Bank's portfolio management under uncertainty.

I. Introduction

Optimal bank-management is a continuous struggle of maintaining a balance between liquidity, profitability and risk. Banks need liquidity because such a large portion of their liabilities are payable on demand but typically an asset more liquid the less it yields. Thus, the decision to choose one combination of assets over another, given the liability size and capital accounts of a bank, would have a direct and significant effect on bank profitability, liquidity and risk.

The pioneering works of Markowitz (1952) and Tobin (1958, 1965) on the theory of portfolio selection have developed into analysis of several distinct areas.(1) One such area has been the theory of banking firm, where banks have been analyzed in a microeconomic firm-theoretical context. In such studies the authors, utilizing the neo-classical theory of the firm, have defined inputs, outputs and appropriate resource cost constraints of a typical profit maximizing bank and derived the optimal resource allocations under perfect or imperfect market conditions under certainty (Klein 1971, Pringle 1973, 1974, Tobin 1982) or uncertainty (Elysiani 1983). An alternative approach treats banks as rational investors or portfolio holders, and therefore, a portfolio choice framework has been employed for analyzing bank behavior (Aigner 1971, Hester and Pierce 1975). The Markowitz-Tobin model of portfolio selection has also been directly applied to financial institutions. Among these works Hart and Jaffer's study (1974) is notable. They, unlike others, have taken into account certain regulatory and institutional characteristics of financial institutions in their model. Although, the modeling of portfolio management is fairly well formulated, these models provide little practical applicability for a commercial bank operating under uncertain environment in the context of new developments in the financial markets such as wide use of federal funds market, repurchase agreement for maintaining the optimal reserve position and the deregulation of the existing deposit rate. The development of the federal fund opportunity and the evolving deregulations of financial markets have sparked interest in the economic implications of portfolio selections of the commercial banks in previously regulated and limited markets. Since the current literature lacks the analysis of optimal decision rules for a commercial bank under such new phenomena, this paper is an attempt to provide such rules, which can be used by other economic agents as well. Specifically, we intend to investigate the optimal asset management policies and examine the impact of the federal funds market and deposit rate deregulation.

The layout of the paper is as follows: Section two explains the model with relevant assumptions. In section three we obtain some comparative static results relating to the optimal decision rules by the bank and provide some economic explanations. Section four explains the impact of the deregulations on the deposit rate and also set up a more general model involving stochastic deposit loss as well as loss in the return from the loans extended by the bank. Concluding remarks are included in section five.

II. The Model

In this paper we examine the optimal strategic (asset and liability management) decision rules derived from the maximization of the expected utility from the net income of the bank. We first identify the important variables of the model given the relevant assumptions. An expected utility functions will be developed and optimal decision rules yielding expected utility maximizing portfolio will be derived.

It is assumed, for simplicity, that the bank has only one type of deposit, namely demand deposit. The type of assets that will be considered here are reserves, government securities and loans. The level of net worth (or equity capital) may be ignored (or included) since it is assumed to be constant in our analysis.

In addition, we assume that the level of demand deposit (or reserve) fluctuates over time due to unforeseen deposits or withdrawals by the public (or depositors). This change is denoted by a non-negative random variable |Alpha~ with a subjectively perceived(2) (by the bank) probability density function |Phi~(|Alpha~) and distribution function |Phi~(|Alpha~). Thus, demand deposit at the end of the period will be |Alpha~ D. If |Alpha~ |is greater than~ 1, it represents a net deposit gain. On the other hand, |Alpha~ |is less than~ 1 implies a net deposit loss. When the bank experiences deposit (reserve) loss, it may borrow from the federal fund market at a higher (penalty) rate to cover that loss. (Note, bank may also use the repurchase agreement to augment the deposit loss, but we ignore that possibility here). When the bank confronts net deposit gain it may very well lend it (thus utilize it profitably) in the federal funds market.

In the market of government securities and federal funds, the bank assumed to be in pure competition, is able to trade without affecting the corresponding interest rates. However, in the case of demand deposit and loan granting, bank may have some degree of monopoly power. The contract rate of interest on loan is assumed to be a decreasing function of the amount lent, i.e. |r.sub.l~ = |r.sub.l~(L) with |r|prime~.sub.l~ |is less than or equal to~ 0 according to the bank behaving as a pure competitor or a monopolist. On the other hand, the bank receives an exogenous quantity D of demand deposits,(3) which depends in part on the exogeneously determined deposit rate |r.sub.d~. The deposit rate is subject to an upper bound because of federal or state regulation.

