uncertainty, risk of default and the savings-consumption decision

Zeitschrift fiir National6konomie 35 (1975), 89--97 �9 by Springer-Verlag 1975

Uncertainty, Risk of Default and the Savings-Consumption Decision

By

Ernst Baltensperger*, Columbus, Ohio, U, S. A.

(Received March 27, 1975)

I. Introduction

The subject of this paper is the effect of uncertainty on optimal savings-consumption behavior. This question has recently been studied by L e l a n d [3] in the context of a simple two-period Fisherian model. H a h n [1], M i r m a n [4] and S a n d m o [8] have investigated closely related models 1. The question is of potential importance for consumption function literature. Is the variability of income an im- portant determinant of consumption behavior? In the above mentioned studies, the nature of the relationship between uncertainty and the optimal savings-consumption decision has been shown to depend crucially on the form of the utility function. For certain utility functions, such as, e. g., the quadratic, the optimal savings- consumption decision has been shown to be independent of uncertainty; that is, the decision which is optimal under certainty is also optimal when uncertainty does exist". It is the purpose of this note to show that these studies of the effects of uncertainty neglect one

:: Ohio State University.

1 A number of other authors have studied the savings-consumption decision under uncertainty, too. See, e.g., Hakansson [2], Phelps [5], Levhari and Srinivasan [6], Sandmo [7], Stiglitz [9]. The studies mentioned above have been singled out here because they focus on the effects of uncertainty itself on Optimal savings-consumption behavior (that is, on a comparison of situations without uncertainty and situations with uncertainty), in the context of a comparatively simple model.

See, e. g., Leland [3], p. 467.

90 E. Bahensperger:

factor which is of potential importance, namely the risk of default. The effects of default risk on savings-consumption behavior are investigated, and it is shown that, if this risk is taken into account, the optimal savings-consumption decision does depend on uncertainty, even in the cases where the above mentioned analyses indi- cate no connection.

L e l a n d considers a simple Fisherian two-period model for an individual with known present income, but stochastic future income. Lending and borrowing is possible at a given interest rate. By ne- glecting the default problem, L e l a n d essentially assumes that borrowing is limited such that debt repayment is possible even if the second period income assumes its smallest possible level. Such a constraint is quite artificial. A meaningful analysis of lending and borrowing must take into account the possibility of default. We will consider a model which does essentially correspond to the one used by Le land , but which does allow the possibility of default. In Sec- tion II, we will review the situation when default risk is not taken into account. In Sections III and IV, we will consider the effects of default risk.

II. Uncertainty of Future Income and the Optimal Savings-Consumption Decision

Consider an individual with a current income yl and a future income y2. The former is known with certainty, but the latter is known in form of a probability distribution only, f (y2). The individual can lend and borrow at an interest rate i. The individual is assumed to maximize expected utility, and his utility function contains Cl and c2 as arguments, where cl and c2 denote current and future consumption, respectively, and c2- -y~+( l+ i ) (y l - c l ) . The possibility of default is neglected, for the moment.

Maximization of expected utility gives the following optimality condition:

b 8

g (cl) = ~ u l f (y2) dy2 - ~ u2 (1 +i) f (y~) dy2 = 0 (1) a

and

g' (cl) < O,

where ul and u2 denote the partial derivatives of total utility with respect to cl and c~, and a and b the minimum and maximum values of y2, respectively.

Uncertainty, Risk of Default and Savings-Consumption Decision 91

It can be easily seen that introduction of uncertainty (i. e., variability of y2 around its mean) leaves the optimum unchanged if

b

tllf (ye) dy2 = ul (-c2) S (2a) a

b

S u2f (y2) dy2 = u2 (32) (2b)

where c2 = E (y2) + (1 + i) (yl - cl). That is, expected marginal utility of current and future consumption must be equal to the respective marginal utilities corresponding to the expected value of y2, and thus of c2. This requires that the marginal utility functions are "symmetric" around 32 (over the relevant range of possible c2's), so that the effects of positive and negative deviations from ca on expected marginal utility just offset each other. Under these conditions, g(cl) would remain zero at the same cl, regardless of the variability of y2 around its mean. The first of the two conditions in (2) is automatically satisfied if, as it is often assumed in this context, we have an additive utility function, which would make ul independent of y2 and thus nonstochastic.

