Modern portfolio theory

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Capital Market Line

Modern portfolio theory (MPT) proposes how rational investors will use diversification to optimize their portfolios, and how a risky asset should be priced. The basic concepts of the theory are Markowitz diversification, the efficient frontier, capital asset pricing model, the alpha and beta coefficients, the Capital Market Line and the Securities Market Line.

MPT models an asset's return as a random variable, and models a portfolio as a weighted combination of assets so that the return of a portfolio is the weighted combination of the assets' returns. Moreover, a portfolio's return is a random variable, and consequently has an expected value and a variance. Risk, in this model, is the standard deviation of return.

Risk and return

The model assumes that investors are risk averse, meaning that given two assets that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher returns must accept more risk. The exact trade-off will differ by investor based on individual risk aversion characteristics. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-return profile – i.e., if for that level of risk an alternative portfolio exists which has better expected returns.

Mean and variance

It is further assumed that investor's risk / reward preference can be described via a quadratic utility function. The effect of this assumption is that only the expected return and the volatility (i.e., mean return and standard deviation) matter to the investor. The investor is indifferent to other characteristics of the distribution of returns, such as its skew (measures the level of asymmetry in the distribution) or kurtosis (measure of the thickness or so-called "fat tail").

Note that the theory uses a parameter, volatility, as a proxy for risk, while return is an expectation on the future. This is in line with the efficient market hypothesis and most of the classical findings in finance such as Black and Scholes European Option Pricing (martingale measure: shortly speaking means that the best forecast for tomorrow is the price of today). Recent innovations in portfolio theory, particularly under the rubric of Post-Modern Portfolio Theory (PMPT), have exposed several flaws in this reliance on variance as the investor's risk proxy:

The theory uses a historical parameter, volatility, as a proxy for risk, while return is an expectation on the future. (It is noted though that this is in line with the Efficiency Hypothesis and most of the classical findings in finance such as Black and Scholes which make use of the martingale measure, i.e. the assumption that the best forecast for tomorrow is the price of today).
The statement that "the investor is indifferent to other characteristics" seems not to be true given that skewness risk appears to be priced by the market^{[citation needed]}.

Under the model:

Portfolio return is the proportion-weighted combination of the constituent assets' returns.
Portfolio volatility is a function of the correlation ρ of the component assets. The change in volatility is non-linear as the weighting of the component assets changes.

Mathematically

In general:

Expected return:-

$\operatorname{E}(R_p) = \sum_i w_i \operatorname{E}(R_i) \quad$

Where $R i$ is return and $w i$ is the weighting of component asset $i$ .

Portfolio variance:-

$\sigma_p^2 = \sum_i w_i^2 \sigma_{i}^2 + \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_{ij}$ ,

where i≠j. Alternatively the expression can be written as:

$\sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_{ij}$ ,

where $ρ i j = 1$ for i=j.

Portfolio volatility:-

$\sigma_p = \sqrt {\sigma_p^2}$

For a two asset portfolio:-

Portfolio return: $\operatorname{E}(R_p) = w_A \operatorname{E}(R_A) + (1 - w_A) \operatorname{E}(R_B) = w_A \operatorname{E}(R_A) + w_B \operatorname{E}(R_B)$

Portfolio variance: $\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_Aw_B \sigma_{A} \sigma_{B} \rho_{AB}$

matrices are preferred for calculations of the efficient frontier. In matrix form, for a given "risk tolerance" $q \in [0,\infty)$ , the efficient front is found by minimizing the following expression:

$\frac{1}{2} \cdot w^T \Sigma w - q*R^T w$

where

$w$ is a vector of portfolio weights. Each $w_i \ge 0$ and

∑ w_i = 1

i

$Σ$ is the covariance matrix for the assets in the portfolio

$q$ is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and $\infty$ results in the portfolio with maximal return

$R$ is a vector of expected returns

The front is calculated by repeating the optimization for various $q \ge 0$ .

The optimization can for example be conducted by optimization that is available in many software packages, including Microsoft Excel, Matlab and R.

Diversification

An investor can reduce portfolio risk simply by holding instruments which are not perfectly correlated. In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification will allow for the same portfolio return with reduced risk.

If all the assets of a portfolio have a correlation of 1, i.e., perfect correlation, the portfolio volatility (standard deviation) will be equal to the weighted sum of the individual asset volatilities. Hence the portfolio variance will be equal to the square of the total weighted sum of the individual asset volatilities.

If all the assets have a correlation of 0, i.e., perfectly uncorrelated, the portfolio variance is the sum of the individual asset weights squared times the individual asset variance (and volatility is the square root of this sum).

