- Consumer Utility
- Consumer Utility Functions
- The Expected Utility Maxim
- Risk Tolerance
- Maximizing Expected Consumer Utility
- Representative Investors
- Market Efficiency
- Betting and Tailoring
- Active and Passive Management

*Normative* financial economics concerns *optimal* decisions made by
individuals, firms and/or institutions. In an important sense, much of the subject matter
of investments deals with optimal choices of investment and consumption.

Thus far we assumed that the investor/consumer makes optimal choices from among alternative combinations of present and contingent future consumption opportunities. Initially, we suggested that the individual picks the combination that he or she likes best. This hardly offers much help. Imagine an Analyst saying to a client: "do what's best".

Our second characterization of investor behavior utilized the concept of *indifference
curves* or, more generally, *indifference surfaces*. The conclusion was
somewhat more elegant, although hardly more useful: pick the combination from the
opportunity line (or plane or hyperplane) on the highest (best) indifference surface.

At this point, we have provided little help for the Analyst seeking to offer direction to individuals or institutions seeking advice on either the optimal amount to be invested or the particular investments that should be undertaken.

As we will see, an Analyst can provide useful advice concerning such decisions. Individuals may differ in preferences, circumstances, constraints and predictions. A rather rich body of analytic methods can be invoked to help take such differences into account. Such techniques provide a core set of normative methods for investment management.

Here we will deal with three aspects that may lead informed individuals to adopt
different strategies: differences in *preferences*, differences in *wealth*
and differences in *predictions*. We leave for later the analysis of differences in
constraints, other circumstances, and so on.

A formal construct that helps to highlight the differences among utility-based,
wealth-based and prediction-based investment decisions uses the concept of *consumer
utility* and the assumption that the goal of the consumer is to *maximize the
expected value* of such utility. In this scheme, (1) consumer utility summarizes an
individual's *preferences*, (2) possible combinations of consumption are related to
*wealth*, and (3) the probabilities utilized to compute expected utility can be
considered *predictions*. In principle, one can thus determine the extent to which
investment decisions differ due to differences in predictions as opposed to differences in
preferences or differences in wealth. In practice, such a neat taxonomy is difficult to
attain. Nonetheless, every investment decision should be scrutinized in an attempt to
determine (as best possible) the role that each such type of difference plays.

Consider an individual trying to select a combination of apples today, apples in the
future if the weather is good, and apples in the future if the weather is bad. We
represent a *consumption plan* as a vector **c** in which the elements
are the levels of consumption in every time and state; in this case: [consumption now,
consumption later if the weather is good, consumption later if the weather is bad].

Consider a particular consumption plan, for example, [80,100,50]. Note that the consumer will, in fact, attain one of two of the mutually exclusive sets of consumption:

if the weather is good: Now: 80 Future: 100 if the weather is bad Now: 80 Future: 50

It is generally assumed that consumer utility functions are such that all types of
consumption are *goods* (i.e., more is preferred to less, other things equal). It
is generally also assumed that in such functions, *marginal utility* (the added
utility from one added unit) decreases as the number of units increases (i.e. there are *decreasing
returns to scale* in consumption).

In some cases the utility associated with a given amount of future consumption will
differ, depending on the state of the world in which the consumption takes place; for
example, 5 apples might give more satisfaction on a rainy day than on a sunny one. In such
cases, we say that the consumer has *state-dependent utility*.

The utility associated with additional consumption in one time period may depend on the
amounts consumed in prior time periods. Consumers may fall into *habits* so that
both the absolute amount of consumption and any change from previous levels may be of
importance.

In many cases Analysts will take neither of these possible complications into account.
Instead, they assume that utility is *separable* and *additive* -- that is,
that there is a utility associated with each time period and state of the world and that
the total utility is simply the sum of these sources of utility. Moreover, they assume
that the utility associated with each time and state is of a particularly simple form.