Bank's revenue is generated through the use of loan extended and government securities purchased; while the bank cost consists of interest rate payment for the demand deposit and the federal fund. We may also include other implicit costs such as servicing, advertising, etc. in the cost structure. For simplicity, we exclude such implicit costs. So, given our assumptions about bank's asset and liability structure, the bank's net income (or profit) is given by the following equation.

|Pi~ = |r.sub.l~L + |r.sub.g~G - |Alpha~|r.sub.d~D - |r.sub.f~F (1)

where

|r.sub.f~ = interest rate on federal funds

|r.sub.d~ = interest rate on deposit

|r.sub.l~ = interest rate on loan

|r.sub.g~ = interest rate on government security

L = loans extended by the bank

G = amount of government securities purchased

D = deposit at the bank

F = net federal fund borrowed

|Pi~ = profit or net income of the bank

As mentioned earlier, in the expression (1), all rates are defined to be net of other administrative or implicit costs.

Let U be the Neumann-Morgenstern utility function,(4) defined over the profit space. Then, the expected utility function of the commercial bank is given by

|Mathematical Expression Omitted~

where E is the expectation operator.

It is assumed that the bank attempts to maximize E|U(|Pi~)~ by choosing L and G accordingly (that means L and G are the decision variables). However, these decision variables are linked by the following relation(5)

L + G = (1 - q)D + W (3)

where q is the required reserve ratio and W is the net worth of the bank, assumed to be constant.

Now making note of the definition of |Pi~ along with the restriction defined in equation (3) we set up the constrained objective function

V = E|U(|Pi~)~ + |Lambda~ (L + G - (1 - q)D - W) (4)

where |Lambda~ is the Lagrangian multiplier parameter.

Differentiating the objective function with respect to L, G and |Lambda~, we get the following first order conditions for the constrained maximization problem.(6)

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

and

|Delta~V/|Delta~|Lambda~ = L + G - (1 - q)D - W = 0.

Solutions of these first order conditions determine the optimal loan and government security purchase decision of the commercial bank. It is evident that the optimal decision variables (L, G and |Lambda~) are functions of the parameters of the model. In the next section, these three conditions are used to derive different comparative static results and hypotheses regarding bank's optimal behavior and attitude towards risk. However, before that, we analyze these optimal conditions in detail comparing those with that of a risk neutral bank. Let |Mathematical Expression Omitted~ is the expected value of the random variable |Alpha~, then the expected profit for the bank can be defined(7) as

|Mathematical Expression Omitted~

Since the objective of a risk neutral bank is to maximize the expected net income (profit) subject to the restriction defined in equation (3), the relevant first order conditions for optimization are:

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Now, the optimal values of G and L obtained from equation (7) can be used to compare with the optimal values of G and L obtained from equation (5).

The effects of changing G and L on expected profit level for a risk averse bank can also be obtained by differentiating the expected profit function (equation 6) with respect to G and L. Thus, we get,

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

The qualitative signs of |Mathematical Expression Omitted~ and |Delta~|Pi~/|Delta~L for a risk averse bank can be derived by using the solutions of the equations (5) and (7). The second relation of equation (5) can be reformulated as

E|U|prime~~ E|y~ + cov(U|prime~,y) = 0 (10)

where y = |r.sub.g~ - ||Alpha~r.sub.d~ + (1 - |Alpha~)(1 - q)(|r.sub.f~~/(1 - q).

Assuming that the bank is risk averse (i.e. U|double prime~(|Pi~} |is less than or equal to~ 0), cov (U|prime~, y) can be shown to be negative in sign. This implies that |Mathematical Expression Omitted~ is positive in sign. In the same fashion, it can also be shown that

|Mathematical Expression Omitted~ q) is also positive. Thus we get

|Mathematical Expression Omitted~

This relationship implies that the optimal amount of loans extended and government securities purchased for a risk averse bank is higher than those of a risk neutral.

Similarly, from the equation (5) we can also get the effect of the changes in L and G on the optimal risk premium paid by the risk averse commercial bank. The risk premium |Theta~, is defined to be the amount by which |Mathematical Expression Omitted~ needs to be adjusted to make decision maker indifferent between the random profit |Pi~ and the certain profit (|Mathematical Expression Omitted~). Thus, |Theta~ is defined as

|Mathematical Expression Omitted~

where

A = -E|U|double prime~(|Pi~)~|r.sub.g~H, (13)

B = -E|U|double prime~(|Pi~)~(|r.sub.l~ + (r|prime~.sub.l~L)H, (14)

C = (1 - q)dD + dW - D dq, (15)

and

H = G d|r.sub.g~ - ||r.sub.d~ - (1 - q)|r.sub.f~~

D d|Alpha~ - ||Alpha~|r.sub.d~ + (1 - |Alpha~)

(1 - q) |r.sub.f~~dD - |Alpha~D d|r.sub.d~

+ (1 - |Alpha~) |r.sub.f~D dq

- (1 - |Alpha~)(1 - q) D d|r.sub.f~.

