Federal Reserve Banl<of San Francisco

Spring 1983 Economic ReviewNumber 2

,MANAGING UNCERTAINTY

Opinions expressed in the Economic Review do not necessarily reflect the views ofthe management ofthe Federal Reserve Bank of San Francisco, or.of the Board of Governors of the· Federal ReserveSystem.

The Federal Reserve Bank of San Francisco's Economic Review is puolished quarterly by the Bank'sResearch am.iPublic Information Department under the supervision of Michael W. Keran, Senior Vicet>resident. Thy publication is edited by Gregory J. Tong, with the assistance of Karen Rusk (editorial) andWilliam Rosenthal (graphics).

For frye copies of this andqther Federal Reserve publications, write or phone the Public InformationDepartment, Federal Reserve Bank of San Francisco. P.O. Box 7702, San Francisco, California 94120.Phone (415)974-3234.

2

Managing Uncertainty

I. Introduction and Summary 4

II. The Baby Boom, the Housing Market and the Stock Market 6Roger Craine

.. .The data support the hypothesis that one or the other or both expected inflation and household formation influenced the rate of return on stocks and housing over the 1965 to 1980 period.

III. The Recent Decline in Velocity: Instability in Money Demand orInflation? 12

John P. Judd

.. .The surprising behavior of velocity in 1982, therefore, appears to be related more to an unexpectedly large decline in inflation and short-term interest rates than to any instability in money demand.

[V. Gap Management: Managing Interest Rate Risk in Banks andThrifts 20

AldenL. Toevs

.. .This model shares a common goal with gap models, that of monitoring and managing the interest rate risk exposure of current bank earnings, while offering several advantages over current gap models.

Editorial Committee:Rose McElhattan, Hang-Sheng Cheng, John P. Judd

3

an ingUnce lin

Unpredictable events surround economic deci­sions and are often critical to their outcome. Forexample, was it uncertain demographic change orinflation that was responsible for the rise in housingprices relative to other assets in the 1970s? In theconduct of monetary policy, the choice of strategiesrelies on the predictability of the behavior of moneydemand. And unexpected fluctuations in interestrates, such as those experienced in the recent past,have had a major impact on depository institutions'net interest income. The three articles in this Eco­nomic Review present methods for identifying theeffects of unpredictable events and for managinguncertainty.

Some housing economists h~ve attributed mostof the increase in the value of housing relative tocorporate stock in the 1970s to increased specula­tive demand. They believe the increased demandresulted from the interaction of inflation and a non­indexed tax system. Roger Craine, in the first ar­ticle, points out that demographic shifts in thedemand for housing also has a significant impact onits relative price. His conclusion has an implicationfor the future of the housing market that differs fromthat commonly held. Even if inflation abates in the1980s and reduces the speculative demand for hous­ing, Craine's findings imply that there will remain astrong demographic demand for housing.

Craine believes that the uncertainty associatedwith the household formation rate of the post-WorldWar II baby boom increased the risk associated withhousing investments. As a result, the rate of returnon houses had to exceed the average rate of returnon other assets to compensate investors. In a seriesof regressions ofexcess returns to housing on short­run anticipated household formation and inflation,he shows that the influence of household formationcannot be dismissed in a statistical sense: "The datasupport the hypothesis that one or the other or bothexpected inflation and household formation influ-

4

enced the rate of return on... housing over the 1965to 1980 period."

John Judd addresses the question of whether thedecline in the velocity ofMl in 1982 was due to anunexpected upward shift in the public's desire tohold money or to a predictable money demand re­sponse to dropping interest rates and inflation. Inthat year, the velocity of Ml declined at a 4.6percent rate (it had risen at an average 2.8 percentannual rate for the last twenty years), and led someto conclude that this was another instance ofmoney­demand "instability"-of the public's demand formoney turning out to be different from what histori­cal relationships would have predicted.

Judd uses the San Francisco Money MarketModel to show that the demand for money was"consistent with available information on the be­havior of widely recognized determinants of Mlgrowth," and thus did not constitute a shift in de­mand. He finds that the rapid Ml-growth in the lasttwo quarters of 1982 can be "explained by themoderate growth in nominal income and the largedecline in short-term interest rates."

Instead of being an unpredictable change in pub­lic behavior, the decline in velocity was a responseto the large drop in inflation which permitted aparallel decline in nominal interest rates althoughnot in real interest rates. Since money demand re­sponds to nominal interest rates, the public waswilling to hold more MI. But since GNP growthresponds to real rates of interest, GNP did not re­ceive the same stimulus: "As a result money growthaccelerated relative to GNP growth, implying adecline in velocity." Judd's analysis suggests thatvelocity should exhibit more "normal" behavior inthe second half of 1983 because inflation appears tohave stabilized at its new lower level.

In the last article, Alden Toevs develops a bettermodel for banks and other depository institutions touse in monitoring and managing the exposure of

bank .earnings to unforeseen changes in interestrates. The model ofcomparison is the popular"gapmanagement model" where the gap refers to thedollar value difference between rate-sensitive assets(i.e., assets whose yields are sensitive to changes inmarket rates of interest) and rate-sensitive liabili­ties. According to the gap model, a bank wouldhedge against earnings being affected by changes ininterest rates (so-called interest rate risk) bykeepingthe gap equal to zero in the time interval concerned.

Toevs, however, notes two serious shortcom­ings. First, he believes that the existing model' 'un­necessarily constrains a bank's choice of assets andliabilities" in creating a hedge. The constraints, intum, reduce "the bank's ability to accommodatecustomer demands for bank services." Second, themodel is unable to generate "a simple and reliableindex of interest-rate risk exposure.' ,

Toimprove on the gap model, Toevs develops a"duration" gap model that, by using more generalconditions for hedging interest rate risk and byincorporating .the timing of repricing decisions bythe bank, "reveals a larger set of asset and liabilitychoices to financial institutions" to hedge net inter­est income. With the model, he is also able todevelop "risk-return frontiers" to quantify thechoices for those institutions that wish to positiontheir balance sheets to profit from interest rate fore­casts. The duration gap model also yields a singlenumber to quantify the risk position of the financialinstitution using it. This number is useful if interestrate risk for the entire bank is to be hedged in thefutures market. Finally, the duration gap model isgeneralized to hedge the market value of bank capi­tal against unexpected change in interest rates.

5

Roger Craine*During most of the past two decades, the housing

market in the· U. S. boomed while the stock marketfaltered. The nominal return on single familyhousing rose fairly steadily from 6.5 percent a yearin 1965 to over 15 percent in 1979. In the sameinterval, stock market returns rose from 3 percent toonly 5.5 percent. Since inflation accelerated from 3percent to almost 12 percent in the meantime, real(inflation-adjusted) returns in the stock market werenegative through most of the 1970s. The real valueof corporate equities declined by 48 percent but thereal value of a single family house increased by 26percent.' As a consequence, the composition ofprivate wealth changed markedly. The total value ofcorporate equities compared to the total value ofowner-occupied housing declined by an astounding

. 150 percent between 1965 and 1980. 2

A number of researchers, e.g., Martin Feldstein,Randall Pozdena, and Lawrence Summers, attrib­ute most of the change in the value of housingrelative to corporate stock to me interaction ofinfla­tion and a non-indexed tax system. Taxable nominalcorporate profits rise more in percentage terms thaninflation because of historical cost depreciation andprevailing (first-in-first-out) inventory accountingpractices. As a result, inflation-adjusted after-taxcorporate profits actually decline with inflation.Furth~rmore, stockholders must pay tax on purelynominal stock market capital gains as inflationpushes them into higher marginal brackets. Home­owners, however, avoid or benefit from many tax"non-neutralities." Owner-occupants consume theflow of services their houses provide. This serviceflow is an imputed rent payment that adds to incomein the National Income Accounts butthe "in kind"payment is not counted as explicit taxable incomeby th~ Internal Revenue Service. In addition, cap­ital •gains taxes on· housing can be deferred or

*Visiting Economist at Federal Reserve Bank ofSaIl Francisco, Professor of Economics, Universityof California, Berkeley.

6

avoided altogether by using rollover provisions andexemptions for those over age 55.

The researchers therefore concluded that the non­neutralities in the tax system were capitalized in theasset prices during the inflationary period of the1970s. Moreover, they believe that the changingrelative asset values induced changes inthe physicalstock of assets and the composition of wealth.

Thus far in the Eighties, inflation has fallenrapidly from 12 percent in 1980 to under 5 percent in1982. The Economic Recovery Act of 1981 reducedindividual and business taxes, and tax indexingslated to begin in 1985 should further reduce taxes.As inflation recedes and the tax system is mademore equitable, the macroeconomic causes of thehousing boom will presumably be eliminated. In the1980s, the U. S. may also face the task of workingoff an excess supply of housing created by the mac­roeconomic climate of the last decade. As a result,economists that attribute the housing boom of the1970sto macroeconomic .causes see a relativelydismal future for the housing indUStry.

Their line of reasoning follows a traditional mac­roeconomic approach in analyzing the changeSinrelative values. The emphasis is on macroeconomicvariables-inflation and taxes-while the compo­sition of consumer demand is assumed constantor relatively unimportalltat the nation-wideJevel.The theoretical and empirical work by Feldstein,Pomena, and Summers shows that macroeconomicvariables should and did affect the value of cor­parate stock relative to owner-occupied housing inthe 1970s.

The 1970s, however, also witnessed majorde­mographic shifts that affected the composition.ofcons~merdemancl' The traditional macroeconomicaggregation assumption that the composition ofunderlying demand is fairly stable was not valid inthat decade. Household formation, for ex.ample,grew much more rapidly than housing starts. Thenumber of households in the 24-35 age cohort

(survey data indicate that half the new home buyersinthe 1970s fell inthis age cohorf) almost doubledbetween 1960 (10 million) and 1980 (18 million).Over the same period, housing starts only increasedabout 20 percent.

An increase in housing demand relative to supplyis a standard microeconomic explanation for the risein housing prices and home construction. However,since the baby boom enters the housing marketthrough a long and supposedly easily observed ges­tation period, many believe the aggregate impact ofthe demographic shift can be anticipated and there­fore should have no significant effects.

Section I presents a brief discussion of the effectof an increase in the demand for housing services onthe relative price of housing services, the relative

aSset price of houses, and investment in housesrelative to corporate capital. It concentrates on thedemographic effects and shows that even if the babyboom had been anticipated by the market and therewere no inflation or tax distortions, demographicchanges would still have led to an increase in therelative value of housing. Section II examines em­pirical evidence from 1965 to 1980. The resultsindicate that either inflation or household formationcan explain the value of houses relative to corporatestock in the 1970s. In fact, both probably influencedthe housing and stock markets in theSeventies.Theresults also indicate that demographic factors willcontinue to exert some·· demand pressure on thehousing market in the Eighties and make the out­look for housing more sanguine.

I. Uncertain Demographics and Rates of ReturnOther things being equal an increase in demand

for a product increases the relative price of thatproduct. The price mechanism sends a signal toindividual decisionmakers to transfer resources tothe high price (high profit) industry from lowerprice (lower profit) industries. The short-run reallo­cation in flow markets is straightforward and quitesimple. When demand shifts, some industries moveup their short-run supply curves by adding variableinputs (labor) and other industries move down theirshort-run supply curves by reducing variable in­puts. If the shift is permanent (or long-lasting) thecapital stock must also be reallocated. Asset prices,which reflect expected discounted future eamings,will change and lead to a change in investment.Both current and unknown future prices affect thepresent value ofassets and capital allocation. More­over, reallocating the capital stock is complex andcostly so the unknown future makes any majorcapital decision risky.

The baby boom led to an obvious increase in thedemand for housing services. As children maturedthey took jobs, left their parents' homes, and de­manded housing. Most married and started familieswhich increased the demandfor housing. The bulgein the age structure of the population created anextraordinary demand for housing that required re­sources to be reallocated toward housing and awayfrom other activities. The adult population grew at arate of 3 million a year in the 1970s; in the 1960s, it

7

grew at 2 million a year. The growth of householdsincreased even more dramatically, from about 1million a year in the 1960s to 1.75 million a year inthe 1970s.

The changing demographic structure of the popu­lation had many economic consequences that can beanalyzed as the microeconomic substitution effectof a change in the mix of consumer goods de­manded, holding everything else constant. In thissection I assume total consumption, investment,and wealth are fixed. This analysis illustrates that achange in the mix of consumer goods demanded canchange relative flow;md asset prices.

The service flow from housing (the serviceshousing provides, such as a place to sleep, eat, andrelax) is a perishable consumption good that can bepurchased by paying rent. Owner-occupants im­plicitly pay themselves rent that equals the value ofthe services they consume. At a fixed level of in­come and saving, an increased demand for housingservices must be matched by a decreased demandfor other goods. The shift in the mix ofconsumptiondemand is reflected in the relative flow prices of thegoods and services. In the simplest case, r~nts

would increase relative to the prices of other con­sumption goods.

Asset prices depend on the current and futureincome stream associated with the asset. Title to theass~t conveys the right to the future income streamto the owner of the title. For example, the ownerofa

house will receive its current and future rent. Thepresent value of a house (PVH) is the stream offuture rents (RT+j) discounted to reflect their currentvalue,

PVH = RT+,/(l+r) + RT+2/(l+d +RTd(l+r)3 ... ,

where r is the discount rate. The prices of titlesreflect the market's evaluation of the future incomestream. If the market expects the rental rate toincrease relative to the price ofconsumer goods, thepresent value of housing and house prices increaserelative to corporate capital. Since the aging of thebaby boom is easily predicted, the increased de­mand for housing was (at least partially) expected.