Vector **c**, which represents a consumption plan, includes entries for at
least two time periods; in our case: now and later. Let **tp** be a vector of
the same length with coefficients indicating the consumer/investor's *time preference*.
For example:

now future future (good weather) (bad weather) 1.0 0.95 0.95

This indicates that a given amount of consumption in the future provides 0.95 times as
much utility as the same amount of consumption now. If there were a third time period, the
entries for that period in vector **tp** would typically be smaller than
those for the second time period, and so on. One possibility would be to make the entries
for the second period some constant *d*, those for the third period d^2, those for
the fourth period d^3, etc.. This would allow the investor's time preference to be
summarized with one number (d). In any event, all entries in vector **tp**
that refer to a given time period will be the same.

Regarding utility itself, Analysts often make another restrictive assumption. They assume that the utility associated with a given time and state can be written as:

tp(ts)*u(c(ts))

where ts is the time and state, tp(ts) is the time preference parameter for the associated time, c(ts) is the consumption planned for the time and state, and u(c(ts)) is the utility associated with that consumption, not taking into account the time at which it is received. This rules out, for example, the possibility that an Investor may have one attitude concerning risk in period 2 and a different attitude concerning risk in period 3.

Only one task remains -- to specify the utility function *u()*. Possible forms
are discussed in subsequent sections. Suffice it to say that at the very least, utility
should increase with consumption, but at a decreasing rate. Equivalently, the *marginal
utility* (change in utility per unit change in consumption) should decrease as
consumption increases.

We have associated an amount of *consumer utility* with each possible level of
consumption. However, in fact some of the levels will not be realized. To take this into
account, we multiply each utility level by the probability that the consumption in
question will be attained. The sum of all such values is the *expected utility* of
the consumption plan. We assume that the consumer's objective is to select from among all *feasible*
plans, the one that provides the *maximum expected utility*. The latter is known as
the *expected utility maxim* principle.

Let **prob** be a vector of the same length as **c** with
probabilities assigned to each time and state by the Investor (consumer) and/or the
Analyst. In this vector, the sum of all entries for a given time period will equal one.
For example:

now future future (good weather) (bad weather) 1.0 0.50 0.50

If there were entries for a third period, the sum of those entries would also be 1.0, and so on.

We are now in a position to write a formula for the expected utility **eu**
of a consumption plan **c**. It is:

eu = sum (prob.*tp.*u(c));

Given a vector of atomic security prices **p**, the optimal
consumption/investment problem can then be stated as:

Select: c to Maximize: eu Subject to: p*c <= W

where W is the consumer/investor's wealth.

The *decision variables* are the levels of planned consumption. The *optimization
problem* is to select values for these variables that maximize the *objective
function* without violating the *inequality constraint*. It can be solved
either through a "search procedure" or, in some cases, by directly finding the
values that satisfy a set of conditions that must obtain when the optimum solution for
such a problem is found. Microsoft's Excel spreadsheet includes a *solver*
procedure that employs an intelligent search method to solve problems of this sort.
MATLAB's *optimization toolbox* also provides functions that can be used for the
purpose.

Given the expected utility maxim, we can see more directly the relationship between the
curvature of the utility function and an Investor's tolerance for risk. To do so, we
utilize a simple function of the following form:

u(c) = c.^k

where k is a positive constant between 0 and 1. The greater the value of k, the
less curved will be the function; if k were to equal 1.0, the curve would become a
straight line.

Consider an investment that offers a probability of 0.50 that consumption will equal 80 apples (Good) and a probability of 0.50 that it will equal 20 apples (Bad). The table below shows the utility associated with each outcome for an investor with k=0.375. The expected utility -- the probability-weighted average of these two utility values -- is shown as well.

k = 0.375 Consumption Utility Good 80 5.172 Bad 20 3.075 -------- Expected Utility 4.124 Certainty Equivalent 43.73

The expected utility maxim assumes that an investor will be indifferent between two
investments if they offer the same expected utility. But the expected utility of a *certain
investment* will equal its utility. The *certainty-equivalent* for a risky
investment can be defined as an amount to be received with certainty that the investor
would just be willing to accept instead of the risky investment. Here, we seek the
value of c for which:

c.^k = 4.124

The answer, also shown in the table above, is 43.73 apples. Thus although
the investment offers an *expected consumption* of 50 apples (0.50*20 + 0.50*80),
this investor considers it only as desirable as 43.73 apples for certain.

