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Portfolio Concepts. Mean – Variance Analysis .

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Mean–Variance Analysis

Mean–variance analysis is the fundamental implementation of modern portfolio theory, and describes the optimal allocation of assets between risky and risk-free assets when the investor knows the expected return and standard deviation of those assets.

Assumptions necessary for mean–variance efficiency analysis:

All investors are risk averse; they prefer less risk to more for the same level of expected return.

Expected returns for all assets are known.

The variances and covariances of all asset returns are known.

Investors need know only the expected returns, variances, and covariances of returns to determine optimal portfolios. They can ignore skewness, kurtosis, and other attributes of a distribution.

There are no transaction costs or taxes.

Efficient portfolios

Efficient portfolios (assets) offer the highest level of return for a given level of risk as measured by standard deviation in modern portfolio theory.

Because investors are risk-averse, by assumption, they will choose to allocate their assets to portfolios that have the highest possible level of expected return for a given level of risk.

These portfolios are known as efficient portfolios.

We can use optimization techniques to determine the necessary weights to minimize the portfolio standard deviation for a specified set of expected returns, standard deviations, and correlations for the assets comprising the portfolio.

Portfolio expected return and risk

We can calculate the expected return and variance of a two asset portfolio as:

We can calculate the expected return and variance of a three asset portfolio as:

Standard deviation is, of course, the positive square root of variance in both cases.

Portfolio Expected return and risk

Focus On: Calculations

You are examining three international indices. What is the expected return and standard deviation of a portfolio composed of 50% French equities, 25% English equities, and 25% German equities?

The E(r) is 3.6758%,

and the standard deviation

is 10.0191.

The efficient frontier

The efficient frontier is a plot of the set of expected returns and standard deviations for all efficient portfolios (assets) above the global minimum-variance portfolio.

The minimum-variance frontier (solid green line) is the set of all portfolios that represent the lowest level of risk that can be achieved for each possible level of return.

The portfolio with the lowest variance of all the portfolios, with the lowest level of risk that can be achieved, is known as the global minimum-variance portfolio.

E(r)

Standard Deviation

The efficient frontier

Portfolios on the efficient frontier provide the highest possible level of return for a given level of risk.

Because portfolios on the efficient frontier use risk efficiently to generate returns, investors can restrict their selection process to portfolios lying on the frontier.

This approach simplifies the risky-asset selection process and reduces selection cost.

The light green portfolios in the figure are inefficient portfolios.

E(r)

Standard Deviation

Diversification and correlation

The trade-off between portfolio risk as measured by standard deviation and portfolio expected return is affected by asset returns, variances, and correlations.

Recall the expected return and variance of a two-asset portfolio.

All the terms in the variance calculation are strictly positive, except the last term, which includes the correlation, which ranges from perfect negative (–1: blue) to perfect positive (+1: purple) with zero correlation in between (0: green).

As a correlation moves from perfect positive toward perfect negative, diversification benefits increase.

E(r)

Standard Deviation

Finding the minimum-variance frontier

We can use an optimizer, such as the Solver in Excel, to solve for the weights in the minimum-variance portfolios and thus the minimum-variance frontier.

Recall that the set of weights in any portfolio must sum to 1 and, if there are no short sales, must all be positive.

The expected return and variance for a given set of weights are

For every return, z, between zmin and zmax,

we solve for the set of weights that

minimizes the portfolio variance subject to E(rp) = z.

If we do so iteratively, we begin at zmin and iterate by a fixed amount of E(rp) until we reach zmax.

Equal-weighted portfolios

The expected return to an equally weighted portfolio is just the sum of the expected returns to the assets divided by the number of assets.

It can be shown that the variance of an equally weighted portfolio is:

where n is the number of assets in the portfolio, is the average variance of those assets, and is the average covariance of the assets.

Consider a 10-asset portfolio with average variance of 0.0225 and average covariance of 0.04. The variance of such a portfolio will be

The capital allocation line

The capital allocation line (CAL) describes the optimal expected return and standard deviation combinations available from combining risky assets with a risk-free asset.

This is a line originating at the expected return–standard deviation coordinates of the risk-free asset and lying tangent to the efficient frontier.

The slope of this line is known as the Sharpe ratio, and it represents the best possible risk–return trade-off by construction.

As can be seen from the equation for the CAL:

The intercept is the risk–return coordinate for the risk-free asset or [RF,0].

The slope is the excess return E(RT) – RF per unit of risk sRT.

The capital allocation line

Focus On: Calculations

Consider an investor facing a 3% risk-free rate with access to a tangency portfolio with a 12% return and an 18% standard deviation.

