THE IMPACT OF THE CAPITAL MARKET
CHAPTER 7 MODELS OF BANKING BEHAVIOUR
7.3 LIQUIDITY MANAGEMENT
Liquidity management involves managing reserves to meet predictable out£ows of deposits.1The bank maintains some reserves and it can expect some loan repayment.
The bank can also borrow funds from the interbank market or at the discount window from the central bank. The management of the asset side of the bank’s balance sheet can be considered as part of a two-stage, decision-making process. At the ¢rst stage the bank decides the quantity of reserves to hold to meet the day-to- day withdrawals of deposits. The remainder of assets can be held as earnings assets.
At the second stage the bank decides how to allocate its earnings assets between low-risk, low-return bills and high-risk, high-return loans.
A simple model of liquidity management will have the bank balancing between the opportunity cost of holding reserves rather than earning assets and the adjust- ment costs of having to conduct unanticipated borrowing to meet withdrawals.
This is a typical tradeo¡, which requires the bank to solve an optimization problem under stochastic conditions. Let the balance sheet of the bank be described by loans (L) plus reserves (R) and deposits (D):
LþR¼D ð7:1Þ
The bank faces a continuous out£ow of deposits over a speci¢c period of time before new deposits or in£ows replenish them at the beginning of the new period. If the withdrawal out£ows are less than the stock reserves, the bank does not face a liquid- ity crisis. If, on the other hand, the bank faces a withdrawal out£ow that is greater than their holding of cash reserves, then they face a liquidity de¢ciency and will have to make the de¢ciency up by raising funds from the interbank market or the central bank. The opportunity cost of holding cash reserves is the interest it could have earned if it was held as an earning asset. Let the deposit out£ow be described by a stochastic variable (x). A reserve de¢ciency occurs ifðRxÞ<0.
1This section of the chapter borrows heavily from Baltensperger (1980). See also Poole (1968).
Let the adjustment cost of raising funds to meet a reserve de¢ciency be proportional to the de¢ciency by a factorp, then it can be shown that a bank will choose the level of liquid reserves such that the probability of a reserve de¢ciency is equal to the ratio of the rate of interest on earning assets (r) to the cost of meeting a reserve de¢ciency (p). The bank chooses the level of reserves such that the marginal bene¢ts (not having to incur liquidation costs) equal the marginal costs (interest income foregone). See Box 7.1.
If the stochastic process describing the deposit out£ow in terms of withdrawals is a normal distribution with a given mean, so that at the end of the period the expected withdrawal is EðxÞ, the optimal stock of reserves held by a bank is described in Figure 7.1. If the cost of obtaining marginal liquidity increases (prises), the ratior=pdeclines and more reserves are held. If the return from earnings assets rise (rise inr), fewer reserves are held. If the probability of out£ows increase (shift in distribution to right) more reserves are held. In Figure 7.1 the ratior=pfalls from 0.6 to 0.4 and cash or liquid reserves rise from 28 to 31.
The model says that, in the absence of regulatory reserve ratios, a bank will decide on the optimal level of reserves for its business based on the interest on earn- ings assets, the cost of meeting a reserve de¢ciency and the probability distribution of deposit withdrawals. However, in reality many central banks operate statutory reserve ratios. But the model is robust to the imposition of a reserve ratio. Box 7.2 shows that the major e¡ect of imposing a reserve ratio is to reduce the critical value of the deposit withdrawals beyond which a reserve de¢ciency occurs. What this means is that the optimality decision relates to free reserves (reserves in excess of the reserve requirement), rather than total reserves.
If adjustments for reserve de¢ciency were costless, the bank would always adjust its portfolio so that it starts each planning period with the optimal reserve position, which would be independent of the level of reserves inherited from the previous period. If adjustment costs exist, an adjustment to the optimal level of reservesR would be pro¢table only if the resulting gain more than o¡sets the cost of the adjustment itself. Suppose that the adjustment costCis proportional to the absolute size of the adjustment, so that:
C¼jRR0j ð7:2Þ
where R0 are beginning period reserves before adjustment and Rare beginning period reserves after adjustment.
