Modern portfolio theory

Modern portfolio theory (MPT) proposes how rational investors will use diversification to optimize their portfolios, and how an asset should be priced given its risk relative to the market as a whole. The basic concepts of the theory are Markowitz diversification, the efficient frontier, Capital Asset Pricing Model and beta coefficient, the Capital Market Line and the Securities Market Line.

MPT models the return of an asset as a random variable and a portfolio as a weighted combination of assets; the return of a portfolio is thus also a random variable and consequently has an expected value and a variance. Risk in this model is identified with the standard deviation of portfolio return. Rationality is modeled by supposing that an investor choosing between several portfolios with identical expected returns, will prefer that portfolio which minimizes risk.

Risk and reward

The model assumes that investors are risk averse. This means that given two assets that offer the same 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. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favourable 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. Note that the theory uses a historical parameter, volatility, as a proxy for risk while return is an expectation on the future.

Under the model:

Portfolio return is the component-weighted return (the mean) of the constituent assets. Return changes linearly with component weightings, $w_i$ .
Portfolio volatility is a function of the correlation of the component assets. The change in volatility is non-linear as the weighting of the component assets changes.

Mathematically:

In general:
Expected return:
$\operatorname{E}(R_p) = \sum_i w_i \operatorname{E}(R_i) \quad$
Portfolio variance:
The variance of the portfolio will be the sum of the product of every asset pair's weights and covariance, $\sigma_{ij}$ - this sum includes the squared weight and variance $\sigma_{ii}$ (or $\sigma_i^2$ ) for each individual asset. Covariance is often expressed in terms of the correlation in returns between two assets $\rho_{ij}$ where $\sigma_{ij} = \sigma_i \sigma_j \rho_{ij}$
$\sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_{ij} = \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_{ij}$
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: $\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_Aw_B \sigma_{AB}$
For a three asset portfolio, the variance is:
$w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + w_C^2 \sigma_C^2 + 2w_Aw_B \sigma_{AB} + 2w_Aw_C \sigma_{AC} + 2w_B w_C \sigma_{BC}$
(As can be seen, as the number of assets (n) in the portfolio increases, the calculation becomes “computationally intensive” - the number of covariance terms = n (n-1) /2. For this reason, portfolio computations usually require specialized software. These values can also be modeled using matrices; for a manageable number of assets, these statistics can be calculated using a spreadsheet.)

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. For diversification to work the component assets must not be perfectly correlated, i.e. correlation coefficient not equal to 1.

Mathematically:

From the formulae above: if any two assets in the portfolio have a correlation of less than 1 (i.e. are not perfectly correlated) the portfolio variance and hence volatility will be less than the weighted average of the individual instruments' volatilities.

Capital Allocation Line

The Capital Allocation Line (CAL) is the line that connects all portfolios that can be formed using a risky asset and a riskless asset. It can be proven that is a straight line and that it has the following equation.

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

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 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 and the Set of Portfolios with Maximum Return.

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

The efficient frontier will be concave – 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 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 bonds. 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:
There is a relatively straightforward way to explain to derivation of the CAPM.
Imagine that you are currently in holding the Market portfolio, which is a well diversified selection of market stocks. This is called 'M'. Now suppose that you see an asset and decide that you want to invest in it. You decide on amount 'a' of this asset which we will call 'I'. So now you would currently expect to see a return on your portfolio of 'expected return on asset 'I' multiplied by the amount 'a' that you invested, plus the expected return on the original market portfolio multiplied by the amount you have invested. First lets get some notation out of the way so that the explanation can be more readable. $E(R_p)=Expected Return on total Portfolio$
$E(R_i) = Expected Return on new asset 'I'$
$E(R_m) = Expected Return on Market Portfolio$
$/sigma^2/ = Variance$
$/sigma/ = Standard Deviation$

Portfolio leverage

An investor can add leverage to the portfolio by holding 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. (A position with zero cash weighting is on the efficient frontier - the market portfolio.)