The determinant of the system (12), denoted by |Delta~, can be simplified as

|Mathematical Expression Omitted~

This relationship clearly implies that |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~. And for a risk averse firm |Delta~|Theta~/|Delta~L, |Delta~|Theta~/|Delta~G are positive. In otherwords, as the amount of loans extended by and the amount of government securities purchased by commercial bank increases for a risk averse bank, so does the risk premium paid by that bank.

III. Comparative Static Results

Total differentiation of the first order conditions (equation 5) also allows for the calculation of the impact on the optimal amount of the decision variables as a result of the parametric changes in our model. We get the system

|Mathematical Expression Omitted~

This determinant is positive if the bank is risk averse and the profit of the bank is a concave function of the loan extended by the bank.(8) Thus, the second order condition of the constrained optimization problem mentioned earlier is, indeed, satisfied under those two conditions.

Now solving the system (12), we get the following table representing the comparative static results.(9)

                                       Government
                    Loan                Security

|r.sub.g~            -                      +
|r.sub.f~            +                      -
q                    +                      -

The economic interpretation of these results are quite straightforward. Taking an expected utility maximization approach to a simple portfolio model of a commercial bank, we find that the commercial bank will buy more government security at the expense of loans extended by it, if the interest rate on the government security increases. On the other hand, if the federal fund rate or the required reserve ratio increases, then the commercial bank will extend more amount of loans but the amount of government securities purchased will decline. That means, commercial bank will substitute the purchase of government security with the loans. All of these results are expected according to the following intuitive explanation. Increase in the security rate makes government security more attractive (because of high liquidity also) compared to the loan, thus the bank buys more government security and grants less amount of loan. However, if the federal fund rate increases, it makes it more costly for the commercial bank to borrow from the federal fund market. The commercial bank reacts to it by adjusting its asset position. It reduces the purchase of government security (which has lower yield rate) and increases the loan. Same result holds in the case of an increase in the required reserve ratio. Under these circumstances, availability of excess reserve to the bank declines, so the commercial in turn reduces the amount of government securities purchased and increases the amount of loan granted. Thus, the commercial bank readjusts its asset side of

|Mathematical Expression Omitted~

where

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

N = 0.

Using Cramer's rule, we obtain the solutions the portfolio in the event of such parametric changes.

It should be noted here that the qualitative signs of these comparative static results are quite determinable in sign without any additional assumptions regarding the behavior of the commercial bank operating under uncertain environment.

Next, we analyze the effect of an increase in riskiness in the stochastic deposit loss or gain in our model. Following Sandmo (1971) we generate another random variable |Alpha~* = |Alpha~|Gamma~ + |Beta~, where |Gamma~ is the multiplicative shift parameter and |Beta~ is the additive one. Initially |Gamma~ = 1, and |Beta~ = 0. An increase of |Gamma~ alone will increase the mean as well as the variance of |Alpha~. To recover the mean, in order to have mean-preserving increase in risk about |Alpha~, we have to reduce |Beta~ simultaneously, so that |Mathematical Expression Omitted~ or |Mathematical Expression Omitted~. Under this assumption, the necessary conditions of the expected utility maximization can be rewritten as

|V.sub.G~(|Gamma~ = = 0,

|V.sub.L~(|Gamma~) = 0, (16)

and

|V.sub.|Lambda~~(|Gamma~) = 0,

where |V.sub.G~(|Gamma~) = |Delta~V/|Delta~G, |V.sub.L~(|Gamma~) = |Delta~V/|Delta~L etc.

Now differentiating these equilibrium conditions (equation 16) with respect to |Gamma~, we obtain

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

where |Delta~ is the same determinant defined as before. The terms |Delta~G/|Delta~|Gamma~ and |Delta~L/|Delta~|Gamma~ now contain stochastic term |Mathematical Expression Omitted~ and hence the qualitative signs are not easily determined. However, in addition to risk averseness on the part of the commercial bank, if we also assume that U|double prime~|prime~ |is greater than~ 0, then it can be shown that(10)

|Delta~G/|Delta~|Gamma~ |is greater than~ 0,

and

|Delta~L/|Delta~|Gamma~ |is less than~ 0.

