- Email: [email protected]
On DFIs' liability management
Journal of Banking and Finance 6 (1982) 533-547. North-Holland Publishing Company
ON DFIs' LIABILITY MANAGEMENT
Deposit Capacity, Multideposit Supply, and Risk-Efficient Rate Setting
J o h n - P e t e r D. C H A T E A U * McGill University, Montreal, Canada H3A 1G5
Received July 1981, final version received June 1982 Beyond reference rate-setting strategies derived in a CAPM framework, the paper addresses two related questions: (i) limited deposit accommodation under capital regulation, and (ii) characterization of rate differentials in correlated-uncorrelated multideposit markets. Under certain circumstances, an optimal rate/capacity choice policy will imply deposit capacity is not fully utilized all the time. Since capacity is not costless, excess liability capacity may be an optimal financial decision when the intermediary must meet adequacy test assorted with financial penalties or with enforceable limitations on its balance-sheet growth. In the other case, the intermediary is likely to introduce risky rate differentials to the extent risk is actually reduced enough to increase its market valuation. For correlated multideposit markets, this obtains whenever pairs of supplies are sufficiently negative to make their weighted covariance sum larger than the segment own variance. Alternatively, for uncorrelated segments, higher rates will be offered to steady depositors whose standard deviation of class supply contributes little to total risk.
1, Introduction The analysis of liability m a n a g e m e n t for depository financial i n t e r m e d i a r i e s ( D F I s ) has n o w gone b e y o n d the riskless r a t e - s e t t i n g m o d e l [cf. B a l t e n s p e r g e r ' s (1980) survey a n d references therein for the deterministic m o d e l ] . By a n d large, recent expected utility m a x i m i z i n g m o d e l s of rate setting u n d e r u n c e r t a i n t y are b a s e d on d i s t r i b u t i o n free a n d (mainly) functional form free d e p o s i t s u p p l y curves c o u p l e d with n o n - l i n e a r risk regimes. W h i l e the expected utility a p p r o a c h is a p p e a l i n g l y general, it yields results t h a t are d e p e n d e n t o n i n d i v i d u a l s ' preferences. M o r e specifically, the generality a n d elegance of using a n expected utility criterion s o m e t i m e s p r o h i b i t s one from o b t a i n i n g u n a m b i g u o u s results. In the case of risky ratesetting for instance, the relations between revenue (loan proceeds) curves, risk *The author is grateful for helpful comments to the members of the Faculty of Management coffee klatch, the Advanced Finance Seminar at McGill, and especially to Etienne Losq, Stylianos Perrakis, William Sealey and an anonymous referee. In addition, several members in the Financial and Monetary Analysis Section of the Bank of Canada provided helpful suggestions. As befits, all responsibility for errors and omissions rests with me. 03784266/82/0000-0000/$02.75
© 1982 N o r t h - H o l l a n d
534
J.-P.D. Chateau, On DFIs' liability management
aversion and deposit uncertainty are not immediately apparent and the situation is not directly comparable to the deterministic solution [cf. contributions by Stigum (1976) or Sealey (1980)-I. By the same token, unambiguous rate differentials between differing states of nature hinge on the critical role of the l i n e a r marginal revenue curve [Chateau (1981)]. In contrast to the very generality of the aforementioned models, it will be shown that the use of a slightly more restrictive framework - - that of the capital asset pricing model (CAPM) - - leads to several simplifications, both in tractability and ease of practical applications. Indeed, several advantages result from the use of a positive theory of market value. To start, the expected utility limitations are overcome by the CAPM stronger assumptions, especially that about investors' homogeneous preferences. Next, the market valuation concept independent of individuals' preferences leads to a mean-standard-deviation (instead of variance) representation as the appropriate criterion for decisions under uncertainty. To continue, the intermediary's valuation in a risk-return framework allows us to introduce directly and explicitly some measure of the market price of risk. Finally, the new focus enables us to determine how and why the risk-return combination presented by the intermediary to the financial market for valuation is affected by decisions variables. Namely, by variables such as single- or multi-deposit rate(s) and/or deposit capacity. The following discussion is developed in four parts. Section 2 describes the reference model to be used in the remainder of the discussion and summarizes the implied rate-setting rules under risky and riskless environments, respectively. Section 3 extends the discussion to incorporate capital regulation and deposit capacity and explains the impact from limited deposit acceptance on equilibrium rate-setting under risk. Section 4 further extends rate setting to the case of multiple deposits; discriminatory rate setting in the face of correlated and uncorrelated deposit supplies leads to a new way of viewing pricing under risk which leads to a rationale for rate discrimination quite different from the traditional one under certainty. Section 5 summarizes the main results.
2. The reference model: assumptions and equilibrium property In the single-period model under consideration, the representative intermediary announces at the beginning of the period the deposit rate, r d, to be set for the duration and accepts all (non-bank) public deposits tendered at that rate. This Cournot-type intermediary faces its own stochastic (~), aggregate and monopsonistic deposit supply, /~, that depends not only on r d but also on other parameters about which it has only incomplete objective and subjective information: namely, rivals' deposit rates, the size and the distribution of public liquid-type financial wealth, the intensity of
J.-P.D. Chateau, On DFIs' liability management
535
depositors' search effort and so on. F o r the sake of expositional clarity, let the functional form of its supply schedule be /) = D(re) + if,
(1)
where the uncertainty t~ enters additively - - could also be a vector comprising the aforementioned r a n d o m variables - - and is n o r m a l l y distributed with E(u')=0 and E(u')2= a~z, the latter being a c o n s t a n t < oo. It then results that E(/))=/)(ra) and v a r ( / ) ) = a ~ with ~a~/Sra=O. F o r c o m p a r i s o n s with riskless settings, the m e a n value of the stochastic supply schedule is assumed to be the riskless supply schedule. We also have ~/~(-)/Srd=/~,>0 where the subscript r refers to the partial derivative of E[D(')] with respect to the a r g u m e n t ra, Since partial equilibrium in its deposit m a r k e t is initially assumed, the total variable cost of actual deposits is given by C[D(')] = taD(" ). This single-rate strategy, which is here representative of liability m a n a g e m e n t , is in general c o n c o m i t a n t with the D F I ' s other basic policies such as liquidity and portfolio (investment and loan) policies. In keeping with the American rate-setting literature, ~ we shall abstract in the reference model from liquidity and investment portfolio considerations; and in its loan policy, we assume that the intermediary accepts the prevailing m a r k e t - d e t e r m i n e d loan rate r~, as in the excess d e m a n d a p p r o a c h [cf. for instance, Jaffee and Modigliani (1969)]. F u r t h e r m o r e , for the sake of clarity, it is assumed that deposit holding exceeds the aggregate a m o u n t of loans, L, thus sidestepping liability borrowing. In due course, this a s s u m p t i o n will be a p p r o x i m a t e d by L = k D ( ' ) , where k < 1 is the (ex ante k n o w n 2) coefficient of deposit-financed loans. It then results that the actual loan proceeds function is written in input form RID(.)] = r t k D ( ' ) and is linear in D(.). G i v e n the above information set, the intermediary's short-run e x post expected profit function and its standard deviation are expressed respectively
1The standard rate-setting position is well stated in Slovin and Sushka (1975) or Stigum (1976). Within this framework, the Canadian near-banks we are considering here - - the trust and mortgage and loans companies - - are characterized as follows. Their main intermediary business is deposit-financed loans, either consumer or mortgage loans; their liquidity is usually insignificant, and their bond portfolio of passive and decreasing importance. Moreover, these intermediaries are regulated by a minimum liability-equity ratio (see footnote 6 for further detail) rather than by deposit reserve requirements as it is the case for the chartered lcommercial) banks. 2When there is an effective rate ceiling, the intermediary's only ex ante control variable is k [cf. Eeckoudt and Caperaa (1977) or Sealey (1980)], the 'percentage invested in the risky asset'. In the absence of rate regulation on the other hand, both ra and k can be chosen as ex ante decision variables. Yet in keeping with the prevalent rate-setting literature, it is here assumed that k is given.