Now consider the investor with a utility function for which k=0.500. The corresponding calculations are shown in the table below. She considers the investment as desirable as 45.00 apples for certain. In a sense she "likes it better" than the first investor. If one had to pay 44 apples to obtain the investment, the first investor would pass the opportunity by, while the second one would seize it with pleasure.

k = 0.500 Consumption Utility Good 80 8.944 Bad 20 4.472 -------- Expected Utility 6.708 Certainty Equivalent 45.00

The greater the value of k in a utility function of the type we have posited, the
greater will be the certainty equivalent for a given risky investment. Hence we can
say that the greater the value of k, the greater the investor's tolerance for risk and the
smaller his or her aversion to risk. More generally, the smaller the curvature of
the utility function, the greater is an investor's tolerance for risk.

Given our simplifying assumptions, the expected utility of a consumption plan
will depend on the consumer's time-preference, risk tolerance, and assessment of the
probabilities of the alternative states of the world. In our example, there are only
two such states. Since the probability of bad weather will equal one minus the
probability of good weather, we may focus on three parameters: two reflecting a consumer's
preferences (time preference and risk tolerance), and one reflecting his or her
predictions.

Given a set of prices **p** and a level of wealth *W*, a
consumer/investor will choose a consumption plan **c** that maximizes
expected utility, taking into account his or her time preference, risk tolerance and
probability assessments. Note that the latter three aspects are not directly observable,
while the former are, at least in principle. Investors with the same wealth facing the
same set of prices can and often will differ in their choices of planned consumption. In
general, those with greater preference for present as opposed to future consumption will
consume more in the present and save less. Those with greater risk tolerance will take
greater risk in their investment portfolio. And, other things equal, those who attach
higher (lower) probabilities to certain events will invest more (less) in securities that
pay off when those events take place.

To illustrate, we consider an investor with the following expected utility function:

eu = cn^k + prg*d*(cg^k) + (1-prg)*d*(cb^k)

Here, the *arguments of the function, *cn, cg and cb, are consumption now,
consumption in the future if the weather is good, and consumption in the future if the
weather is bad, respectively. The three *parameters* of the function are k (a
measure of risk tolerance), d (a measure of time preference), and prg (the estimated
probability of good weather). As in earlier examples, we assume that the prices are
[1.00 0.285 and 0.665] for the three types of consumption (cn, cg and cb, respectively).

Investors whose preferences can be described with this type of utility function will react to increases in wealth by adjusting their plans proportionately. Thus, compared with an Investor with a wealth of 100, an investor with a wealth of 200 will consume twice as many apples today, and select a consumption plan involving twice as many apples if the weather is good and twice as many apples if the weather is bad. The savings rate and portfolio composition will be the same for any two investors that have (1) the same risk tolerance (more precisely, the same value of k) and (2) the same time-preference (more precisely, the same value of d), and (3) the same probability assessment (more precisely, the same value of prg) . Such invariance with respect to wealth is not a generally observed relationship, indicating that this form of an expected utility function does not capture the preferences of all investors. However, for now allows us to avoid issues associated with the effects of differences in wealth.

Consider an Investor with a wealth of 100 for whom k=0.375, d=0.96, and prg=0.50. Her
optimal consumption plan **c** in *units* will be [48.76 112.27
28.94]. The *values* of the components (*p.*c*) will be:

Now Good Weather Bad Weather 48.76 32.00 19.24

She will spend 48.76% of her wealth on present consumption and invest 51.24%. Her
investment portfolio will consist of claims on apples if the weather is good with a value
of 32.00 and claims on apples if the weather is bad with a value of 19.24. Thus the
proportion of the portfolio's value invested in good weather apples is 32.00/51.24, or
62.45%.