Figure I shows the relationship between rent (R),other prices (P), house prices (H) and stock prices(S) assuming a one-time demand shift that is per­fectly anticipated. (The stock of housing and cor­porate capital is assumed fixed.)

The top panel shows the flow prices-rent andthe price of other consumer goods. The demograph­ic shift, which is assumed to occur in year T, in­creases the demand for housing services and re­duces the demand for other goods. As a result therent for houses rises, and other prices fall in year T.Since the change is permanent, rents exceed theprices of other goods thereafter (RT+i>PT+i)' Whenthe flow prices change, asset values also change.After rents increase, the asset value of a house

Rent (R)Other Prices (P)

Time

House Prices (H)Stock Prices (S)

Time

8

(HT+i) must exceed the asset value of corporatestock (ST+i) as shown in the bottom panel. How­ever, asset prices depend on the entire streamof future earnings and, therefore, change prior toyearT.

The rate of return to an asset is the sum of flowincome and capital gains expressed as a percent ofasset price. After year T, when asset prices areconstant, the rates of return are,

In a world ofcertainty, rates of return on all assetsare equal, that is, rH = rs; otherwise, riskless ar­bitrage opportunities exist. For example, if the rateof return on housing exceeds the rate of return oncorporate stock, speculators (in theory) can sellstock short and use the proceeds to buy houses. Inthe process, they make a riskless profit. However,as agents buy one asset and sell another, the assetprices change to equalize the rates of return.

Prior to year T, the flow income from the twoassets (RT-i> PT- i) is equal but the asset priceschange in anticipation of the demand shift. Whenthe future is perfectly anticipated, asset priceschange over time so that the rates of return arealways equal, i.e.

RT-i+.lHT- i _ r _ PT-i+.lST- irH = HT- i - S - ST-i

House prices gradually rise (.lHT- i) to givehomeowners capital gains that offset the lower cur­rent rents, while stock prices (.lSr-J gradually fallto give equity holders capital losses that offset cur­rent higher profits.

After period T, asset and flow prices are constantbut not equal. Prior to period T, flow prices wereconstant and equal, but asset prices were changing.Throughout the period, however, rates of return areequal.

Figure I illustrates the relationship between theflow and asset prices in a stylized form. In thisexample, the rates of return had to be equal becausethe investors saw the future with perfect clarity. Theactual relationship between flow and asset pricesis much more complicated. Even though one canaccurately predict the aging of the baby boomgeneration, its demand for housing and its rate ofhousehold formation is much more uncertain.

Household fonnation depends on complex socialand.economic factors. The quality and quantity ofhousing services can be varied and home purchasesdelayed. Furthennore, home builders add to an ex­isting supply which feeds back on house prices andthe rental rate. Construction has always been aboom and bust industry precisely because structuresare long-lived durables and the future is uncertain.

In an uncertain environment, major shifts, suchas the aging of the baby boom, provide increased

opportunities for profit but only at the cost of bear­ing additional risk. Building too far in advanceresults in high vacancies, low rents, and sometimesbankruptcy. The current glut of commercial officespace in many cities exemplifies the risky nature ofreal estate speculation.

In this risky environment, the rate of return onassets is likely to diverge as investors try to gainaccess to the uncertain future.

II. Empirical Evidencesury bill rate (see the Appendix for details). Ex­pected inflation (OPE) and household fonnation(HFE) are three-year averages for forecasts offuture rates of change of the consumer price indexand household fonnation from ARIMA models. 4

Summers tested the hypothesis that the short-runexpected inflation can "explain" the divergence inthe excess returns by regressing the excess returnson expected inflation. Table 1 shows the resultsfrom estimated equations of the fonn,

El= bo + bpPI; + ~

where E is the excess return and u is an error, oromitted effects, and OPE is expected inflation.

Table 1*Excess Returns and Expected

Household Formation:Stock and Housing Markets

Dependent variable bo b1 R2

ESTOCK .14 -66.02 .44(.96) (19.56)

*Standard errors in parentheses.

The regressions, based on annual data from 1965to 1979, indicate that expected inflation has a statis­tically significant depressing effect on the stockmarket and a positive, although not statisticallysignificant, effect on housing. These results aresimilar to Summers' who used a different data setand measure of the change in expected inflation totest the hypothesis. The results weakly support thehypothesis that expected inflation increased thedemand for housing relative to other goods.

To test the hypothesis that household fonnation

The real price of homes increased by 26 percentbetween 1965 and 1980. Over the same period, therate of return to housing was over twice the rate ofreturn on corporate stock. These numbers are con­sistent with an increased demand for housing due toan uncertain but expected rapid growth in house­hold fonnation. They are also consistent with anincreased speculative demand for housing due toaccelerating inflation and distortions in the taxsystem. The consequences of these two sources ofdemand, however, have very different implicationsfor the 1980s. If demographics caused the change,demand will continue to grow but less rapidly thanin the 1970s and the prices of housing relative toother assets will stabilize. On the other hand, if theshift in asset values was due only to inflation and taxdistortion, and we have disinflation and tax changesin the 1980s, then relative home prices and homeconstruction will decline.

To test the proposition that anticipated householdfonnation and/or anticipated inflation increased therate of return to housing and decreased the rate ofreturn on stocks in the short-run, I regressed theexcess return in each market on these variables.This test extends the work of Summers who onlytested for the effect of inflation.

The excess return in the stock market (ESTOCK)is defined as the sum of capital gains plus dividendsas a percent of the beginning-of-period value lessthe beginning-of-period Treasury bill rate. This isthe difference between the one-period holding yieldon stock and the return on an alternative "safe"asset-Treasury bills. The excess return to housing(EHOUSE) is the rent plus capital gains as a percentof the beginning-of-period value minus the Trea-

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EHOUSE 2.08(2.23)

55.56 .10(51.27)

"explains" the divergence in returns, I also esti­mated equations of the form,

E! = c + clHFE + v.o ·t t

where HFE is expected household formation. Table2 gives the results.

Table 2*Excess Returns and Expected Inflation:

Stock and Housing Markets

Dependent variable Co Cl R2

ESTOCK -3.09 .002 .37( .45) (.0006)

EHOUSE 4.75 .0003 .49( .60) (.0001)

*Standard errors in parentheses.

The regression results indicate that expectedhousehold formation had a statistically significantdepressing effect on the stock market and a statisti­cally significant positive effect on the housingmarket. These results provide somewhat strongerstatistical support for the hypothesis that the demo­graphic changes increased the demand for housingservices relative to other goods.

Obviously, there is no need to have an either/orhypothesis. Economic data are not generated by acontrolled experiment and many factors changesimultaneously in the actual economy. To test thehypothesis that expected inflation and householdformation "caused" the divergence in the rates ofreturn, I estimated equations of the form:

E, = do + d,DPE + d2HFE + w,.

Table 3 gives the results.

The results are consistent with the hypothesis thatboth variables help explain the short-run divergencein the rates of return. However, they are not strongin statistical terms. The only statistically significantcoefficient of interest (atthe 5 percent or even 20percent level) is the coefficient on expected house­hold fonnation in the excess return to housing equa­tion. Expected inflation in this equation has a nega­tive sign, contradicting the expected inflationhypothesis at least for· housing; the coefficient,however, is statistically insignificant.

It is not terribly surprising that the data cannotcleanly separate the effects. The data are annual(household formation is only reported on an annualbasis) and both household formation and inflationaccelerated in the 1970s.5 But an F test of the nullhypothesis that neither expected inflation nor ex­pected household formation affected the rates ofreturn can be rejected at the 95 percent confidencelevel.

In summary, the data support the hypothesis thatone or the other or both expected inflation andhousehold formation influenced tha rate of return onstock and housing over the period from 1965 to1980, although the statistical evidence is not pre­cise. However, economic theory and common sensebolster the conclusion that inflation was bad for thestock market and probably good for the housingmarket, while the rapid increase in household for­mation was good for the housing market and prob­ably bad for the stock market.

Table 3*Excess Returns, Expected Inflation, and Expected Household Formation:

Stock and Housing Markets

Dependent variable do d1 ds R2

ESTOCK .74 -52.78 .0005 .45(1.73) (37.78) (.0011)

EHOUSE 7.02 -53.03 .0004 .55(2.28) (51.41) (.0001)

*Standard errors in parentheses.

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III. ConclusionDuring the past fifteen years, .the total value of

corporate equities relative to the total value ofowner..occupiedhousing fell by an incredible 150percent. The 1960s and 1970s witnessed a massivechange in the value and composition of privatelyheld wealth. The macroeconomic explanation forthe change in the value of housing relative to cor­porate stock is that accelerating inflation in the1970s coupled with a non-neutral tax system in­creased the.effective corporatetax rate. This expla­nation implies that the housing boom was not basedon fundamental demand factors but peculiar fea­tures in the tax system and inflation. These factorscan be reversed, so as we look forward to lowerinflation and taxes in the 1980s, we might also lookwith trepidation to falling house prices and a stag­nant homebuilding industry.

APPENDIXThe rate of return to stocks was calculated as

follows:

(Stock Return), = SD'+I + (SP'+I - SP,)/SP,

where:

SD the dividend yield on the Standard andPoor's 500 composite common stockindex.

SP = the price of the Standard and Poor's 500composite common stock index.

The rate of return to housing was calculated asfollows:

(Housing Return), = [RENT, + (HP'+I - HP,)]/HP,

where:

RENT = the rental return to housing calcu­lated by using the rent component ofthe CPI normalized by the rent forresidences in 1972.

HP = the price of housing calculatedby using the Department of Com­merce's price index for new one­family houses sold, normalized bythe median home price in 1972.

II

This paper couples the macroeconomic explana­tion with a more fundamental microeconomic ex­planation. that the demand. for housing increasedbecause of demographic change. The large increasein households during the last decade also can ex­plain house and stock prices over the period. Theaging baby boom and the rapid rate of householdformation swelled the real demand for housing.While household formation in the 1980s shouldgrow less rapidly than in the 1970s, thedemograph­ic factors will continue to exert demand pressure onthe housing market through most of this decade.

The evidence in this paper indicates that bothinflation and household formation affected the re­turns to stock and housing. For the future, thismeans that while disinflation should help the stockmarket and reduce the speculative tax-induced de­mand for housing, the fundamental demographic­based demand for housing will remain strong.

FOOTNOTES

1. These calculations use the Standard and Poor's stockindex, the CPI, and data on home prices from CensusReports G-25 and G-27.

2. See Lawrence H. Summers, p. 429.

3. Michael Sumichrast, et ai, Profile of a New HomeBuyer: 1979 Survey of New Home Buyers (Washington,D.C.: National Association of Home Builders.)

4. ARIMA models are a statistical forecasting procedure inwhich future values are forecast from past values. Actualvalues, or perlect foresight, give similar results.

5. The correlation between the series is .86.

REFERENCES

Feldstein, Martin, "Inflation, Tax Rules and the Accumula­tion of Residential and Nonresidential Capital;' NationalBureau of Economic Research, Working Paper #753.

Hendershott, Patric H., "Income and Housing in the 1980's"and "The Price of Housing Services and TenureChoice in the 1980's" Quarterly Review, The FederalHome Loan Bank of Cincinnati, 1982.

Pozdena, Randall J., "Inflation Expectations and the Hous­ing Market" Federal Reserve Bank of San Francisco,Economic Review, Fall 1980.

Russell, Louise B., The Baby Boom Generation andthe Economy, Brookings Institution, Washington,D.C. 1982.

Summers, Lawrence, "Inflation, the Stock Market andOwner Occupied Housing" American EconomicReview, Vol. 71 No.2, May, 1981.

Recent

John P. Judd*In 1979, the Federal Reserve embarked on a

long-run strategy of monetary policy designed toreduce the rate of inflation gradually over a numberof years. The idea behind this "gradualist" policywas to reduce growth in the monetary aggregates,especially M 1, slowly enough over several years towin the battle against inflation in the long-run withthe smallest possible adverse effects on output andemployment in the interim period. To this end, theFederal Open Market Committee gradually reducedits annual growth-rate target ranges for the mone­tary aggregates each year from 1980 through 1982.The range for Ml, for example, reached 2VZ-5V2percent in 1982 in comparison to actual Ml growth0f7V2 percent in 1979.

For this approach to work as intended, the veloc­ity of Ml (the ratio of nominal income to Ml) mustgrow at a relatively constant rate on a year-by-yearbasis. If, for example, velocity growth were abso­lutely constant, a I-percent reduction in Ml growtheach year would translate into a I-percent reductionin growth in the aggregate demand for goodsservices, as measured by nominal GNP. Thissmooth, gradual reduction in nominal GNP wouldbe consistent with the goal of reducing inflationwithout creating substantial unemployment and idlecapacity. However, gradual reductions in moneygrowth rates would not necessarily be consistentwith these macroeconomic goals if the growth invelocity fluctuated widely on a year-to-year basis.In such a case, aggregate demand also would fluc­tuate widely.