536
J.-P.D. Chateau, On DFIs' liability management
as E(Tr) = (rlk-- rd)E(D ) - B, a~ = r f l a
with
and
8tr~/drd = - ao,
(2) (3)
where rs - ( r z k - r d ) denotes the usually positive deposit-financed i n t e r m e d i a t i o n s p r e a d a n d B represents fixed outlays, a i.e., the cost of the i n t e r m e d i a r y ' s n a t i o n w i d e b r a n c h - b a n k i n g n e t w o r k (as in C a n a d a , for instance). M o d e r n securities m a r k e t e q u i l i b r i u m t h e o r y p r o v i d e s a f o r m a l basis, as well as a clearly t r a c t a b l e format, for the t h e o r y of the i n t e r m e d i a t i n g firm u n d e r uncertainty. W i t h i n the C A P M f r a m e w o r k , the i n t e r m e d i a r y ' s value, V, can be expressed 4 as a c a p i t a l i z e d c e r t a i n t y equivalent; n a m e l y , V=
p
- ' [E(n)
--
Rflaj,
(4)
where p is the risk-free interest factor, E0z ) a n d tr~ a r e the e x p e c t e d value and s t a n d a r d d e v i a t i o n of profit respectively, R d e n o t e s p a r t of the m a r k e t price of risk, a n d fl the c o r r e l a t i o n between the i n t e r m e d i a r y ' s r e t u r n s a n d the r e t u r n on a p o r t f o l i o of all securities in the m a r k e t . In essence, the e q u i l i b r i u m m a r k e t v a l u a t i o n has led here to a m e a n - s t a n d a r d d e v i a t i o n (instead of variance) r e p r e s e n t a t i o n as the a p p r o p r i a t e c r i t e r i o n for decision u n d e r uncertainty. 5 W i t h this f r a m e w o r k , the i n t e r m e d i a r y m u s t set its aAssume for instance, that the overall operating cost function, defined as h(A,D,L), depends on the number of agencies (A) - - especially relevant in a nationwide branch banking system - the level of deposits (D) and the demand for loans (L). If in single-period setting deposit and loan (marginal) operation costs are deemed proportional (economies of scale are long-run by definition) to their respective holding, namely 6D and yL, the operating cost function reduces to short-term fixed costs of branches in existence, that is h(A=Ao)=B. It also follows that ra and r, comprise both input and resource costs of deposits and loans; they then refer to effective deposit and loan rates, respectively. A more thorough discussion of product characteristics of intermediaries can be found in Sealey and Lindley (1977), or Baltensperger (1980). 4The analysis of capital market behaviour expresses the present value of the ith intermediary as V~=p-l[E(ni)-;~cov(ni, R.,)], where R,, is the rate of return on the market portfolio of all, say M, firms, and where 2=(R.,-p)/var(R,.) is the CAPM measure of systematic risk, the market price of risk (MPR). Since fl=cov(ni, R,,) [var(R.,).var(ni)] -½, one obtains eq. (4) by letting R=(R.,-p)[var(Rm)] -½. To avoid confusion, it should be emphasized that R is part of the MPR and thus related to 2. Since in the single-period model no change in the number of shares outstanding occurs, V~ (or V in (4)) represents the equilibrium market value of the intermediary. The same mean-standard-deviation framework is used by Meyer (1980) in his analysis of the regulated financial firm. It qualifies his (1976) previous market valuation criterion used for the regulated, multiproduct firm. 5Beyond tractability, the CAPM and mean-standard-deviation frameworks emphasize that under uncertainty the intermediaries' valuations are interdependent. The expected utility approach, on the other hand, is free of the constant reference, through r, to 'the portfolio of all firms in the market'. While this permits a wider scope of risk regimes [Stigum (1976)], it sometimes also leads to less tractable results I-Sealey(1980), Chateau (1981)].
J.-P.D. Chateau, On DFIs' liability management
537
deposit rate, in the absence (as in Canada) of rate ceiling, prior to the revelation of the true public supply conditions. Formally, its unconstrained rate problem can be stated as: max V = p - l [ r , / 5 - B - R f l r s a o ]
subject to
(5)
rd>O.
rd
The first-order condition valuation requires that
for an interior
maximum
of intermediary's
d V/dr d = 0 = p - x [(rl k _ rd)/5, _ / 5 + Rflao].
(6)
To the extent d2V/dr~ < 0 is assumed satisfied for all ra, expression (6) yields the optimality condition that the intermediary equals marginal revenue to marginal cost; namely that p - ~ [r~k/5,]
= p -'
[rd/5,+ / 5 -
(7)
Rl~ao],
where the le•hand side term refers to the riskless marginal revenue, M R ~ (superscript c denotes certainty along the mean supply schedule) evaluated at the risky (") and optimal (*) deposit rate r~*, and capitalized at the riskless discount rate p. The right-hand side, also capitalized at the same rate, comprises two terms in the bracket: the first one, rd/5,+/5, represents the riskless marginal cost of deposits, M C ~, and the second, Rflao, constitutes the 'marginal cost of risk', M C R . To interpret further the deposit cost terms, and hence outline the implied rate policies, we divide expression (7) by p-1/5,, rearrange and collect the terms. This yields the condition rtk=rd(l + l/~l-)--Rflao(/5,) -1
or
M R C = M C C - M C R ( f l ) = - M C ~,
(8)
where q, the expected deposit supply elasticity, is normally larger than unity for a 'well behaved' supply function. Since R, a o and I5, are non-negative, it turns out that the sign of the marginal risk-adjustment term in (8) hinges on that of ft. If the intermediary's deposit flows, and hence profits, are positively correlated with the returns on the market portfolio, then risky marginal cost, MC", is less than the riskless marginal cost. This in turn implies that the intermediary would post a higher rate, and hence elicit more deposits, than would be the case in a riskless setting. When the DFI's variability of returns is not affected by variations in returns in the rest of the economy (i.e., fl = 0), then the optimal rate-setting rule for a risky setting is the same as if the decision environment is riskless. With f l < 0 illustrating a countercyclical variation in returns, deposit marginal cost under risk will be greater than its
J.-P.D. Chateau, On DFIs" liability management
538
riskless counterpart. And the optimal rate under risk will be lower than that under certainty, with a corresponding lower level of deposit. By the same token, expression (8) can be developed one step further to arrive at a final relation that focuses on the equilibrium deposit rate. Namely, rtk
Rfl~yD
r~* =(1 + l/r/-)q/)r(1 + l/r/-)
or
r~a*=~*+MCR(fl).
(9)
Reviewing in (9) the aforementioned assumptions about fl, one again obtains optimal state-dependent deposit-pricing assertions, now stated in inequality form as r]*<(=)>~a*
iff f l < ( = ) > 0
or when
MCU*>(=)
The rate-setting rules of (10) reflect optimality in the traditional economic sense. Yet a second type of efficiency suggests itself: risk efficiency. This becomes evident in the next section, once one adds risk constraints to the reference model.
3. Rate setting under capital regulation The case of limited deposit accommodation In the foregoing section, the slightly restrictive CAPM setting was used to glean penetrating insights into the DFI's basic rate-setting strategy. If one does not insist on complete generality, this market equilibrium framework permits to model explicitly, and to arrive at tractable solutions to, many rate-setting related questions. The first to be considered now is that of capital adequacy and its impact on the intermediary's rate-setting policy. The institutional or regulatory arrangement of several, if not most, countries calls for minimal capital ratio requirements for depository intermediaries. This is to say that DFIs are legally or informally required to maintain their own funds (or equity) at a level equal to at least a minimum percentage - - usually in the 4-5~o range - - of their total asset or total balance sheet. Such requirement, in turn, determines a liability structure and hence, put a ceiling on the DFI's deposit accommodation as will now be shown the case. At this point, it is worth mentioning that the model to be presented below is not a mere theoretical construction, but corresponds to the actual regulatory environment governing Canadian near-banks. 6 [See 6The intermediary business of trust and mortgage and loans companies is regulated as follows. Their liabilities - - deposits plus investment certificates - - designated as 'guaranteed funds' cannot be more than twenty times their shareholders' equity defined as capital account, reserves and surplus. On the other hand, the institutions are not required to hold specific cash reserves for liquidity consideration. By controlling balance-sheet size, the restriction controls earnings risk in a way that is intended to prevent that risk from growing too large in relation to the capital available to absorb the risk. [See also Neave (1981).] Finally, these near-banks do not generally hold significant proportions of any kinds of liability whose behavioural mode is quantity setting, i.e., Advances, CDs or Eurodollars.