Of course, conditions (2) need not be true. Then, introduction of uncertainty may change the optimal cl ~. L e l a n d has shown that Pratt's "principle of decreasing absolute risk aversion", (implying a positive third derivative of the utility function, i. e. a marginal utility function which falls with decreasing slope), together with an additive utility function, guarantees a reduction of cl when uncertainty is introduced (as compared to a situation with no uncertainty). S a n d m o arrives at the same conclusion. Both L e l a n d and S a n d m o have also obtained the same result for certain extensions of the P r a t t principle to the case of a nonadditive utility function. However, note that risk aversion by itself is not sufficient to explain an effect of uncertainty on the optimal decision. If the utility function is quadratic (over the relevant Iange), the marginal utility functions are linear, and conditions (2) will hold. The introduction of uncertainty then would in no way affect the optimal decision. In terms of a complete exchange system, the conclusion would be that under these conditions the introduction of uncertainty would have no effect on the equilibrium of the system, since nobodies' behavior is changed.

3 Conditions (2) are sufficient, but not necessary for an unchanged optimum. If uncertainty changes the left hand sides of both (2a) and (2b), but in proportion, the optimum will still be the same. An unchanged optimum requires that the expected marginal utility of future consumption remains constant relative to that of present consumption.

92 E. Baltensperger:

However, note that all this is based on the assumption of no default risk. This means to assume that a > - (1 +i) (yl-cl), or, in other words, borrowing (cl-yl) cannot exceed the present value of the smallest possible second period income, a/(1 +i). Obviously, this does leave rather limited room for borrowing only, particularly if y2 can assume relatively low values. And since one person's borrowing is another person's lending, it does not leave much room for lending either. In fact, if the lowest possible values for the second period incomes are zero, no lending and borrowing at all is possible, without risk of default. A meaningful analysis of lending and borrowing must take into account the risk of default.

III. Risk of Default

One of the crucial aspects of default risk is that uncertainty about the debtor's income does affect not only his, but also the lender's situation. Let us consider a simple system including two individuals. We will use small letters to denote variables referring to the first individual, and capital letters for those referring to the second. To simplify the situation further, assume that only the first individual's second period income (y2) is uncertain, while the second individual's (Y2) is known with certainty. Suppose that without uncertainty (i. e. for y2=E(y~), with probability one), the first individual does end up, in market equilibrium, as the debtor, i.e. cl > yl (and thus C1 < Y1). What is the effect of introducing uncertainty (i. e. variability of y2 around its mean) on optimal behavior if we allow for default risk?

The introduction of default risk means that now it is possible that the first individual's (the debtor's) second period income is not sufficient to cover his debt: y ~ < ( l + i ) ( c l - y l ) . In this case, the second period consumptions for both individuals are not as defined in Section II above. The first individual's (the debtor's) second period consumption, in this case, is zero, since he has to pay his total second period income to the lender. However, it may be realistic to assume that an individual does care or "worry" about the amount of debt he ends up with, i. e. assigns utility (or disutility) to his terminal debt. We can capture this by allowing the individual to experience negative levels of c2 (occuring if y2 < x), and defining his utility function over the whole range of possible c~'s, including negative ones. If the individual dislikes ending up in debt, u~ > 0 over the whole range of c~, including the range of negative values. In this case, the first individual's situation (his expected utility function and his optimality condition) is formally precisely as described in section II, and thus


his reaction to the introduction of uncertainty governed by the form of his utility function alone. If conditions (2) hold, which neutralize this effect, uncertainty will not affect his behavior.

However, the same is not true for the second individual (the lender), since he is facing risk of default. His second period consumption C2 is

Y2 + (1 +i) (Y1 - C1) if y ~ > X - = ( l + i ) (Y1-C1)

but only

Y2 + y_~ if y2 < Y 4

As long as y2 > X, the lender's second period consumption depends on his first period consumption in the usual way ( -dC2/dC1 = 1 +i), while this is not the case anymore if y2 < X (where d CJdC1 =0). Furthermore, as long as y2 > X, the lender's second period consumption is independent of y2 and thus nonstochastic (due to our simplify- ing assumption that Y2 is nonstochastic). But for y2 < X, C2 does depend on y2 and thus is stochastic.