Prior to 1982, a case could be made that yearlyMl-velocity growth was sufficiently stable to sup­port a gradualist policy. However, in 1982 Ml­velocity unexpectedly declined at a 4.7 percent rate;this compares to its 2.8 percent average rate ofincrease over the previous twenty years. In re­sponse, the Federal Reserve chose to depart from

*Research Officer, Federal Reserve Bank of SanFrancisco. Research assistance was provided byThomas Iben and David Murray.

12

its long-run strategy of gradual reductions in thegrowth of monetary aggregates. It allowed Ml toaccelerate sharply to an average growth rate of 8V2percent in 1982, well above the 5Vz percent upperboundary of its 1982 target range. Even at thishigher Ml growth rate, nominal income increasedby only 3.5 percent and real income declined 0.9percent.

The purpose of this paper is to assess what went"wrong" with velocity in 1982. One possible ex­planation is that the public's demand to hold moneybalances "shifted" upward in the sense that, forgiven interest rates, income, and prices, the publicwanted to hold more money than historical relation­ships would predict. Evidence based on data fromthe 1970s, however, suggests that the demand forMl was stable, and that the declines in velocity in1982 are explained mainly by the sharp drop innominal short-term interest rates in that year. Thisdrop in nominal rates was roughly equal in size tothe surprisingly sharp decline in inflation, andmeant that inflation-adjusted, or real short-terminterest rates remained high. These high real inter­est rates helped depress total spending in the econ­omy and caused GNP to grow very slowly or todecline. At the same time, lower nominal interestrates increased money demand, causing Ml growthto surge. The combination of fast Ml-growth andslow income growth meant that velocity actuallyfell. The surprising behavior of velocity in 1982,therefore, appears to be related more to an unex­pectedly l,arge decline in inflation and short-terminterest rates than to any instability in moneydemand.

The remainder at this paper is organized as fol­lows: Section I presents the empirical evidence con­cerning the behavior of money demand. Section IIdescribes how the behavior of inflation and interestrates may have accounted for the surprising move­ments in velocity and other economic variables in1982; and Section III presents the policy implica­tions of this finding.

I. Did the Demand for Money Shift?The problem faced by policymakers in 1982 is

amply illustrated by Chart 1, which shows annualgrowth rates in the velocity of MI. The averagegrowth rate from 1960 through 1981 was 2.8 per­cent, with a standard deviation of 2.3 percent. In1982, velocity fell sharply at a 4.7-percent rate.This decline is over 3 standard deviations from theaverage and represents highly unusual behavior forthe series.

One possible explanation for this unexpectedchange in velocity is that there was an upward shiftin the public's demand for money, that is, thatincreasing quantities of Ml were demanded by thepublic for given levels of prices, real GNP andinterest rates. This alleged shift has been attributedto a precautionary motive for holding money causedby the economic uncertainty of the recession. I Pro­ponents of this view argue that some precautionarydemands that showed up in an increase in passbooksavings accounts in previous business cycles mostlikely appeared as an increase in the NOW-accountcomponent of MI in 1982. Authorized for offeringon a nationwide basis in January 1981, NOW ac­counts are counted as Ml and, because they paypassbook rates of interest with checking privileges,most likely attracted some of the precautionarybalances that used to be held in savings accounts. 2

This would be a plausible hypothesis if the evi­dence showed that the demand for Ml did shiftupward in 1982. However, the evidence presentedin this paper argues that the demand for Ml was

Chart 1

Annual Changes in Velocity: 1960·1982

Percent Change8

6

o-2

-4

-6 ......I-......&.....I-.................~................A1960 1964 1968 1972 1976 1982

(Fourth quarter over fourth quarter)

13

stable, that is, that growth in MI was consistentwith the observed relationships in the 1970s be­tween money, on the one hand, and prices, incomeand interest rates on the other.

The evidence is based upon simulations using anMI-demand equation similar to the one in the SanFrancisco money market model (Table IY Theequation specifies MI as a function ofthe six-monthcommercial paper rate, nominal personal income,and the change in total commerical bank loans out­standing. The first two arguments in the equationare commonly-used representations of the interestrate, price, and income variables suggested by theconventional theory of the demand for money.(Prices and income are combined in nominal per­sonal income.)

The third variable-the change in bank loans-isnot used in conventional specifications.4 It reflectsthe view that transactions money balances (check­ing accounts) act as buffer stock between receiptsand spending and that unplanned receipts and dis­bursements cause checking accounts to rise and falltemporarily. Although in principle these temporaryimbalances could be immediately removed, in prac­tice, they may persist for a time because of portfolioadjustment costs. Changes in the supply oftransac­tions deposits created as banks extend or call loansare therefore a potentially important source of fluc­tuations in observed money balances. The estima­tion results of the San Francisco model suggest thatthis is in fact an empirically important effect.

The evidence presented below, however, doesnot depend on this difference from conventionalspecifications. The buffer-stock variable plays aninsignificant role in explaining the events of 1982 asa whole because growth in bank loans was relativelyslow and steady that year. 5

The equation in Table 1 was used to determine ifMl growth in the period from January 1982 toMarch 1983 was consistent or inconsistent withhistorical relationships between Ml and the deter­minants of MI demand. This was done by estimat­ing the equation from July 1976 through December1981, and then dynamically simUlating it over theperiod in question. (A similar experiment was con­ducted with an equation estimated with data fromJanuary 1971 through December 1981.) We then

LAGoI2345

A1-1.31

Table 1M1 Demand Equation*

InMI = Al + A2*CHBL + A3*InPI + A4*InCPRT+ A5*TIME + A6*TIME2 + A7*TIME3

A2 A3 A4 AS

0.71 0.11 ~0.059 0.00700.65 0.22 -0.0410.56 0.27 -0.0270.45 0.25 -0.0150.32 0.15 -0.0070.17 -0.002

A6

~0.0017

A7

0.000061

SUM -1.31(-51.2)

2.86(3.28)

1.00** -0.15(-13.83)

0.007(1.29)

-0.0017(-1.41)

0.000061(0.92)

TIME2TIME3

PICPRTTIME

R2 = 0.998SER = 0.0044DW = 1.96AUTOI = 1.43 (13.67)AUT02 = -0.58 (-5.95)Estimation Period: August I976-December 1981.

Definitions of Variables

CHBL change in the log of total loans of commercial banks, including loan sales to affiliates, and adjusted for the introduction ofinternational banking facilities.nominal personal income.six-month commercial paper rate.zero in August 1976-December 1980; 1,2,3, ... 12 in January-December 1981. (Frozen at 12 for simulation in Table 2.)Included to capture the effects of the introduction of nationwide NOW accounts.(TIME)2(TIME)3

* Second-degree Almon lag distributions used for A2, A3, A4. Instrumental variables used for InCPRT. Student"t statistics inparentheses.

** Sum of lag distribution restricted to unity. Unrestricted estimates of coefficients on log of prices and log of real income bothinsignificantly different from unity at 95 percent level.

Table 2M1 Growth at Annual Rates*

Dynamic Ex AnteActual Simulation Forecast

1982/QI 7.3 5.5 6.51982/Q2 4.3 5.1 6.11982/Q3 8.6 11.9 7.11982/Q4 13.0 15.9 17.41983/QI 16.1 10.5 12.5

Averagefor theperiod 10.3 10.2 9.9

* Calculated as the annualized percent change of the last month in a quarter over the last month in the previous quarter.

** Three-month ahead forecasts made in the middle of the first month of forecast period using the San Francisco money market model.See John P. Judd, "A Monthly Model of the Money and Bank Loan Markets," Working Papers in Applied Economic Theory andEconometrics, Number 830I, Federal Reserve Bank of San Francisco, May 1983.

14

compared the simulated Ml-growth over the periodto actual growth. If the demand for MI shiftedupward during this period, as some observers havesuggested, the equation .shollld "underforecast"Ml growth.

The results of the simulation experiment are pre­sented in Table 2. Column I shows that actual Mlgrowth over the simulation period was 10.3 percent(at an annual rate). The MI equation predictedgrowth of 10.2 percent, suggesting that rapid MIgrowth can nearly all be "explained" by the deter­minants of Ml demand. (When the equation inTable I is estimated over January 1971 throughDecember 1981, the average growth simulated forthe period from the first quarter of 1982 throughthe first quarter of 1983 is 10.9 percent.) Moreover,this simulation accurately captured the pattern ofgrowth over the period. Ml grew at a moderate5.8-percent rate in the first two quarters of 1982,then accelerated to 12.6 percent in the next threequarters. The simulated growth ofMl for these twoperiods is 5.3 and 12.8 percent, respectively.

The final column of Table 2 shows ex ante Mlforecasts made with the full San Francisco moneymarket model. This model includes equations forMl and the markets for bank reserves. and bankloans. It is a set of simultaneous equations thatforecast M1, the commercial paper rate, bank loansand other variables for given levels of income,prices, the discount rate and nonborrowed reserves.

Each entry in column 3 represents a three-monthahead forecast made prior to the availability of datapertaining to the quarter being forecast. For ex­ample, the forecast for the first quarter of 1982 wasmade in mid-January 1982, while that for the sec­ondquarter of 1982 was made in mid-April 1982;Column 3 thus contains forecasts of Ml based onforecasted values of interest rates, income, andbank loans, whereas the simulations in column 2take these explanatory variables at their actual val­ues. The ex-ante forecasts put average MI growthover the five-quarter period at 9.9 percent, makingitpossible to have predicted the rapid Ml growth inthat period. Thus on a three-month-ahead basis, MIgrowth from the first quarter of 1982 to the firstquarter of 1983 was not a surprise. It was consistentwith available information on the behavior of wide­1y recognized determinants of M1growth.

If the demand for Ml did not shift, what explainsthe rapid growth of that aggregate in the periodbeing considered? An answer is provided in Table3, which separates the simulated MI growth inTable 2 into three categories: growth due to changesin the commercial paper rate, personal income, andbank loans. The figures in column 3 suggest thatbank loans had little to do with average M1 growthover the period, and that on balance they caused asmall decline in MI. Changes in nominal personalincome contributed a fairly steady 4.9 percent toaverage Ml growth. The largest contributions are

Table 3Decomposition of Dynamic Simulation of M1 in Table 2

(Annualized Growth Rates)*M1 Growth Due to"

1982/QI1982/Q21982/Q31982/Q41983/QI

Average for the period

CommercialPaper Rate

1.70.39.3

14.94.4

5.3

NominalPersonalIncome

3.15.06.94.84.5

4.9

Change inBank loans

2.4-0.5-2.8-2.9

1.2

-0.5

* Calculated as last month in quarter over last month in previous quarter.

** The three columns below do not add up to the simulated values for MI in any given quarter (Table 2) because the simulated equation

(Table I) has an auto-correlation correction.

15

made by the declines in the commercial paper rate inthe third and fourth quarters. These drops by them­selves caused Ml to grow at an annual rate of about9Vz percent between the third quarter of 1982 andthe first quarter of 1983.6 Apparently, most of thesharp decline in velocity in this period is explainedby the drop in interest rates.

These results raise a question as to why velocitydid not decline sharply in the past when interestrates fell. For example, short-term interest rates fellsharply from 1974 through 1975, but velocity didnot decline. (See Chart 1.) A partial answer lies inthe widely documented shift in the demand for Mlin the period 1974 to 1976.7 Apparently in responseto financial innovation, the public's demand formoney shifted down by about 10 percent betweenmid-1974 and 1976. This downward money de­mand shift raised velocity growth by roughly 11/2

percent in 1974, 4V2 percent in 1975, and 3 percentin 1976.8 After making a downward adjustment forthis money demand instability, velocity growthwould have been between -0.7 and +0.3 percent inthe three years. Although a full analysis ofepisodes

prior to 1982 is beyond the scope of this article, it isreasonable to conclude that velocity also wouldhave behaved "strangely" following the 1974-75interest rate decline had it not been for the coinci­dental occurrence of a large downward shift in Mldemand.9 It is aninteresting "twist" of the conven­tionalwisdom on the relationship between moneydemand and velocity that velocity was "stable" in1974-75 when money demand shifted, whereasvelocity was "unstable" in 1982-83 when moneydemand apparently did not shift.

On the basis of the analysis in this section, itseems fair to reach the following conclusions. First,the public's demand for money did not appear toshift in the period from the first quarter of 1982 tothe first quarter of 1983. Second, the rapid Mlgrowth in that period is explained by the moderategrowth in nominal income and the large decline inshort-term interest rates. Moreover, the moneydemand estimates indicate that without the largedecrease in interest rates, Ml growth most likelywould have stayed within the 2Vz-5 1/2 percent targetrange established for 1982.