If correlation is less than zero, i.e., the assets are inversely correlated, the portfolio variance and hence volatility will be less than if the correlation is 0.

Capital allocation line

The capital allocation line (CAL) is the line of expected return plotted against risk (standard deviation) that connects all portfolios that can be formed using a risky asset and a riskless asset. It can be proven that it is a straight line and that it has the following equation.

$\mathrm{CAL} : E(r_{C}) = r_F + \sigma_C \frac{E(r_P) - r_F}{\sigma_P}$

In this formula P is the risky portfolio, F is the riskless portfolio, and C is a combination of portfolios P and F.

The efficient frontier

Efficient Frontier. The hyperbola is sometimes referred to as the 'Markowitz Bullet'

Every possible asset combination can be plotted in risk-return space, and the collection of all such possible portfolios defines a region in this space. The line along the upper edge of this region is known as the efficient frontier (sometimes "the Markowitz frontier"). Combinations along this line represent portfolios (explicitly excluding the risk-free alternative) for which there is lowest risk for a given level of return. Conversely, for a given amount of risk, the portfolio lying on the efficient frontier represents the combination offering the best possible return. Mathematically the Efficient Frontier is the intersection of the Set of Portfolios with Minimum Variance (MVS) and the Set of Portfolios with Maximum Return.

The efficient frontier is illustrated above, with return μ_p on the y-axis, and risk σ_p on the x-axis; an alternative illustration from the diagram in the CAPM article is at right.

The efficient frontier will be convex – this is because the risk-return characteristics of a portfolio change in a non-linear fashion as its component weightings are changed. (As described above, portfolio risk is a function of the correlation of the component assets, and thus changes in a non-linear fashion as the weighting of component assets changes.) The efficient frontier is a parabola (hyperbola) when expected return is plotted against variance (standard deviation).

The region above the frontier is unachievable by holding risky assets alone. No portfolios can be constructed corresponding to the points in this region. Points below the frontier are suboptimal. A rational investor will hold a portfolio only on the frontier.

The risk-free asset

The risk-free asset is the (hypothetical) asset which pays a risk-free rate. It is usually proxied by an investment in short-dated Government securities. The risk-free asset has zero variance in returns (hence is risk-free); it is also uncorrelated with any other asset (by definition: since its variance is zero). As a result, when it is combined with any other asset, or portfolio of assets, the change in return and also in risk is linear.

Because both risk and return change linearly as the risk-free asset is introduced into a portfolio, this combination will plot a straight line in risk-return space. The line starts at 100% in cash and weight of the risky portfolio = 0 (i.e., intercepting the return axis at the risk-free rate) and goes through the portfolio in question where cash holding = 0 and portfolio weight = 1.

Mathematically

Using the formulae for a two asset portfolio as above:

Return is the weighted average of the risk free asset, $f$ , and the risky portfolio, p, and is therefore linear:

Return = $w_{f} \operatorname{E}(R_{f}) + w_p \operatorname{E}(R_p) \quad$

Since the asset is risk free, portfolio standard deviation is simply a function of the weight of the risky portfolio in the position. This relationship is linear.

Standard deviation = $\sqrt{ w_{f}^2 \sigma_{f}^2 + w_p^2 \sigma_{p}^2 + 2 w_{f} w_p \sigma_{fp} }$

= $\sqrt{ w_{f}^2 \cdot 0 + w_p^2 \sigma_{p}^2 + 2 w_{f} w_p \cdot 0 }$

= $\sqrt{ w_p^2 \sigma_{p}^2 }$

= $w_p \sigma_p \quad$

Portfolio leverage

An investor adds leverage to the portfolio by borrowing the risk-free asset. The addition of the risk-free asset allows for a position in the region above the efficient frontier. Thus, by combining a risk-free asset with risky assets, it is possible to construct portfolios whose risk-return profiles are superior to those on the efficient frontier.

An investor holding a portfolio of risky assets, with a holding in cash, has a positive risk-free weighting (a de-leveraged portfolio). The return and standard deviation will be lower than the portfolio alone, but since the efficient frontier is convex, this combination will sit above the efficient frontier – i.e., offering a higher return for the same risk as the point below it on the frontier.
The investor who borrows money to fund his/her purchase of the risky assets has a negative risk-free weighting – i.e., a leveraged portfolio. Here the return is geared to the risky portfolio. This combination will again offer a return superior to those on the frontier.