In other words, as the uncertainty about the deposit loss increases, a commercial bank with increasingly partial risk aversion will react to it by readjusting its asset side of the portfolio again. This specific risk averse attitude will cause the bank to reduce its investment in risky and less liquid assets. the bank will decrease the amount of loans which are more risky and less liquid than the government securities that the bank now will purchase more. (Banks will be more reluctant and rigid to extend loan now and will be demanding more information about collaterals and other aspects of the loans).

IV. Effect of Deregulation

In this section, we explore the impact of regulation over the deposit rate in our model. As we have modeled the deposit rate in general without any specification, we can analyze the relaxation of the restriction on deposit rate in a straightforward fashion. Deregulation can be described as an increase in the legally established upper bound of the deposit rate |Mathematical Expression Omitted~. We investigate what happens to the loan rate and the total deposit as deposit rate deregulation occurs. Again differentiating the optimal equations we obtain the following comparative static results.(11)

|Mathematical Expression Omitted~ and |Mathematical Expression Omitted~.

These results can be interpreted as follows. Because of the deregulation, the deposit rate increase leads to higher amount of deposit gain by the bank. In that situation, a commercial bank will extend more loan which is likely to pull down the loan rate. However, lower loan rate is still economically profitable, since the bank no longer needs borrowing from the federal funds market, rather they can sell excess reserve. Thus, deregulation increases the loan rate of interest in the economy.

We also can extend our model incorporating uncertainty about return on loans extended by the commercial bank. However in this case, the predictive content of our model seriously reduces (as explained briefly below) because of uncertainties originating from two distinct sources.

Let the random variable |Delta~ |Epsilon~ |0, 1~ denote the stochastic nature of the returns on loan extended by the bank. If |Delta~ = 0, then complete loss on the loan and when |Delta~ = 1, no loss at all. Under these circumstances, the net income of the bank can be defined as

|Pi~ = |Delta~|r.sub.l~L + |r.sub.g~G - |Alpha~|r.sub.d~D - |r.sub.f~F, (23)

and the expected utility function can be expressed as

|Mathematical Expression Omitted~

where f(|Delta~) is the probability density function of the random variable |Delta~.

The optimal values for G and L can be obtained from the relevant first order conditions of the expected utility maximization problem. Unfortunately, the determinative qualitative signs of the comparative static results following the parametric changes cannot be obtained as two random variables |Delta~ and |Alpha~ are working simultaneously in this generalized model.

IV. Concluding Remarks

The major objective of this paper was to derive a set of inventory (asset management) rules for an individual commercial bank operating under uncertain environment subject to stochastic deposit (reserve) loss. To achieve this goal we assume that the bank is maximizing the expected utility derived from its net income. So, an expected utility function and a budget constraint for the bank have been specified for this purpose.

Analysis of the relevant marginal conditions provide us with the inventory decisions that are needed by an individual banker. It also provides directions of changes in the composition of the bank portfolio in response to changes in the parameters of the model. Increase in the government security interest rate, the federal fund rate and the required reserve ratio lead to the readjustment in the asset side of the portfolio of the commercial bank. When the government security rate increases, the commercial bank increases the amount of government securities purchased at the expense of loans. However an increase in the federal fund rate and in the required reserve ratio force the commercial bank to adopt the opposite route to readjust its portfolio. In these cases, the commercial bank will reduce its purchase of government securities while the loan extended by it will increase. The increase result does not hold if the commercial bank faces an increase in uncertainty about the deposit (reserve) loss, and the bank's attitude toward risk is one of increasingly partial risk aversion. Then the commercial bank will increase the purchase of government securities and decrease the amount of loans.

Furthermore, this study also shows that the deregulation over the bank's deposit rate, in turn, lowers the loan rate of interest charged by the bank. Finally, a more general model incorporating uncertainty about deposit loss as well as the loss of the return on the loans extended by the bank has been specified. Unfortunately this generalized model has lower predictive power than our restrictive model has. However, it should be conceded here that the comparative static results obtained here are model specific and may be non-robust to decomposition of loans and deposit

Notes

1. For a detailed survey of the literature see Santomero (1984) and Baltensperger (1980).

2. It is assumed that the bank knows the moments of the random variable |Alpha~.

3. More reasonable assumption will be, demand deposit is an increasing function of the deposit rate, i.e. D = D(|r.sub.d~) with |Mathematical Expression Omitted~. However both will lead to the same results.

4. Following convention, it is assuming that U(|Pi~) is thrice differentiable function with U|prime~(|Pi~) |is greater than~ 0 and U|double prime~(|Pi~) |is less than~ 0.

5. This can be conceived as a budget constraint for the bank.

6. Second order condition is also satisfied as shown later.

7. In the definition of expected profit |Mathematical Expression Omitted~, the mean of the random variable is assumed to be known.