II. The Decline in InflationGiven that the demand for Ml does not appear tohave shifted, an alternative explanation of the de­cline in velocity in 1982 is required. The researchstaff at the Federal Reserve Bank of San Franciscohas argued that the unusually rapid decline in infla­tion provides a partial explanation. lo This explana­tion draws on the conventional distinction betweennominal, or market interest rates, and real, or infla­tion-adjusted interest rates. Economic theory arguesthat the level of spending on goods and servicesdepends .on the real rate of interest, that is, thenominal interest rate minus the expected rate ofinflation. In contrast, as theory also argues, thepublic's demand for Ml depends on the nominalrate of interest. To illustrate the significance of thisdichotomy for developments in 1982, assume thatthe rate of inflation falls and that the Federal Re­serve allows this to be reflected in an equal declinein nominal interest rates. In this circumstance, thereal rate of interest would be unchanged, implyingthat the decline in nominal interest rates would notstimulate additional growth in the aggregate de-

16

mand for goods and services. However, the public'sdemand for money would grow more rapidly, for atime, in response to the drop in nominal interestrates. As a result, money growth would acceleraterelative to GNP growth, implying a decline in thegrowth of velocity. 1

I

This stylized scenario is a rough approximationto the events that occurred in 1982 as a whole. TheGNP deflator rose at an 8.9 percent rate in 1981 (seeTable 4), then fell suddenly to a 4.4 percent rate in1982, for a decline of 4.5 percent in the rate ofinflation. The commercial paper rate fell by aboutthe same amount, dropping from 12.9 percent in thefourth quarter of 1981 to 8.8 percent in the fourthquarter of 1982 for a decline of 4.1 percent. Thevery rapid growth in Ml associated with the dropin nominal interest rates, however, did not providea great deal of stimulus to the economy becausereal interest rates were not reduced subst<}ntially.Thus, real GNP in 1982 fell on average at a 0.9percent rate.

The preceding analysis discussed developmentsover 1982 as a whole. The explanation for thepattern of developments within the year is morecomplex. The year can be divided into two seg­ments: the first half, when short-term interest ratesstayed at a high plateau 003 to 14 percent, and thesecond half, when rates fell to a lower plateau.Velocity declined in both periods for somewhatdifferent, but related reasons.

The sharp decline inthe rate of inflation in 1982occurred early in the year. At that time, Ml wasabove its annual range and the Federal Reserve wasgradually bringing that aggregate back toward itsupper boundary. Nominal short-term interest rateswere therefore relatively high; combined with lowinflation, they produced high real short-term inter­est rates that contributed to a continuation of theweakness in the economy that had prevailed in1981. A fall in nominal income in the first quarter of1982 contributed to the decline in velocity in thatquarter.

In the second half of 1982, in response to theweak economy, the Federal Reserve adopted a moreaccommodative posture toward supplying reserves.Nominal interest rates (which also benefitted fromreductions in the discount rate) declined, and Mlaccelerated. As explained earlier, velocity fell inthe next three quarters in a predictable response tolower nominal interest rates, and GNP remainedweak despite the rapid Ml growth.

Given that the 1982 decline in velocity seemsconsistent with standard macroeconomic theory,why was this decline so surprising as 1982 un­folded? The Federal Reserve clearly did not antici­pate the events of 1982 or it would not have set a2V2 __51/2 percentannual target range for the year.The major economic forecasters were also surprisedas is evident in a survey by the FRBSF staff of tenmacroeconomic forecasts made early in 1982 for theyear 1982.12 On average, these forecasters believedthat Ml growth of about 5 to 6 percent in 1982would produce nominal income growth in the 9 toII percent range. Their forecasts implied a growthin velocity of around 4 to 5 percent.

What went wrong with these forecasts? One pos­sibility lies in their over-predictions of inflation.The predictions of the ten forecasters were that therate of inflation (as measured by the GNP deflator)would decrease by about I to 2 percentage points in1982 compared to 1981. As noted earlier, inflationactually fell by 4.5 percentage points. If the fore­casters had known that inflation would fall so sharp­ly, they may have anticipated that there would bestrong pressure for nominal interest rates to fall,which in turn would imply lower growth in velocity.The events in 1982 were a surprise, therefore, notbecause the demand for Ml shifted but at leastpartially because the rate of inflation droppedsuddenly and by a large amount.

Table 4Selected Economic Data

Six-monthCommercial

Growth in Growth in Growth in PaperReal GNP" GNP Deflator" Velocity" Rate""

1981/QI 7.7 10.5 13.3 14.5Qll 1.5 6.7 3.7 15.4QllI 2.2 8.7 7.7 16.2QIV -5.4 8.5 - 0.2 12.9

1982/QI -5.2 4.2 11.3 13.7Qll 2.1 4.5 3.4 13.5QllI 0.7 4.9 - 0.5 1l.6QIV l.l 3.7 10.1 8.8

1983/QI 3.1 5.7 - 5.1 8.3

* Annual rates of change calculated from average of monthly figures.

**Averages of monthly figures.

17

III.Policy ImplicationsThe. conclusion that the surprising behavior of

vel?city in 1982 lllllY have been relllte~Joa~~~

drop in inflation and nominal interest r~tes, and nott?ashift .in.the~emandJorMl,has .aniInportantimpliclltion forpo!icy.in 1983. If the dtlllandforllloney hadb~en unstable between the first .quartersof 1982.and1983,ther~.wouldbe goodrtason.forc()ncem that. the instability. would continueiforanindefmite period into the future. However,underthe in~ation/interest rateexplallation, thereisgoodreason tobelieve that velocity will return to morenormal behavior at le~stby mid-year'

It is important to recognize that the 1982 declinein interest rates should affect Ml growth (and thusvelocity growth) only temporarily. Money growthwill rise relative to GNP growth only as long as thepublic's demand for money is stimulated by de­clines in interest rates. Once interest rates stabilizeat their new lower levels, the effects on moneygrowth should dissipate according to the lags in thedemand for money.

The equation in Table 1 suggests that interestrates affect Ml demand for six months. A one-timedecline in the commercial paper rate in any givenmonth causes Ml to accelerate relative to GNP (thatis, causes velocity growth to fall) contemporane-

18

ously and for the next five .months.• This resultsuggests that Ml-growthinduced by the decline ininterest rates in 1982 should play itself out in thesecond9~<Uterof1~83.• As shown in Table 4, thec?U1U1ercialpaper. rate .fellsh~ly in the. third andfourth quartersof 1982. By the second quarter of1983, these interest rate ~hanges should be. having0llly milloreffects onMlgrowth. Thi~conclusionimplies.thatthevelocity ofMl shouldbehavelllorenOrmally aQdthat Ml should be. take.Qlllore serious­ly as an indicator in thesecondhalf()f 1983.

Thisconclusioll also raises th~issll~ofwhether itwould be advisable for the Federal Open MarketCommittee to return to the strict targeting of U1one­tary aggregates. Unfortunately, this is too broad aquestion tobe answered in this article. The answerdepends not only on the considerations discussedabove, but also on possible distorting effects ofrecent interest rate deregulation on money de­mand. 13However, the discussion abovedoes implythat the factors causiQg the unusual behavior. ob­served in velocity between the first quarters of 1982and. 1983 are not likely to continue into the secondhalf of 1983. It would be risky, therefore, to ignoreMl-growth when setting monetary policy for theremainder of the year.

FOOTNOTES

1. This possibility is raised, Jar example, in "Record.ofPolicy Actions of the Federal Open Market Committee,"meeting held on August 24, 1982.

2. This perceived shift of ~IJnds led the Federal Reserve to"shift-adjust" M1 in 1981. See Barbara A. Bermett, "Shift­Adjustments to the Monetary Aggregates," EconomicReview, Federal Reserve Bank of San Francisco, Spring1982,pp.6-18.

3. The equation in Table 1 uses M1 as the dependentvariable, whereas the SF model has separate equations fortransactions deposits and currency in the hands of thepublic. See John P. Judd, "A Monthly M()del of the Moneyand Bank Loan Markets," Working Papers in Applied Eco­nomic Theory and Econometrics, Number 8301, FederalReserve Bank of San Francisco, May 1983.

4. See John P. Judd and John .L. Scadding, "What DoMoney Market ModelsTeU Us About How To ImplementMonetary Policy?-Reply," Journal of Money, Credit andBanking, November 1982, Part 2, pp. 868-877.

5. Since the San Francisco model predictions depend onits conventional arguments, conventional money marketmodels may produce similar predictions in 1982.

6. The simulations in Table 3 were repeated with theesti­mated interest elasticity raised by one standard error andlowered by one standard error. The higher (in absolutevalue) elasticity yielded average growth for the period of11.6 percent, while the lower elasticity yielded growth of 9.1percent.

7. See John P. Judd and John L. Scadding, "The Search fora Stable Money Demand Function: A Survey of the Post­1973 Literature," Journal of Economic Uterature, Sep­tember 1982, pp. 993-1023.

8. See Richard D. Porter, Thomas D. Simpson and EileenMauskopf, "Financial Innovation and the Monetary Aggre­gates," Brookings Papers on Economic Activity, 1979:1,p.214.

19

9. Another possibility is that the interest sen~itivity of M1demand has increased since 1974. Estimatesofthe long­run interest elasticity of M1 for the June .1965 to May 1974sampleperiodareonly __ .05, about V3the post 1975 esti­mates reported in Table 1. On the other hand, when theeCjuation in Table 1 is estimated over .a sample periodincluding the 1970s (January 1971-December1982), thelong"run interest elasticity is - .143, very close to the resultsin Table 1. The issue of possible changes in the interestelasticity of M1 demand appears to be unresolved. Thesalient point for the analysis in this paper is that on the basisof data for the 1970s through 1981 ,behavior of velocity in1982 i$ consistent with a stable M1 demand equati()n.

10. ThiS explanation is advanced in Michael W. Keran,"Velocity and Monetary Policy in 1982,'; Weekly Letter,Federal Reserve Bank of San Francisco, March 18,1983.

11. In formal terms, I have in mind an IS/LM model, derivedin terms of the nominal rate of interest, in which the LMcurve is infinitely elastic at a given nominal interest rate. Adrop in the rate of expected inflation would cause (ceterisperibus) the IS-curve to shift to the left, reducing real GNP.An equal drop in the nominal interest rate would move downalong the IS curve until the original level of real income wasrestored. At the same time, the public's demand for M1would rise in response to the drop in nominal interest rates,and velocity would faU.

12. The forecasters surveyed include Data Resources,Chase Econometrics, UCLA Business Forecasting Project,Bank of America, Evans Econometrics, Georgia State Uni­versity Forecasting Project, Security National Bank,Wharton Econometrics, Claremont Economics Institute,and the Reagan Administration.

13. For a discussion of these issues, see John P. Judd,Weekly Letter, Federal Reserve Bank of San Francisco:"New Deposits," January 21, 1983; "Is M1 Ruined?-PartI," March 25,1983; Is M1 Ruined?-Part II," April 1, 1983.

Alden L Toevs*

Bankers use many different asset/liability man­agement models. Each focuses on the types andamounts of assets and liabilities needed to attain aparticular goal. Gap models are concerned with theexposure of net interest income-interest incomeless interest expense-to changes in interest rates.These models are currently popular because recentinterest rate variability has increased the uncertaintyof net interest income, which currently constitutes60 percent to 80 percent of total bank earnings. I Asan example, net interest income in a recent surveyof larger commercial banks had a quarter to quarteraverage variation of 5.5 percent for 1977 and 1978.From 1979 through 1980, the a~erage variation wasthree times higher.

Existing gap models, however, have seriousshortcomings, and several of these are revealed forthe first time in this paper. One shortcoming is thatthese models impede banks and thrifts (henceforth,banks) from hedging the interest rate risk of theirearnings by unnecessarily constraining the bank'schoice of assets and liabilities that create the hedge.This inflexibility also reduces the bank's ability toaccommodate customer demands for bank services.The increased competition created by financial de­regulation makes customer loss particularly threat­ening to banks. 2 Indeed, the banks that survive inthe new financial environment will be those thatlearn to reduce their risks while meeting customerdemands for financial services.

*Visiting Scholar 1982, currently Associate Pro­fessor of Economics, University of Oregon. The.author wishes to thank Jack Beebe, Chris James,Rose McElhattan, Charles Pigott, and Randy Poz­dena for helpful comments.

20

Section I presents the fundamentals of traditionalgap management models and discusses the assump­tions that underlie them. Section II develops a newinterest rate risk management model. This modelshares a common goal with extant gap models, thatof monitoring and managing the rate risk exposureof current bank earnings, while offering severaladvantages over current gap models. First, it pro­vides a measure of interest rate risk that can beexpressed as a single index number. Current gapmodels provide only" scenario modelling" throughelaborate computer simulation. Second, the newmodel removes unnecessary restrictions imposedby extant gap models on the bank's choices ofassetsand liabilities to hedge bank earnings. This addedflexibility makes interest rate risk management lesscumbersome as it more completely accommodatesbank customers. Finally, while not depending ontheir use, the new model can straightforwardly in­corporate financial futures.

Sections I and II treat interest-rate-risk manage­ment models as useful only for hedging bank earn­ings against changes in interest rates. In Section III,these models are extended to consider how a bankcan structure its balance sheet to adopt a prespeci­fied level of interest rate risk in bank earnings. Wealso show that the new model is capable of hedg­ing the market value of bank equity. This interestrate hedge can be constructed simultaneously withor independently from a hedged position of bankearnings.

I. The Gap Management ModeP

The Basic Gap ModelThe gap model derives its name from the dollar

gap (Gap$) that is the difference between the dollaramounts of rate-sensitive assets and rate-sensitiveliabilities.

As an introduction. to the reader, a basic gapmodel is described first below. After the descrip­tion, a "state of the art" gap model is presentedalong with the assumptions that underlie gap mod­els in general.

To use the model, a bank must supply fourpieces of information. First, the bank must selectthe length of time over which net interest income isto be managed.4 One year is usually chosen for this"gapping period." Second, the bank must decidewhether to preserve the currently expected net inter­est income (NIl) for the gapping period or to attemptto better it. For the former, the gap model is usedto hedge NIl against changes in interest rates; forthe latter, an "active" (speculative) strategy isadopted. Third, if the bank adopts an active stra­tegy, an interest rate forecast for the gapping periodis required. Finally, the bank must determine thedollar amounts of the rate-sensitive assets and therate-sensitive liabilities.

