A note on discount rate policy and the variability of discount window borrowing

Journal of Banking and Finance 11 (1987) 563470. North-Holland

A NOTE ON DISCOUNT RATE POLICY AND THE VARIABILITY OF DISCOUNT WINDOW BORRQWING*

David D. VanHOOSE Indiana Uniwrsity, Bloomington, IN 47405, USA Received March 1986, final version received December 1986 The purposeof this note is to evaluate the appropriate discount rate policy rules consistent with minimization of the variability of borrowing at the Federal Reserve discount window. In the context of Goodfriend’s (1983) model of the bank borrowing decision, it is demonstrated that either a penalty rate or a subsidy rate policy will produce minimized variability of borrowing, so long as the subsidy rate adjusts point for point to changes in the value of the Federal funds rate. These policy rules are compatible with a policy procedure designed to target borrowed or nonborrowed reserves. If the Fed does not adhere to one of these specilic rules, minimization of borrowing variability requires an open market procedure in which the Fed pegs the Federal funds rate.

1. Introduction

Since October 1979, increasing attention has been given, both inside and outside the Federal Reserve, to the issue of discount rate policy [e.g., Kier C IlOSl),Taylor (1983), Roth and Seibert (1983) and Santomero (1983)]. The initial impetus for a renewal of interest in this issue developed from the Fed’s intended implementation of a policy procedure designed to s@ttarget growth paths for total non-borrowed bank reserves. However, subsequent actions and statements [e.g., Wallich (1984)] have convinced some observers, such as Poole (1982) and Gilbert (1985), that the Fed has informally adopted a procedure that targets the level of discount window borrowing. Although the ultimate consequences of either type of operating procedure for monetary control or for macroeconomic stability continue to be debated, large variability in the demand for discount window borrowing impacts adversely upon the Fed’s chosen policy procedure in either instance. For this reasca, the present policy of an ‘administered’ discount rate is coming under increasing scrutiny, and alternative proposals, such as a penalty rate policy, *This paper has benefited from helpful suggestions by C. Waller and W. Witte and from particularly useful conx~PF”tsby the referee, although I am solely responsible for its contents. I am grateful for financial support received from the Indiana University Ofice of Research and Graduate Development. 03784266/87/$3.50 0 1987 Elsevier Science Publishers B.V. (NorthlHoltand)

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D.D. VanHoose,Discount rate policy, discount window borrowing

have been suggested as preferable replacements for the present policy. The purpose of this note is to evaluate the appropriate policy rules to minimize the variability of borrowing at the discount window. 2. The bank borrowing decision and discount rate policy

A basic foundation for the discount window borrowing decision has been provided by Goodfriend (1983), who assumes that a bank maximizes the discounted value of the net stream of dollar benefits derived from borrowing from the Fed, given by

2

K= j=O

'Cfr+jB~+j-cOB~+j-2ClB,+jB~+j_l-d~+jB~+j],

(1)

where B,,j is the quantity of discount window borrowing at time t+ j, d,,j is the discount rate at time t + j, ft+j is the Federal funds rate at time t + jS b is a constant rate of time discount, and co and cl are positive constants. This formulation takes into account the implicit cost of current borrowing, which is related to the current level of borrowing and, due to the non-price rationing mechanism used at the discount window, lagged borrowing. The ideal level of borrowing in period t that is consistent with an optimally chosen intertemporal sequence of decisions is shown by Goodfriend to be

(2) where h=(2bc,)-’ and it is assumed that the values of co, cl, and 6 are consistent with the restriction on the characteristic roots, iz, and il,, such that -l
replaces (2) as the applicable decision rule, where E,(X, +j) is the conditional expectation Of X,+ based w information available at time t(. In his analysis, Goodfriend considers ihe case where the spread ft+j-dt+j follows a firstorder autoregressive piocess, and he concludes that ‘...if the Fed wants to estimate and utilize a borrovving fijnction, it should make its po!icy intentions toward the spread as simple and ur&rstandable as possible’, so that a preannounced pattern, or rule, for Fed policy with respect to the spread is j

D.D. VanHoose, Discount rate policy, discount window borrowing

565

preferable to a policy of ‘deliberate obfuscation’ [Goodfriend (1983, p. 352)]. TO expand upon Goodfriend’s model, consider the following general rule for the Fed’s setting of the discount rate: dt+j=6ft+j+Y+Et*j9