J.-P.D. Chateau, On DFIs' liability management
539
also Meyer (1980) and the references therein regarding other facets of the capital adequacy question.] In the single-period, aggregate model at hand, assume that the intermediary's equity base, E, and the minimal capital ratio, ~ (say 59/o), are both ex ante (beginning-of-period) known or determined. Their combination, then, determines an e x ante upper bound on the DFI's deposit accommodation D ( ' ) < D M = ~ E , where ]~ is the multiplier (here 20) defined as the inverse of the minimal capital requirement. Therefore, when the minimum capital ratio is (legally) given, the maximum deposit or third-party (liability) capacity DM is a single-valued function of the DFI's capital base. To illustrate, the intermediary can sell at any point in time additional equity so as to enlarge its capital base and thereby its optimal deposit capacity. Correspondingly, there is a cost of capital, and hence a cost of capacity attached to the potential D M. Assume here, for analytical convenience, that this capacity investment is done at a cost of c per unit (dollar) of capacity per period of time. 7 The notion of capacity, as well as the coming chanceconstraint 'reliability of service' standard, adopted in this section is that discussed in Meyer (1975, 1976). Next, this deterministic or risk-free constraint has to be adapted to the CAPM risky setting: its most direct and clearly tractable generalization is provided by a chance-constraint of the form prob I-D(') > DM] < ~> 0 which has a risk-free equivalent that can be expressed as p r o b ( D -¢r/ ~ > D ua- / ) ) ~ <=e'e~DM> D + Ntr'
(11)
where N denotes the number of standard deviations above the mean, /), needed to reduce the area in the upper tail of the probability distribution to e, the latter e measuring the standard or risk of excess supply permitted. Before proceeding further, the implications of this stochastic constraint will be clarified and indicated in a diagram. The chance-constraint (11) allows us to directly introduce a 'reliability of deposit service or risk standard', namely here a 'deposit accommodation standard' specified exogenously by regulatory agencies. Suppose the agency set e very small at, say, 1~. Which means that the intermediary will have to accommodate any public deposit supply, given its current equity base, with a probability of 99%. 7Total deposit cost is now explicitly composed of two parts; an input cost of ra per unit (dollar) of deposit accommodatedand a resource (capacity)cost of c per unit ($) of capacity.The intermediary may adjust deposit accommodationafter public supply is known, but only up to the limit imposedby capacity which must now be chosen ahead of time, namelyex ante.
540
J.-P.D. Chateau, On DFIs' liability management
Diagrammatically, the intermediary may set its rate with risk limitations as /f supply is not random by using an augmented supply schedule satisfying (11). In fig. 1, the new supply schedule /9 is constructed in such a manner that, for each rate, constraint (11) is satisifed, s DM is the deterministic optimal capacity and one changes only rates in order to meet the risk constraint. From the figure, it is clear that the rate r] has to be lowered b e l o w the one that would prevail in the deterministic case, r °. The rate decrease would also be more pronounced, the more the supply schedule rotates anticlockwise (implying a relatively inelastic supply) or the lower is the degree of excess supply risk permitted. In general, the rates found in this manner will not be stochastic optimal rates, but m i n i m u m rates consistent with the risk constraint for any given level of capacity. By the same token asymmetric probability distribution will be attractive since a symmetric probability density about the mean /) would imply a 50?/0 risk of 'accommodation failure' for rates such as r °. I I
% rd
I
~
5
= D(.)
r~
I
I D
Fig. 1
Appending the above constraint (11) to the initial objective function (5), we obtain a two-decision model which can be characterized by the Lagrangean, W = p - 1[ r ~ D - c D M - B -
Rflr:ro] +
2(D M- / ) -
Na).
8In practice, it is well to think of aggregate deposit supply as a cumulative curve: at zero and low rates, demand and savings deposits are collected, followed by term deposits fetching higher yields and capped by financial paper and large CDs with negotiated rates. Whenever the intermediary comes close to its deterministic overall capacity, it is likely to stop offering CDs and/or simultaneously adjust its whole rate structure. For Canadian nearbanks, the CD's or Eurodollar option is not available, though.
J.-P.D. Chateau, On DFIs' liability management
541
Positive optimal rate setting and deposit capacity for the liability-regulated intermediary require that 0 W/Or a = p - 1[(rlk -- rd)D, -- D + Rflao] -- 2D. = O,
(12)
OW/aDu=~,--p-ac=O
(13)
with inequalities replacing the equality conditions in (12) and (13) if the optimal value of the variable is zero. The additional conditions involving 2 and duality are omitted, since they are not currently needed. One immediately reckons that in (13), 2 = p - l c stands for the capitalized marginal valuation of deposit capacity as implied by the intermediary's capital base and/or minimal requirement. The capacity multiplier measures the implicit economic cost (per unit) of risk reduction - - essentially the maximum insurance premium for risk removal. On the other hand, to capture the effect of deposit capacity on rate setting, we need to evaluate and interpret the risky marginal cost of deposits. To do this, we divide (12) b y / ) r and rearrange terms to obtain = p-
},
{MR - [ M C
(14)
where all terms are already defined. Since M C R ( f l ) is a function of fl in (14), one reckons that the rate-setting rules of (10) still hold when deposit capacity is non-saturated (2"=0). In the absence of a binding capacity constraint, the intermediary still adjusts deposit rate until marginal profit is driven to zero. By contrast, even for fl = 0, the riskless marginal revenue is greater than the riskless marginal cost to the extent 2 is non-null. The latter observation implies that full-capacity rates are not necessarily lower than off-capacity rates. At full deposit capacity, the optimal rate-setting rules implied by (14) explicitly obtains once one substitutes for 2 using (13), and rearrange terms. That is r]* = r t k / H + R f l a o / D r H - c/H,
r]* =~a* + M R C ( f l ) - M C C ( 2
or
or c),
(15)
where H = ( 1 + l/r/3 and M C C refers to the marginal cost of (full) capacity, which depends on 2 or c, with 2 or c > 0. One immediately notices that when fl > 0 the sign of R f l a o / D , H , or equivalently of the M C R ( f l ) , will be opposite that for capacity cost, c / H - - or M C C ( 2 ) - - at full capacity; hence the conditions for signing r~* and comparing it to r]* will only be sufficient. Thus, granted (15), we can derive all state-dependent rate inequalities that
542
J.-P.D. Chateau, On DFIs" liability management
focus on the intermediary's dependence with the other firms in the market. Pricing assertions are as follows: (i)
2 or c = 0 coupled with fl > ( = ) < 0 results in rules for off-capacity rates, namely condition (10), (ii) 2 > 0 coupled with flO coupled with fl>O and M C R ( f l ) > ( = ) < M C C ( 2 ) implies that r]*> (=) < ~* at full capacity. From the foregoing development one can clearly infer that the risky rate will unambiguously be less than the riskless rate only when fl < 0 or when the marginal cost of risk is less than capacity cost, for fl > 0. In these cases, as already shown by Meyer (1975, 1976), the optimal capacity plan will generally not be characterized by using capacity fully; since capacity is not costless, high 'accommodation' standards (reflected in a small s) may imply heavy resource cost as the gap between D M and the (lower) expected supply widens. So far, risky and riskless rates or rate differentials are obtained on the basis of the intermediary's return relationship (fl) with the other firms in the securities market, for the single-deposit case. Yet, the moment one considers multideposit markets, rate gifferentials obtain on other bases, regardless of the intermediary's across-firms (in)dependence. This approach is taken up in the next section. 4. Discriminatory rate-setting in the face of correlated and uncorrelated supplies This section will address another key feature of deposit rate setting in liability markets - - the intermediary's practice of first-degree rate discrimination, namely multipart rate-seRing. This strategy occurs in general when the intermediary has both the informational and legal capabilities for identifying and segmenting depositors into different markets. Suppose the representative intermediary decides to set differentiated rates so as to appeal to heterogeneous deposit clienteles, with the ultimate goal of stabilizing its global deposit resources (a goal sometimes coupled with the attendant finality of liability cost reduction...). To this end, it engages into discrimination on diverse bases, i.e., it identifies and segments the market by type of depositors (household, corporate, government...), by way of time horizons (call, demand, time deposits...), as well as on the basis of amount (block rate-setting according to deposit size) or origin (foreign, domestic), and so forth. Furthermore, such market segments are often likely to display a certain degree of correlation. It ensures that extending the intermediary's
J.-P.D. Chateau, On D F I s ' liability management
543
valuation criterion to allow for market segmentation with or without correlated deposit supplies requires several changes to section 2's basic assumptions. For instance on the DFI's liability side, the assumption of multi-markets, and hence non-uniform rate setting, alters the expression for expected total variable costs - - and accordingly for expected profit - - to
E{C[D(')]} = ~ r,E(d,),
(16)
i = 1..... n,
i=1
where, for each deposit market with additive uncertainty, di refers to the ith segment and ri to the corresponding differentiated rate, the latter being constant over depositors in each class. Consequently, the profit variance becomes a 2 = ~ r2a 2 + 2 Z rirjaiJ, i=1
(17)
i>j
where a u is the covariance between the ith and jth deposit segments. In the following analysis also, fl will be treated as a (positive) constant and emphasis will be placed on the influence of small changes in a~. Both the small character of changes in a, and the [ - 1, 1] bounded, scale invariant nature of make this approach a tractable and reasonable approximation. 9 On the asset side, on the other hand, while the basic excess demand assumption is kept, the aggregate loan supply now has to be expressed either as a fixed dollar amount of or as a percentage of total deposit supply, namely L=kD(.), irrespective of deposit origin. This implies that there is no segment-induced uncertainty regarding loan supply: indeed, the intermediary's ability to tap several deposit markets stabilizes its global deposit requirement for credit coverage. The actual loan proceeds function is written in input form:
(18)
R = RED(')] = r,kE(D).