Rate-sensitive assets (RSA) are those that canexperience contractual changes in interest rates dur­ing the gapping period. All financial assets thatmature within the gapping period are rate-sensitive.Variable rate assets "repriced" during the gappingperiod are also rate-sensitive regardless of theirmaturity dates. Interest income and the periodicreturn of principal, as on a mortgage, are also rate­sensitive if these flows are invested in new instru­ments during the period. Rate-sensitive liabilities(RSL) are similarly defined. 5 CD's maturing duringthe gapping period, Fed Funds borrowed, Super­NOW and money market accounts are all rate-sensi­tive. Because Regulation Q ceiling interest rates arecurrently binding, regular checking and time depos­its are not considered to be rate-sensitive. 6

If the bank wishes to hedge NIl against changesin interest rates, then the basic gap model recom­mends setting Gap$ = O. It is argued that the Gap$

Gap$ = RSA$ - RSL$ (1)

21

causes a rate. change to influence interest incomeand interest expense equally.

Those banks wishing to be more aggressive mayactively place NIl at risk. As.one Citibank officialnoted, "if we don't gap we can't make enoughmoney."7 An active gap strategy requires the for­mation of a mismatch between RSA$ and RSL$.The direction of this desired mismatch depends onthe interest rate forecast. If rates are expected torise, Nil can be enhanced (should the rate forecastcome to pass) by setting Gap$ greater than zero. Inthis case, more assets than liabilities shift into high­er earning accounts during the gapping period. As aresult, the NIl realized exceeds the NIl that wouldhave been earned had either rates not increased orGap$ been set at zero. 8 These recommendations,and similar ones for when rates decline, are consis­tent with the following formula:

E(LlNlI) = RSA$'E(Llr) - RSL$'E(Llr)= Gap$'E(Llr) (2)

where Ll means "change in," E(LlNIl) is the ex­pected change in net interest income and E(Llr) isthe expected change in interest rates. Thus, to get anexpected NIl greater than the hedged NIl, i.e., apositive expected change in NIl, one contructs apositive Gap$ when E(Llr) is positive and a negativeGap$ when E(Llr) is negative.

One issue remains to be addressed: how are assetsand liabilities that are automatically repriced anumber of times in the gapping period--such asmonthly variable rate loans- treated in measuringGap$? A liability or asset is said to be repriced whenthe contractual interest rate changes, as when amaturing account is rolled over into a new accountwithin a bank or when rates change contractually, asin a variable rate account. Each such account isincluded in the values of either RSA$ or RSL$once, corresponding to its first repricing date pro­vided this date is within the gapping period. Thistreatment is logically consistent with that given tomaturing rate-sensitive accounts. Moreover, it isconsistent with an important but not often discussedassumption made in ALL gap models that each inter­est rate change is treated separately and in sequence.

As an example, suppose the goal is to hedge NIl.The Gap$ initially constructed may hedge NIl only

•'State of the Art" Gap ModelA major problem with the basic gap model is that

it computes Gap$ as the difference between RSA$and RSL$ regardless of when the assets and liabili­tiesarFrepricedwithin the gapping period. All thatcounts in measuring Gap$ is that repricing occursduring the gapping period; it does not matter whenduring the period the repricing occurs or when itoccurs first, as in the case of a variable rate instru­ment. As an extreme example, suppose all the rate­sensitive assets are repriced on day 1 while all therate-sensitive liabilities are repriced but only on thelast day of the year. Should RSA$ = RSL$ in thisinstance the basic gap model would falsely indi­cate, by Gap$ = 0, that NIl is protected from ratechanges. 9

The newer literature attempts to solve this intra­period problem by using a "maturity bucket" ap­proach. The approach calls for the dollar gap to bemeasured for each of several subintervals (maturitybuckets) of the gapping period. Most authors rec­ommend that Gap$ be measured over each 30- to90-day time increment. These separate dollar gapvalues are called incremental Gap$s and they sum tothe total that is measured by the basic gap model.From now on, this total will be referred to as thecumulative Gap$. Take, for example, a bank with acumulative Gap$ of $12 for the year. This Gap$could arise from any number of different incremen­tal (intrayear) gaps. Several of these incrementalgap patterns, each of which has a +$12 cumulativeGap$, are depicted in Figures lA and lB. Thevertical axis in· these graphs measures the net assetrepricing for the associated month, a month beingthe assumed maturity bucket.

Suppose that one interest rate shock occurs oncebefore any repricing occurs, pattern (a) in Figure lAwould have a net interest income realized at year­end that differs from the originally expected NIl bymore than that for pattern (b). Similarly, pattern (b)is more· NIl risky than pattern (c). The NIl riskexposure of gap patterns like those in Figure 1B aremore difficult to assess-an issue we will addresslater. Nevertheless, it should be clear that anycumulative Gap$ can arise from a large number ofdifferent incremental gap patterns and, therefore,many different levels of net interest income risk canbe associated with one measured cumulative Gap$. 10

The current gap literature recommends that NIl

(c)

10 12Months

8

b

6

[B]

42

(a)

o

-3-4

+6+5

IncrementalGap $

Figure 1Alternative Incremental Gap Patterns

Incremental [AIGap $

+12

against the first interest rate change. As time passesin the gapping period and the first repricing date onthe assetand/or liability is reached, the funds mustbe redeployed to make the Gap$ for the remainderofthegapping period zero once again. This proce­dure positions the bank to protect NIl expected atthe beginning of the year against the next ratechange, and after that rate change, against the next,and soon.

The last. is an important observation because anunderstanding of the influence of Gap$ on NIl forone interest rate shock implies an understanding ofits operation on multiple rate changes. We need,then, explicitly consider only one rate change pergapping period to illustrate any gap model. Theexposition is simplified further by having the ratechange come before the first repricing date in thegapping period.

22

be hedged by setting each incremental gap equal tozero. If rates are expected to rise, positive gapsshould be put into place; the opposite holds forexpected rate declines. The use of incrementalGap$ rather than just the cumulative Gap$ increasesthe probability that NIl will tum out to be asexpected.

There are, however, two common simplifica­tions made in the measurement of incrementalGap$s that can distort the purported accuracy ofthe incremental gap approach. First, incrementalGap$s are normally measured such that cash flowsof interest and periodic principal payments are ig­nored or attributed to the wrong maturity bucket.For banks that use book values to compute incre­mental Gap$s, it is common for pre-maturity inter­est and principal cash flows (i.e., payments orreceipts) to be attributed to the maturity bucket thatincludes the maturity date of the instrument. II Thiscan falsely attribute pre-maturity cash flows tomaturity buckets that occur after these flows arereceived. For banks that use maturity values ratherthan book values to compute incremental Gap$s,intervening cash flows are often completely ignored.

The second commonly encountered measure­ment error in computing incremental Gap$s resultsfrom the use of large maturity buckets. Just as withthe cumulative Gap$s, each incremental Gap$ willfail to reveal perverse intraperiod repricing. Forexample, suppose each 90-day incremental Gap$ iszero. If rates change once, as much as one quarter ofthe annual NIl can be exposed to a single ratechange. 12 If 30-day gaps are used, as much as onetwelfth of the annual NIl can be exposed. Continu­ing along these lines, one can construct exampleswherein each bucket has a positive incrementalGap$ but in which if rates increase, contrary to ourexpectations, NIl decreases.

Binder (1982) has described a gap model basedon daily incremental gap measurements. 13 Hismodel relies heavily on the use of computer simula­tions which are essential when the number of incre­mental gaps to be monitored is large. Even theeffects of fairly simple gap patterns, as in FigureIB, on NIl are difficult to predict without computersimulations. We argue later in this paper that suchsimulations are not needed because there existsa summary measure of a bank's incremental gappattern.

23

Another major criticism of the basic gap model isthat it assumes rate changes for assets and liabilitiesof all maturities are of the same magnitude whenthere is overwhelming evidence that rate changesoccur in varying magnitudes. The gap literature hashandled this issue of different rate change magni­tudes by assuming that the volatility of the rates inquestion is in constant proportion to the volatility ofsome standard interest rate. If this assumption isreasonably accurate, it should be incorporated intothe gap model to improve its performance.

One can use historical interest rate change data onvarious RSA and RSL to estimate rate change pro­portionality.14 These proportional factors measurethe rate volatility of RSA and RSL relative to oneparticular account. This standard account can beanything, but interest rate futures contracts makeconvenient benchmarks. Suppose a 90-day futurescontract is selected as the standard. Furthermore,assume that the proportionality factors for 90-daycommercial paper (CP) and 90-day certificates ofdeposit (CD) are .95 and l.05, respectively. Thesenumbers indicate that, on average, the CP rate is 95percent as volatile as the rate on the deliverablecontract underlying the futures contract while theCD rate is 105 percent as volatile. If the bank has a$100 obligation in 90-day CD and $500 lent in90-day CP, then the apparent 90-day Gap$ is+ $400. But taking into account the relative volatili­ties, the "standardized" Gap$ is $370. 15 The bankcan hedge its current asset sensitivity (NIl increaseswith rate increases) by buying $370 in futures. 16

The remaining substantive criticism of the basicgap model, that it pays too little attention to theevolution of NIl risk exposure as time passes, hasalso been corrected in the current gap literature. Asasset and liabilities mature, they can be reissued indenominations, maturities and repricing intervals toalter incremental Gap$s that remain in the gappingperiod. That is, the incremental gap pattern can bedynamically reshaped during the gapping periodeither towards a more hedged position or a moreactive one. Suppose the bank has to start the gap­ping period with a +$1000 Gap$ on day 270. If allother daily Gap$ equal zero and the bank wishes tohedge, it should attempt to reissue maturing RSL tore-mature on day 270 and maturing assets to re­mature after year-end. If the Gap$270 is completelyeliminated before rates change, then the NIl com-

puted at the beginning of the year will have to behedged. If rates change before the Gap$270 is com­pletely eliminated, then NIl will not have been fullyhedged but the risk will have been reduced. Similartreatment can be accorded to expected deposit in­flows and the like.

In summary, the "state of the art" gap modelcomputes incremental Gap$s daily or weekly. Theoutcomes of various interest rate forecasts, givenspecific incremental gap patterns, are simulated us­ing computers. These incremental Gap$s may havebeen adjusted or standardized to reflect the relativeinterest rate volatilities of various RSA and RSL.The dynamic evolution of the gap pattern is con­sidered by allowing banks to interrupt computersimulations during the gapping period to restructurerate risk with maturing RSA and RSL and newaccounts. As such, the model is more complete andtheoretically pleasing than the basic gap model.

Remaining CriticismsThe "state of the art" gap model, hereafter called

the gap model, itself suffers from five deficienciesthat are directly or indirectly addressed later in thispaper. First, others have implied that the model canhedge NIl only by equating each incremental gap tozero. We will show that this hedging condition isunnecessarily strong by revealing the existence ofnonzero incremental Gap$ that hedge NIl. This isan important point because in Section II we derive asystematic means to discover all hedging incre­mental gap patterns. This increased number of gappatterns yields more flexibility in simultane­ously hedging NIl and accommodating customerdemands for bank products. Furthermore, what welearn about flexibility in hedging also applies toactive NIl management.

An example will help illustrate that non-zeroincremental gap patterns can hedge NIl. Suppose,for simplicity, that all assets and liabilities are cur­rently earning 10 percent. If the RSA repriced oneach day of the year (360 days) equals the RSLrepriced on the same day, then net interest income(NIl) would be zero whether or not rates change.Consider now daily Gap$s that equal zero on everyday but three: Gap$30 = $1000, Gap~ = -$2000,and Gap$152 = $1000. The cumulative Gap$ for theyear is zero. The basic gap model would have usbelieve we are hedged but the more detailed incre-

24

mental gap model would not. However, in thisinstance the incremental gap model is misleading.Suppose that on day one, just after this incrementalgap pattern has been acquired, rates on all RSA andRSLincrease to 12 percent. I? NIl can change onlybecause of the non-zero Gap$s on days 30, 90, and152. The first such gap causes NIl for the gappingperiod to rise by $18.17. 18 The second and thirdnon-zero Gap$s causes NIl to fall by $29.23 and torise by $11.05, respectively. These three influencescancel. Moreover, this netting out of the threeeffects is independent of the rate change assumed.An infinite number of other gap patterns also hedgeNIl. Like the example above, some have non-zeroincremental Gap$ values but a cumulative Gap$ ofzero. Others, perhaps like those in Figure IB, havenon-zero values for both incremental and cumula­tive Gap$. 19

A second problem with the gap model is thatwhen many maturity buckets are used, the modeldoes not generate a single number index of theinterest rate risk of the bank. The basic gap modelprovides one in the cumulative Gap$, but we haveshown that this measure tells us very little. The gapmodel would be more appealing if such indexnumbers existed. These numbers could, in tum, beused to derive risk-return trade-offs or frontiers, ahelpful concept elaborated on in Section III.

A third problem arises out ofthe second problem.Because the gap model does not generate a singlenumber for risk exposure, it cannot easily be usedto determine the number of futures contracts thatwould hedge the overall rate risk of a bank, a calcu­lation of current interest to many bankers. In itscurrent form, the gap model incorporates financialfutures in one of two ways, neither of which isparticularly appealing. First, it can use financialfutures to hedge specific instruments. These indi­vidual hedges are then removed from incrementalgap computations. Second, one can simulate theeffectofa futures contract on NIl in the same way orat the same time the incremental gap pattern's influ­ence is simulated. Bytrial and error, the appropriateaggregate hedge can be discovered.