If the investor requires a 10% return, how much risk will she have to bear?

14%

The capital market line

When all investors share identical expectations about the expected returns, variances, and covariances of assets, the CAL becomes the CML.

The capital market line (CML) represents the case in which all investors have the same expectations and, therefore, hold the same risky portfolio as the tangency portfolio.

In equilibrium, this will be all risky assets in their market value weights; hence, all investors will hold the market portfolio as part of their portfolio.

The slope of the CML is known as the market price of risk and is the Sharpe ratio for the market portfolio.

Capital asset pricing model

The capital asset pricing model, or CAPM, describes the expected return to any asset as a linear function of its “beta.”

The CAPM proposes that all security expected returns can be broken down into two components:

A risk-free component (in red).

A component received for bearing market risk (in blue).

This component is the amount of risk, bi, times the price of risk, E(RM) – RF.

biis a measure of the asset’s sensitivity to market movements (market risk).

bi = 1 is the beta for the market, or bM.

bi > 1 is greater than the beta for the market and we would expect returns in excess of market returns.

bi < 1 is less than the beta for the market and we would expect returns lower than market returns.

bi = 0 is zero market risk (risk free) and we would expect the risk-free return.

E(RM) – RF is known as the market risk premium.

CAPM assumptions

Investors need only know the expected returns, the variances, and the covariances of returns to determine which portfolios are optimal for them.

This assumption appears throughout all of mean–variance theory.

Investors have identical views about risky assets’ mean returns, variances of returns, and correlations.

Investors can buy and sell assets in any quantity without affecting price, and all assets are marketable (can be traded).

Investors can borrow and lend at the risk-free rate without limit, and they can sell short any asset in any quantity.

Investors pay no taxes on returns and pay no transaction costs on trades.

The security market line

The graphical depiction of the CAPM is often known as the security market line, or SML.

SML

E(r)

E(rm)

bm=1

rf

b

Markowitz decision rule

The Markowitz decision rule provides several principles by which investors can determine how to allocate their assets.

When choosing to allocate all of your money to Asset A or Asset B, choose A when

The mean return on A is greater than or equal to that of B, but A has a smaller standard deviation than B, or

The mean return of A is strictly larger than that of B, and A and B have the same standard deviation.

When either of these is the case, we say that A “mean–variance dominates” B.

If we can borrow and lend at the risk-free rate, then

The portfolio with the higher Sharpe ratio mean–variance dominates the asset with the lower Sharpe ratio and should be chosen.

Adding an asset class

We will add a new asset class to our existing portfolio when making that addition provides a higher Sharpe ratio for the resulting portfolio.

In order to determine whether we will have a higher Sharpe ratio, we need

The Sharpe ratio of the new asset class;

The Sharpe ratio of the existing portfolio, p; and

The correlation between the new investment’s returns and those of our existing portfolio.

If this condition is true, our risk–return relationship is improved by adding the new asset class.

Limitations of mean–variance analysis

Historical estimates of model parameters involve two potential problems: (1) the large number of estimates needed, and (2) the quality of such estimates.

The number of parameters needed for mean–variance efficiency analysis is n2/2 + 3n/2.

For even a small set of assets, this number is very large.

For example, if we have 28 assets, that is 434 parameters that must be used in the optimization process.

The quality of the estimates themselves is generally low.

The estimate of mean return has a large variance, and small changes can dramatically effect mean–variance estimation outcomes.

The estimate of the variance has a smaller relative variance, but is also measured with error.

The estimates of the covariance also have a large amount of measurement error.

The market model

The market model can be estimated via linear regression and is often used to estimate unadjusted firm betas.

The estimated regression equation for the market model is:

From this, we can calculate the expected return, variance, and standard deviation of any stock as:

Using the market model to determine the necessary mean–variance parameters reduces the set of parameter estimates to 3n + 2.

The market model

Focus On: Calculations

We are examining two industry indices, one of which has a beta of 1.87 and a residual standard deviation of 14.56. The other has a beta of 1.24 and a residual standard deviation of 9.46.

If the market portfolio has a variance of 30.2, what is the correlation between the two assets?

Beta: to adjust or not adjust

Historical betas may not be as useful for predicting future behavior because we know that betas change over time.

We can model beta itself from past values of beta.

Beta can be modeled as an AR(1) process as in Chapter 10, where

We know that adjusted betas predict better than unadjusted betas, and that they are typically mean reverting (see Chapter 10).

One common adjustment is

which can easily be shown to mean revert to a beta of 1.

Adjusted beta

Focus On: Calculations

Use the beta adjustment model:

What is the adjusted beta for a firm whose unadjusted beta is 1.87?