Using the above information in conjunction with the objective function (4), allows us to formally state the intermediary's multideposit problem as
riE(di)-B-R fl
m a x Z = p -~ R [ D ( ' ) ] ri
i=1
r~a~ +2 X r,rja u i
,
i>j (19)
9In other words, intersegmentrate differentialswill be explained in terms of supply risk rather than on the basis of intermediaries'interdependencethrough the securitiesmarket. JBF
C
544
J.-P.D. Chateau, On DFIs' liability management
subject to r~>0,Vi. We also characterize its optimal rate settings as
c3Zc~r__~ =p_ IL[ ,t3E(d~)_E(di)_R~{(r~aa~ cr~ +~>jr,a~,)/an}l =O Vi with inequalities replacing strict equalities for zero optimal values of ri. We now show that the previous rate rules - - risky and riskless alike - - take on a much different character when multiple and correlated supplies are present. For this purpose, divide expression (20) throughout by c3E(di)/Sri-- with E(di) -di and multiply it by (-1); the expression below results, where optimal rates are implicitly defined, 1° -
~ri
2
Vi,
where ri(l+,l/qi)=MC ~ is the ith riskless marginal cost that depends primarily on the corresponding rate expected elasticity, ql. In the absence of the risk adjustment term (second term on the left-hand side) in (21), the intermediary equates the ith riskless, capitalized marginal cost to the certain, common to all segments and also capitalized marginal revenue, p-l[R'(')]. As expected in riskless settings, the intermediary mainly relies on rate elasticities for determining equilibrium rate differentials as well as the optimal deposit holding in each submarket. In risky situations by contrast, the depositors' 'risk class' or market segment becomes all important in characterizing rate differentials. Given that R, /~ and OrJOa7 i are positive by assumption, we have to sign in (21) the lefthand side term in braces that reflects the marginal risk contributed by the ith group of depositors. Since rates are non-negative and a~ is positive, the sign hinges on that of tro. Marginal risk will be negative whenever the correlation(s) between pairs of supplies are sufficiently negative to make their weighted (by rjs) sum larger than the ith segment own weighted variance. The latter condition, in turn, implies that the risky marginal cost will be lower than its riskless counterpart. Granted the model's constant and riskless marginal revenue, p-l[R'(.)], depositors whose marginal risk is negative are more likely to be offered higher risky rates than depositors whose supplies increase the variability of total variable cost. By contrast, a positive weighted sum of covariances only adds to the segment variance and hence, raises the t°Alternatively, (21) can be written MC~* =-MC~ * + MCR~crij ) = M R c*
Vi
(21')
as we have done in previous sections. T h u s the analysis can be extended to cover the case of jq > ( = ) < 0, although only at the cost of extraneous notational complexity.
J.-P.D. Chateau, On DFIs' liability management
545
risky marginal cost above its riskless counterpart. This segment will then be characterized by a lower risky rate. So far, the foregoing discussion has pointed out that the key attribute that distinguishes risky rates to one class of depositors vis-/t-vis another is not individual (negative) covariances but their negative sum relative to the segment own variance. While the actual sign of correlations is an empirical question, one can think a priori of negatively paired deposit classes. At first glance, the supplies of consumer demand deposit and the corresponding government accounts (federal, provincial and so forth) are likely to be negatively correlated. The same perhaps applies for one group supplying sight deposit and another international or domestic certificates of investment (5 years plus). Nevertheless, many correlations are more likely to be positive, and hence reinforce the own variance effect in determining the risky rate differentials. The fact that negative marginal risk terms may not be so prevalent suggests that the intermediary should pay more attention in risk-efficient rate setting to the magnitude of the class' own variance than to the several crosseffects with other segments. This can be illustrated by re-formulating the problem for the uncorrelated case in which the covariance terms are assumed to be zero. Assume the intermediary wishes to maximize expected profit, but with an upper limit ~ on the standard deviation of total profit. That is max E(n)= p -11R[D(.)]rr OM
I
~ r,e(d,)-cDM- B},
(22)
i= 1
subject to the (optional) capacity constraint
Pr{~= d~(r~)>DM}<=~>O,
and
a~ =
r i > O,
(23) (24)
i = 1.... , n;
DM > O.
(25)
From the necessary conditions for optimal rates and capacity in (22) to (25), we isolate the term corresponding to the risk constraint
-2(Oa~jar~)
Vi,
(26)
where 2 > 0 represents the marginal profit for a small change in total standard deviation at the constraint boundary. Expression (26) measures the financial cost of the marginal risk contributed by the ith class of depositors.
546
J.-P.D. Chateau, On DFIs' liability management
Those with stable supplies may contribute little to total risk and thus the rates they will be offered will in general be higher, while those with unstable supply will be required to bear a negative risk premium - - essentially an insurance premium for exercising an option on deposit capacity. The foregoing provides a partial explanation for the intermediary's practice of offering higher rates to 'steady' or term depositors, i.e., the ones with low standard deviation of segment supply. The key to risk-efficient rate-setting strategies is a positive market value for risk reduction, so as to increase the DFI's market valuation. The standard deviation of aggregate deposit supply, as well as that of cost or profit risk, may change with rate for any class of depositors (uncorrelated case) or there may be covariances between the supplies of various segments (correlated case). Consider the uncorrelated case, for instance. When the cr~ are different, then (even though they do not depend on rate in additive uncertainty) the choice of different rate sets changes the proportions of total supply - - not unlike the segment proportions in a deposit portfolio. It then results that the standard deviation of aggregate supply - - and tr~ - - may be altered by the mix of rates. The need for such a risk-efficient concept seems rather clear since the intermediary accepting deposits from multiple segments could raise the same level of total supply from a wide array of rate choices along its isocost surfaces, but even though total variable cost remains constant, total risk would not.
5. Concluding comments This research has extended the basic rate-setting strategy expounded in section 2, in two directions: (i) limited deposit accommodation under capital regulation, and (ii) characterization of rate differentials in correlateduncorrelated multideposit markets. Yet this set of results under risk were arrived at only because a tractable setting - - that of CAPM - - was chosen, which incorporates the market price of risk directly. Under certain circumstances, an optimal rate/capacity choice policy will imply deposit capacity is not fully utilized all the time. This is not inefficiency, but rather is the optimal response on the part of an intermediary which seeks to limit input and resource costs. Since capacity is not costless, high 'accommodation' standards (reflected in small e's) may imply heavy costs as the gap between capacity and expected supply widens; the intermediary will accordingly adjust its rate policy. Thus, excess liability capacity may be simply an optimal financial decision when the intermediary must meet adequacy tests assorted with financial penalties and/or with enforceable limitations on its balance-sheet growth. The concept of risk efficiency in multipart rate-setting focuses on the depositors' class and the marginal risk contributed by each group, either
J.-P.D. Chateau, On DFIs' liability management
547
through its own segment variance or through cross-effects with other segments. The intermediary is likely to introduce risky rate differentials to the extent it is economically optimal to sacrifice some expected profit if risk is actually reduced enough to increase its market valuation. For instance, the DFI's value is maximized by setting multiple rates that minimize total risk, for any level of expected profit; and there are two possible solutions. In the correlated case, higher risky rates (versus riskless rates) obtain whenever correlation(s) between pairs of supplies are sufficiently negative to make their (negative) weighted sum larger than the segment own variance. In the uncorrelated case by contrast, higher rates are offered to steady or term depositors whose standard deviation of class supply contributes little to total risk. Yet, given the dearth of (empirically verified) negative cross effects, the intermediary is more likely to rely on the explanation of the uncorrelated multideposit case in setting its risky rate differentials. References Baltensperger, E., 1980, Alternative approaches to the theory of the banking firm, Journal of Monetary Economics 6, no. 1, 1-37. Chateau, J.-P.D., 1981, On the theory of financial intermediaries: Deposit rate-setting under supply uncertainty, in: H. G6ppl and R. Henn, eds., The Proceedings of the Karlsruhe Symposium on Money, Banking and Insurance, Vol. II (Athen/ium, Kfnigstein) 603-616. Eeckoudt, L. and Ph. Caperaa, 1977, Interest-bearing demand deposits and bank portfolio behaviour: Comment, Southern Economic Journal 44, no. 4, 395-398. Jaffee, D.M. and Fr. Modigliani, 1969, A theory and test of credit rationing, American Economic Review 59, no. 5, 850-872. Meyer, R.A., 1975, Monopoly pricing and capacity choice under uncertainty, American Economic Review 65, no. 3, 326-337. Meyer, R.A., 1976, Risk efficient monopoly pricing for the multiproduct firm, Quarterly Journal of Economics 90, no. 3, 461-474. Meyer, R.A., 1980, The regulated financial firm, Quarterly Review of Economics and Business 20, no. 4, 44-57. Neave, Ed. H., 1981, Canada's financial system (Wiley, Toronto). Sealey, C.W., Jr., 1980, Deposit rate-setting, risk aversion and the theory of depository financial intermediaries, Journal of Finance 35, no. 5, 1139-1154. Sealey, C.W., Jr. and J.T. Lindley, 1977, Output, input and a theory of production and cost at depository financial institutions, Journal of Finance 32, no. 4, 1251-1266. Slovin, M.B. and M.E. Sushka, 1975, Interest rates on savings deposits: Theory, estimation and policy (D.C. Heath, Lexington, MA). Stigum, M.L., 1976, Some further implications of profit maximization by a savings and loan association, Journal of Finance 31, no. 5, 1405 1426.