A fourth problem is that stockholders could quar­rel with exclusive concern over NIl by bank man­agement. Stockholders are interested in sharevalues, an important determinant of which is themarket value of bank assets and liabilities. They

wish to position their capital based upon their atti­tudes towards risk and expected return, where ex­pected return is the expected earnings for a givenperiod plus the expected change in market value ofequity over this period. The gap literature paysattention to expected earnings but not to the influ­ence of interest rates on the market value of assetsand liabilities.

A fifth problem lies not with the model per se; itarises because proponents of the model almost com­pletely ignore the theory of the term structure ofinterest rates. This theory suggests that currentlyavailable information on how rates are expectedto change have already been incorporated in the"yield curve" or term structure. If these ratechanges, which represent the market's forecasts,come to pass, any active NIl strategy will not im­prove NIl relative to that available by NIl hedging.20

To be successful in actively managing NIl, onemust have a better interest rate forecast than themarket's. 21

Consider the following example: The marketexpects interest rates, expressed on an annualizedbasis, to be 10 percent for the first quarter, IIpercent for the second, 12 percent for the third, and13 percent for the fourth. This interest rate patterngives a one-year rate of 11.49 percent22 and indi­cates that the market expects rates to rise. A bankmight incorrectly infer that it will profit from a gapcon: "Ucted, say, by booking a one-year loan of

$1000 at 11.49 percent and a $1000 90-day CD thatwill roll over with interest every quarter. Shouldrates rise as the market expects, NIl earned for theyear will be zero. Given our assumption of the samerate structure for assets and liabilities, this is the NIlobtained by hedging. (The initial negative carryswitches mid-year to a positive carry and it does soin such a way that there is no time-value benefit tothe initial negative carry. 23) Only if rates are fore­casted by the bank to fall by more than the marketforecast would this incremental gap pattern yield aNIl in excess of that originally promised. If rates areforecasted by the bank to rise or to fall by less thanthe above market forecast, then it would be appro­priate to construct a negative Gap$ on net for theyear.

The final problem with the gap model is its in­attention to unexpected deposit withdrawals or loanprepayments. Predictable withdrawals and pre­payments, e.g., deposit reductions as bank cus­tomers meet seasonal requirements, can easily beaddressed in the gap model. These" maturities" arebest matched with similarly maturing assets or lia­bilities. A more difficult problem arises when theamount and timing of withdrawals and prepaymentsdepends upon the spread between their contractualrates and current market rates. Unexpected changesof this sort can substantially affect realized net inter­est income, yet the gap literature is silent on theappropriate treatment.

II. A Generalized "Duration" Gap ModelWe have developed a generalized "duration"

gap model to show that the "state of the art" gapmodel described in Section I is a special and con­straining model for measuring and monitoring theinterest-rate risk exposure of NIl. The technicalaspects of the duration gap model are contained inAppendix I but we will briefly review its mainfeatures and derivation here.

We start the derivation of the generalized modelwith a definition of NIl when interest rates do notchange unexpectedly within a year; this net earningsfigure is referred to as NII". We then restate netinterest income in general terms for cases whenrates change unexpectedly. The last step involvesfinding the overall combination ofassets and liabili­ties that would balance changes in interest income

25

with those in interest expense. The result is that NUnwill be realized, even if rates change unexpectedly,provided that the weighted sum of the market valueof all rate-sensitive assets equals the weighted sumof the market value of the rate-sensitive liabilitieswhere the weights are equal to the fraction of theyear from repricing to the end of the year:

(3)

where MVAj (MVLk) is the market value at thebeginning of the year of a single asset (liability)payment that will be repriced during the year; tj (tk )

is the fraction of the year until this asset (liability)payment is repriced or is first repriced if repricing

MVRSA (I-DRSA) = MVRSL (I-~sd (4)

DRSA and DRSL are, respectively, the Macaulay'sdurations of the RSA and RSL. Duration is a cashflow timing statistic of financial instruments thathas been used in bond portfolio management formany years.25 Duration in our context is the weightedaverage time to repricing, where individual weightsin this average are MVA/MVRSA for RSA and

occursmore than once during the year; and N (M) isthe number of separate rate-sensitive asset (liabil­ity) p.ayments}4

It is important to note that NIlo and the NIlassociated with rate changes does not presume thatas~ets.andJiabilities are valued by the bank atmarlcetvalue. Rather, .as shown in Appendix I, allaccounts are carried at book value and interest in­come and expense are computed using these values.The generalized hedging condition stated inequa­tion (3) is in market value terms because of mathe­matical.conditions needed for a NIl hedge ratherthan an assumed accounting convention.

The generalized model in its simplest form relieson the following assumptions: (I) that the gappingperiod is one year, (2) that the term structure ofinterest rates is flat (constant) for each type of assetand liability, (3) that the unbiased expectations hy­pothesis of the term structure governs the expectedevolution of interest rates (given the flat term struc­ture assumption, this means that all rate changes areunexpected), (4) that all asset and liability interestrates are affected equally when any unexpectedchange in rates occurs, and· (5) that no depositwithdrawals or loan prepayments take place. Eachof these assumptions can be relaxed, and in Appen­dix II, we do so for several of them.

Many .incremental gap patterns are consistentwith equation (3). Ifeach repriced asset is matchedin timing and amount with a liability, equation (3)will be upheld. But, item by item matching is notnecessary. For example, let the market values of allRSA and all RSL be represented as MVRSA andMVRSL, respectively. Equation (3) can be re­expressed as

M

DRSL = L (MVLk/MVRSL) tkk=\

(6)

The sign of DG indicates the type of rate risk towhich the bank is currently exposed. The larger isDG in absolute value, the greater is the risk. Sup­pose DG>O. Thisinequality indicates that a fall(rise) in rates will cause realized NIl to be less than(greater than) NII".This is analogous to a "net posi­tive gap" in the conventional literature .26 The con­verse is true when DG<O, in which case, we have insome sense a "net negative gap."

The duration gap thus defined yields a single­valued risk index that is not only a convenientstatistic but also an indicator of risk as accurate asthe risk level derived from computer simulations ofthe incremental gap pattern embedded in the currentvalue for DG. This is an important point. The use ofduration in the gap model results in a gap-typemeasure of risk that can be as intuitively under­standable .as the outmoded cumulative Gap$ mea­sure and as accurate as using the measured incre­mental gap pattern in a computer simulation to

MV4/MVRSL for RSL.Equation (4) reveals an interesting alternative to

the NIlhedging condition as typically expressed inthe gap literature. Sufficient, but not necessary,hedging conditions are rnat MVRSA= MVRSL andD~SA;:;:DRSL. The first of these is somewhat likeequating RSA$ and RSL$, i.e., somewhat like set­ting the cumulative Gap$ of the basic gap modelequal to zero. The second equates the "average"repricing date of the RSA with the "average" re­pricing date of the RSL.

Reconsider the example given in Section I ("Re­maining Criticisms") of a gap pattern that hedgesNIl even though all incremental gaps are not equal.The details are given in Example 1ofTable I, whereequation (4) is shown to be met by setting MVRSA=MVRSL and D RSA =DRSL . Example 2 in Table 1provides a case where neither market values nordurations are equal yet equation (4) is satisfied.Example 3 will be discussed momentarily.

The full value of equation (4) becomes clearwhen the accounts currently on the books of thebank violate ·this hedging condition. To illustrate,define the duration gap (DG) as the size of thedeparture from the equality given in equation (4), or

DG = MVRSA (1-DRSA) -

MVRSL (1-DRSL)

(5)

N

L (MVA/MVRSA) tj and)=1

whereDRsA

26

27

(8)

detennine the effect of an interest rate change onNIl.

Furthermore, the duration gap can be used toselect the appropriate adjustment in RSA and/orRSL to remove NIl risk:

If))(}<::Q,thento ~chieveaNIlhedge add$X inrU<ifket value ofnetrate.-sensitive assets with a dura­tionYwhere

X "",<absolute value ofDG/(I-Y) (7)

If))(}?,O, then to.achieve a NIl hedge,add $X inmarketvalue of netrate-sensitive liabilities with aduration of Y where X and Y are defined above.

Exanlple J in Table 1 can be used to illustrate thepoint. In it,we have taken some of the RSL ofEXample 1 away from the bank. This makes thebankuet asset sensitive, i.e., MVRSA>MVRSL,and the bank will do well if rates rise unexpectedlybut it will do poorly if rates fall .. Equation (7) can beuseg to rediscover the amount of RSL we had inExample 1. Alternatively, this equation can be usedto restructure RSA or RSL in other ways. Severaloptions are given at the end of Example 3.21 Theasset/liability manager can use the type of flexi­bility illustrated in Example 3 to achieve the NIlhegge •with •minimal disruption of existing bankaccounts .and/or maximum accommodation of cus­tomer demands.

Fed funds and interest rate futures contracts areparticularly useful hedging instruments to use inequ(ltiQn (7).13oth can be used to alter, in eithergirection, NIl sensitivity to rate changes ofaccountsalreadyon.the books. Because daily and term Fedfunds contracts are paid in lump sum at maturity, theduration of any Fed funds contract is its maturityd~te.The"duration" of an interest rate futurescQntract is more complicated. No scheduled c.ashp~)'rnents wise from this contract so evaluation of adllrationformula like that in equation (5) is impos­sibl~.Nevertheless, one can obtain a duration for afutures contract; it has been shown to be the dura­tionofthe underlying deliverable contract. 28 Notethattheuse of futures in the context of equation (7)

28

provides a NIl hedge for the entire bank, not just aninterest rate risk hedge for a single cash instrument.

As time passes during the. gapping period, therewill be changes in market values and durations ofRSA and RSL. There will also arise a need torestructure thehalanceisheet as maturing accountsare re-booked. Thus,one must.periodicallyrebal­ance the assets and.liabilities to re~establish theequality in •equation (4). At the end of the firstmonth, for example,the hedging condition for theremainder of the yearis

MVRSA' (11/12 - DRsA) =

MVRSL' (11/12 --DRSL)

where the primes indicate market values and dura­tions, measured after the month has passed, for theaccounts that are rate-sensitive during the monthsremaining in the year. If the NIl hedge is in placethroughout the year, any number of unexpectedinterest rate changes can occur and yetNIl willequal NIlo at year-end. These re-balancing costs areapt to be low.29 Re-balancing will be taking placeanyway as unexpected inflows and outflows occurduring the year. Also, the disposition of maturingcontracts alone may provide sufficient flexibility inaltering asset and liability durations to re-establishthe NIl hedge.

Several assumptions made for simplicity's sakeat the beginning of this section are actually toostrong. The more important generalizations are:

1. Term structures of interest rates are not alwaysor even normally flat, and unexpected changes inrates may be term specific.

2. Rate changes for certain instruments are morevolatile than others with like maturities.

3. Deposit withdrawals and loan prepayments arefunctions of the spread between the rates paid onthese accounts and current market rates.

Appendix II addresses each of these generaliza­tions in a preliminary form to show that the modelcan be reworked to inCOrPorate more realisticassumptions.

nl. U~e ()f C:;ap.MOdelsforPurposesOther than Nil Hedging

number. The one chosen from the available set can,then, depend solely on the current set of assets andliabilities and on maximizing the accommodationofnew customer demands.

A Nil risk-return frontier can be derived asfol­lows; .An accurate approximation to a one-yearholding period return on an asset or liability re­priced during the year is30

where ~ is the realized one-year return on a RSAorRSL that has a Macaulay duration of D. (D has to beless than one year by the definition of rate sensitiveity.) The one-year rate of interest at the beginning ofthe year is io and the forecasted one-year rate at therepricing date is i*. An intuitive interpretation ofthis fonnula is that io is attained for D portionoftheyear and i* is experienced for the remaining (I-D)portion of the year. This equation can be rewrittenas i, = io + (1-D)Lli, where Lli is the bank forecastof how the rate will change in comparison to themarket forecast. (To simplify the presentation, wehave assumed that the market forecasts no change.)

Through very simple substitutions in the equa­tions given in Appendix I, one can obtain the resultthat

where NIl, is the NIl realized over the one-yeargapping period and NIIo is the NIl realized if rateschange only as expected by the market. Equation(10) can be used to compute the gains in NIl forvarious 00 values. If Lli > 0, then the bank shouldset 00 > 0 and NIl, will increase with the magni­tude of 00. (If Lli < 0, then DG < 0.)

It is unlikely that the value or even the sign of Lliwill be known with certainty, but we assume thebank can identify all possible outcomes and theirprobabilities of occurrences. 31 The expected (prob­ability weighted) NIl in excess of NIIo becomes

E(NII, - NIIo ) = E(LlNII) = DG Lli (II)

where Lli is the probability weighted interest ratechangeY If either Lli = 0 or DG = 0, then theexpected NIl is NIIo ' otherwise, E(LlNII) > 0 when­ever 00 and Lli have the same sign.

