Portfolio optimization is the process of selecting the best
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 c ...
(
asset
In 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 of ownership that can ...
distribution), out of the set of all portfolios being considered, according to some objective. The
objective
Objective may refer to:
* Objective (optics), an element in a camera or microscope
* ''The Objective'', a 2008 science fiction horror film
* Objective pronoun, a personal pronoun that is used as a grammatical object
* Objective Productions, a Brit ...
typically maximizes factors such as
expected return
The expected return (or expected gain) on a financial investment is the expected value of its return (of the profit on the investment). It is a measure of the center of the distribution of the random variable that is the return. It is calculated b ...
, and minimizes costs like
financial risk. Factors being considered may range from tangible (such as
asset
In 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 of ownership that can ...
s,
liabilities,
earnings Earnings are the net benefits of a corporation's operation. Earnings is also the amount on which corporate tax is due. For an analysis of specific aspects of corporate operations several more specific terms are used as EBIT (earnings before intere ...
or other
fundamentals
Fundamental may refer to:
* Foundation of reality
* Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental"
* Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" ide ...
) to intangible (such as selective
divestment).
Modern portfolio theory
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 diversificati ...
was introduced in a 1952 doctoral
thesis
A thesis ( : theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: ...
by
Harry Markowitz
Harry Max Markowitz (born August 24, 1927) is an American economist who received the 1989 John von Neumann Theory Prize and the 1990 Nobel Memorial Prize in Economic Sciences.
Markowitz is a professor of finance at the Rady School of Management ...
;
see
Markowitz model
In finance, the Markowitz model ─ put forward by Harry Markowitz in 1952 ─ is a portfolio optimization model;
it assists in the selection of the most efficient portfolio by analyzing various possible portfolios of the given securities.
Here ...
.
It assumes that an investor wants to maximize a portfolio's expected return contingent on any given amount of risk. For portfolios that meet this criterion, known as efficient portfolios, achieving a higher expected return requires taking on more risk, so investors are faced with a trade-off between risk and expected return. This risk-expected return relationship of efficient portfolios is graphically represented by a curve known as the
efficient frontier
In modern portfolio theory, the efficient frontier (or portfolio frontier) is an investment portfolio which occupies the "efficient" parts of the risk–return spectrum.
Formally, it is the set of portfolios which satisfy the condition that no ...
.
All efficient portfolios, each represented by a point on the efficient frontier, are
well-diversified. While ignoring higher moments can lead to significant over-investment in risky securities, especially when volatility is high, the optimization of portfolios when return
distributions are non-
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
is mathematically challenging.
Optimization methods
The portfolio optimization problem is specified as a
constrained utility-maximization problem. Common formulations of portfolio
utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
functions define it as the expected portfolio return (net of transaction and financing costs) minus a cost of risk. The latter component, the cost of risk, is defined as the portfolio risk multiplied by a
risk aversion
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 c ...
parameter (or unit price of risk). Practitioners often add additional constraints to improve diversification and further limit risk. Examples of such constraints are asset, sector, and region portfolio weight limits.
Specific approaches
Portfolio optimization often takes place in two stages: optimizing weights of asset classes to hold, and optimizing weights of assets within the same asset class. An example of the former would be choosing the proportions placed in equities versus bonds, while an example of the latter would be choosing the proportions of the stock sub-portfolio placed in stocks X, Y, and Z. Equities and bonds have fundamentally different financial characteristics and have different
systematic risk
In finance and economics, systematic risk (in economics often called aggregate risk or undiversifiable risk) is vulnerability to events which affect aggregate outcomes such as broad market returns, total economy-wide resource holdings, or aggreg ...
and hence can be viewed as separate asset classes; holding some of the portfolio in each class provides some diversification, and holding various specific assets within each class affords further diversification. By using such a two-step procedure one eliminates non-systematic risks both on the individual asset and the asset class level.
For the specific formulas for efficient portfolios, see
Portfolio separation in mean-variance analysis.
One approach to portfolio optimization is to specify a
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 ...
defined over final portfolio wealth; the expected value of utility is to be maximized. To reflect a preference for higher rather than lower returns, this objective function is
increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
in wealth, and to reflect risk aversion it is
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity of a ...
. For realistic utility functions in the presence of many assets that can be held, this approach, while theoretically the most defensible, can be computationally intensive.
Harry Markowitz
Harry Max Markowitz (born August 24, 1927) is an American economist who received the 1989 John von Neumann Theory Prize and the 1990 Nobel Memorial Prize in Economic Sciences.
Markowitz is a professor of finance at the Rady School of Management ...
developed the "critical line method", a general procedure for
quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constr ...
that can handle additional linear constraints and upper and lower bounds on holdings. Moreover, in this context, the approach provides a method for determining the entire set of efficient portfolios. Its application here was later explicated by
William Sharpe.
Mathematical tools
The complexity and scale of optimizing portfolios over many assets means that the work is generally done by computer. Central to this optimization is the construction of the
covariance matrix for the rates of return on the assets in the portfolio.
Techniques include:
*
Linear programming
*
Quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constr ...
*
Nonlinear programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or s ...
*
Mixed integer programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming i ...
*
Meta-heuristic methods
*
Stochastic programming for multistage portfolio optimization
*
Copula based methods
*
Principal component-based methods
*
Deterministic global optimization
Deterministic global optimization is a branch of numerical optimization which focuses on finding the global solutions of an optimization problem whilst providing theoretical guarantees that the reported solution is indeed the global one, within som ...
*
Genetic algorithm
Optimization constraints
Portfolio optimization is usually done subject to constraints, such as regulatory constraints, or illiquidity. These constraints can lead to portfolio weights that focus on a small sub-sample of assets within the portfolio. When the portfolio optimization process is subject to other constraints such as taxes, transaction costs, and management fees, the optimization process may result in an under-diversified portfolio.