The market portfolio

The efficient frontier is a collection of portfolios, each one optimal for a given amount of risk. A quantity known as the Sharpe ratio represents a measure of the amount of additional return (above the risk-free rate) a portfolio provides compared to the risk it carries. The portfolio on the efficient frontier with the highest Sharpe Ratio is known as the market portfolio, or sometimes the super-efficient portfolio; it is the tangency-portfolio in the above diagram. This portfolio has the property that any combination of it and the risk-free asset will produce a return that is above the efficient frontier—offering a larger return for a given amount of risk than a portfolio of risky assets on the frontier would.

Capital market line

When the market portfolio is combined with the risk-free asset, the result is the Capital Market Line. All points along the CML have superior risk-return profiles to any portfolio on the efficient frontier. Just the special case of the market portfolio with zero cash weighting is on the efficient frontier. Additions of cash or leverage with the risk-free asset in combination with the market portfolio are on the Capital Market Line. All of these portfolios represent the highest possible Sharpe ratio. The CML is illustrated above, with return μ_p on the y-axis, and risk σ_p on the x-axis.

One can prove that the CML is the optimal CAL and that its equation is

$\mathrm{CML} : E(r_{C}) = r_F + \sigma_C \frac{E(r_M) - r_F}{\sigma_M}.$

Asset pricing

A rational investor would not invest in an asset which does not improve the risk-return characteristics of his existing portfolio. Since a rational investor would hold the market portfolio, the asset in question will be added to the market portfolio. MPT derives the required return for a correctly priced asset in this context.

Systematic risk and specific risk

Specific risk is the risk associated with individual assets - within a portfolio these risks can be reduced through diversification (specific risks "cancel out"). Specific risk is also called diversifiable, unique, unsystematic, or idiosyncratic risk. Systematic risk (a.k.a. portfolio risk or market risk) refers to the risk common to all securities - except for selling short as noted below, systematic risk cannot be diversified away (within one market). Within the market portfolio, asset specific risk will be diversified away to the extent possible. Systematic risk is therefore equated with the risk (standard deviation) of the market portfolio.

Since a security will be purchased only if it improves the risk / return characteristics of the market portfolio, the risk of a security will be the risk it adds to the market portfolio. In this context, the volatility of the asset, and its correlation with the market portfolio, is historically observed and is therefore a given (there are several approaches to asset pricing that attempt to price assets by modelling the stochastic properties of the moments of assets' returns - these are broadly referred to as conditional asset pricing models). The (maximum) price paid for any particular asset (and hence the return it will generate) should also be determined based on its relationship with the market portfolio.

Systematic risks within one market can be managed through a strategy of using both long and short positions within one portfolio, creating a "market neutral" portfolio.

Security characteristic line

The security characteristic line (SCL) represents the relationship between the market return (r_M) and the return r_i of a given asset i at a given time t. In general, it is reasonable to assume that the SCL is a straight line and can be illustrated as a statistical equation:

$\mathrm{SCL} : r_{i,t} = \alpha_i + \beta\, r_{M,t} + \epsilon_{i,t} \frac{}{}$

where α_i is called the asset's alpha coefficient and β_i the asset's beta coefficient.

Capital asset pricing model

The asset return depends on the amount for the asset today. The price paid must ensure that the market portfolio's risk / return characteristics improve when the asset is added to it. The CAPM is a model which derives the theoretical required return (i.e., discount rate) for an asset in a market, given the risk-free rate available to investors and the risk of the market as a whole.

Securities market line

The Security Market Line

The SML essentially graphs the results from the capital asset pricing model (CAPM) formula. The x-axis represents the risk (beta), and the y-axis represents the expected return. The market risk premium is determined from the slope of the SML.

The relationship between β and required return is plotted on the securities market line (SML) which shows expected return as a function of β. The intercept is the nominal risk-free rate available for the market, while the slope is E(R_m − R_f). The securities market line can be regarded as representing a single-factor model of the asset price, where Beta is exposure to changes in value of the Market. The equation of the SML is thus:

$\mathrm{SML}: E(R_i) - R_f = \beta_i (E(R_M) - R_f).~$

It is a useful tool in determining if an asset being considered for a portfolio offers a reasonable expected return for risk. Individual securities are plotted on the SML graph. If the security's risk versus expected return is plotted above the SML, it is undervalued since the investor can expect a greater return for the inherent risk. And a security plotted below the SML is overvalued since the investor would be accepting less return for the amount of risk assumed.

Applications to project portfolios and other "non-financial" assets

Some experts apply MPT to portfolios of projects and other assets besides financial instruments.^[1] When MPT is applied outside of traditional financial portfolios, some differences between the different types of portfolios must be considered.