Maximizing expected utility gives the following optimality conditions:

b b G (C1) = ~ 81[ (y2) dy2 - S U2 (1 +i) f (y~) dyz =0 (3)

X

and G ' (C1) < O,

where U1 and U2 stand for the marginal utilities of C1 and C~, respectively, and X is defined above. For y~ > X, C~ and thus U., are independent of y2 and nonstochastic. The second term of G (C1) in

b (3) can, therefore, be rewritten as U2 (1+i)J" [ (y2)dy25. Without

x b

risk of default, .f [(y2)dy2 = 1. Introduction of default risk reduces x

this expression below unity, and thus raises G(C1), ceteris paribus. If our individual's utility function is additive, U1 is independent of

4 For simplicity, assume y2~0. If we would allow negative y2's, we should have written C2 = Y2 + y2 if 0 < y2 < X, Ca = Y2 if y2 < 0, unless we want to assume that the second individual is responsible for the first individual's debts beyond what he (the second) lent to the first in the first period, which seems unreasonable.

5 Assuming independence of utilities between individuals, of course.


C2 and thus of y2, and therefore nonstochastic. (3) then can be rewritten as

b

G ( C 1 ) = 8 1 - 8 2 (1+i) ]" f (yg_) dy2 =0. (3')

Introduction of uncertainty and default risk then unambigously raises G(C1), at any given C1. At the level of C1 which is optimal under certainty, therefore, we would clearly have G(C1)> 0, if default risk is present. The optimal C1 will increase under these conditions, since optimality also requires G' (C1)<0. Given the interest rate i, the introduction of default risk will make the lender unwilling to supply the same amount of credit as before (i. e., without default risk). The lender's behavior, at any given interest rate i, will be changed. At the interest rate which did provide market equilibrium before introduction of default risk, this tends to create - - ceteris paribus - - an excess of desired borrowing over desired lending, and presumably market equilibrium would - - ceteris paribus - - require a higher interest rate, in order to close this gap. Note that this result is quite independent of the shape of the utility function.

IV. Interest Rate Depending on Risk of Default

We can look at the effect of default risk in a somewhat different way. In the preceding section, it was assumed that individuals treat the contractual interest rate i as parametric to their optimization decisions. This can be justified as representing competitive (price taking) behavior. However, it can be argued that, even in an atom- istic environment, it would be more reasonable to expect that individuals take into account in their optimization procedures that the rate of interest on a loan contains a risk premium reflecting the risk of default of the specific loan, rather than general market conditions (the state of excess demand for the credit market as a whole). A change in the risk of default represents a change in the "quality" of the loan, and the borrower cannot expect to obtain a loan of lesser quality at the same contractual interest rate, even in a competitive market. The assumption of a functional relationship between the contractual rate of interest and the risk of default does also seem to reflect institutional behavior.

Suppose that our first individual (the one with the stochastic second period income) faces an institutional lender who is able to diversify his portfolio sufficiently (among many different borrowers), so that all he cares about is his expected return. Consequently, we propose to consider the situation where individuals treat the expected


rate of interest (rather than the contractual rate) as parametric to their decision process. That is, our individual acts as a price taker in the sense of assuming that he can borrow any amount, as long as the (parametrically given) expected rate of return to the lender is held constant. As long as default risk is zero, the expected rate r is equal to the contractual rate i. However, in the range of cl where default risk exists [i. e. if x = ( l + i ) ( c l - y l ) > a ] an increase in cl will, given the probability distribution of y2, raise the probability of default and lower the lender's expected rate of return r, unless the contractual rate i is adjusted appropriately. That is, we assume that individuals take the "basic" risk free rate of interest as given from the market, but expect that in the presence of default risk, the interest rate will have to be adjusted upwards to compensate the lender for the loss in expected return he would otherwise experience.

