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Intertemporal portfolio choice is the process of allocating one's investable wealth to various
asset In financial accountancy, financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value ...
s, especially
financial asset A financial asset is a non-physical asset whose value is derived from a contractual claim, such as bank deposits, bonds, and participations in companies' share capital. Financial assets are usually more liquid than other tangible assets, such as ...
s, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines a
portfolio Portfolio may refer to: Objects * Portfolio (briefcase), a type of briefcase Collections * Portfolio (finance), a collection of assets held by an institution or a private individual * Artist's portfolio, a sample of an artist's work or a ...
. Since the returns on almost all assets are not fully predictable, the criterion has to take
financial risk Financial risk is any of various types of risk associated with financing, including financial transactions that include company loans in risk of default. Often it is understood to include only downside risk, meaning the potential for financial ...
into account. Typically the criterion is the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of some
concave function In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued function f on an in ...
of the value of the portfolio after a certain number of time periods—that is, the
expected utility The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
of final wealth. Alternatively, it may be a function of the various levels of
goods and services Goods are items that are usually (but not always) tangible, such as pens, physical books, salt, apples, and hats. Services are activities provided by other people, who include architects, suppliers, contractors, technologists, teachers, doctor ...
consumption Consumption may refer to: *Resource consumption *Tuberculosis, an infectious disease, historically * Consumption (ecology), receipt of energy by consuming other organisms * Consumption (economics), the purchasing of newly produced goods for curren ...
that are attained by withdrawing some funds from the portfolio after each time period.


Discrete time


Time-independent decisions

In a general context the optimal portfolio allocation in any
time period The categorisation of the past into discrete, quantified named blocks of time is called periodization.Adam Rabinowitz. And kingIt’s about time: historical periodization and Linked Ancient World Data'. Study of the Ancient universe Papers, 2014 ...
after the first will depend on the amount of wealth that results from the previous period's portfolio, which depends on the asset returns that occurred in the previous period as well as that period's portfolio size and allocation, the latter having depended in turn on the amount of wealth resulting from the portfolio of the period before that, etc. However, under certain circumstances the optimal portfolio decisions can be arrived at in a way that is separated in time, so that the shares of wealth placed in particular assets depend only on the stochastic asset return distributions of that particular period.


Log utility

If the investor's utility function is the risk averse
log utility In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more ce ...
function of final wealth W_T, :\text = \ln W_T, then decisions are intertemporally separate. Let initial wealth (the amount that is investable in the initial period) be W_0 and let the
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
portfolio return in any period (the imperfectly predictable amount that the average dollar in the portfolio grows or shrinks to in a given period ''t'') be R_t. R_t depends on the portfolio allocation—the fractions w_ of current wealth W_ inherited from the previous period that are allocated at the start of period ''t'' to assets ''i'' (''i''=1, ..., ''n''). So: :W_T=W_0R_1R_2 \cdots R_T where :R_t=w_r_+w_r_+\cdots + w_r_, where r_ refers to the stochastic return (the imperfectly predictable amount that the average dollar grows to) of asset ''i'' for period ''t'', and where the shares w_ (''i''=1, ..., ''n'') are constrained to sum to 1. Taking the log of W_T above to express outcome-contingent utility, substituting in for R_t for each ''t'', and taking the expected value of the log of W_T gives the expected utility expression to be maximized: :\ln _0+ \sum_^T \text\ln _r_+w_r_+\cdots + w_r_ The terms containing the choice shares w_ for differing ''t'' are additively separate, giving rise to the result of ''intertemporal independence of optimal decisions'': optimizing for any particular decision period ''t'' involves taking the derivatives of one additively separate expression with respect to the various shares, and the
first-order condition In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about ...
s for the optimal shares in a particular period do not contain the stochastic return information or the decision information for any other period.


=Kelly criterion

= The
Kelly criterion In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet), is a formula that determines the optimal theoretical size for a bet. It is valid when the expected returns are known. The Kelly bet size is found by maximizing the expec ...
for intertemporal portfolio choice states that, when asset return distributions are identical in all periods, a particular portfolio replicated each period will outperform all other portfolio sequences in the long run. Here the long run is an arbitrarily large number of time periods such that the distributions of observed outcomes for all assets match their ex ante probability distributions. The Kelly criterion gives rise to the same portfolio decisions as does the maximization of the expected value of the log utility function as described above.


Power utility

Like the log utility function, the
power utility function In economics, the isoelastic function for utility, also known as the isoelastic utility function, or power utility function, is used to express utility in terms of consumption or some other economic variable that a decision-maker is concerned with ...
for any value of the power parameter exhibits
constant relative risk aversion In economics, the isoelastic function for utility, also known as the isoelastic utility function, or power utility function, is used to express utility in terms of consumption or some other economic variable that a decision-maker is concerned with ...
, a property that tends to cause decisions to scale up proportionately without change as initial wealth increases. The power utility function is :\text = aW_T^a with positive or negative, but non-zero, parameter ''a'' < 1. With this utility function instead of the log one, the above analysis leads to the following expected utility expression to be maximized: :a \cdot W_0^a \cdot \text _1^a \cdot R_2^a \cdots R_T^a where as before :R_t=w_r_+w_r_+\cdots + w_r_ for each time period ''t''. ''If there is serial independence of the asset returns''—that is, if the realization of the return on any asset in any period is not related to the realization of the return on any asset in any other period—then this expected utility expression becomes :a \cdot W_0^a \cdot \text R_1^a \cdot \textR_2^a \cdots \textR_T^a; maximization of this expected utility expression is equivalent to separate maximization (if ''a''>0) or minimization (if ''a''<0) of each of the terms \text R_t^a. Hence under this condition we again have intertemporal independence of portfolio decisions. Note that the log utility function, unlike the power utility function, did not require the assumption of intertemporal independence of returns to obtain intertemporal independence of portfolio decisions.