ON DFIs' LIABILITY MANAGEMENT
Deposit Capacity, Multideposit Supply, and Risk-Efficient Rate Setting
J o h n - P e t e r D. C H A T E A U * McGill University, Montreal, Canada H3A 1G5
Received July 1981, final version received June 1982 Beyond reference rate-setting strategies derived in a CAPM framework, the paper addresses two related questions: (i) limited deposit accommodation under capital regulation, and (ii) characterization of rate differentials in correlated-uncorrelated multideposit markets. Under certain circumstances, an optimal rate/capacity choice policy will imply deposit capacity is not fully utilized all the time. Since capacity is not costless, excess liability capacity may be an optimal financial decision when the intermediary must meet adequacy test assorted with financial penalties or with enforceable limitations on its balance-sheet growth. In the other case, the intermediary is likely to introduce risky rate differentials to the extent risk is actually reduced enough to increase its market valuation. For correlated multideposit markets, this obtains whenever pairs of supplies are sufficiently negative to make their weighted covariance sum larger than the segment own variance. Alternatively, for uncorrelated segments, higher rates will be offered to steady depositors whose standard deviation of class supply contributes little to total risk.
1, Introduction The analysis of liability m a n a g e m e n t for depository financial i n t e r m e d i a r i e s ( D F I s ) has n o w gone b e y o n d the riskless r a t e - s e t t i n g m o d e l [cf. B a l t e n s p e r g e r ' s (1980) survey a n d references therein for the deterministic m o d e l ] . By a n d large, recent expected utility m a x i m i z i n g m o d e l s of rate setting u n d e r u n c e r t a i n t y are b a s e d on d i s t r i b u t i o n free a n d (mainly) functional form free d e p o s i t s u p p l y curves c o u p l e d with n o n - l i n e a r risk regimes. W h i l e the expected utility a p p r o a c h is a p p e a l i n g l y general, it yields results t h a t are d e p e n d e n t o n i n d i v i d u a l s ' preferences. M o r e specifically, the generality a n d elegance of using a n expected utility criterion s o m e t i m e s p r o h i b i t s one from o b t a i n i n g u n a m b i g u o u s results. In the case of risky ratesetting for instance, the relations between revenue (loan proceeds) curves, risk *The author is grateful for helpful comments to the members of the Faculty of Management coffee klatch, the Advanced Finance Seminar at McGill, and especially to Etienne Losq, Stylianos Perrakis, William Sealey and an anonymous referee. In addition, several members in the Financial and Monetary Analysis Section of the Bank of Canada provided helpful suggestions. As befits, all responsibility for errors and omissions rests with me. 03784266/82/0000-0000/$02.75
© 1982 N o r t h - H o l l a n d
534
J.-P.D. Chateau, On DFIs' liability management
aversion and deposit uncertainty are not immediately apparent and the situation is not directly comparable to the deterministic solution [cf. contributions by Stigum (1976) or Sealey (1980)-I. By the same token, unambiguous rate differentials between differing states of nature hinge on the critical role of the l i n e a r marginal revenue curve [Chateau (1981)]. In contrast to the very generality of the aforementioned models, it will be shown that the use of a slightly more restrictive framework - - that of the capital asset pricing model (CAPM) - - leads to several simplifications, both in tractability and ease of practical applications. Indeed, several advantages result from the use of a positive theory of market value. To start, the expected utility limitations are overcome by the CAPM stronger assumptions, especially that about investors' homogeneous preferences. Next, the market valuation concept independent of individuals' preferences leads to a mean-standard-deviation (instead of variance) representation as the appropriate criterion for decisions under uncertainty. To continue, the intermediary's valuation in a risk-return framework allows us to introduce directly and explicitly some measure of the market price of risk. Finally, the new focus enables us to determine how and why the risk-return combination presented by the intermediary to the financial market for valuation is affected by decisions variables. Namely, by variables such as single- or multi-deposit rate(s) and/or deposit capacity. The following discussion is developed in four parts. Section 2 describes the reference model to be used in the remainder of the discussion and summarizes the implied rate-setting rules under risky and riskless environments, respectively. Section 3 extends the discussion to incorporate capital regulation and deposit capacity and explains the impact from limited deposit acceptance on equilibrium rate-setting under risk. Section 4 further extends rate setting to the case of multiple deposits; discriminatory rate setting in the face of correlated and uncorrelated deposit supplies leads to a new way of viewing pricing under risk which leads to a rationale for rate discrimination quite different from the traditional one under certainty. Section 5 summarizes the main results.
2. The reference model: assumptions and equilibrium property In the single-period model under consideration, the representative intermediary announces at the beginning of the period the deposit rate, r d, to be set for the duration and accepts all (non-bank) public deposits tendered at that rate. This Cournot-type intermediary faces its own stochastic (~), aggregate and monopsonistic deposit supply, /~, that depends not only on r d but also on other parameters about which it has only incomplete objective and subjective information: namely, rivals' deposit rates, the size and the distribution of public liquid-type financial wealth, the intensity of
J.-P.D. Chateau, On DFIs' liability management
535
depositors' search effort and so on. F o r the sake of expositional clarity, let the functional form of its supply schedule be /) = D(re) + if,
(1)
where the uncertainty t~ enters additively - - could also be a vector comprising the aforementioned r a n d o m variables - - and is n o r m a l l y distributed with E(u')=0 and E(u')2= a~z, the latter being a c o n s t a n t < oo. It then results that E(/))=/)(ra) and v a r ( / ) ) = a ~ with ~a~/Sra=O. F o r c o m p a r i s o n s with riskless settings, the m e a n value of the stochastic supply schedule is assumed to be the riskless supply schedule. We also have ~/~(-)/Srd=/~,>0 where the subscript r refers to the partial derivative of E[D(')] with respect to the a r g u m e n t ra, Since partial equilibrium in its deposit m a r k e t is initially assumed, the total variable cost of actual deposits is given by C[D(')] = taD(" ). This single-rate strategy, which is here representative of liability m a n a g e m e n t , is in general c o n c o m i t a n t with the D F I ' s other basic policies such as liquidity and portfolio (investment and loan) policies. In keeping with the American rate-setting literature, ~ we shall abstract in the reference model from liquidity and investment portfolio considerations; and in its loan policy, we assume that the intermediary accepts the prevailing m a r k e t - d e t e r m i n e d loan rate r~, as in the excess d e m a n d a p p r o a c h [cf. for instance, Jaffee and Modigliani (1969)]. F u r t h e r m o r e , for the sake of clarity, it is assumed that deposit holding exceeds the aggregate a m o u n t of loans, L, thus sidestepping liability borrowing. In due course, this a s s u m p t i o n will be a p p r o x i m a t e d by L = k D ( ' ) , where k < 1 is the (ex ante k n o w n 2) coefficient of deposit-financed loans. It then results that the actual loan proceeds function is written in input form RID(.)] = r t k D ( ' ) and is linear in D(.). G i v e n the above information set, the intermediary's short-run e x post expected profit function and its standard deviation are expressed respectively
1The standard rate-setting position is well stated in Slovin and Sushka (1975) or Stigum (1976). Within this framework, the Canadian near-banks we are considering here - - the trust and mortgage and loans companies - - are characterized as follows. Their main intermediary business is deposit-financed loans, either consumer or mortgage loans; their liquidity is usually insignificant, and their bond portfolio of passive and decreasing importance. Moreover, these intermediaries are regulated by a minimum liability-equity ratio (see footnote 6 for further detail) rather than by deposit reserve requirements as it is the case for the chartered lcommercial) banks. 2When there is an effective rate ceiling, the intermediary's only ex ante control variable is k [cf. Eeckoudt and Caperaa (1977) or Sealey (1980)], the 'percentage invested in the risky asset'. In the absence of rate regulation on the other hand, both ra and k can be chosen as ex ante decision variables. Yet in keeping with the prevalent rate-setting literature, it is here assumed that k is given.