(9)

(10)NIl, - NIL = DG Lli

Active Strategies for Net Interest IncomeEarlier sections of this paper assumed that a bank

wished to hedge its NIlcompletely.This may notbetrue. Some banks may believe that they can dobetter than the market in forecasting interest rates,and decide to adopt interestrate risk. For them, theduration gap model described in Section II, can bequite usefuL

AImostall that the basic gap model offers foractive strategies is contained in equation (2). Theequation implies that if the bank expects interestrates to increase (that is, to increase more than themarket forecast), it should set the cumulative Gap$greater than zero in order to increase expected NIl;it should set the cumulative Gap$ less than zero forthe opposite expectation. As was shown in Example2 in Table I, non-zero cumulative Gap$s can beconstructed such that NIl is hedged. This line ofreasoning can be extended to note that a non-zerocumulative Gap$ can have no rate risk. Thus, thebasic gap model is as deficient in active NIl man­agement as it was in passive NIl management.

The extended traditional gap models rely oncomputer simulations to detennine NIl under vari­ous .·scenarios ·of rate changes. Conditional upondifferent plausible rate forecasts, users of tradi­tional models seek incremental gap patterns thatgenerate computer stimulations with the highestextra NIl return. Simulations might also be con­ducted to determine the downside risk should ratesmove against the bank.

The duration gap model developed in Section IIcan easily be used to systematize active NIl strate­gies through the construction of a risk-return trade­off or "frontier." Such a frontier would represent amenu ofchoices ofexpected gains ofNIl above NII.,accompanied by associated risks. With this frontier,bankers· can select a duration gap strategy that isconsistent with their risk-return preferences.

Implementation of the chosen duration gap for anactive NIl strategy would be accompanied by thesame asset/liability choice freedom discussed ear­lier when NIl was hedged by setting the durationgap (DG) equal to zero. That is, what ever is re­vealed to be the optimal DO, there are thousandsof .incremental gap patterns consistent with this

29

or

MVANS(DNSA -I) + MVBC = (14)

MVLNS(DNSL -I) + DG

where MVA (MVL) is the market value of all bankfimmcial assets (liabilities) and DA(DL) is the dura­tionof these assets (liabilities) and MVBC = MVA- MVL.If MVBC is greater than zero, as is nor­mallythe case, then equation (12) can hold only ifDLexceeds DA'

A bank may wish to puts its MVBC at risk. If thebank expects rates to rise more than the marketconcensus, then it will increase MVBC if MVA DA> MVL DL. One can easily use the developments inthe first part· of Section III to derive MVBC-riskfrontiers analogous to those for NIl-risk illustratedin Figure 2.

One can tie the immunization or active strategiesforMVBC together with the earlier work in thispaper on NIl. Equation (12) can be rewritten as36

MVANs(DNsA - 1) '+ MVA - MVRSA(I-DRSA) =(13)

MVLNS(DNSL -I) + MVL - MVRSL(l- DRSL)

(12)MVADA= MVLDL

.,r()~~ti~~.",,~r~~tVallJeBank shareholders are ultimately interested in the

market value of their shares. The market value ofbank capital i(MVBC), which is obtained by sub­tractingthe market value of liabilities from themarketvalue.ofiassets,. is a prime determinant ·ofshareivallle. 33 In general, shareholders have re­centlybecomeconcerned over their investmentsbecause past interest rate movements have causedMVBCtodepart substantially from book values inmany institutions. Recent empirical work by Flan­neryaridJames(1982) reveals that bankstockpricesare also sensitive to the degree of current interestraterisk.exposure. Under pressure .from share­holders onane side, bank management has alsorecently seen several regulatory proposals concern­ingtheinterest rate risk of MVBC These proposalsinclude mark to market accounting, risk relatedFDIC/FSLIC premiums, and can report disclosureof gap positions. 34

The duration analysis presented here can be ex­tended to hedge MVBC The condition needed forMVBC to be hedged (in the literature, "immu­nized' ') is that35

6i=O

Figure 2

Risk·Return Frontier

Nllo R' kDG=O ~

The figure sketches several risk-return frontiers; each oneof whi.ch is represented as a solid line.Higher SUbscripted DG entries have larger positive valuesthan lower subscripted duration gaps. DG > 0 since wehaveassumedthatLSI2> XI, > o.Dashed lines indicate the change in expected return andrisk, holding DG constant, as interest rate forecasts change-moving us from one frontier to another.

E(l> Nil)

Equation (11) tells us that if rates are expected torise (fall) on average, then it is better to set RSAshorter (longer) than RSL. This can be done byhaving more dollars of RSA than RSL or by settingthe duration of the RSA less than that of RSL orboth, Notice the close similarity to the intuitivelyappealing, but incorrect, equation (2).

One can solve for the standard deviation of ,iNIIand substitute it into equation (II). The standarddeviation is related to the variance of ,iNII out­comes and as such can be viewed as a measure oftheriskiness of the NIl strategy in force. The result ofthis substitution is an expression of E(,iNII)-therisk frontier. A set of such frontiers is given inFigure 2 for ,ii>O. Note that there is a frontier foreach probability distribution of ,ii's. The more cer­tain a rote increase, the steeper is the frontier. Thatis, increases in forecast certainty generate largerE(,iNII) for every risk level. The position on therelevant frontier depends upon DG. The more DGdeparts from zero, the greater the risk and expectedreturn. Note that for any DG, E(,iNII) increases andrisk decreases as ,ii certainty increases. The bankmanager's and/or shareholders' preferences for NIlrisk and return determine the desired position onthe frontier, i.e., the duration gap. This is a subjec­tive choice; the frontiers are, themselves, objec­tively determined from the interest rate forecasts ofthe bank.

30

Here, NSstands for non-rate sensitive. If onewishes to hedge NIl and immunize MVBC, one setsDG=OandMVANs(DNsA-1) + MVAequaltoMV4,s(DNSL -1) + MVL. Alternatively, •• ifonewishes to immunize MVBC but to take a fling on

NIl, one selects a nonzero DG but then fulfills theequality in equation (14). Finally, one can hedgeNIlbut put MVBCatrisk by setting DG = 0 but notfulfilling the equality in equation (14).

IV. SummaryThispaperbeg~nb)'reviewing how b<lI1ks .and

thrifis.>Ill.anagethe .i9terest rate risk e)(p()sure .ofcurrent bank earnings. Many do sousingtlie "gap"asset/liabilit)' ID.a.t1ageJ:Uent .J:U0del. SeVeral short­comings. in this wodel ",ere revealed .in Section I.Chie.faJ110m~th~se were theillabilitypfthe Ill()del togenerate a simple and reliableindex ()fthe interest­rate risk exposure of.bank and thrift net •interestincome and the unnecessarily contraining set ofassets and liabilities allowed by the model. SectionII used more general conditions for hedging bankand thrift net interest income to determine a "dura­tion gap" model that generates a single number toquantifY the risk position of the financial institu-

tions. The model also reveals a larger set of assetand liability choices to financial institutions to en­able them to establish net interest income hedgesnot currently in place.

Section III expanded the duration gap model toconsider strategies that actively place earnings atrisk. Risk-return frontiers were developed to quan­tifY the choices for "better earnings" based on rateforecasts. Finally, it was shown that the durationgap model can be generalized to hedge the marketvalue ofbank capital, should the bank wish to do so,against unexpected changes in interest rates. Weshowed that net interest income and the marketvalue of bank capital can be either simultaneouslyOf independently hedged.

Appendix IAssume the bank has a one year gapping period.

Furthermore, assume the unbiased expectations hy­pothesis of the term structure holds and that yieldcurves are flat. Any change in interest rates is,therefore, unexpected. Let NIIo represent net inter­est income for the coming year should rates notchange unexpectedly. This level of net interest in­come is the goal of the hedging strategy. Mathemat­ically, NIIo can be expressed as:

Aj is the asset book value at the beginning of theyear ofa cash inflow that will occur at time tj , wheretj is expressed asafraction of a year. This asset hasan associated contractual interest rate of rj, ex­pressed as an annualized rate. Up()n repricing, thiscash..inflow is expected to earn a new rate of ij .

There are N repriced asset flows .• (A long-termmortgage with monthly payments of $500, a con-

31

tractual interest rate of 5 percent and a new interestrate of 10 percent-the rate on new mortgages­would have twelve flows represented in the abovesummation. The Xth month flow has an Aj of5OO/1.05XfI2

, tj=X/12, rj =.05 and ij=.IO.) If anasset does not generate any flows during the oneyear gapping period, then it influences net interestincome only in an accrual sense and tj= I. Similardefinitions apply to Lk, one of M liability repricingflows.

Consider the impact on NIl of an unexpectedchange in all current rates by an additive amount A,where Acan be p()sitive or negative. For mathemat­icalconvenience, assume the interest rates changebefore any cash inflows or outflows occur. NIl nowbecomes functionally dependent on A..

N(b) NIl(A.) =~ Aj[(l+r)tj(l+ij+A.)l-tj 1]-

J~I

M

~ Lk[(l +rkrk(l +ik +A.)l-tk - I]k~1

If NU(A.) is to be hedged, then a set of asset andliability flows must be found that leaves NIl(A.)

equal to NII., . for this to occur, it must be true thatthere be no change in NII(A) as Adeparts from zeroby some small amount. Mathematically, this meansthat the derivative ofNII(A) with respect to Aequalszero in the neighborhood of 11.=0. Now,

N(c) iJNII(A)/iJA = L Aj (l+r/j(l+ij+A)-tj(l-tj)

J~I

ML Lk(l +rk)tk(l +ik+A)-tk(l+tk)k~1

If we evaluate this derivative at 11.=0 and set theresult equal to zero, we obtain

N(d) L A{l+r)'j(l+iytj(l-t) =

j~1 J J J J

But Ail +rj)'i/(l+i/j is the current market value{MY} of a contractual flow of Ap +r/j dollars tjperiQ<lsJrom now. Thus, the first order conditionfora NIl hedge is that

N N(e)L MVA(I-t) = L MVLk(l-tk)

j~1 J J k~1

This is equation (3) in the text from which equation(4) was derived. The second order conditions in thisdev¢lopm~lltprove to be more complicated thaninfonnativeand are not treated here.

A remaining question is whether or not the hedg­ing5~nditionexpressedin. equation (e) holds fornon-infinitesimal changes in A. Fulfillment of thehe?~ing conditions does not exactly equate NII(A)with NIIo for very large changes in A, say of ± 300basis points. These errors are, however, smalL

AppendixUThree strong assumptions made in Section II are

that (1) the term structure is flat and unexpectedchanges in rates keep it so, (2) the rate change on aninstrument is as volatile as on any other instrumentof similar maturity, and (3) deposit withdrawals andloan prepayments are not interest dependent. One ata time each of these assumptions is relaxed in thisAppendix to determine how the hedging conditionpreviously expressed will change.

A. Nonflat term structures that shift in a nonpar­allel fashion:

If term structures are not flat, each instrument hasan annualized interest rate of h(O,tn), where h(O,!.)is an element in a term structure for the nth type ofinstrument with repricing at date tn' The NIIo forthis model is the same as equation (a) in Appendix I,exceptthath(O,tj) replaces ijand h(O,tk) replaces ik.The stochastic process affecting interest rates mustbe specified. Suppose, as did Fisher and Wei!(1971),<that l+h*(O,t) = (1 +h(O,t»(l +11.), whereh*(O,t) is the new term structure after an unexpectedinterest rate shock. NII(A) is now

32

Differentiate NII(A) with respect to A, evaluate at11.=0, and set the result equal to zero. This gives

(t) LAj(1 +rj)(1 +h(O,tjW-tj(l-tj) =

LLk(1 +rk)(1 +h(O,tk»l-tk(l-tk)or

LMVAj(l+h(O,tj»(1-tj) =

LMVLk(1 +h(O,tk»(I-~)

A reexpression of this hedging condition using aduration measure evolves in a less direct mannerthan>before.Nevertheless, one can rewrite this lastequation as

(g) MVRSA TRSA = MVRSL TRSL

where TRSA =LMVAp +h(O,tj»(l-t)/MVRSA,etc. Equality of TRSA with TRSL is equivalent tosetting a weighted average repricing date of RSAwith that of RSL. The weights differ from thoseused in equations-(4) and (5). This reflects the moregeneral. assumptions on the term structure and thestochastic process affecting it.

B. Relative Interest Rate Changes:Return to the assumption of flat term structures

thatshift in a parallel fashion, but relax the assump­tionthat the sizes of the unexpected rate changesare equal for instruments with the same maturity.

Assume instead that all unexpected rate changesareperfectly cOrrel~ted but have differing magnitudesacross securities.Thll~' Aj=PjA and Ak=PkA, wherePj andpkareconstants. These values may be foundby examining historical series. NII(A) becomes

(h) NII(A)=2:Aj [(l +rj)lj(l +ij+PjA) 1~Ij -.1J­

2:Lk[(l +rk)lk(l +ik+PkA)I-tk - IJ

Differentiate this equation with respect to A, evalu­atetheresultforA.=0aIld set it equal to zero. Thisgives

(i)2:MVAj(l ..... tj)pj = 2:MVLk(l---tk)Pk

Again, implicit in this equation is the possibility ofhedging by equating MVRSA with MVRSL and aweighted average repricing date of RSA with that oftheRSL.