The assets in financial portfolios are, for practical purposes, continuously divisible while portfolios of projects like new software development are "lumpy". For example, while we can compute that the optimal portfolio position for 3 stocks is, say, 44%, 35%, 21%, the optimal position for an IT portfolio may not allow us to simply change the amount spent on a project. IT projects might be all or nothing or, at least, have logical units that cannot be separated. A portfolio optimization method would have to take the discrete nature of some IT projects into account.
The assets of financial portfolios are liquid can be assessed or re-assessed at any point in time while opportunities for new projects may be limited and may appear in limited windows of time and projects that have already been initiated cannot be abandoned without the loss of the sunk costs (i.e., there is little or no recovery/salvage value of a half-complete IT project).

Neither of these necessarily eliminate the possibility of using MPT and such portfolios. They simply indicate the need to run the optimization with an additional set of mathematically-expressed constraints that would not normally apply to financial portfolios.

Furthermore, some of the simplest elements of MPT are applicable to virtually any kind of portfolio. The concept of capturing the risk tolerance of an investor by documenting how much risk is acceptable for a given return could be and is applied to a variety of decision analysis problems. MPT, however, uses historical variance as a measure of risk and portfolios of assets like IT projects don't usually have an "historical variance" for a new piece of software. In this case, the MPT investment boundary can be expressed in more general terms like "chance of an ROI less than cost of capital" or "chance of losing more than half of the investment". When risk is put in terms of uncertainty about forecasts and possible losses then the concept is transferable to various types of investment.^[1]

Applications of Modern Portfolio Theory in Other Disciplines

In the 1970s, concepts from Modern Portfolio Theory found their way into the field of regional science. In a series of seminal works, Michael Conroy modeled the labor force in the economy using portfolio-theoretic methods to examine growth and variability in the labor force. This was followed by a long literature on the relationship between economic growth and volatility.^[2]

More recently, modern portfolio theory has been used to model the self-concept in social psychology. When the self attributes comprising the self-concept constitute a well-diversified portfolio, then psychological outcomes at the level of the individual such as mood and self-esteem should be more stable than when the self-concept is undiversified. This prediction has been confirmed in studies involving human subjects.^[3]

Comparison with arbitrage pricing theory

The SML and CAPM are often contrasted with the arbitrage pricing theory (APT), which holds that the expected return of a financial asset can be modeled as a linear function of various macro-economic factors, where sensitivity to changes in each factor is represented by a factor specific beta coefficient.

The APT is less restrictive in its assumptions: it allows for an explanatory (as opposed to statistical) model of asset returns, and assumes that each investor will hold a unique portfolio with its own particular array of betas, as opposed to the identical "market portfolio". Unlike the CAPM, the APT, however, does not itself reveal the identity of its priced factors - the number and nature of these factors is likely to change over time and between economies.

References

Notes

^ ^a ^b Hubbard, Douglas (2007). How to Measure Anything: Finding the Value of Intangibles in Business. Hoboken, NJ: John Wiley & Sons. ISBN 9780470110126.
^ Chandra, Siddharth (2003). "Regional Economy Size and the Growth-Instability Frontier: Evidence from Europe". Journal of Regional Science 43 (1): 95–122. doi:10.1111/1467-9787.00291.
^ Chandra, Siddharth; Shadel, William G. (2007). "Crossing disciplinary boundaries: Applying financial portfolio theory to model the organization of the self-concept". Journal of Research in Personality 41 (2): 346–373. doi:10.1016/j.jrp.2006.04.007.

Bibliography

Markowitz, Harry M. (1952). "Portfolio Selection". Journal of Finance 7 (1): 77–91.
Lintner, John (1965). "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets". The Review of Economics and Statistics 47 (1): 13–39. doi:10.2307/1924119.
Sharpe, William F. (1964). "Capital asset prices: A theory of market equilibrium under conditions of risk". Journal of Finance 19 (3): 425–442.
Tobin, James (1958). "Liquidity preference as behavior towards risk". The Review of Economic Studies 25: 65–86.

External links

References

Macro-Investment Analysis, Prof. William F. Sharpe, Stanford
An Introduction to Investment Theory, Prof. William N. Goetzmann, Yale School of Management

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Modern portfolio theory

Contents

Risk and return

Mean and variance

Mathematically

Diversification

Capital allocation line

The efficient frontier

The risk-free asset

Mathematically

Portfolio leverage

The market portfolio

Capital market line

Asset pricing

Systematic risk and specific risk

Security characteristic line

Capital asset pricing model

Securities market line

Applications to project portfolios and other "non-financial" assets

Applications of Modern Portfolio Theory in Other Disciplines

Comparison with arbitrage pricing theory

References

Notes

Bibliography

See also

Contrasting investment philosophy

External links