Regulation and taxes
Investors may be forbidden by law to hold some assets. In some cases, unconstrained portfolio optimization would lead to
short-selling
In finance, being short in an asset means investing in such a way that the investor will profit if the value of the asset falls. This is the opposite of a more conventional "long" position, where the investor will profit if the value of the ...
of some assets. However short-selling can be forbidden. Sometimes it is impractical to hold an asset because the associated tax cost is too high. In such cases appropriate constraints must be imposed on the optimization process.
Transaction costs
Transaction cost
In economics and related disciplines, a transaction cost is a cost in making any economic trade when participating in a market. Oliver E. Williamson defines transaction costs as the costs of running an economic system of companies, and unlike pro ...
s are the costs of trading in order to change the portfolio weights. Since the optimal portfolio changes with time, there is an incentive to re-optimize frequently. However, too frequent trading would incur too-frequent transactions costs; so the optimal strategy is to find the frequency of re-optimization and trading that appropriately trades off the avoidance of transaction costs with the avoidance of sticking with an out-of-date set of portfolio proportions. This is related to the topic of
tracking error In finance, tracking error or active risk is a measure of the risk in an investment portfolio that is due to active management decisions made by the portfolio manager; it indicates how closely a portfolio follows the index to which it is benchmarked ...
, by which stock proportions deviate over time from some benchmark in the absence of re-balancing.
Improving portfolio optimization
Correlations and risk evaluation
Different approaches to portfolio optimization measure risk differently. In addition to the traditional measure,
standard deviation, or its square (
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
), which are not
robust
Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
risk measures, other measures include the
Sortino ratio The Sortino ratio measures the risk-adjusted return of an investment asset, portfolio, or strategy. It is a modification of the Sharpe ratio but penalizes only those returns falling below a user-specified target or required rate of return, while t ...
,
CVaR (Conditional Value at Risk), and
statistical dispersion
In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile ...
.
Investment is a forward-looking activity, and thus the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ...
s of returns must be forecast rather than observed.
Portfolio optimization assumes the investor may have some
risk aversion
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 c ...
and the stock prices may exhibit significant differences between their historical or forecast values and what is experienced.
In particular, financial crises are characterized by a significant increase in correlation of stock price movements which may seriously degrade the benefits of diversification.
In a mean-variance optimization framework, accurate estimation of the
variance-covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
is paramount.
Quantitative techniques that use
Monte-Carlo simulation
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
with the Gaussian copula and well-specified marginal distributions are effective.
Allowing the modeling process to allow for empirical characteristics in stock returns such as
autoregression
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
, asymmetric volatility,
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal ...
, and
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
is important.
Not accounting for these attributes can lead to severe estimation error in the correlations, variances and covariances that have negative biases (as much as 70% of the true values).
Other optimization strategies that focus on minimizing tail-risk (e.g.,
value at risk
Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...
,
conditional value at risk
Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the wor ...
) in investment portfolios are popular amongst risk averse investors.
To minimize exposure to tail risk, forecasts of asset returns using Monte-Carlo simulation with vine copulas to allow for lower (left) tail dependence (e.g., Clayton, Rotated Gumbel) across large portfolios of assets are most suitable.
(Tail) risk parity
Risk parity (or risk premia parity) is an approach to investment management which focuses on allocation of risk, usually defined as volatility, rather than allocation of capital. The risk parity approach asserts that when asset allocations are a ...
focuses on allocation of risk, rather than allocation of capital.
More recently, hedge fund managers have been applying "full-scale optimization" whereby any investor utility function can be used to optimize a portfolio.
It is purported that such a methodology is more practical and suitable for modern investors whose risk preferences involve reducing
tail risk
Tail risk, sometimes called "fat tail risk," is the financial risk of an asset or portfolio of assets moving more than three standard deviations from its current price, above the risk of a normal distribution. Tail risks include low-probability ...
, minimizing negative skewness and
fat tail
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are somet ...
s in the returns distribution of the investment portfolio.
Where such methodologies involve the use of higher-moment utility functions, it is necessary to use a methodology that allows for forecasting of a
joint distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
that accounts for asymmetric dependence.
A suitable methodology that allows for the joint distribution to incorporate asymmetric dependence is the Clayton Canonical Vine Copula. See .
Cooperation in portfolio optimization
A group of investors, instead of investing individually, may choose to invest their total capital into the joint portfolio, and then divide the (uncertain) investment profit in a way which suits best their
utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
/risk preferences. It turns out that, at least in the expected utility model,
and mean-deviation model,
[Grechuk, B., Molyboha, A., Zabarankin, M. (2013)]
"Cooperative games with general deviation measures"
Mathematical Finance, 23(2), 339–365. each investor can usually get a share which he/she values strictly more than his/her optimal portfolio from the individual investment.
See also
* for related articles
*
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 ...
, for the formulas
*
Asset allocation
Asset allocation is the implementation of an investment strategy that attempts to balance risk versus reward by adjusting the percentage of each asset in an investment portfolio according to the investor's risk tolerance, goals and investment t ...
*
Merton's portfolio problem
*
Intertemporal portfolio choice Intertemporal portfolio choice is the process of allocating one's investable wealth to various assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines ...
*
Marginal conditional stochastic dominance, a way of showing that a portfolio is not efficient
*
Mutual fund separation theorem In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in ap ...
, giving a property of mean-variance efficient portfolios
*
*
Universal portfolio algorithm, giving the first online portfolio selection algorithm
*
*{{sectionlink, Machine learning, Applications
References
Financial economics
Portfolio theories