The lender's (gross) income from lending the amount ( c l - y l ) is (1+i) ( c l - y l ) if ye>x=(l+i) (cl-yl), but only y~ if y.,<x. His

expected (net) return is i (cl - y l ) - j~" (X- y2) [ (y~) dy2. a

His expected rate of return r is

l * i - - a L cl-Y13

From this we get, by implicit differentiation, the adjustment in i which is necessary to keep the lender's expected rate of return r unchanged when cl is increased (we call this a "fair" interest rate adjustment):

X

J" Y2 f (y,2) dy2 ( d~ "i dr=O a ~-~ ! = b ( 4 )

( c l - y l ) 2 ]" [ (y~) dya X

This expression is positive if default risk is present (x > a), and zero otherwise.

Let us consider now the borrower's behavior when he treats the expected rate of interest r as a datum. His situation can be described in the same way as before, with the difference that he has to take into account the fact that the contractual rate of interest i which he has to pay depends on his probability of default, and thus on his current consumption el. If we assume that the borrower does care about his terminal debt, and therefore allow negative values of c2, as suggested at the end of our discussion of debtor behavior in Sec-


tion Ill, maximization of his expected utility yields the following condition for optimality:

g(cl)=SUl[(y2)dy2-~u2a 1 + i + (cl-yl) [(y2) dy2=O, (5)

d~- and g' (Cl) < O; where ~ is given in (4) above 6.

d~ d~ Without uncertainty, ~ = 0. With uncertainty, ~ > 0 (if

y2 rain= a < x, i. e. if default risk is present). It is clear thus that, compared to the situation discussed before (where di/dcl =0), introduction of uncertainty about y2 and default risk will lower g(cl), at any level of Cl. Since optimality requires g(cl)=O and g ' (c l )<0, this means that the introduction of uncertainty will tend to lower the optimal cl. Given the expected rate of interest r, introducing default risk will raise the contractual rate of interest i and reduce borrowing and thus present consumption cl 7.

V. Conclusion

The relationship between uncertainty and the optimal savings- consumption decision has been discussed. A simple model has been used to show that the existence of default risk has potentially im- portant effects on the nature of this relationship and should not be neglected in its analysis.

References

[1] F. H. Hahn: Savings and Uncertainty, Review of Economic Studies, 37 (1970), pp. 21--24.

[2] N. K. Hakansson : Optimal Investment and Consumption Strat- egies under Risk for a Class of Utility Functions, Econometrica 38 (1970), pp. 587--607.

[3] Hayne E. Leland: Saving and Uncertainty: The Precautionary Demand for Savings, Quarterly Journal of Economics 82 (1968), pp. 465--473.

[4] L.J. Mirman: Uncertainty and Optimal Consumption Decisions, Econometrica 39 (1971), pp. 179--185.

6 If the individual does not care about his terminal debt, the integration in the second component of (5) would be between x and b, rather than between a and b.

7 It may be noted that the effect of default risk would be even stronger if there are costs associated with the event of default (costs of collection, adjustment, etc.), against which the lender wants to protect himself, too.


[5] E. Phelps: The Accumulation of Risky Capital: A Sequential Utility Analysis, Econometrica 30 (1962), pp. 729--743.

[6] D. Levhar i and T. N. Sr in ivasan: Optimal Savings under Un- certainty, Review of Economic Studies 36 (1969), pp. 153--163.

[7] A. Sandmo: Capital Risk, Consumption and Portfolio Choice, Econometrica 37 (1969), pp. 586--599.

[8] A. S a n d m o : The Effect of Uncertainty on Saving Decisions, Review of Economic Studies 37 (1970), pp. 353--360.

[9] J. E. St igl i tz : A Consumption-Oriented Theory of the Demand for Financial Assets and the Term Structure of Interest Rates, Review of Economic Studies 37 (1970), pp. 321--351.

Address of author: Prof. Dr. Ernst Bal tensperger , The Ohio State University, Department of Economics, 1775 South College Road Columbus, OH 43210, U. S. A.

Zeitschr. f. National6konomie, 35. Bd., Heft 1-2 7

uncertainty, risk of default and the savings-consumption decision

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