Instability in the efficient frontier

When small changes in the input values lead to large changes in the efficient frontier, it is called “instability in the efficient frontier.”

Instability arises because we use parameter estimates as inputs rather than the true underlying parameter values.

If the differences in parameters are small (statistically or economically insignificant), the optimization process will likely overfit the model.

Large negative weights in the absence of short-selling restrictions may be indicative of this problem.

The model may indicate frequent rebalancing in response to only small variable changes.

Instability may also arise across time because of true changes in the underlying parameters or because of the same estimation problem as already noted.

Multifactor Models

Models of asset returns that use more than one underlying source of risk, known as a factor, are known as multifactor models.

Features of multifactor models:

The underlying sources of risk are known as systematic factors and referred to as priced risks.

Multifactor models explain asset returns better than the market model.

Multifactor models provide a more detailed analysis of risk than single-factor models.

Categories of multifactor models:

Macroeconomic The factors are surprises in macroeconomic variables.

Fundamental The factors are attributes of stocks or companies.

Statistical The factors are determined statistically and are often the return on differing portfolios.

Macroeconomic factor models

A macroeconomic factor surprise is the component of the factor’s return that was unexpected.

The surprise is generally measured as the difference between the realized value and the predicted value prior to realization.

A k-factor macroeconomic model is expressed as:

where “a” is the expected return to the asset, the “b” terms are factor sensitivities, and the “F” terms are the surprises in the macroeconomic factors.

Macroeconomic factor models

Focus On: Calculations

Suppose I believe a multifactor asset pricing model is a correct description of the risk–return relationship for equity returns. The model takes the following form:

Ri= ai + bi,fForex +bi,dDefault+ bi,sSize+ ei

I plan on buying two stocks with the following factor sensitivities:

What is the expected return to a portfolio of 25% Stock X and 75% Stock Y?

ri= 0.0290 + 0.908Forex +1.292Default + 0.9180Size

Arbitrage pricing theory

The APT, as it is known, describes the expected return to an asset as a linear function of the risk of the asset with respect to a set of factors.

The APT is an equilibrium model

where the bs represent factor sensitivities and the ls represent risk premiums.

The APT relies on three assumptions:

A factor model describes asset returns.

There are many assets, so investors can form well-diversified portfolios that eliminate asset-specific risk.

No arbitrage opportunities exist among well-diversified portfolios.

In contrast to multifactor models, the APT models the expected return in equilibrium (the first term of the equation), in essence restricting the first term in the general multifactor expression to the APT value for that term.

Arbitrage pricing theory

Focus On: Calculations

You are considering purchasing shares in Cleveland Corp., and you believe the APT with three priced risk factors is an accurate description of the expected return to Cleveland Corp. The first risk factor, Macro, has a risk premium of 3% and Cleveland Corp. has a b for this risk factor of 1.1. The second risk factor, Term, has a risk premium of 2% and Cleveland has a b of 0.74. Finally, the last risk factor, Inflation, has a risk premium of 1.3% and Cleveland has a b of 0.27.

If the current risk-free rate is 3.5%, what is the APT three-factor expected return to Cleveland Corp. shares?

The apt and arbitrage

Focus On: Calculations

Consider the following stock returns and factor sensitivities for a single factor APT.

Can we combine X and Y to achieve an arbitrage possibility with Z?

What weights create a portfolio with equal sensitivities so that the sensitivity of the portfolio = the sensitivity of Z?

Is the expected return to this portfolio the same as the expected return to Z?

The apt and arbitrage

Focus On: Calculations

What weights create a portfolio with equal sensitivities so that the sensitivity of the portfolio = the sensitivity of Z?

Is the expected return to this portfolio the same as the expected return to Z?

No. Therefore, if we go short Z, we can use the proceeds to go long X and Y in weights 25% and 75%, respectively. We will generate a risk-free profit of 2% = 13% – 11%.

Fundamental factor models

In contrast to macroeconomic models, fundamental models use expected returns (instead of surprises) as factors.

Because the expected returns no longer have an expected value of zero, as do the surprises in macroeconomic factor models, the intercept, ai, is no longer an expected return but the intercept term from a regression.

The bi terms are typically factor sensitivities that have been standardized by the sensitivity across all stocks.

This is done by subtracting the average sensitivity across all stocks and then dividing the result by the standard deviation of the attribute across all stocks.

Doing this enables us to interpret all factor sensitivities as unitless and by comparison with the “typical” stock.

A factor sensitivity of 0.75 would then be interpreted as a sensitivity that is ¾ standard deviations above average.