C. Rate Sensitive Withdrawals and PrepaymentsThe current interest rates on mortgages, con­

sumer CD's, etc. help determine the rate of loanprepayments and early deposit withdrawals. Let Ajand 4 become functionally dependent on the unex­pected change in interest rates. The hedging condi­tion becomes

(j) MVRSA(l-DRsA) + 2:0AjlOAh~o(l +i'j) =

MVRSL(l-DRsL) + 2:oLk/oAIA~o(l+i\)

where (1 +i')=(l +rWl +i)I-I.

FOOTNOTES

1. Rose (19El2b) p. 1.

2. Rose (1980) notes that banks' requirement of variablerates on lOans offered institutional custe>mers wasaprimemotivi:\tion for their shifting to the commercial paper market.

3. Among the first authors in this area are Baker (1978),Binder (1979), and Clifford (1975).

4. Most e>f the gap literature focuses on managing netinterestmargin (NIM) rather than net interest income. Netinterest margin is Nil -;- Earning Assets. Since there arevery few instances when growth in earning a$sets isexplic­itly treated and because it eases mathematicaldevele>p­ments ti'Jre>ughe>ut th~pi:\per, Nil not NIMwiUbeused. Ife>neunderstands the model in terms of Nil, one also under­stands it with respect to NIM.

5. One can COmpute thedollar amount of theJate-sensitiveassets (RSA$) and the dollar amount of the rate-sensitiveliabilities (RSL$) using either book values at the beginningof the year or the dollar values as•of the repricing dates.Both have been used in the literature. On both expositionaland theoretical grounds, the second method is preferableand will be used here. The qualitative conclusions, how­ever, donot <iepend on Which of these meth()ds of valuationis used.

6. These two accounts may well be rate-sensitive because,should rates rise, they may be (1) withdrawn from the bank,(2)d~POsited in higherearoingaccounts, or (3) paid ahigh~rimplicitreturnby the pre>vision ofadditioni:\I"free"banking.services.Thegap m()delne>rmally de>es npt takethese effects into account.

7. Rose (1980) p. 90.

8. When Gap$=Q, the Nil earned equals the Nil for nochange in rates because Gt:lp$ '= 0 hedgesNII.

9. The influences of dissimilar repricing schedules probablydid notgil/eprop()l1ents e>f th~gaprT1()del much in the way ofuSurprises" in realized n~tillterest inc()me ov.er the1.9?4....7Q perie><iduringVihich tirn~ the gap 11'l()d~1 wi:\sintre>­duced. More recently, however, the increase in interest rate

33

variability has made the timing of asset and liability repric­ing within the gt:lpping period a significantinfluenceon netinterest income.

10. As time Passes within the gapping perie><i, incrementalgi:\pschange .in ways that depend upon.the characteristicsof the rate-sensitive accounts. This becomes importantwhen there are multiple rate changes. For example, pattern(a)inFigure1acould arise either because there is a $12assetmaturing in one month or because there is aone yearloan with an interest rate set monthly equal to the marketrate.lfitisthe maturing asset, then the bank in month two isas susceptible to rate changes in month two as in thevariable rate loan case onlyifthe maturingassetis rolledoverinto a 30 day loan. Should the maturing asset be rolledover into a one year fixed rate loan, the bank's exposure toadditional rate changes this year will be zero unlike thecontinued exposure under the variable rate loan scenario.AsmentionedinS~ion I each rate change is conceptuallyaddressed.Separately and, as such, multiple rate changesdo not greatly complicate the analysis.

11. See footnote 5.

12. Suppose the bank has only a$100 loan maturing on dayone and an equal ame>untin a90-day deposit. The 90-dayincremental Gap$ is zerO yetNil is at riskfor 1/40f a year.Notet/'Jatthe Jisk. would be even higher if this situationexisted every quarter arld there were multiple changes inrates.

13.• Accurate . information on the repricing structures ofassets and liabilities iscostly to come by. Once the repricingdates are recorded, however,it would seem to matter littlein terms of costs on how this information is grouped intomaturitybuckets.

14. See Baker (1981) or Dew (1981) for the evidence in gt:lpliterature on relative rate volatilities and the methods ofincorpprating this type ofinformation into the gap model.

15. The $500 in CP is equiValent to the volatility of .95 x$500 = $475 in 90 day futures. The $100 in CD is equivalent

to the yolatility 01$105 in 90 day futures. This method ofstandardization can be extended to measure the incre­mental gaps at dates other than the maturity length associ­atedwiththe standard contract. It is beyond the scope ofthis paper, however, to provide the details here. Dew (1981)hints at how such a procedure might work.

1E)./fratEYsrisei\JUwillalso because of the + $370standard­iled Gap$. Therise will be less than the naive +$400 Gap$would halfe/usbelielfe.ltispossiblefor a positilfenaive gaptq bE! cqnsi§tl:1nt with a.negative standardized gap. If $370 infutures ""ElrEl p~rchased, the.. rate increase will cause acompletely offsetting fall in the futures contract value.

17. We assume equal rates and rate changes for assetsand liabilities to simplify the exposition.

18. Given that $1000 more assets than liabilities are re­priced on day 30, this bank earns on netforthe remainder ofthe year $1000 x (1.12)330/360 rather than$1QOOx (1.10)330/360.

19. The example in the text has a cumulative Gap$ of zero.This does not mean that the basic gap model is superior tothe incremental gap model. The following provides an ex­ample where neither the basic nor the "state of the art" gapmodel would indicate a Nil hedge but where in fact sucha hedge exists. Let Gap$90 +$1000 and Gap$180 =~$1536. The cumulative Gap$ of ~$536 indicates, accord­ing to the basic gap model, that Nil rises when rates fall. Letrates fallfrqm1Q.percentto 8 percent. The influence of theratephangl:1 <;incl ~ap$90 on Nil is $1000[1.08.75 - 1.10.75]=$14.70. The magnitUde of the second Gap$'s influenceonNllis __ $1536[1.08.5 1.10.5] = -$14.70.

20. The term structure may contain a liquidity premium.This can conceptually be treated as an issue separate fromusing the current term structure to predict future ratechanges. If such a.liquiditypremium exists and is positive,one would wish to be somewhat shorter in the times toliabilityrl:1pricing than otherwise would be the case. Never­theless,conditionalon the current value of the liquiditypremium,thebank'sasset and liability position is still onethat depends.on a difference between the bank's interestrate forecast and the market's.

21. For information on the ability of forecasters to outper­form the market in interest rate predictions, see Prell (1973)and Throop (1981).

22. The one year interest rate of 11.49% is found by eval­uating 1.10.25 1.11-25 1.12.25 1.13.25 - 1.

2;3.•TheCP repriced on d<;iY..90 has a value of $1000 x1.10·~5,at.day 1.80 it will be. $1000 x 1.1025 x 1.11-25.Continuing in thisfa~hiongivesa Yl:1ar end CP value of$1114.90. This equals the one year $1000 loan with accu­mulated interest. Assets return 1.49% more than liabilities,expressed on an annualized basis, during the first quarter.ThisfallstoA9 then -.51 percent and then -1.51 percentinthenextthreequarters, respectively. The initial negativecarryon the asset (the positive spread between earningsand costs) eventually turns into a positive one. See Kauf­m<;in (1~72) .and Rose (19132b) for more details.

24. Note that repricing amounts that occur after the end ofthe gapping period (one year) do not enter equation (3).Thisisasit should be. These flows are not rate-sensitiveancl' jyst as in thl:1 conventional gap models, do not influ­ence banker decisions on structuring Nil rate risk for theassumed one year gapping period.

34

25. A convenient introduction to duration is provided byWeil (1974) and Hopewell and Kaufman (1973).

2fj.. \/fle can. assume .that MVRSA=MV8SL•.Under thisassumption, a positive DG is equivalent to DRSA < DRSL;thafis,the average repricing date on RSA is shorter thanthe average repricing date on RSL. Another way of inter­preting<apositive DG is to assume DRSA=DRSL' NowDG < obecause MVRSA < MVRSL; that is, at the averagerepricing date moreRSA are repriced than RSL. Undereither interpretation, the bank is "net asset sensitive" whenthe duration gap is positive.

27. Example 3 (c) has us book an additional $488 in RSLwith 8 = .25. This. monl:1yh<;iS to alter the balance Sheet bymqrf:jthan thisl:1ntry' The $488 could '.'refinance" anotherliability with duration in excess of one year,·· or it couldfinance a new asset with cash flows beyond one year, etc.The same consideration applies to Example 3 (d). Noticethat as one tries to hedge NII, one may alter the Nil to behedged or may push some asset and liability choices intoperiods outside the current gapping period, potentiallyexacerbating problems associated with hedging Nil infuture years.

28. Kolb and Chiang (1981) develop the reasoning on thisduration issue and show how futures contracts which haveno defined net present value would enter equations (4) and(5)'See also, Bierwag, Kaufman and Toevs (1983b).

29.l?implesimulations in the bond portfolio literaturesu~gest that rebalancing need not be undertaken compul­sively;· once a .month is probably more than sufficient.Bierwag, Kaufman and Toevs (1983a) provide, in the con­tf:j)(t of <lJ>ond portfolio, a graphical interpretation on howthis sequential hedging protects against multiple ratechanges.

30iBabcock(1976)was the first to point out this relation·shipintermsof duration.

31 .. Fpr rl:1<;ilism, at least one .ii must be of the opposite signof the others.

3g...Ll:1teac:h of V ppssible interest rate outcomes have an~sociatl:1d probability Vj, where 2:j~1vj=1 and 2:jY1vj.iij.ii.

33. Market participant expectations on future earnings, ex­pectedmonopoly rents, expected changes in regulations,etc., can cause bank stockvalues to depart from the propor­tionateshareofthemarket value of bank capital.

34. Gap reports can be vieWed as more fUlly disclosing theinterest rate risk exposure of the bank. As such, bank stockvalues may be affected as could be the value of future newstbckissues.

35..Jhi$immunization condition was developed by Reding­tPI'l(W!;i2) and l:1xpandedu pon by Grove (1974). This for­mula is derived under the assumption that term structuresare flat and shift due to unexpected causes to new positionsparallel to the original ones.

36. DA # (MVANS DNSA + MVRSA DRSA)/MVA andDL (MVLNSONSL + MVRSL DRSLl/MVL. Thus,MVA,NSDNSA +MVRSA ORSA = MVLNS DNSL +

MVRSLDRsLwhen equation (12) .holds. Add and subtract MVA on theriQtlth?i1d§iclEl of this .I<lst •equation (MVA = MVANS +MVRSA) and do the. same for MVL on the lefthand side(MVL == MVLNS + MVRSL). This gives equation (13).

REFERENCES

Babcock, Guilford C., "A Modified Measure of Duration,"Working Paper, University of Southern California,1976.

Baker, James V., "To Meet Accelerating Change: Why YouNeed a Formal Assetll,.iability Management Policy,"Banking, June, 1978. This is the first of five install­ments appearing in Banking from June throughOctober.

___. Asset/Liability Management. Washington:American Bankers Association, 1981.

Bierwag, G. D., George G. Kaufman and Alden L. Toevs,Immunization for Multiple Planning Periods," Journalof Financial and Quantitative Analysis, March 1983.

____, ''The Uses of Duration in Bond Portfolio Strategy: ASurvey Review," Financial Analysts Journal, July/August, 1983.

Bierwag, G. O. and Alden L. Toevs, "Immunization of Inter­est Rate Risk in Commercial Banks and Savings andLoan Associations," Bank Structure and Competi­tion 1982 Chicago: Federal Reserve Bank of Chicago.

Binder, Barret, "New Initiatives in Asset/Liability Manage­ment," Bank Administration, June 1979_

Binder, Barret and Thomas W. F. Lindquist. Asset/Liabil­ity and Funds Management at U.S. CommercialBanks. Rolling Meadows, Illinois: Bank AdministrationInstitute, 1982.

Clifford, John T., "A Perspective on Asset/Liability Manage­ment," Bank Administration, March 1975.

Dew, James Kurt, "The Effective Gap: A More AccurateMeasure of Interest Rate Risk," American Banker,June 10, 1981, September 19, 1981 and December 9,1981.

35

Flannery, Mark J. and Christopher M. James, ''An Analysisof the Effect of Interest Rate Changes on CommonStock Returns," Working Paper, University of Oregon,1982.

Grove, Myron A., "On 'Duration' and the Optimal MaturityStrategy of the Balance Sheet," The Bell Journalof Economics and Management Science, Autumn1974.

Hopewell, Michael H. and George G. Kaufman, "BondPrice Volatility and Term to Maturity: A GeneralRespecification," American Economic Review, Sep­tember 1973.

Kaufman, George G., "The Thrift Institution ProblemReconsidered," Journal of Bank Research, Spring1972.

Kolb, R. and R. Chiang, "Improving Hedging PerformanceUsing Interest Rate Futures," Financial Manage­ment, Autumn 1981.

Prell, Michael J., "How Well Do the Experts Forecast Inter­est Rates?" Federal Reserve Bank of Kansas City,Monthly Review, September-October, 1973.

Redington, Frederick M., "Review of the Principle of LifeOffice Valuations," Journal of the Institute of Actu­aries, 1952.

Rose, Stanford, "Dark Days Ahead for Banks," Fortune,June 30, 1980.

__, "Profits and the Yield Curve," American Banker,November 16,1982.

Throop, Adrian w., "Interest Rate Forecasts and MarketEfficiency," Federal Reserve Bank of San Francisco,Economic Review, Spring 1981.

Weil, Roman L., "Macaulay's Duration: An Appreciation,"Journal of Business, October 1973.