duration will alwaysbe equal to the number of years to maturity,which is 90 days or .25 years. 14

Solving the above equation for the num-ber of futures contracts yields —64, which in-dicates that the number of Treasury billsfutures contracts to sell short at the beginningof the planning period is 64. Because no cashchanges hands at the time the futures contractsare originated and no deliveries are made, thefutures contracts per se do not change the cur-rent cash value of the portfolio, which remains$3,221.50. 15 However, as time passes and in-terest rates change, the futures contracts mustbe marked to market and any changes in theprice settled in cash on the day they occur.Thus, changes in the value of the futures con-tracts change the cash value of the portfolio.

Suppose the bank sells 64 (hypothetical)90-day Treasury bills futures contracts at aprice of 97.21 to hedge its net interest rate ex-posure. Now assume that interest rates riseunexpectedly by 200 basis points immediatelyfollowing this transaction and remain 200 basispoints higher indefinitely. 16 Assume also thatall cash flow receipts during the 360-day plan-ning period can be reinvested at 14 percent.''Table 6 presents the effect of the interest rateshift on asset and liability values, the futurescontracts, and the asset and liability values atthe end of the planning period (360 days).Table 7 presents the same result without fi-nancial futures. As Table 6 reveals, the bankwas able to earn 14 percent on the asset port-

Federal Reserve Bank of Chicago

Table 6Effects of a 200-basis point increase

in yields on realized interest ratesspread (with futures)

Assets Liabilities

Original portfolio value $3,221.50 $3,221.50

New portfolio value 3,180.31 3,207.02

Gain/loss on futures 27.52 0.00

Total portfolio change (13.67) (14.48)

Beginning portfolio value 3,207.83 3,207.02

Value of all accountsat day = 360 3,656.92 3,591.86

Annualized $3,656.92—$3,591.86yield spread $3,221.50over 360 days $65.06= 2.0 %

$3,221.50 ' =2.0%

folio and paid 12 percent on its large CDs. Theunexpected increase in interest rates causes thepresent value of the loans to fall more ($41.19)than the present value of the liabilities ($14.48).By itself, this would cause a reduction in thebank's equity and in the spread between therate earned on the loan portfolio and the ratepaid on the large CDs (see Table 7). At thesame time, however, the increase in interestrates generates a gain of $27.52 from the fu-tures contracts. Other things the same, this

Table 7Effects of a 200-basis point increase

in yields on realized interest ratesspread (without futures)

Assets Liabilities

Original portfolio value $3,221.50 $3,221.50

New portfolio value 3,180.31 3,207.02

Total portfolio change (41.19) (14.48)

Beginning portfolio value 3,180.31 3,207.02

Value of all accountsat day = 360 3,625.55 3,591.86

Annualized $3,625.55—$3,591.86yield spread $3,221.50over 360 days $33.71= = 1.05%'$3,221.50

causes equity to rise, and allows the bank tomaintain its 2 percent spread between the rateearned on assets and the rate paid on largeCDs. The effects of a 200-basis-point declinein yields on realized interest rate spread aresummarized in Table 8. 18

Thus the use of financial futures enablesthe bank to eliminate its exposure to interestrate risk. The formulation of a bank futuresposition in light of its entire mix of assets andliabilities helps to balance the interest sensitiv-ity of duration-mismatched assets and liabil-ities. These macro financial futures hedges arean effective means for banks to reduce the var-iability of net interest margin and improve thestability of bank profits.

While macro hedges are important gapmanagement tools, they must be used with agreat deal of care and attention. Due to thenature of banks' assets and liabilities, gap posi-tions can change rapidly. Therefore, the sizeof the interest-sensitive gap being hedged mayalso vary significantly from day to day. Be-cause of this, when a macro hedge is employed,it must be monitored continuously and some-times modified, if a target gap or interest sen-sitivity is to be maintained. The value of thefutures contracts employed in macro hedges isalso marked to market and the associated in-come or expense shown on the income state-ment in each reporting period.