HARA utility

Hyperbolic absolute risk aversion In finance, economics, and decision theory, hyperbolic absolute risk aversion (HARA) (Chapter I of his Ph.D. dissertation; Chapter 5 in his ''Continuous-Time Finance'').Ljungqvist & Sargent, Recursive Macroeconomic Theory, MIT Press, Second Edition ...
(HARA) is a feature of a broad class of
von Neumann-Morgenstern utility function The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
s for choice under risk, including the log and power utility functions dealt with above. Mossin showed that under HARA utility, optimal portfolio choice involves partial time-independence of decisions if there is a risk-free asset and there is serial independence of asset returns: to find the optimal current-period portfolio, one needs to know no future distributional information about the asset returns except the future risk-free returns.


Time-dependent decisions

As per the above, the expected utility of final wealth with a power utility function is :a \cdot W_0^a \cdot \text _1^a \cdot R_2^a \cdots R_T^a If there is not serial independence of returns through time, then the expectations operator cannot be applied separately to the various multiplicative terms. Thus the optimal portfolio for any period will depend on the probability distribution of returns for the various assets contingent on their previous-period realizations, and so cannot be determined in advance. Moreover, the optimal actions in a particular period will have to be chosen based on knowledge of how decisions will be made in future periods, because the realizations in the present period for the asset returns affect not just the portfolio outcome for the present period, but also the
conditional probability distribution In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the co ...
s for future asset returns and hence future decisions. These considerations apply to utility functions in general with the exceptions noted previously. In general the expected utility expression to be maximized is :\textU(W_T) = \textU(W_0R_1R_2 \cdots R_T), where ''U'' is the utility function.


Dynamic programming

The mathematical method of dealing with this need for current decision-making to take into account future decision-making is
dynamic programming Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. I ...
. In dynamic programming, the last period decision rule, contingent on available wealth and the realizations of all previous periods' asset returns, is devised in advance; then the next-to-last period's decision rule is devised, taking into account how the results of this period will influence the final period's decisions; and so forth backward in time. This procedure becomes complex very quickly if there are more than a few time periods or more than a few assets.


Dollar cost averaging

Dollar cost averaging Dollar cost averaging (DCA) is an investment strategy that aims to apply value investing principles to regular investment. The term was first coined by Benjamin Graham in his book ''The Intelligent Investor''. Graham writes that dollar cost averag ...
is gradual entry into risky assets; it is frequently advocated by investment advisors. As indicated above, it is not confirmed by models with log utility. However, it can emerge from an intertemporal mean-variance model with negative serial correlation of returns.Balvers, Ronald J., and Mitchell, Douglas W., "Efficient gradualism in intertemporal portfolios", ''
Journal of Economic Dynamics and Control The ''Journal of Economic Dynamics and Control ''(JEDC) is a peer-reviewed scholarly journal devoted to computational economics, dynamic economic models, and macroeconomics. It is edited at the University of Amsterdam and published by Elsevier ...
'' 24, 2000, 21-38.


Age effects

With HARA utility, asset returns that are independently and identically distributed through time, and a risk-free asset, risky asset proportions are independent of the investor's remaining lifetime. Under certain assumptions including
exponential utility In economics and finance, exponential utility is a specific form of the utility function, used in some contexts because of its convenience when risk (sometimes referred to as uncertainty) is present, in which case expected utility is maximized. For ...
and a single asset with returns following an ARMA(1,1) process, a necessary but not sufficient condition for increasing conservatism (decreasing holding of the risky asset) over time (which is often advocated by investment advisors) is negative first-order
serial correlation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as ...
, while non-negative first-order serial correlation gives the opposite result of increased risk-taking at later points in time. Intertemporal portfolio models in which portfolio choice is conducted jointly with intertemporal labor supply decisions can lead to the age effect of conservatism increasing with age as advocated by many investment advisors. This result follows from the fact that risky investments when the investor is young that turn out badly can be reacted to by supplying more labor than anticipated in subsequent time periods to at least partially offset the lost wealth; since an older person with fewer subsequent time periods is less able to offset bad investment returns in this way, it is optimal for an investor to take on less investment risk at an older age.


Continuous time

Robert C. Merton Robert Cox Merton (born July 31, 1944) is an American economist, Nobel Memorial Prize in Economic Sciences laureate, and professor at the MIT Sloan School of Management, known for his pioneering contributions to continuous-time finance, especia ...
(Chapter I of his Ph.D. dissertation; Chapter 5 in his ''Continuous-Time Finance''). showed that in
continuous time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
with hyperbolic absolute risk aversion, with asset returns whose evolution is described by
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
and which are
independently and identically distributed In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
through time, and with a risk-free asset, one can obtain an explicit solution for the demand for the unique optimal portfolio, and that demand is linear in initial wealth.


See also

*
Decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
*
Intertemporal budget constraint In economics and finance, an intertemporal budget constraint is a constraint faced by a decision maker who is making choices for both the present and the future. The term intertemporal is used to describe any relationship between past, present an ...
* Intertemporal capital asset pricing model *
Intertemporal choice Intertemporal choice is the process by which people make decisions about what and how much to do at various points in time, when choices at one time influence the possibilities available at other points in time. These choices are influenced by the ...
*
Investment strategy In finance, an investment strategy is a set of rules, behaviors or procedures, designed to guide an investor's selection of an investment portfolio. Individuals have different profit objectives, and their individual skills make different tactics an ...
*
Modern portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversificatio ...
* Two-moment decision model


References

{{reflist Financial economics Portfolio theories