The information ratio

Often denoted IR, the information ratio takes a form similar to the Sharpe ratio.

The information ratio can be used to capture the mean active return per unit of active risk.

The historical IR is

where the subscripts p and B indicate the portfolio being evaluated and the benchmark, respectively.

The information ratio is the difference in mean return for the portfolio and the benchmark divided by the standard deviation of the difference in return for the portfolio and the benchmark.

This can be used to set guidelines for the amount by which the portfolio performance can deviate from its benchmark (tracking risk).

Assessing Active return

Focus On: Calculations

Returning to our previous example, consider a firm that uses the following asset pricing model to determine expected return:

This is an empirical model such that the factor sensitivities used are determined via regression and are not standardized.

If the portfolio factor sensitivities, benchmark sensitivities, and factor returns are as follows, how would you decompose the sources of active return for the portfolio?

Assessing Active return

Focus On: Calculations

If the manager achieved 3.4% active return from asset selection, the active return sources are then:

This is an active asset selection manager, as seen by the large proportion of return attributable to asset selection. The manager also had a positive contribution from a macro tilt, but did poorly with the inflation and term tilts.

Assessing active risk

Focus On: Calculations

Recall that Active risk squared = Active factor risk + Active specific risk

Consider the following portfolios and their risk calculations:

Assessing active risk

Focus On: Calculations

Portfolio D is effectively a passive portfolio with little or no tracking risk (last column).

Portfolio A gets most of its active risk from an industry component followed by a stock-specific component, then a risk-index component.

Portfolios B and C have similar levels of tracking error, but C has more from risk factor selection and B from a stock-specific component.

Active Risk: Multifactor models

Focus On: Calculations

Factor marginal contribution to active risk squared is

Recall our three-factor model with active factor exposures of 0.1, –0.36, and –0.63.

You have calculated the variance–covariance matrix as:

Active Risk: Multifactor models

Focus On: Calculations

What are the factor marginal contributions to active risk squared if total active risk squared is 158.87?

Ex. calculation: NumFMCAR(Term) = –0.36[0.1(16) + –0.36(89) + –0.63(24)]

Ex. calculation: FMCAR (Inflation) = 30.9015/158.87

If 29.50% of the active risk is attributable to factor tilts, then (1 – 0.2950) = 0.705, or 70.5% of the active risk is attributable to security selection.

Tracking Risk

Focus On: Calculations

Consider that a mutual fund and its relevant benchmark have the returns and tracking error shown in the table.

The client is a foundation that wants to earn an active return above the cost of managing its account and keep tracking risk below 5%. We currently receive 1.5% for managing the account.

Evaluate the performance of the fund, calculate the IR, and interpret it.

Tracking Risk

Focus On: Calculations

Evaluate the performance of the fund.

The fund is currently earning slightly in excess of its benchmark, but it is currently not meeting its active return objective because its average tracking error is below current management fees (1.19 < 1.5).

It is also not meeting its tracking risk objective because the tracking risk calculated as the active risk of 7.5% in the prior example is greater than 5%.

The information ratio for this portfolio is

The client is earning 15.83 bps per unit of active risk accepted.

Forming a tracking portfolio

Focus On: Calculations

Tracking portfolios are portfolios with factor sensitivities that match those of the benchmark portfolio.

We can formulate the weights for a tracking portfolio of n factors as long as we have n + 1 well-diversified portfolios.

Consider the following three well-diversified portfolios and target benchmark weights:

Forming a tracking portfolio

Focus On: Calculations

What are the weights for a tracking portfolio that has the benchmark’s sensitivities? Solve this set of equations:

which gives weights of w1 = 0.4117, w2 = –0.0882, and w3 = 0.6765.

Confirming this matches the desired factor sensitivities:

Market risk and nonmarket risk premiums

Investors can earn substantial premiums from exposure to risks unrelated to market risk when his or her factor risk exposures to other sources of income and his or her risk aversion differs from the average investor.

In such cases, tilts away from indexed investments may be optimal.

For example, human capital risk increases the factor sensitivity of an investor who relies on earned employment income to recession risk. Such an investor will bid up the price of countercyclical stocks and sell down the price of countercyclical stocks, causing a recession risk premium to exist for procyclical stocks.

summary

Portfolio management has a host of quantitative techniques that are used to

Select assets

Assess expected returns and risks

Track performance

Mean–variance efficient analysis forms the foundation of modern portfolio theory and describes how investors will choose between risky assets and how they will weight a portfolio of risky and risk-free assets.

Asset pricing models generally describe the expected return to assets (portfolios) as a function of the types and levels of risk they bear and the rewards due for bearing each type of risk.

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