536
J.-P.D. Chateau, On DFIs' liability management
as E(Tr) = (rlk-- rd)E(D ) - B, a~ = r f l a
with
and
8tr~/drd = - ao,
(2) (3)
where rs - ( r z k - r d ) denotes the usually positive deposit-financed i n t e r m e d i a t i o n s p r e a d a n d B represents fixed outlays, a i.e., the cost of the i n t e r m e d i a r y ' s n a t i o n w i d e b r a n c h - b a n k i n g n e t w o r k (as in C a n a d a , for instance). M o d e r n securities m a r k e t e q u i l i b r i u m t h e o r y p r o v i d e s a f o r m a l basis, as well as a clearly t r a c t a b l e format, for the t h e o r y of the i n t e r m e d i a t i n g firm u n d e r uncertainty. W i t h i n the C A P M f r a m e w o r k , the i n t e r m e d i a r y ' s value, V, can be expressed 4 as a c a p i t a l i z e d c e r t a i n t y equivalent; n a m e l y , V=
p
- ' [E(n)
--
Rflaj,
(4)
where p is the risk-free interest factor, E0z ) a n d tr~ a r e the e x p e c t e d value and s t a n d a r d d e v i a t i o n of profit respectively, R d e n o t e s p a r t of the m a r k e t price of risk, a n d fl the c o r r e l a t i o n between the i n t e r m e d i a r y ' s r e t u r n s a n d the r e t u r n on a p o r t f o l i o of all securities in the m a r k e t . In essence, the e q u i l i b r i u m m a r k e t v a l u a t i o n has led here to a m e a n - s t a n d a r d d e v i a t i o n (instead of variance) r e p r e s e n t a t i o n as the a p p r o p r i a t e c r i t e r i o n for decision u n d e r uncertainty. 5 W i t h this f r a m e w o r k , the i n t e r m e d i a r y m u s t set its aAssume for instance, that the overall operating cost function, defined as h(A,D,L), depends on the number of agencies (A) - - especially relevant in a nationwide branch banking system - the level of deposits (D) and the demand for loans (L). If in single-period setting deposit and loan (marginal) operation costs are deemed proportional (economies of scale are long-run by definition) to their respective holding, namely 6D and yL, the operating cost function reduces to short-term fixed costs of branches in existence, that is h(A=Ao)=B. It also follows that ra and r, comprise both input and resource costs of deposits and loans; they then refer to effective deposit and loan rates, respectively. A more thorough discussion of product characteristics of intermediaries can be found in Sealey and Lindley (1977), or Baltensperger (1980). 4The analysis of capital market behaviour expresses the present value of the ith intermediary as V~=p-l[E(ni)-;~cov(ni, R.,)], where R,, is the rate of return on the market portfolio of all, say M, firms, and where 2=(R.,-p)/var(R,.) is the CAPM measure of systematic risk, the market price of risk (MPR). Since fl=cov(ni, R,,) [var(R.,).var(ni)] -½, one obtains eq. (4) by letting R=(R.,-p)[var(Rm)] -½. To avoid confusion, it should be emphasized that R is part of the MPR and thus related to 2. Since in the single-period model no change in the number of shares outstanding occurs, V~ (or V in (4)) represents the equilibrium market value of the intermediary. The same mean-standard-deviation framework is used by Meyer (1980) in his analysis of the regulated financial firm. It qualifies his (1976) previous market valuation criterion used for the regulated, multiproduct firm. 5Beyond tractability, the CAPM and mean-standard-deviation frameworks emphasize that under uncertainty the intermediaries' valuations are interdependent. The expected utility approach, on the other hand, is free of the constant reference, through r, to 'the portfolio of all firms in the market'. While this permits a wider scope of risk regimes [Stigum (1976)], it sometimes also leads to less tractable results I-Sealey(1980), Chateau (1981)].
J.-P.D. Chateau, On DFIs' liability management
537
deposit rate, in the absence (as in Canada) of rate ceiling, prior to the revelation of the true public supply conditions. Formally, its unconstrained rate problem can be stated as: max V = p - l [ r , / 5 - B - R f l r s a o ]
subject to
(5)
rd>O.
rd
The first-order condition valuation requires that
for an interior
maximum
of intermediary's
d V/dr d = 0 = p - x [(rl k _ rd)/5, _ / 5 + Rflao].
(6)
To the extent d2V/dr~ < 0 is assumed satisfied for all ra, expression (6) yields the optimality condition that the intermediary equals marginal revenue to marginal cost; namely that p - ~ [r~k/5,]
= p -'
[rd/5,+ / 5 -
(7)
Rl~ao],
where the le•hand side term refers to the riskless marginal revenue, M R ~ (superscript c denotes certainty along the mean supply schedule) evaluated at the risky (") and optimal (*) deposit rate r~*, and capitalized at the riskless discount rate p. The right-hand side, also capitalized at the same rate, comprises two terms in the bracket: the first one, rd/5,+/5, represents the riskless marginal cost of deposits, M C ~, and the second, Rflao, constitutes the 'marginal cost of risk', M C R . To interpret further the deposit cost terms, and hence outline the implied rate policies, we divide expression (7) by p-1/5,, rearrange and collect the terms. This yields the condition rtk=rd(l + l/~l-)--Rflao(/5,) -1
or
M R C = M C C - M C R ( f l ) = - M C ~,
(8)
where q, the expected deposit supply elasticity, is normally larger than unity for a 'well behaved' supply function. Since R, a o and I5, are non-negative, it turns out that the sign of the marginal risk-adjustment term in (8) hinges on that of ft. If the intermediary's deposit flows, and hence profits, are positively correlated with the returns on the market portfolio, then risky marginal cost, MC", is less than the riskless marginal cost. This in turn implies that the intermediary would post a higher rate, and hence elicit more deposits, than would be the case in a riskless setting. When the DFI's variability of returns is not affected by variations in returns in the rest of the economy (i.e., fl = 0), then the optimal rate-setting rule for a risky setting is the same as if the decision environment is riskless. With f l < 0 illustrating a countercyclical variation in returns, deposit marginal cost under risk will be greater than its
J.-P.D. Chateau, On DFIs" liability management
538
riskless counterpart. And the optimal rate under risk will be lower than that under certainty, with a corresponding lower level of deposit. By the same token, expression (8) can be developed one step further to arrive at a final relation that focuses on the equilibrium deposit rate. Namely, rtk
Rfl~yD
r~* =(1 + l/r/-)q/)r(1 + l/r/-)
or
r~a*=~*+MCR(fl).
(9)
Reviewing in (9) the aforementioned assumptions about fl, one again obtains optimal state-dependent deposit-pricing assertions, now stated in inequality form as r]*<(=)>~a*
iff f l < ( = ) > 0
or when
MCU*>(=)
The rate-setting rules of (10) reflect optimality in the traditional economic sense. Yet a second type of efficiency suggests itself: risk efficiency. This becomes evident in the next section, once one adds risk constraints to the reference model.
3. Rate setting under capital regulation The case of limited deposit accommodation In the foregoing section, the slightly restrictive CAPM setting was used to glean penetrating insights into the DFI's basic rate-setting strategy. If one does not insist on complete generality, this market equilibrium framework permits to model explicitly, and to arrive at tractable solutions to, many rate-setting related questions. The first to be considered now is that of capital adequacy and its impact on the intermediary's rate-setting policy. The institutional or regulatory arrangement of several, if not most, countries calls for minimal capital ratio requirements for depository intermediaries. This is to say that DFIs are legally or informally required to maintain their own funds (or equity) at a level equal to at least a minimum percentage - - usually in the 4-5~o range - - of their total asset or total balance sheet. Such requirement, in turn, determines a liability structure and hence, put a ceiling on the DFI's deposit accommodation as will now be shown the case. At this point, it is worth mentioning that the model to be presented below is not a mere theoretical construction, but corresponds to the actual regulatory environment governing Canadian near-banks. 6 [See 6The intermediary business of trust and mortgage and loans companies is regulated as follows. Their liabilities - - deposits plus investment certificates - - designated as 'guaranteed funds' cannot be more than twenty times their shareholders' equity defined as capital account, reserves and surplus. On the other hand, the institutions are not required to hold specific cash reserves for liquidity consideration. By controlling balance-sheet size, the restriction controls earnings risk in a way that is intended to prevent that risk from growing too large in relation to the capital available to absorb the risk. [See also Neave (1981).] Finally, these near-banks do not generally hold significant proportions of any kinds of liability whose behavioural mode is quantity setting, i.e., Advances, CDs or Eurodollars.