(4)

where S 20 and y 20 are policy parameters determined by the Fed and Et+j-i(E,+j)=O and E,+j-1[&~+j-Et+j-1(&~+j)12=~~), Various types of policy rules emerge as special cases of (4); possible cases are summarized as follows: y =0 and 0~ 6 < 1. On average, the discount rate is a percentage subsidy mark-down from the Federal funds rate. (11) y 1. On average, the discount rate is a percentage penalty mark-up over the Federal funds rate. (Iv) y 20 and S = 1. On average, the discount rate is a lumpsum penalty mark-up over the Federal funds rate. 09 y>O and 0~6~1, or yc0 and 6>1. The Fed responds in part to Federal funds rate variations in such a way that the discount rate may be either a penalty rate or a subsidy rate on average, depending upon the actual values of y and 6 and upon the observed value of ft+P (For instance, the joint setting y =0.05 and 6 = 0.5 yields a penalty rate on average for ft + 0.10.)

(1)

j

=

The Euler equation for the dynamic optimization problem of eq. (1) is, for Bt+jgO, given by

For B,+j>O, eq. (5) holds as an equality; for Bt+j=C, (5) holds as an inequality.’ On average, the first situation unambiguously arises for case (I) or (II) above; that is, a risk-neutral bank will borrow positive quantities from the discount window when the discount rate is a subsidy rate. This is the situation considered by Goodfriend (1983, p. 348, n. 10). However, (5) always holds as an inequality in the penalty rate cases (III) and (IV) and may hold as an inequality at some points in time in case (V). Hence, a risk-neutral bank will not borrow in this model if a penalty rate always holds [case (III) or case (IV)] and generally will fail to borrow during some time intervals if the Fed follows neither a pure penalty rate rule nor a pure subsidy rate rule [case (V)]. *This is the dynamic analogue of the static Kuhn-Tucker conditions. See Kamien and Schwartz (1981, ch. 13).

-

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D.D. VanHoose, Discount rate policy, discount window borrowing

3. Discount rate policies to minimize borrowing variability Based on the discussion above, there are three general situations that should be considered: ‘pure subsidy rate rules’ [cases (I) and (II)], ‘pure penalty rules’ [cases (III) and (IV)] and ‘ambiguous policy rules’ [case (V)]. In this section, an effort is made to evaluate optimal discount rate policy under each type of regime. From a monetary policy perspective, discount rate policy should be formulated in such a way as to minimize variations in bank borrowing at the window, so that actual borrowing does not depart significantly from the level projected in advance by the Fed’s staff. Hence, the criterion considered below is minimization of the forecast varian.ce of discount window borrowing, defined as (6) where E,_ r(BJ is the period t - 1 forecast of period t borrowing. 3.1. The case of pure subsidy rate policies In the case of pure subsidy rate policies, the Goodfriend model is unaltered, so that eq. (3) unambiguously describes the bank borrowing decision. If the Fed’s projection in any period t - 1 is made by estimating a function such as (3), then this projected value is given by

-h f ~~‘E,-I{E,[(1_6)f,_l+i-~]~. i=2

Use of (3) and (7) in (6) indicz.tes that the forecast variance is

+h2(6-1)2if= n;2iE,-,{E,(S,-,+i)-Er-ICEt(f,-1+i)l}2,(8) 2 where

+Et-1Ch-k1(S,)12. Eq. (8) indicates that, under a pure subsidy rate policy rule, unpredictable variability in discount window borrowing may arise from three potential sources. First, unpredictable variability in the Federal funds rate directly affects the variability of the spread between the Federal funds rate and the