This example might seem extreme, since the investor spends only 48.76% of her wealth
and invests the remaining 51.24% -- a seemingly extremely high savings rate. However,
recall that we are dealing here with total wealth, including the present value of future
income. It is important to recognize that an individual's wealth prior to retirement
includes the value of his or her *human capital*. It should be included, along with
financial and physical capital, when considering total wealth and when making plans for
savings and risk-taking. Among other things, this suggests that younger investors (for
whom human capital is likely to represent a majority of wealth) may choose to invest their
physical and financial capital rather differently than older investors (for whom human
capital may represent a minority of wealth). It also suggests that the nature of one's
human capital should be taken into account when determining the appropriate investment of
the non-human capital that is not consumed.

The next table shows the relationship between k and the decisions of interest. Each row portrays the optimal choice for a different investor. The first three columns indicate the parameters used in the analysis. The final columns show the values of, respectively, the amount consumed in the present, the amount planned to be consumed if the weather is good and the amount planned to be consumed if the weather is bad. The eighth row contains the results obtained earlier. The other rows show results for Investors that are alike with regard to time-preference and prediction but differ in risk tolerance.

prg k d consumed good bad 0.5000 0.2000 0.9600 50.2676 27.4901 22.2422 0.5000 0.2250 0.9600 50.1184 27.9926 21.8890 0.5000 0.2500 0.9600 49.9600 28.5286 21.5114 0.5000 0.2750 0.9600 49.7729 29.1149 21.1121 0.5000 0.3000 0.9600 49.4887 29.7447 20.7666 0.5000 0.3250 0.9600 49.3278 30.4329 20.2394 0.5000 0.3500 0.9600 49.0605 31.1812 19.7583 0.5000 0.3750 0.9600 48.7561 31.9981 19.2458 0.5000 0.4000 0.9600 48.4089 32.8932 18.6978 0.5000 0.4250 0.9600 48.0107 33.8783 18.1109 0.5000 0.4500 0.9600 47.5525 34.9661 17.4814 0.5000 0.4750 0.9600 47.0219 36.1728 16.8053 0.5000 0.5000 0.9600 46.4062 37.5155 16.0783 0.5000 0.5250 0.9600 45.6875 39.0175 15.2951 0.5000 0.5500 0.9600 44.8436 40.7054 14.4510 0.5000 0.5750 0.9600 43.8488 42.6102 13.5410 0.5000 0.6000 0.9600 42.6711 44.7683 12.5606

As the table shows, investors with greater tolerance for risk will invest a greater proportion of their portfolios in the good pure security. Recall that it has a larger expected return but greater underperformance in bad times. Thus investors whose utility decreases at a slower rate (higher k) with decreases in wealth are more willing to take the risk of doing badly in bad times. Note that such investors also devote a slightly smaller portion of wealth to present consumption, and hence a larger portion of wealth to investment, since future prospects are somewhat more attractive to them than to those who are more concerned with the risk such investments entail.

The next table provides the same type of analysis for a group of investors who differ in time-preference but are, in other respects, like our original investor.

prg k d consumed good bad 0.5000 0.3750 0.9000 51.3375 30.3861 18.2765 0.5000 0.3750 0.9100 50.8964 30.6614 18.4422 0.5000 0.3750 0.9200 50.4588 30.9350 18.6062 0.5000 0.3750 0.9300 50.0265 31.2052 18.7684 0.5000 0.3750 0.9400 49.5981 31.4720 18.9299 0.5000 0.3750 0.9500 49.1757 31.7361 19.0883 0.5000 0.3750 0.9600 48.7561 31.9981 19.2458 0.5000 0.3750 0.9700 48.3429 32.2557 19.4014 0.5000 0.3750 0.9800 47.9336 32.5117 19.5547 0.5000 0.3750 0.9900 47.5279 32.7655 19.7066

As the table shows, investors with greater preference for future consumption will consume less and invest more. While the absolute values of the claims for consumption in the good and bad states of the world are affected, their relative values are not. Investors with this type of utility function will change only their savings rate when their time-preference changes. The composition of their portfolios will not be affected.

