Portfolio Theory Lecture Notes PDF

Week 3 Portfolio Theory: Part II 1 © 2024 ICMA Centre, Henley Business School  Excel Solver and Quantitative Methods Let's use Solver to obtain the weights of the port...

Week 3 Portfolio Theory: Part II 1 © 2024 ICMA Centre, Henley Business School  Excel Solver and Quantitative Methods Let's use Solver to obtain the weights of the portfolio of Coca Cola and Intel that has the lowest possible risk – Solver steps – Minimize the standard deviation – By changing the weight cells – Subject to the constraint that the weights sum up to 1 Note: Solver is available in the Data tab – If not installed: File, Options, Add-Ins, Manage: Excel – Add-Ins, Press Go, Select Solver, Press Ok 2 © 2024 ICMA Centre, Henley Business School  Portfolio of Many Stocks Imagine that we now hold n stocks in our portfolio, labeled i = 1 to N The expected return of the portfolio is given by: The variance of the portfolio is a lot more complicated: which can be represented via the “box method" 3 © 2024 ICMA Centre, Henley Business School  Computing Portfolio Variance Add up all boxes in the weighted variance-covariance matrix: 4 © 2024 ICMA Centre, Henley Business School  Equally Weighted Portfolio Imagine that we hold a portfolio of N stocks with the following assumptions – Equally weighted: for all assets – The variance of all assets is the same σ2 – The covariance between all pairs of assets is the same : σi,j What is the variance of our portfolio? We need to sum all the boxes – N variance boxes (the diagonal ones) – N2 boxes total – Therefore: N2 - N covariance terms (the off-diagonal ones) 5 © 2024 ICMA Centre, Henley Business School  Determinants of Portfolio Risks Note that if N is very large, the variance of the portfolio is equal to the average covariance 6 © 2024 ICMA Centre, Henley Business School  Unsystematic and Systematic Risk Note Assume that standard deviation of each stock is 20% 7 © 2024 ICMA Centre, Henley Business School  Efficient Frontier with Three Stocks 8 © 2024 ICMA Centre, Henley Business School  Efficient Frontier with Three Stocks Cont’d 9 © 2024 ICMA Centre, Henley Business School  Efficient Frontier with Many Stocks 10 © 2024 ICMA Centre, Henley Business School  Efficient Frontier To get the best set of risk and return opportunities, we should keep adding stocks until all risky assets are represented The optimal portfolio for any investor must lie on the top half of the minimum variance frontier – This is called the efficient frontier – No other portfolios offer a higher expected return for a given standard deviation = No other portfolios offer a lower standard deviation for a given expected return – Different investors may choose different efficient portfolios How would you find the frontier in practice? Use Solver to find portfolio weights that minimize the variance for a fixed expected return Appendix provides theoretical proof 11 © 2024 ICMA Centre, Henley Business School  Efficient Frontier: N Risky Assets 12 © 2024 ICMA Centre, Henley Business School  Efficient Frontier: N Risky Assets Cont’d All the portfolios on and within AZ’Z’’P*B are feasible N+ is not feasible Portfolios on P*Z’’Z’A dominate the others; i.e., they offer: – a higher E(R) for a given risk (Z’ dominates Z) – a lower risk for a given E(R) (Z’’ dominates Z) Define Markowitz's mean variance efficient frontier with only risky assets; i.e., set of portfolios that offer the highest E(R) for any risk – P*Z’’Z’A 13 © 2024 ICMA Centre, Henley Business School  Choice Between a Risky Asset and a Risk-free Asset So far, we have not considered risk-free asset with rate of return Rf A portfolio with one risky asset and one risk-free asset: Slope of CAL line: Sharpe Sharpe Ratio – Excess return: – Sharpe ratio represents the excess return per unit of risk 14 © 2024 ICMA Centre, Henley Business School  Efficient Frontier with Risk-free Asset Capital Market Line 15 © 2024 ICMA Centre, Henley Business School  Efficient Frontier with Risk-free Asset Capital Market Line Cont’d 16 © 2024 ICMA Centre, Henley Business School  Two Fund Separation Theorem Every investor will invest in M, the tangency portfolio Two fund separation theorem: In the presence of capital markets, the portfolio of two assets (the risk-free asset Rf and tangency portfolio M) dominates all other possible combinations. Therefore, the mean-variance optimizing investors will only hold efficient portfolios that lie on the CML 17 © 2024 ICMA Centre, Henley Business School  What is M? The tangency portfolio has the highest Sharpe ratio (slope) of all portfolios – i.e. the highest excess return per unit of risk According to the two fund separation theorem, the mean-variance optimizing investors will only hold a portfolio of two assets: risk-free asset and the tangency portfolio M In equilibrium, demand must equal supply in financial markets: – Demand: All investors hold the optimal (tangency) portfolio – Supply: All shares of all companies = the market Therefore, the tangency portfolio must be the market portfolio – i.e. If we only discuss about the stock market, the market portfolio