J.-P.D. Chateau, On DFIs' liability management
539
also Meyer (1980) and the references therein regarding other facets of the capital adequacy question.] In the single-period, aggregate model at hand, assume that the intermediary's equity base, E, and the minimal capital ratio, ~ (say 59/o), are both ex ante (beginning-of-period) known or determined. Their combination, then, determines an e x ante upper bound on the DFI's deposit accommodation D ( ' ) < D M = ~ E , where ]~ is the multiplier (here 20) defined as the inverse of the minimal capital requirement. Therefore, when the minimum capital ratio is (legally) given, the maximum deposit or third-party (liability) capacity DM is a single-valued function of the DFI's capital base. To illustrate, the intermediary can sell at any point in time additional equity so as to enlarge its capital base and thereby its optimal deposit capacity. Correspondingly, there is a cost of capital, and hence a cost of capacity attached to the potential D M. Assume here, for analytical convenience, that this capacity investment is done at a cost of c per unit (dollar) of capacity per period of time. 7 The notion of capacity, as well as the coming chanceconstraint 'reliability of service' standard, adopted in this section is that discussed in Meyer (1975, 1976). Next, this deterministic or risk-free constraint has to be adapted to the CAPM risky setting: its most direct and clearly tractable generalization is provided by a chance-constraint of the form prob I-D(') > DM] < ~> 0 which has a risk-free equivalent that can be expressed as p r o b ( D -¢r/ ~ > D ua- / ) ) ~ <=e'e~DM> D + Ntr'
(11)
where N denotes the number of standard deviations above the mean, /), needed to reduce the area in the upper tail of the probability distribution to e, the latter e measuring the standard or risk of excess supply permitted. Before proceeding further, the implications of this stochastic constraint will be clarified and indicated in a diagram. The chance-constraint (11) allows us to directly introduce a 'reliability of deposit service or risk standard', namely here a 'deposit accommodation standard' specified exogenously by regulatory agencies. Suppose the agency set e very small at, say, 1~. Which means that the intermediary will have to accommodate any public deposit supply, given its current equity base, with a probability of 99%. 7Total deposit cost is now explicitly composed of two parts; an input cost of ra per unit (dollar) of deposit accommodatedand a resource (capacity)cost of c per unit ($) of capacity.The intermediary may adjust deposit accommodationafter public supply is known, but only up to the limit imposedby capacity which must now be chosen ahead of time, namelyex ante.
540
J.-P.D. Chateau, On DFIs' liability management
Diagrammatically, the intermediary may set its rate with risk limitations as /f supply is not random by using an augmented supply schedule satisfying (11). In fig. 1, the new supply schedule /9 is constructed in such a manner that, for each rate, constraint (11) is satisifed, s DM is the deterministic optimal capacity and one changes only rates in order to meet the risk constraint. From the figure, it is clear that the rate r] has to be lowered b e l o w the one that would prevail in the deterministic case, r °. The rate decrease would also be more pronounced, the more the supply schedule rotates anticlockwise (implying a relatively inelastic supply) or the lower is the degree of excess supply risk permitted. In general, the rates found in this manner will not be stochastic optimal rates, but m i n i m u m rates consistent with the risk constraint for any given level of capacity. By the same token asymmetric probability distribution will be attractive since a symmetric probability density about the mean /) would imply a 50?/0 risk of 'accommodation failure' for rates such as r °. I I
% rd
I
~
5
= D(.)
r~
I
I D
Fig. 1
Appending the above constraint (11) to the initial objective function (5), we obtain a two-decision model which can be characterized by the Lagrangean, W = p - 1[ r ~ D - c D M - B -
Rflr:ro] +
2(D M- / ) -
Na).
8In practice, it is well to think of aggregate deposit supply as a cumulative curve: at zero and low rates, demand and savings deposits are collected, followed by term deposits fetching higher yields and capped by financial paper and large CDs with negotiated rates. Whenever the intermediary comes close to its deterministic overall capacity, it is likely to stop offering CDs and/or simultaneously adjust its whole rate structure. For Canadian nearbanks, the CD's or Eurodollar option is not available, though.
J.-P.D. Chateau, On DFIs' liability management
541
Positive optimal rate setting and deposit capacity for the liability-regulated intermediary require that 0 W/Or a = p - 1[(rlk -- rd)D, -- D + Rflao] -- 2D. = O,
(12)
OW/aDu=~,--p-ac=O
(13)
with inequalities replacing the equality conditions in (12) and (13) if the optimal value of the variable is zero. The additional conditions involving 2 and duality are omitted, since they are not currently needed. One immediately reckons that in (13), 2 = p - l c stands for the capitalized marginal valuation of deposit capacity as implied by the intermediary's capital base and/or minimal requirement. The capacity multiplier measures the implicit economic cost (per unit) of risk reduction - - essentially the maximum insurance premium for risk removal. On the other hand, to capture the effect of deposit capacity on rate setting, we need to evaluate and interpret the risky marginal cost of deposits. To do this, we divide (12) b y / ) r and rearrange terms to obtain = p-
},
{MR - [ M C
(14)
where all terms are already defined. Since M C R ( f l ) is a function of fl in (14), one reckons that the rate-setting rules of (10) still hold when deposit capacity is non-saturated (2"=0). In the absence of a binding capacity constraint, the intermediary still adjusts deposit rate until marginal profit is driven to zero. By contrast, even for fl = 0, the riskless marginal revenue is greater than the riskless marginal cost to the extent 2 is non-null. The latter observation implies that full-capacity rates are not necessarily lower than off-capacity rates. At full deposit capacity, the optimal rate-setting rules implied by (14) explicitly obtains once one substitutes for 2 using (13), and rearrange terms. That is r]* = r t k / H + R f l a o / D r H - c/H,
r]* =~a* + M R C ( f l ) - M C C ( 2
or
or c),
(15)
where H = ( 1 + l/r/3 and M C C refers to the marginal cost of (full) capacity, which depends on 2 or c, with 2 or c > 0. One immediately notices that when fl > 0 the sign of R f l a o / D , H , or equivalently of the M C R ( f l ) , will be opposite that for capacity cost, c / H - - or M C C ( 2 ) - - at full capacity; hence the conditions for signing r~* and comparing it to r]* will only be sufficient. Thus, granted (15), we can derive all state-dependent rate inequalities that
542
J.-P.D. Chateau, On DFIs" liability management
focus on the intermediary's dependence with the other firms in the market. Pricing assertions are as follows: (i)
2 or c = 0 coupled with fl > ( = ) < 0 results in rules for off-capacity rates, namely condition (10), (ii) 2 > 0 coupled with fl
J.-P.D. Chateau, On D F I s ' liability management
543
valuation criterion to allow for market segmentation with or without correlated deposit supplies requires several changes to section 2's basic assumptions. For instance on the DFI's liability side, the assumption of multi-markets, and hence non-uniform rate setting, alters the expression for expected total variable costs - - and accordingly for expected profit - - to
E{C[D(')]} = ~ r,E(d,),
(16)
i = 1..... n,
i=1
where, for each deposit market with additive uncertainty, di refers to the ith segment and ri to the corresponding differentiated rate, the latter being constant over depositors in each class. Consequently, the profit variance becomes a 2 = ~ r2a 2 + 2 Z rirjaiJ, i=1
(17)
i>j
where a u is the covariance between the ith and jth deposit segments. In the following analysis also, fl will be treated as a (positive) constant and emphasis will be placed on the influence of small changes in a~. Both the small character of changes in a, and the [ - 1, 1] bounded, scale invariant nature of make this approach a tractable and reasonable approximation. 9 On the asset side, on the other hand, while the basic excess demand assumption is kept, the aggregate loan supply now has to be expressed either as a fixed dollar amount of or as a percentage of total deposit supply, namely L=kD(.), irrespective of deposit origin. This implies that there is no segment-induced uncertainty regarding loan supply: indeed, the intermediary's ability to tap several deposit markets stabilizes its global deposit requirement for credit coverage. The actual loan proceeds function is written in input form:
(18)
R = RED(')] = r,kE(D).