8. |Mathematical Expression Omitted~. This can be written as |Mathematical Expression Omitted~, where

|Alpha~ = |r.sub.g~,

b = |r.sub.l~ + |r.sub.l~L.

|(|Alpha~ - b).sup.2~ |is greater than or equal to~ 0, U|double prime~ |is less than~ 0, so the first term is positive second term is also positive assuming profit is a concave function of the amount of loan extended.

9. To get these comparative static results, we assume |r.sub.d~ |is greater than~ |r.sub.f~ and |r.sub.g~ |is less than~ |r.sub.l~ + |r|prime~.sub.l~L, which are quite reasonable.

10. See the Appendix.

11. |Mathematical Expression Omitted~

by assumption |Mathematical Expression Omitted~ and dL/dD can be shown positive from the system (12).

Appendix

The objective is to find the qualitative signs of the expressions

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~.

By assumption |r.sub.d~ |is less than~ |r.sub.f~ and |r.sub.g~ |is less than~ |r.sub.l~ + |r|prime~.sub.l~L but the sign of (|Mathematical Expression Omitted~) is not known. To evaluate the signs of these two expressions, we determine the sign of |Mathematical Expression Omitted~ at first, since others can be determined immediately in the same way.

To evaluate the sign of |Mathematical Expression Omitted~ we define the profit function in a special way.

|Pi~ = |r.sub.l~L + |r.sub.g~G - (|r.sub.d - |r.sub.f~)|Alpha~D - |r.sub.f~D

can be rewritten as

|Pi~ = W + Z

where W is the risk free profit and Z is the risky part of the total profit. Now W and Z can be defined in the following way

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

We assume that |Mathematical Expression Omitted~ is known to bank as explained in footnote 2.

Thus defining |Mathematical Expression Omitted~ as Z we get the absolute risk aversion function (as developed by Arrow (1965) and others)

|R.sub.|Alpha~~ = - U|double prime~(W + Z)/U|prime~(W + Z).

Let us take a specific value of |Mathematical Expression Omitted~ for which Z = 0 and |Mathematical Expression Omitted~. Now assuming that the bank's attitude towards risk is one of decreasing absolute risk aversion, we get

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

So, it is true for all values of Z, and we can take expectation. Now multiplying both sides by |r.sub.g~ we get

|Mathematical Expression Omitted~

or

|Mathematical Expression Omitted~

Since the righthand side is positive, we get

E|ZU|double prime~(|Pi~)|r.sub.g~~|is greater than~0. Similarly it can be shown that |Mathematical Expression Omitted~.

Thus we get

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

and consequently

|Delta~G/|Delta~|Gamma~ |is greater than~ 0,

and

|Delta~L/|Delta~|Gamma~ |is less than~ 0.

References

Aigner, Dennis J. and W. T. Bryan, "A Model of Short-Run Bank Behavior." Quarterly Journal of Economics, February 1971, pp. 97-118.

Baltensperger, E. B. 1980, "Alternative Approaches to the Theory of the Banking Firm." Journal of Monetary Economics, pp. 1-37.

Elyasiani, Elyas, "The Two Product Banking Firm under Uncertainty." Southern Economic Journal, April 1983, pp. 1002-17.

Hart, O. D., and D. M. Jaffee, 1974, "On the Application of Portfolio Theory to Depository Financial Intermediaries", Review of Economic Studies 4, pp. 129-147.

Hester, Donald and James Pierce. Bank Management and Portfolio behavior. New Haven, Conn.: Yale University Press, 1975.

Klein, M. A., 1971. "A Theory of the Banking Firm", Journal of Money, Credit, and Banking, pp. 205-218.

Markowitz, H. M., Portfolio Selection, Efficient Diversification of Investment, N. Y., John Wiley and Sons, 1959.

Pringle, J. J., 1973, "A Theory of Banking Firm: A Comment", Journal of Money, Credit, and Banking, pp. 990-996.

-----, 1974, "The Imperfect Market, Model of Commercial Bank Financial Management", Journal of Financial and Quantitative Analysis, pp. 69-87.

Sandmo, A., "On the Theory of Competitive Firm under Price Uncertainty." American Economic Review, March 1971, pp. 65-73.

Santomero, A. M., "Modeling The Banking Firm: A Survey", Journal of Money, Credit and Banking, November 1984, Part 2, pp. 576-616.

Tobin, James, 1958, "Liquidity Preference as Behavior Towards Risk", Review of Economic Studies, 26, 1, pp. 65-86.

-----, "The Commercial Banking Firm: A Simple Model." Scandinavian Journal of Economics, 1982, 84(4), pp. 495-530.