includes all shares of all public companies 18 © 2024 ICMA Centre, Henley Business School  Tobin’s Separation Theorem How do investors with different risk aversion find their optimal portfolios on the capital market line? Tobin's Separation Theorem: – You can separate the portfolio construction problem into two steps – First, find the tangency portfolio; i.e., the unique mean variance efficient combination of risky assets that is tangent to the line emanating from the risk-free asset – Then, decide whether your client should borrow or lend depending on his/her attitude towards risk Every investor chooses a portfolio on the CML that matches his/her preference toward risk and E(R) 19 © 2024 ICMA Centre, Henley Business School  The Investor’s Optimal Portfolio 20 © 2024 ICMA Centre, Henley Business School  Different Lending and Borrowing Rates 21 © 2024 ICMA Centre, Henley Business School  Different Lending and Borrowing Rates Cont’d Usually investors face borrowing restrictions If they can't borrow at all, optimal portfolio is Q instead of Q’ If RB > RL, the mean variance efficient frontier will be kinked (RLLBC) – Risky portfolio selected by all defensive investors L : Tangency point between Markowitz and line originating at RL – Risky portfolio selected by all aggressive investors B : Tangency point between Markowitz and line originating at RB The market portfolio is somewhere between L and B 22 © 2024 ICMA Centre, Henley Business School  Some Definition Reminders Minimum variance opportunity set: Locus of risky portfolios with the lowest possible risk for a given E(R) Mean-variance efficient portfolio: Portfolio that offers the highest possible E(R) for a given level of risk: – In the absence of risk-free asset, efficient frontier is the upper half of the minimum variance opportunity set – In the presence of risk-free asset, efficient frontier is the CML (CAL with the highest expected Sharpe ratio) Investor's optimal portfolio: Mean-variance efficient portfolio that maximises investors expected utility; i.e. tangent to his indifference curve 23 © 2024 ICMA Centre, Henley Business School  Portfolio Construction Problem Solved in three steps Find Markowitz's mean-variance efficient frontier with only risky assets Introducing the risk-free asset into Markowitz's framework: Sharpe's analysis – Two fund separation theorem – Choice between the market portfolio and a risk-free asset: The capital market line Introducing investors' preferences – Find the tangent point between investors' indifference curve and CML 24 © 2024 ICMA Centre, Henley Business School  Markowitz’s Optimasation Puzzle Puzzle: Given the success of the MVE frontier as a theoretical construct and its availability for more than 40 years, why is the investment community reluctant to use MV optimisers? Theoretical limitations – Mean-variance optimisers assume normal returns and quadratic utility functions – Utility functions are also difficult to define – Fat tail distributed returns and non-quadratic utility function: Empirical limitations – Historical means, standard deviations and correlations do change. The optimiser assumes that the past will repeat itself – Market frictions: short-sell constraints and other portfolio weights constraints; liquidity; transaction costs; etc – Error maximisation. “Garbage in, garbage out" 25 © 2024 ICMA Centre, Henley Business School  Standard Deviations Are not Constant 26 © 2024 ICMA Centre, Henley Business School  Correlations are not Constant Either 27 © 2024 ICMA Centre, Henley Business School  Market Frictions Missing factors: such as liquidity, transaction costs, short selling restrictions have dramatic impacts on the efficient frontier 28 © 2024 ICMA Centre, Henley Business School  Key Points Diversification: as the number of assets in a portfolio increases towards infinity, the portfolio variance will tend to the (weighted) average covariance between pairs of component assets The CML will be the CAL with the highest slope – This will occur when a line is drawn in risk-return space from the risk-free return to be a tangent with the mean-variance efficient frontier Two fund separation theorem – In equilibrium, all investors will hold the same combination of risky assets. – Investors with different risk tolerances will hold different weightings of the risky asset versus risk-free asset – Investors maximise their utility when they chose an allocation to risky assets so that their indifference curve is tangential to the CML Under a set of assumptions, Markowitz's portfolio theory ensures an optimal portfolio in theory but has a number of serious limitations 29 © 2024 ICMA Centre, Henley Business School  Appendix 30 © 2024 ICMA Centre, Henley Business School  Revision of Random Variables X is a random variable that takes value of x1, x2 ….. xN with probabilities: p1, p2…pN Mean: Variance: Standard Deviation: Covariance: Correlation: 31 © 2024 ICMA Centre, Henley Business School  Revisions of Random Variables Cont’d X1, X2….XM are M random variables Mean of Sum: Variance of Sum: Covariance of Sum: 32 © 2024 ICMA Centre, Henley Business School