The CML is illustrated above, with return $\mu_p$ on the y axis, and risk $\sigma_p$ on the x axis.

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

CAL : 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"). Systematic 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) is the set of all combinations of the market return (r_M) and the return of a given asset i (r_i) at a given time. It can be proven that the SCL is a straight line and that its equation is:

SCL : r_{it} = \alpha_i + \beta_i  r_{Mt} + \epsilon_{it} \frac{}{}

$\alpha_i$ is called the asset's alpha coefficient and $\beta_i$ the asset's beta coefficient.

Capital Asset Pricing Model

The asset return depends on the amount paid 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.

The CAPM is usually expressed:

\operatorname{E}(R_i) = R_f + \beta_i (\operatorname{E}(R_m) - R_f)

$\beta$ , Beta, is the measure of asset sensitivity to a movement in the overall market; Beta is usually found via regression on historical data. Betas exceeding one signify more than average "riskiness"; betas below one indicate lower than average.

$(\operatorname{E}(R_m) - R_f)$ is the market premium, the historically observed excess return of the market over the risk free rate.

Once the expected return, $E(r_i)$ , is calculated using CAPM, the future cash flows of the asset can be discounted to their present value using this rate to establish the correct price for the asset. (Here again, the theory accepts in its assumptions that a parameter based on past data can be combined with a future expectation.)

A more risky stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. In theory, an asset is correctly priced when its observed price is the same as its value calculated using the CAPM derived discount rate. If the observed price is higher than the valuation, then the asset is overvalued; it is undervalued for a too low price.

Mathematically:

(1) The incremental impact on risk and return when an additional risky asset, a, is added to the market portfolio, m, follows from the formulae for a two asset portfolio. These results are used to derive the asset appropriate discount rate.
Risk = $(w_m^2 \sigma_m ^2 + w_a^2 \sigma_a^2 + 2 w_m w_a \rho_{am} \sigma_a \sigma_m)$
Hence, risk added to portfolio = $w_a^2 \sigma_a^2 + 2 w_m w_a \rho_{am} \sigma_a \sigma_m$
but since the weight of the asset will be relatively low, $w_a^2 \approx 0$
i.e. additional risk = $\quad$
Return = $( w_m \operatorname{E}(R_m) + w_a \operatorname{E}(R_a))$
Hence additional return = $w_a \operatorname{E}(R_a)$
(2) If an asset, a, is correctly priced, the improvement in risk to return achieved by adding it to the market portfolio, m, will at least match the gains of spending that money on an increased stake in the market portfolio. The assumption is that the investor will purchase the asset with funds borrowed at the risk free rate, $R_f$ ; this is rational if $\operatorname{E}(R_a) > R_f$ .
Thus: $w_a ( \operatorname{E}(R_a) - R_f ) / w_m w_a \rho_{am} \sigma_a \sigma_m = w_a ( \operatorname{E}(R_m) - R_f ) / w_m w_a \sigma_m \sigma_m$
i.e. : $= R_f + - R_f * \rho_{am} \sigma_a \sigma_m / \sigma_m \sigma_m$
i.e. : $= R_f + - R_f * / [ \sigma_{mm}$
$\quad$ is the “beta”, $\beta$ -- the covariance between the asset and the market compared to the variance of the market, i.e. the sensitivity of the asset price to movement in the market portfolio.

Securities Market Line

The relationship between Beta & required return is plotted on the Securities Market Line (SML) which shows expected return as a function of $\beta$ . The intercept is the risk free rate available for the market, while the slope is $(\operatorname{E}(R_m) - R_f)$ . 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:

SML: E(r_i) - r_f = \beta_i (E(r_M) - r_f) \frac{}{}

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

Markowitz, Harry M. (1952). Portfolio Selection, Journal of Finance, 7 (1), 77-91.
Sharpe, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19(3), 425-442.
Lintner, J. (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.
Tobin, James (1958). Liquidity preference as behavior towards risk, The Review of Economic Studies, 25, 65-86.