The final table completes the analysis by showing a group of investors with different assessments of the probabilities associated with the alternative future states of the world, but who are like our investor in other respects.

prg k d consumed good bad 0.4000 0.3750 0.9600 50.3109 23.1037 26.5854 0.4200 0.3750 0.9600 50.0734 24.8632 25.0634 0.4400 0.3750 0.9600 49.7994 26.6357 23.5648 0.4600 0.3750 0.9600 49.4864 28.4204 22.0932 0.4800 0.3750 0.9600 49.1391 30.2092 20.6517 0.5000 0.3750 0.9600 48.7561 31.9981 19.2458 0.5200 0.3750 0.9600 48.3435 33.7809 17.8756 0.5400 0.3750 0.9600 47.8998 35.5540 16.5462 0.5600 0.3750 0.9600 47.4280 37.3134 15.2586 0.5800 0.3750 0.9600 46.9306 39.0545 14.0150 0.6000 0.3750 0.9600 46.4086 40.7727 12.8187

Note the dramatic effects of differences in predictions. Optimists, who assign a higher probability to good weather, will invest considerably larger portions of their portfolios in good weather apples (securities with higher expected returns and possibilities for greater underperformance). They invest more of their wealth as well. Differences in opinions really do make horse races (as has been said).

Recall that an Investor who assigned a probability of 0.50 to good weather, and had a
utility function with k=0.375 and d = 0.96 and wealth of 100 would choose a consumption
plan (in units) of [48.76 112.27 28.94]. Now, assume that in the aggregate
social product, the *proportions* of the three types of consumption are precisely
the same. If so, our candidate can be considered a *representative Investor*.
Why so? Because a society with aggregate consumption of [48.76*z 112.27*z
28.94*z] (where z is a positive constant) and prices [1.00 0.285 0.665] could be populated
entirely by a set of identical investors, each of whom had this specific utility function.
The existence of such preferences and predictions on the part of every investor
would be consistent with the attributes of equilibrium that would be observed in such a
society. Since preferences and predictions cannot generally be observed by the
outside analyst (financial or other), it is helpful to determine at least some possible
attributes for such elements from observed magnitudes.

By construction, a representative investor will find it optimal to (1) save at the
societal savings rate, and (2) hold the market portfolio. Hence, any investor who assigns
the same probabilities to states of the world , has the same risk tolerance, the same
impatience and (in the general case) the same wealth as a representative investor should
also save at the societal savings rate and hold the market portfolio. The concept of a
representative investor thus provides a useful *benchmark* against which one can
compare oneself. Other things equal, if an Investor makes different probability
assessments from the representative investor, it will be optimal to "tilt"
holdings towards securities that pay more in states that the Investor feels are more
likely than does the representative Investor. Other things equal, if an Investor has
greater (less) tolerance for risk than the representative Investor, he or she should hold
a portfolio with a higher (lower) expected return than the market portfolio. Other things
equal, if an investor is more (less) patient, he or she should save more (less) than is
typical in the society.

This type of comparison is complicated by the fact that the representative Investor may not be unique. For example, the world we have described could instead be populated by some other set of representative investors. In one sense, this does not matter. Any one of the possible set of representative investors can be used as a benchmark with which one can compare oneself. However, most analyses of optimal consumption and investment decisions go farther, as we will see.

We say that a securities market is *efficient relative to a given set of information*
if the *prices* of securities are the same as they would be if all participants had
that information and processed it appropriately. Note that this definition does not
require the *holdings* to be the same as they would be if all the participants had
the information and processed it appropriately. Consider the situation in which
Investor A is overly optimistic about the prospects for a firm., while Investor B is
overly pessimistic. Under these conditions, the price of the firm's stock might be
precisely the same as it would be if A and B had each made informed predictions. If
so, we would say that the market was efficient because the average of Investor's opinions
was, in a rather broad sense, correct. Note, however, that under the posited
conditions, Investor A would hold "too much" of the security, and investor B
"too little", relative to the amounts they would hold (and should have held) had
they obtained the same information and processed it appropriately.