D.D. VanHoose, Discount rate policy, discount window borrowing

567

discount rate, hence the term h21i2(6 - 1)2$. Second, any unsystematic fluctuations in the Fed’s setting of the discount rate contribute directly to the forecast variance of discount window borrowing by altering the variance of the spread; this source of variability is captured by the term h2il;2a,Z. Deviation between ex ante and ex post forecasts of future values of ft arising from information acquisition contribute to the forecast variancm by producing variability in banks’ forecasts of future spreads, as captured by the last t&m in (7). Clearly, tl,e second source of variability can be dealt with by the Fed in a straightforward manner. Formulation of disco&n’:raie policy in a systematic fashion, by adhering closely to a preannounced subsidy rate policy, lowers 0: and, therefore, & As reference to (8) makes clear, the first and third causes of borrowing variability can be eliminated by the pure subsidy rule (II), where 6 = 1 ant! y ~0. Note that, for rule (I), where 0 4 c 1 and y =0, exclusion of the first and third sources of variability requires ;331open market procedure which pegs the Federal funds rate over time and thereby sets 6: =0 and EJf, _ 1+J = )] for all i. This result coincides with the argument that, Et-lCEtM-l+i under a subsidy rate policy, targeting on borrowed reserves may entail a day-to-day operating procedure which is qualitatively similar to targeting the value of the Federal funds rate [see Poole (1982, p. 578) and Gilbert (1985, P*20)1In the absence of a procedure that pegs the Federal funds rate over time, the above discussion indicates that borrowing variability can be eliminated via discount rate policy alone by following rule (II) in setting the discount rate, so that

Under this rule, the discount rate is a subsidy rate that adjusts point for point to any variation in the Federal funds rate. Therefore, the spread between the two rates is realigned automatically to any movement in the Federal funds rate, countering any impact that otherwise would be produced by variability either in the actual value of the Federal funds rate or in bank forecasts of its future values. The economic intuition behind this result is illustrated in fig. 1, which relates the spread between A and dt to the level of borrowing B,. [Fig. 1 is drawn under the assumptions that 6: = 0 and that E,_ ,Cfi)= -yO( l-S)- ’ > 0 when rule (I) is in effect. In addition, it has been assumed that A1B,_1hC~21;iEt(ft_l+i--d,_l+i)=0. Under these assumptions, Br=hAzl~o. These assumptions clearly are very restrictive, but they permit the simple graphical comparison that is illustrated.] A borrowing function for rules (I) and (II) is graphed in the upper quadrant where J 4, > 0. On the diagram

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D.D. Vanlpoose, Discoulrt rate policy, discount window .boTowing

Fig. 1

the origin is assumed to be the intercept coefkient of eq. (3). Under rule (I), the slope of the borrowings function, as measured against positive spreads that arise under this subsidy rate rule, is given by a(&-dJ/aB,= -il&- ’ >O. Under rule (I), if it happens that E,_,(jJ is such that E,_,(f,-dJ= -yo>O, then E,- JB,)= I$‘. However, if ft is actually above E,_ I(fr),then the realized value of Bz under rule (I) is B:, so that deviations in Bt from E,_,(B,) may occur. Under rule (II), ft -& = - y. for all t, and Et- ,(B,) = Br= BF, hence, CT;= 0. Therefore, rule (II) is illustrated diagrammatically by a single print on the rule (I) borrowings function. Alternatively, &=O under either rule (I) or rule (II) if ft is pegged by the Fed, so that the spread never varies and Bt= E,_ ,(B,) for all t.

3.2. TIae case of pure penalty rate policies Under penalty rate rule (III) and (IV), Bt=0 for all t. It follows that, in this simple formulation, E,_ ,(B,) = Bt=O, so that C; =0 under either of the pure penalty rate rules. As illustrated in fig. 1, the borrowing function is simply the .non-positive portion of the vertical axis in either case. While this rule is compatible with a non-borrowed reserves targeting procedure, it would of course be inconsistent with a policy procedure in which the Fed varies a borrowed reserve target so as to affect other goal variables, such as the money stock. If Goodfriend’s model were extended to a more complex environment in which banks also incurred implicit costs in borrowing at the Federal funds market, the case of a penalty rate policy would not be so trivial: since it would then be possible for positive borrowing to occur for sufficiently small