Using the above information in conjunction with the objective function (4), allows us to formally state the intermediary's multideposit problem as
riE(di)-B-R fl
m a x Z = p -~ R [ D ( ' ) ] ri
i=1
r~a~ +2 X r,rja u i
,
i>j (19)
9In other words, intersegmentrate differentialswill be explained in terms of supply risk rather than on the basis of intermediaries'interdependencethrough the securitiesmarket. JBF
C
544
J.-P.D. Chateau, On DFIs' liability management
subject to r~>0,Vi. We also characterize its optimal rate settings as
c3Zc~r__~ =p_ IL[ ,t3E(d~)_E(di)_R~{(r~aa~ cr~ +~>jr,a~,)/an}l =O Vi with inequalities replacing strict equalities for zero optimal values of ri. We now show that the previous rate rules - - risky and riskless alike - - take on a much different character when multiple and correlated supplies are present. For this purpose, divide expression (20) throughout by c3E(di)/Sri-- with E(di) -di and multiply it by (-1); the expression below results, where optimal rates are implicitly defined, 1° -
~ri
2
Vi,
where ri(l+,l/qi)=MC ~ is the ith riskless marginal cost that depends primarily on the corresponding rate expected elasticity, ql. In the absence of the risk adjustment term (second term on the left-hand side) in (21), the intermediary equates the ith riskless, capitalized marginal cost to the certain, common to all segments and also capitalized marginal revenue, p-l[R'(')]. As expected in riskless settings, the intermediary mainly relies on rate elasticities for determining equilibrium rate differentials as well as the optimal deposit holding in each submarket. In risky situations by contrast, the depositors' 'risk class' or market segment becomes all important in characterizing rate differentials. Given that R, /~ and OrJOa7 i are positive by assumption, we have to sign in (21) the lefthand side term in braces that reflects the marginal risk contributed by the ith group of depositors. Since rates are non-negative and a~ is positive, the sign hinges on that of tro. Marginal risk will be negative whenever the correlation(s) between pairs of supplies are sufficiently negative to make their weighted (by rjs) sum larger than the ith segment own weighted variance. The latter condition, in turn, implies that the risky marginal cost will be lower than its riskless counterpart. Granted the model's constant and riskless marginal revenue, p-l[R'(.)], depositors whose marginal risk is negative are more likely to be offered higher risky rates than depositors whose supplies increase the variability of total variable cost. By contrast, a positive weighted sum of covariances only adds to the segment variance and hence, raises the t°Alternatively, (21) can be written MC~* =-MC~ * + MCR~crij ) = M R c*
Vi
(21')
as we have done in previous sections. T h u s the analysis can be extended to cover the case of jq > ( = ) < 0, although only at the cost of extraneous notational complexity.
J.-P.D. Chateau, On DFIs' liability management
545
risky marginal cost above its riskless counterpart. This segment will then be characterized by a lower risky rate. So far, the foregoing discussion has pointed out that the key attribute that distinguishes risky rates to one class of depositors vis-/t-vis another is not individual (negative) covariances but their negative sum relative to the segment own variance. While the actual sign of correlations is an empirical question, one can think a priori of negatively paired deposit classes. At first glance, the supplies of consumer demand deposit and the corresponding government accounts (federal, provincial and so forth) are likely to be negatively correlated. The same perhaps applies for one group supplying sight deposit and another international or domestic certificates of investment (5 years plus). Nevertheless, many correlations are more likely to be positive, and hence reinforce the own variance effect in determining the risky rate differentials. The fact that negative marginal risk terms may not be so prevalent suggests that the intermediary should pay more attention in risk-efficient rate setting to the magnitude of the class' own variance than to the several crosseffects with other segments. This can be illustrated by re-formulating the problem for the uncorrelated case in which the covariance terms are assumed to be zero. Assume the intermediary wishes to maximize expected profit, but with an upper limit ~ on the standard deviation of total profit. That is max E(n)= p -11R[D(.)]rr OM
I
~ r,e(d,)-cDM- B},
(22)
i= 1
subject to the (optional) capacity constraint
Pr{~= d~(r~)>DM}<=~>O,
and
a~ =
r i > O,
(23) (24)
i = 1.... , n;
DM > O.
(25)
From the necessary conditions for optimal rates and capacity in (22) to (25), we isolate the term corresponding to the risk constraint
-2(Oa~jar~)
Vi,
(26)
where 2 > 0 represents the marginal profit for a small change in total standard deviation at the constraint boundary. Expression (26) measures the financial cost of the marginal risk contributed by the ith class of depositors.
546
J.-P.D. Chateau, On DFIs' liability management
Those with stable supplies may contribute little to total risk and thus the rates they will be offered will in general be higher, while those with unstable supply will be required to bear a negative risk premium - - essentially an insurance premium for exercising an option on deposit capacity. The foregoing provides a partial explanation for the intermediary's practice of offering higher rates to 'steady' or term depositors, i.e., the ones with low standard deviation of segment supply. The key to risk-efficient rate-setting strategies is a positive market value for risk reduction, so as to increase the DFI's market valuation. The standard deviation of aggregate deposit supply, as well as that of cost or profit risk, may change with rate for any class of depositors (uncorrelated case) or there may be covariances between the supplies of various segments (correlated case). Consider the uncorrelated case, for instance. When the cr~ are different, then (even though they do not depend on rate in additive uncertainty) the choice of different rate sets changes the proportions of total supply - - not unlike the segment proportions in a deposit portfolio. It then results that the standard deviation of aggregate supply - - and tr~ - - may be altered by the mix of rates. The need for such a risk-efficient concept seems rather clear since the intermediary accepting deposits from multiple segments could raise the same level of total supply from a wide array of rate choices along its isocost surfaces, but even though total variable cost remains constant, total risk would not.
5. Concluding comments This research has extended the basic rate-setting strategy expounded in section 2, in two directions: (i) limited deposit accommodation under capital regulation, and (ii) characterization of rate differentials in correlateduncorrelated multideposit markets. Yet this set of results under risk were arrived at only because a tractable setting - - that of CAPM - - was chosen, which incorporates the market price of risk directly. Under certain circumstances, an optimal rate/capacity choice policy will imply deposit capacity is not fully utilized all the time. This is not inefficiency, but rather is the optimal response on the part of an intermediary which seeks to limit input and resource costs. Since capacity is not costless, high 'accommodation' standards (reflected in small e's) may imply heavy costs as the gap between capacity and expected supply widens; the intermediary will accordingly adjust its rate policy. Thus, excess liability capacity may be simply an optimal financial decision when the intermediary must meet adequacy tests assorted with financial penalties and/or with enforceable limitations on its balance-sheet growth. The concept of risk efficiency in multipart rate-setting focuses on the depositors' class and the marginal risk contributed by each group, either
J.-P.D. Chateau, On DFIs' liability management
547
through its own segment variance or through cross-effects with other segments. The intermediary is likely to introduce risky rate differentials to the extent it is economically optimal to sacrifice some expected profit if risk is actually reduced enough to increase its market valuation. For instance, the DFI's value is maximized by setting multiple rates that minimize total risk, for any level of expected profit; and there are two possible solutions. In the correlated case, higher risky rates (versus riskless rates) obtain whenever correlation(s) between pairs of supplies are sufficiently negative to make their (negative) weighted sum larger than the segment own variance. In the uncorrelated case by contrast, higher rates are offered to steady or term depositors whose standard deviation of class supply contributes little to total risk. Yet, given the dearth of (empirically verified) negative cross effects, the intermediary is more likely to rely on the explanation of the uncorrelated multideposit case in setting its risky rate differentials. References Baltensperger, E., 1980, Alternative approaches to the theory of the banking firm, Journal of Monetary Economics 6, no. 1, 1-37. Chateau, J.-P.D., 1981, On the theory of financial intermediaries: Deposit rate-setting under supply uncertainty, in: H. G6ppl and R. Henn, eds., The Proceedings of the Karlsruhe Symposium on Money, Banking and Insurance, Vol. II (Athen/ium, Kfnigstein) 603-616. Eeckoudt, L. and Ph. Caperaa, 1977, Interest-bearing demand deposits and bank portfolio behaviour: Comment, Southern Economic Journal 44, no. 4, 395-398. Jaffee, D.M. and Fr. Modigliani, 1969, A theory and test of credit rationing, American Economic Review 59, no. 5, 850-872. Meyer, R.A., 1975, Monopoly pricing and capacity choice under uncertainty, American Economic Review 65, no. 3, 326-337. Meyer, R.A., 1976, Risk efficient monopoly pricing for the multiproduct firm, Quarterly Journal of Economics 90, no. 3, 461-474. Meyer, R.A., 1980, The regulated financial firm, Quarterly Review of Economics and Business 20, no. 4, 44-57. Neave, Ed. H., 1981, Canada's financial system (Wiley, Toronto). Sealey, C.W., Jr., 1980, Deposit rate-setting, risk aversion and the theory of depository financial intermediaries, Journal of Finance 35, no. 5, 1139-1154. Sealey, C.W., Jr. and J.T. Lindley, 1977, Output, input and a theory of production and cost at depository financial institutions, Journal of Finance 32, no. 4, 1251-1266. Slovin, M.B. and M.E. Sushka, 1975, Interest rates on savings deposits: Theory, estimation and policy (D.C. Heath, Lexington, MA). Stigum, M.L., 1976, Some further implications of profit maximization by a savings and loan association, Journal of Finance 31, no. 5, 1405 1426.