Portfolio Management

Portfolio Management Grenoble Ecole de Management MSc Finance 2011

Learning Objectives Mastering the principles of the portfolio management process: • The minimum variance and the efficient frontier • Capital Market Line • Capital Asset Pricing Model • Multi-factor models • Asset Pricing Theory

Portfolio Management The minimum variance and the efficient frontier

Mean variance analysis Ideas come from an article of 1952 by Harry Markowitz where one can find the mathematic formalization of the common practice of diversification to reduce risk. Harry Markowitz wrote the basic principles of portfolio construction and propose the first answers to the question: how to determine the optimal weights of each asset in the portfolio ? This is the mean variance analysis. It is a normative framework.

Mean variance analysis • Markowitz’s mean variance analysis is based on the following assumptions: • all investors are risk averse; they prefer less risk to more for the same level of expected return. • They are able to form expectations onreturns for all assets. • They are able to form expectations onvariances and covariances for all assets. • Investors need 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. In practice, to form expectations on returns, variances and covariances investors may use estimation techniques. The estimation of those quantities may be a source of mistakes in decision making when we use mean-variance analysis.

Portfolio of assets: Diversification Variability for selected stocks larger than the variability of the whole portfolio: this the essence of diversification.

The portfolio possibilities curve – 2 assets Expected Return Variance

The portfolio possibilities curve – 2 assets • If Energy and Materials were perfectly correlated, the standard deviation of the portfolio would be the weighted average of the two securities standard deviation: portfolios would lie on the dashed line. • Since correlation is less than 1, the standard deviation of the portfolio of assets is less than the weighted average. The difference between portfolio A and C is the reduction of risk from diversification. A dominates C. It is also true of B that offers larger return for comparable risk.

The portfolio possibilities curve – 2 assets ExpectedReturns Variance The curvature of the portfolio possibilities curve depends on correlation. Lower correlations increase the curvature and generate higher diversification gains.

Efficient Portfolios – 2 assets ExpectedReturns Variance Portfolios which dominates are said to be efficient. The location of these portfolios is the efficient frontier or minimum variance frontier. It runs from the global minimum variance portfolio (70% Energy – 30% Materials) to the farthest on the right.

Efficient Portfolios – 2 assets The minimum variance portfolio

Efficient Portfolios – n assets Expected return Energy + Materials + AA + BB portfolio BB AA Materials Energy Standard deviation Often a new asset permits us to move to a superior minimum variance frontier.

Efficient Portfolios – n assets Efficient portfolios use risk efficiently: every portfolio on the efficient frontier has either a higher rate of return for equal risk or a lower risk for an equal rate of return. Investors making portfolio choices in terms of mean return and variance of return can restrict their selections to portfolios lying on the efficient frontier: this is the selection process

Min Subject to Efficient Portfolios – n assets • Determining the minimum-variance frontier for many assets: • we first determine the minimum and maximum expected returns possible with the set of assets. These are rmin and rmax • we must determine the portfolio weights that will create the minimum-variance portfolio for values of expected returns. Between rmin and rmax. • Therefore we must solve the following problem for specified values of r comprise in between rmin and rmax. Min Subject to This optimization problem says that we solve for the portfolio weights that minimize the variance of return for a given level of expected return r subject to the constraint that the weights sum to 1. The weights define a portfolio and the portfolio is the minimum variance portfolio for its level of expected return.

Min Subject to Efficient Portfolios – n assets The Lagrangian is:

Min Subject to Efficient Portfolios – n assets First order conditions:

Min Subject to Efficient Portfolios – n assets Then loading w into the other two conditions yields: Solving this system yields:

Min Subject to Efficient Portfolios – n assets Then reloading in the equation of w we are able to determine the vector w of weights that solve the program for given level of E(Rp): This result defines the location of efficient portfolios that solve the optimization program. This location is an hyperbola.

Corner Portfolios • Corner portfolio are portfolio in which one weight is nil. • As the minimum variance frontier passes through a corner portfolio an asset weight either changes from zero to positive or from positive to zero. • Adjacent corner portfolios define a segment of the minimum variance frontier within which portfolios hold identical assets and the rate of change of asset weights in moving from one portfolio to another is constant. • In a sign constrained optimization, the asset weights of any minimum variance portfolio are a positive linear combination of the corresponding weights in the two adjacent corner portfolios that bracket it in terms of expected return.

Which efficient portfolio ? • If an investor can quantify her/his risk tolerance in terms of variance, the efficient portfolio for that level of variance will represent the optimal mean variance choice. • However in real life, it is difficult to really quantify risk tolerance then risk scales are often used: 1 no tolerance for risk; 2 low tolerance; 3 medium; 4 high; 5 very high; 6 risk taker. • In the theoretical framework investors’ preferences are represented by mean of utility functions which associate an ordinal measure of utility to the pair risk/return. Risk adverse investors have convex utility functions. One example of utility function is the quadratic utility function:

Indifference/Utility curves Larger utility Lower utility As long as the utility function of the investors is defined, one can derive the indifference curves which set the location of the portfolios providing a constant utility to the investor. The investor looks at maximizing her/his utility.