Table 8Effects of a 200-basis point decrease

in yields on realized interest ratesspread (with futures)

Assets Liabilities

Original portfolio value $3,221.50 $3,221.50

New portfolio value 3,264.05 3,236.31

Gain/loss on futures (28.16) 0.00

Total portfolio change (14.39) (14.81)

Beginning portfolio value 3,235.89 3,236.31

Value of all accountsat day = 360 3,559.48 3,495.21

Annualized $3,559.48—$3,495.21yield spread $3,221.50over 360 days _ $64.07 = 2 ' 0%

$3,221.50

Economic Perspectives

Accounting rules and futures contractsCurrent accounting procedures for

futures contracts are set out in a uniformpolicy on bank contract activity issued bythe Federal Reserve Board, the FederalDeposit Insurance Corporation, and theComptroller of the Currency on November15, 1979, revised March 12, 1980. Federalregulations give banks the option of car-rying futures contracts on a mark-to-market or lower-of-cost-or-market basis.Other rules require all open contract posi-tions be reviewed at least monthly, atwhich time market values are determined.Futures contracts are valued on either themarket or lower-of-cost and marketmethod, at the option of the bank, exceptthat the accounting for trading accountcontracts and cash positions should beconsistent. Underlying securities commit-ments relating to open futures contractsare not reported on the balance sheet; theonly entries are for margin deposits, unre-alized losses and, in certain instances, un-realized gains related to the contracts. Inaddition, banks must maintain generalledger memorandum accounts or commit-ment registers to identify and control allcommitments to make or take delivery ofsecurities. Following monthly contractvaluation, unrealized losses would be rec-ognized as a current expense item, andbanks that value contracts on a marketbasis would also recognize unrealized gainsas current income. Acquisition of securi-ties under futures contracts are recordedon a basis consistent with that applied tothe contracts, either market or lower-of-cost-or-market.

The Financial Accounting StandardsBoard (FASB), in its ruling effective De-cember 31, 1984, introduced new guide-lines for futures contracts. The new rulesallow firms to use hedge accounting forfuture transactions.* In hedge accounting,a futures position is defined as a hedgingtransaction if it can be linked directly withan underlying asset or liability and if theprice of the futures contracts is highly cor-related with the price of the underlyingcash position. If these conditions are met,

and if the underlying cash position is notcarried at market, futures gains or lossescan be deferred until the position is closedout. The gains or losses can then becomepart of the accounting basis of the under-lying cash position, to be amortized over theremaining life of the asset or liability, andtherefore taken into income gradually.

The FASB standards require thatbanks and other firms formulate theirhedged positions in light of their entiremix of assets and liabilities so that macrointerest rate exposure is reduced by microhedges. By insisting that all futures hedgesbe linked to an identifiable instrument "orgroup of instruments, such as loans thathave similar terms" to qualify for hedgeaccounting, the FASB is encouragingbanks to analyze thoroughly their overallexposure to interest rate risk as well as thecomponents that make up that risk. TheFASB standards, however, do not allowhedge accounting for the macro hedgingof an overall gap on a bank's balance sheetthat cannot be identified with a specificitem.

The FASB statements call for theclassification of deferred gains and lossesas an adjustment to the carrying amountof the hedged items. Bankers should beaware that if such an adjustment is madeto appropriate general ledger accounts, thecomputation of average daily balances forthe purpose of determining average yieldswill be distorted unless special provisionsare made. In addition, other FASB state-ments require that the amortization of thedeferred futures gains or losses to interestincome or expense start no later than thedate that a particular contract is closedout. Profits or losses from the futures po-sition must be taken into the incomestream over that time period when thebank expected an adverse impact from in-terest rates.

*Bank regulators reactions to FASB statement, if any,are yet to be determined. As a result, banks futurestransactions are still governed by federal bankingregulations.