D.D. VanHoose, Discount rate policy, discount window borrowing

569

penalties and sufficiently large marginal Federal funds borrowing costs.2 In this situation, rule (IV) (a lump-sum penaity rate) would dominate rule (III) (a percentage penalty rate) for the same reasons that rule (II) dominates rule (I) in the case of a pure subsidy rate. 3.3. The case of an ambiguouspolicy rule If rule (V) is applicable, so that either the policy settings y CO and S > 1 or the settings y > 0 and 0~ 6 < 1 are utilized by the Fed, then the discount rate may be either a pehalty rate or a subsidy rate depending upon actual values of the two policy parameters and upon realized values of the Federaii funds rate over time. Explicit solution of the bank’s optimization problem in this case would be a complex undertaking, because the Euler equation in (5) could alternate between holding as an equality versus an inequality at varying points in time. Indeed, the transversality condition for an optimal borrowing decision path need not hold as an equality in this situation. However, note that if the Euler equation can alternate between an equality or an inequality depending upon Federal Funds rate variations, the implication is that the level of borrowing could alternate between positive and zero quantities as a result. Hence, use of an ambiguous policy rule will, in general, produce positive borrowing variability over time, lessening the predictability of borrowing as compared to the optimal pure rules [(II), (III), and (IV)] discussed above. It may be concluded, then, that rule (Vj is inferior to these alternative policy rules. 4. Conclusion

It has been demonstrated that minimization of the variability of bank borrowing at the discount window requires that the Federal Reserve utilize either a penalty rate policy that is binding at all points in time or a subsidy rate policy in which the subsidy rate is adjusted automatically, and in equal measure, to upward or downward movements in the Federal funds rate. An additional requirement for the latter policy rule to be optimal is that the Fed should not unsystematically vary the spread between the Federal funds rate and the subsidy rate. Under either a penalty rate policy or this specific 21mplicit co sts of borrowing could result, for instance, from a lack of managerial expertise necessary to monitor and organize adequately day-to-day liability management strategies; such costs are mentioned often as the reason that small banks are not active borrowers in the Federal funds market. It is also possible to visualize banks which are known to face signif&antliquidity shortfalls being rationed by the market in certain instances. Such resource costs or rationing mechanisms would, if modeled, give rise to an upward-sloping marginal cost schedule for Federal funds borrowing whose shape and position, like that of discount window borrowing, depends upon lagged borrowing. The simple zero-discount window borrowing result obtained from the model used here arises from the assumption that such implicit Federa! funds borrowing costs are zero, so that the marginal cost schedule for Federal funds is horizontal at the Federal funds rate.

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D.D. VanHoose, Discount rate policy, discount window borrowing

subsidy rate polidy, unpredictable variations in discount window borrowing are eliminated. The choice between the alternative optimal rules therefore would depend upon other criteria, such as the amounts of borrowin produce or equity and eticiency consideraticns. The chosen policy rule then could be used in conjunction with a policy procedure designed to target borrowed or non-borrowed reserves. On the other hand, systematic adherence to any particular subsidy rate .policy, coupled with an open market procedure under which the Fed consistently pegs the Federal funds rate, also will produce minimum variability of discount window borrowing. Of course, this approach is inconsistent with any procedure designed to target the growth path of non-borrowed reserves. However, it would represent a feasible mode of policy administration under a procedure in which the level of borrowed reserves is the Fed’s operating target. References Gilbert, R, Alton, 1985, Operating procedures for conducting monetary policy, Review 67, Feb. (Federal Reserve Bank of St. Louis, St. Louis, MO) 13-21. Goodfriend, Marvin, 1983, Discount window borrowing, monetary policy, and the post-October 6, 1979 Federal Reserve operating procedure, Journal of Monetary Economics 12.343356. Kamien, Morton and Nancy Schwartz, 1981, Dynamic optimization (North-Holland, Amsterdam). Kier, Peter, 1981, Impact of discount policy procedures on the effectiveness of reserve targeting, New Monetary Control Procedures 1, Feb. (Board of Governors of the Federal Reserve System, Washington, DC). Poole, William, 1982, Federal Reserve operating procedures, Journal of Money, Credit and Banking 14,575-596. Roth, Howard and Diane Seibert, 1983, The effect of alternative discount rate mechanisms or monetary control, Economic Review 68, March (Federal Reserve Bank of Kansas City, Kansas City, MO) 1629. Santomero, Anthony, 1983, Controlling monetary aggregates: The discount window, Journal of Finance 38,827-844. Taylor, Herb, 1983, The discount window and monetary control, Business Review, May/June (Federal Reserve Bank of Philadelphia, Philadelphia, PA) 3-12. Wallich, Henry, 1984, Recent techniques of monetary policy, Economic Review 69, May (Federal Reserve Bank of Kansas City, Kansas City, MO) 21-30.