Which efficient portfolio ? The investor looks at maximizing her/his utility under the constraint of the “best” risk/return portfolios: those lying on the efficient frontier. The tangency point between the efficient frontier and the utility function define the optimal portfolio. Also optimization under VaR or shortfall constraints in the research section (Moodle) and the document Utility functions and optimization.

How to determine optimal weights We are in a two-asset world: stock AA and stock BB. Stock AA has a mean return of 6% and a standard deviation of 18%. Stock BB has a mean return of 12% and a standard deviation of 27%. Correlation is 0.4. Graph the efficient frontier and point the Global Minimum Variance portfolio. Your customer, Miss Jones, would like a portfolio with a return of 9%. Which portfolio (weights) do you propose ? What do you say about risk to Miss Jones ? Mr Jones would like 11% of return but with risk below 20%. Which portfolio (weights) do you propose ?

How to determine optimal weights We are in a two-asset world: stock AA and stock BB. Stock AA has a mean return of 6% and a standard deviation of 18%. Stock BB has a mean return of 12% and a standard deviation of 27%. Correlation is -0.3. Graph the efficient frontier and point the Global Minimum Variance portfolio. Your customer, Miss Jones, would like a portfolio with a return of 9%. Which portfolio (weights) do you propose ? What do you say about risk to Miss Jones ? Mr Jones would like 11% of return but with risk below 20%. Which portfolio (weights) do you propose ?

Summary • When correlations are less than 1, they offer diversification opportunities • Diversified portfolios lie on the possibilities curve. • Portfolios which dominates are said to be efficient. The location of these portfolios is the efficient frontier or minimum variance frontier. • If an investor can quantify his risk tolerance in terms of variance, the efficient portfolio for that level of variance will represent the optimal mean variance choice.

Lending and Borrowing We introduce another possibility: we can also lend and borrow money at some risk-free rate rf. If we invest some money in T-bill (lend money) and place the remainder in common stock portfolio we can obtain any combination of expected return and risk along the straight line joining rf and the S portfolio.

Lending and Borrowing Since borrowing is negative lending we can extend the range of possibilities to the right of the market portfolio

Lending and Borrowing S has an expected return of 15% and a Sd-dev of 25%. T-bill offer a risk-free rate (rf) of 5%. If you invest half your money in T-bill and half in S. What is the expected return of your portfolio ? Its st-dev ? Then you borrow at rf an amount initial to your original wealth and you invest everything in portfolio S. What is the expected return of your portfolio ? Its st-dev ?

Capital allocation line The capital allocation line describes the combinations of expected return and standard deviation of return available to an investor from combining her optimal portfolio of risky assets with the risk free asset. The CAL is the line starting at the risk free rate of return that is tangent to the efficient frontier of risky assets.

Lending and Borrowing • If we can invest in a risk-free asset, then the CAL represents the best risk return trade-off achievable. • The CAL has a y-intercept equal to the risk-free rate. • The CAL is tangent to the efficient frontier of risky assets.

CAL Equation Suppose an investor facing a risk-free rate of rf has expectations for the S portfolio of E(Rt) and σ(Rt). The return of his portfolio is: And st-dev is: Then we can rewrite wt as: If we substitute wt in the expected return equation:

CAL Equation This equation shows the best possible trade-off between expected risk and return, given this investor’s expectations

In the mean variance analysis, investors are able to form expectations about return and variance for all assets. They have the same expectations. Thus they all hold the same portfolio. This is the Market portfolio M. • The proportion of each asset in that portfolio are the same for all investors. Thus the efficient portfolio is a market capitalization weighted basket of securities. • The sum of all portfolios held by investors constitute the market portfolio, of which every investor holds a proportionate share • when investors share identical expectations, the CAL for all investors is the same and is known as the Capital Market Line – CML - Capital Market Line

Capital Market Line The slope of the CML, is called the market price of risk. It is the amount of return in excess of the risk free rate per unit of risk. Also we can define this ratio for any portfolio lying on the CAL: This ratio is known as the Sharpe ratio

Capital Market Line • The market portfolio is the core component of asset pricing as well as the theoretical foundation for index investing. If the market portfolio is efficient allocation, then there is no role for active investing. All investors can satisfy their investment needs by combining the risk-free asset with the market portfolio. • But if you think you have better information than your rivals you will want the portfolio to include relatively large investments in the stocks you think are undervalued. This is active investment.

Which efficient portfolio ?