Federa l Reserve Bank of Chicago

In managing its asset and liability posi-tions in the financial futures'markets, a bank islimited by federal guidelines to transactions re-lated to the bank's business needs and its ca-pacity to meet its obligations. By taking aposition in the futures market, a bank shouldreduce its exposure to loss through interest ratechanges affecting its investment portfolio.

Conclusions

The recent increases and broad fluctu-ations in interest rates have led many banks toa better understanding of interest rate risk andhow to manage it. The use of gap managementcan be particularly important to bank fundsmanagement as a technique to manage interestrate risk. A bank can reduce the risk of loss dueto unfavorable changes in interest rates byhedging its duration gap. The use of financialfutures and the duration approach to gapmanagement enables the bank to maintain apredetermined spread and to lock in an antic-ipated rate of return.

1 See Sanford Rose, "Dark Days Ahead forBanks," Fortune (June 30, 1980), pp. 86-90.2 The length of time over which net interest marginis to be managed.

3 See Paul L. Kasriel, "Interest Rate Volatility in1980." Economic Perspectives, Federal Reserve Bankof Chicago (January/February 1981), pp. 8-17.

4 These numbers were taken from James Kurt Dew,"The Effective Gap II: Two Ways to MeasureInterest Rates Risk," American Banker (September18, 1981), p. 4.

5 The historical volatility of an entire spectrum ofassets and liabilities relative to a benchmark finan-cial instrument can be calculated using regressionanalysis. Dew (1981) shows how such calculationsare made.

6 This means that when interest rates change, in-terest income changes more than interest expense.

7 See Lawrence Fisher and R. Weil, "Coping withthe Risk of Interest Rate Fluctuations: Returns toBondholders from Naive and Optimal Strategies."Journal of Business (October 1971).

8 The market

+ iCsa)

value of a contractual flow of)Trsa)Trsa dollars Trsa periods from now is

RSA(1 + r rsa)Trsa

Trsa. Similarly, the mar-ket value of a contractual flow of RSL(1 Y;si)Trsidollars T,s1 periods from now is RSL(1 Trsi)Trsil(1 + Kr,i)Trsi. It is assumed that all asset and liabilityinterest rates are affected equally when any move-ment in rates occur.

9 See Alden Toevs, "Gap Management: ManagingInterest Rate Risk in Banks and Thrifts," EconomicReview, Federal Reserve Bank of San Francisco(Spring 1983), pp. 20-35.

113 For a discussion of this point see Albert E.Burger, Richard W. Lang, and Robert H. Rasche,"The Treasury Bill Futures Market and MarketExpectations of Interest Rates," Review, FederalReserve Bank of St. Louis (June 1977), pp. 2-9.

II As a futures contract moves toward maturity, thefutures price and cash price tend to converge.Therefore, basis narrows predictably over time.

12 A flat yieldA fiield curve is assumed for ease of exposi-tion.

13 See Alden Toevs, op. cit.

14 For further details, see G. 0. Bierwag, GeorgeG. Kaufman and Alden Toevs, "Duration: Its De-velopment and Use in Bond PortfolioManagement," Financial Analysts Journal, (July-August 1983), pp. 15-35 and Gerald D. Gay andRobert W. Kolb, "Interest Rate Futures as a Toolfor Immunization," The Journal of Portfolio Man-agement (Fall 1983), pp. 65-70.

15 G. 0. Bierwag, George G. Kaufman, and AldenToevs, op. cit.16 A cash or liquid security margin requirement isgenerally maintained.

17 Zero basis risk is assumed.

18 Actually the reinvestment rate is not certain.However, the assumption of a 14 percent reinvest-ment rate simplifies the example with no loss ingenerality.19

is iIt s nteresting to note that if a bank were toengage in the type of hedge in these examples whenit was exposed to loss from increases in interestrates, it would not only limit the potential rise inbank costs from unfavorable shifts in interest ratesbut agree implicitly to limit the potential of itslower costs from favorable shifts in interest rates.The bank must be content with the usual profitsfrom lending.

Economic Perspectives