Key to the notion of market efficiency is that of what we will call *fully-informed
probabilities*. Such probabilities would be assessed by a sophisticated Analyst with
access to a defined set of information. In effect, such probabilities "take the
information into account" in an efficient manner.

Using this construct, we can say that a market is efficient relative to a given set of information if security prices are the same as they would be if every Investor utilized fully-informed probabilities. Under these conditions, it makes sense to concentrate on a representative Investor who uses fully-informed probabilities. The risk tolerance of such an Investor is likely to be roughly (or exactly) equal to a wealth-weighted average of the risk tolerances of the Investors in the society (the latter is sometimes termed the societal risk tolerance). Similarly, the impatience of this representative Investor is likely to be roughly (or exactly) equal to a wealth-weighted average of the degrees of impatience of the consumer/investors in the society (sometimes termed the societal degree of impatience).

There are three possible reasons why Investors X and Y may wish to hold different
portfolios. Two relate to preferences and one to predictions.

First, Investors' *attitudes* toward risk and return may differ. An outside
expert cannot argue that such differences (if fully informed) are "wrong", any
more than an outside expert can tell a consumer whether he or she should prefer beer to
wine. If Investors understand the characteristics of alternative investment vehicles and
agree on their prospects and probabilities, differences in holdings can be said to be *utility-based*
or *consistent with market efficiency*. Adjusting portfolio holdings (and/or
consumption/investment decisions) to suit differences in utility can be considered *tailoring*.

The second reason for differences in holdings is associated with differences in levels of wealth. Wealthier individuals generally invest a greater absolute amount of money. Some may invest a greater percentage of their wealth, others a smaller percentage, and yet others the same percentage. Some wealthier individuals choose investment portfolios that are riskier, others portfolios with the same risk, and yet others portfolios with less risk. Differences in holdings that arise due to differences in Investor wealth are, like differences due to underlying preferences, consistent with efficient markets. Implementing strategies designed to accommodate such differences can thus also be considered tailoring.

The third reason is different. Investors (or their financial advisors) may disagree
about the *probabilities* of alternative future states of the world. (Here we
assume that all agree about the possible states of the world and the payments associated
with various securities in each of the states; in this setting, the only disagreements
about the future relate to the probabilities associated with alternative states.) Such
disagreements can arise when investors utilize disparate sets of information and/or
process a given set of information differently (although at a more profound level, the
latter can be considered a variant of the former).

Despite this, the wealth-weighted average probabilities assessed by investors may be equivalent to fully-informed probabilities, since the probabilities assessed by those with greater amounts invested (and hence greater incentive to gather information and process it well) are weighted more heavily in computing the averages. This notion underlies the often-made assumption that consensus probabilities are equal to fully-informed probabilities.

Loosely speaking, we can term an investor's probability assessments to be *deviant*
if they differ from those of the consensus of investors. If security prices do not reflect
efficient-market probabilities, at least some investors whose holdings are based on
deviant predictions may profit from such differences. On the other hand, if markets are
efficient, differences in holdings based on deviant predictions will generally prove
undesirable in the long run, leading only to added transactions costs and lack of
appropriate diversification. In either event, adjusting portfolio holdings (and/or the
consumption/investment decision) to suit differences from consensus estimates of the
likelihoods of alternative scenarios can be considered *prediction-based*, and *inconsistent
with market efficiency*. In the vernacular, such choices can simply be termed *betting*.

Those who concentrate on tailoring holdings to suit an investor's preferences and/or
circumstances generally employ investment approaches that the investment industry
classifies under the heading *passive management*. Those who bet on
differences in predictions generally employ approaches classified under the heading *active
management*. Tailoring decisions tend to be designed to implement a *strategy*
that requires only small changes over time. Such decisions typically involve small
and relatively infrequent changes, and hence can be considered "passive".

Bets, however, are likely to change rather dramatically, as new information is revealed and investors react differently to such information. Investors making decisions based on differential predictions are likely to generate a significant number of trades, and are thus appropriately termed "active".