The separation theorem • First investors form expectations and invest in portfolio M that lies on the efficient frontier. If returns are known M is unique. • Second investors estimate their risk tolerance and decide to lend or borrow money to reach a certain point of risk on the Capital Market Line. • Tobin(1958) has called this separation of the investment decision from the financing decision the separation theorem.

CAL calculations The risk-free rate is 5%, the expected return to an investor’s tangency portfolio is 15% and the St-dev of the tangency portfolio is 25%. How much return does this investor demand in order to take on an extra unit of risk? The investors wants a portfolio sd-dev of 10%. Which are the weights of the risk-free rate and the tangency portfolio in his own portfolio ? The investor wants to put 40% of the portfolio in the risk free asset. What is the return and the sd-dev of this portfolio ? What return can expect the investor for a portfolio with sd-dev of 35% ? If the investor has EUR10 million to invest, how much she borrow at the risk-free rate to have a portfolio with an expected return of 19% ?

Summary • The Capital Allocation Line describes the combinations of expected return and standard deviation of return available to an investor from combining her optimal portfolio of risky assets with the risk free asset. • The CAL is tangent to the efficient frontier of risky assets. • When investors share identical expectations, the CAL for all investors is the same and is known as the Capital Market Line – CML. The tangency portfolio is the Market portfolio. • The slope of the CML, is called the market price of risk. • If an investor can quantify his risk tolerance in terms of variance, the portfolio lying on the CML for that level of variance will represent the optimal mean variance choice.

Portfolio Management Capital Asset Pricing Model Multifactor models APT

Capital Asset Pricing Model • The CML states a relationship between risk and returns at the portfolio level when expected returns are known: • more risk calls for more return • the relationship is linear • it enables one to choose an allocation according to ones risk tolerance in a two-step procedure (investment – financing). • Mean-variance optimization is a tool to estimate the location of the efficient frontier and the M portfolio . • However it needs to be fed with large quantities of data mainly because the number of covariances increases in the square of the number of securities.

Capital Asset Pricing Model • To simplify this analysis, Sharpe has proposed to consider that expected returns for any asset i depends on: • a common unique factor (which has turned to be the expected return of the market portfolio) • the sensitivity of the return on asset i to the return on the market returns βi • the part of returns on asset i that are independent of market returns αi • This greatly reduce the computational task of providing the inputs to a mean-variance optimization. • The most convenient way to estimate these parameters is to estimate a linear regression using time-series data on the returns to the market and returns to each asset.

Capital Asset Pricing Model • The number of parameters a portfolio manager needs to estimate to determine the minimum-variance frontier depends on the number of potential stocks in the portfolio. • In a n-stock portfolio, the manager must estimate n variances and n(n-1)/2 covariances, hence a total of n(n+1)/2 parameters. • Using the beta representation of the universe, the manager has to estimate n variances, the n covariances of the n stocks to the market portfolio and the variance of the market portfolio. Hence 2n+1 parameters. • For a portfolio of 100 stocks the number of parameters to be estimated decreases from 50050 to 201. • The beta representation aggregates all the relationship between assets (covariances) in a single parameter: β

Capital Asset Pricing Model The use of the CAPM simplifies the estimation of the variance-covariance matrix. It has further implications: it is a single factor equation describing the expected return on any asset as a linear function of its beta. i 1 Security market line: the relation between risk and returns for any asset

Capital Asset Pricing Model Less security-more volatility Absence of residual value risk Default risk Interest rate risk Inflation risk 1 Security market line: the relation between risk and returns

CAPM: simple message The message is simple: in a competitive market, the expected risk premium varies in direct proportion to the beta: T-bill have a beta of 0 and a premium of 0 The market portfolio has a beta of 1 and a risk premium of The expected risk premium on an investment with a beta of 0.5 is therefore half the expected risk premium on the market.

CAPM assumptions • The CAPM shares most of the assumptions of the mean variance analysis: • Investors need only know the expected returns, the variances and the covariances of returns to determine which portfolio are optimal for them. • Investors have identical views about risky assets’ mean returns, variances and correlation. • Investors can buy and sell assets in any quantities without affecting price and all assets are marketable. • Investors can borrow and lend at the risk free rate without limit and they can sell short any assets in any quantity. • Investors pay no taxes on returns and pay no transaction costs

CAPM Equation The CML represents the efficient frontier when the assumptions of the CAPM hold The CAPM says that expected return has two components: first, the risk-free rate, rf and second an extra return equal to With The market risk premium In a well functioning market nobody will hold a stock that offers an expected risk premium of less than

CAPM: origin of the equation At the tangency point between the possibility curve and the Capital Market Line, the slope of the two are equal

CAPM: origin of the equation This is the equation of the Security Market Line (SML)

Portfolio Management