This type of model (shown in Box 7.1) allows for reserves to £uctuate within a range and triggers an adjustment only if the level of reserves goes above or below the limits. WhenR<R, an increase in reserves lowers costs. The marginal gain from a reserve adjustment is greater than the marginal cost de¢ned by the parameter . In other words, when@C=@R >0 it is pro¢table to make an adjustment.
When the marginal gain from an adjustment is equal to the marginal cost, in other words, when@C=@¼0, a further adjustment in Ris no longer pro¢table.
AlthoughCis reduced, it would do so only by an amount smaller than. When R>R, a reduction inRis pro¢table because that also lowers costs. Again, when
@C=@¼0, any further adjustment does not cover marginal net adjustment
LIQUIDITY MANAGEMENT 93
BOX 7.1
The optimal reserve decision
Letxdenote the outflow of deposits,fðxÞthe probability distribution function ofx andr is the interest earned on the bank’s earnings assets. The balance sheet of the bank is as described by equation (7.1.1). Let the expected adjustment cost of a reserve deficiency be denoted byA. This would be the cost of funding a reserve shortfall. The opportunity cost of holding reserves is rR. For simplicity assume that the expected adjustment cost is proportional to the size of the reserve deficiency and thepr. Then:
A¼ ð1
R pðxRÞfðxÞdx ð7:1:1Þ For a given set of parameters, the bank can optimize its holding of reserves by minimizing the expected net cost function:
C¼rRþA ) rRþ
ð1
R pðxRÞfðxÞdx ð7:1:2Þ Minimizing (7.1.2) with respect toR:
@C
@R¼rp ð/
R fðxÞdx¼0 ) r
p¼ ð1
R fðxÞdx
ð7:1:3Þ
The bank chooses the level of reserves such that the probability of a reserve deficiency is just equal to the ratior=p.
When the adjustment cost is proportional to the absolute size of the adjustment, the optimal position for the bank is given by:
TC¼CðRR0Þ
@TC
@R ¼@C
@R¼0 ) rp
ð1
RfðxÞdx¼0
; r p ¼
ð1
RfðxÞdx
The final equation defines a lower and upper bound for R. As long as R is bounded by upper and lower limits RL<R<RU, no adjustment takes place.
LIQUIDITY MANAGEMENT 95
FIGURE 7.1
Cumulative distribution of deposit outflow
BOX 7.2
Reserve requirements
Without legal reserve requirements, the critical level of deposit outflowx is the beginning period level of reservesR. Let the reserve requirement be that the end period reserves (Rx) should be a fractionkof end period deposits:
Rx¼kðDxÞ ð7:2:1Þ A reserve deficiency occurs when:
Rx<kðDxÞ ð7:2:2Þ Solving the inequality forxgives a critical value, which defines a new critical outflow that marks a reserve deficiency:
x>RkD
1k xx^ ð7:2:3Þ The size of the reserve deficiency is given by:
xð1kÞ ðRkDÞ ¼ ðxxxÞð1^ kÞ ð7:2:4Þ The expected value of the adjustment cost is now defined as:
AA~¼ ð1
^
xx pðxxxÞf^ ðxÞdx )
ð1
^
xx pðxð1kÞ ðRkDÞÞfðxÞdx ð7:2:5Þ The optimality condition gives:
r p¼
ð1
^
xx fðxÞdx ð7:2:6Þ The difference with the result obtained in Box 7.1 is that the probability gives the likelihood ofxexceedingxx^rather thanR.
0 0.2 0.4 0.6 0.8 1 1.2
0 10 20 30 40 50 60 70
Reserves
r/p
costs. WhenR0<RL, reserves increase toRL. Similarly, whenR0>RU, reserves decrease toRU. Figure 7.2 illustrates.