The document provides an overview of key concepts in portfolio optimization and the Markowitz model. It defines terms like random variables, mean, variance, correlation, asset returns, portfolio return, risk, and diversification. It then describes how the Markowitz model uses a mean-variance approach to optimize a portfolio by maximizing expected return for a given level of risk. The efficient frontier shows the minimum risk portfolio for each level of expected return.

1. Panos Parpas Single Period Markowitz Model 381 Computational Finance Imperial College London

2. Topics Covered Notation and Terminology Random variables, mean, (co)variance, correlation Asset return, portfolio return and risk Portfolio Optimisation Optimal asset allocation, risk management Short sale, diversification Mean-Variance model, efficient frontier

3. Terminology Random Variable y is a random variable, takes finite number of values, y j for j=1,2,…,m. a probability (associated with each event) represents the relative chance of an occurrence of y j. such that Expected Value (Mean value or mean) average value obtained by regarding probabilities as frequencies Variance measure of possible deviation from the mean relative frequency finite number of possibilities Expected value of squared variable how much y tends to vary from its mean

4. Terminology Covariance of two random variables Correlation between two random variables sign defines direction of the relationship

5. Asset Returns Asset: investment instrument that can be bought and sold uncertain asset prices – the return is random uncertainty described in probabilistic terms Asset return : you purchase an asset today and sell it next year Total return on this investment is defined as Rate of return: Rate of return acts much like an interest rate

6. Returns

7. Portfolio Return Consider n risky assets to form a portfolio and invest among n assets Select an amount invested in the i th asset The amounts invested can be expressed as fractions of the total investment is fraction or weight of asset i in portfolio Let the total return of asset i be . The amount of money gained at the end of the period on asset i is The total return of the portfolio The rate of return of portfolio

8. Portfolio Return Both the total return and the rate of return of the portfolio of assets are equal to the weighted sum of the corresponding individual asset returns, with the weight of an asset being its relative weight in the portfolio. Expected return of the portfolio is the weighted sum of the individual expected rates of return

9. Example Consider a portfolio of a risk-free asset and two risky stocks and three equally likely states Find the expected return of the portfolio formed by 3 assets Scenario tree with 3 states root represents today future uncertainty is discretised by 3 events (states)

10. Example: Expected Returns The expected return of risk free asset is For stock A and B, expected return of the portfolio The expected return of the portfolio For equally weighted portfolio

11. Risk Risk: a chance that investment’s actual return will be different than expected –includes losing some or all of original investment risk averse and risk-seeking (loving) Systematic risk: influences a large number of assets, such as political events – impossible to protect yourself against this risk Unsystematic risk: (specific risk) affects very small number of assets. For example, news that affects a specific stock such as a sudden strike Diversification is the only way to protect yourself Others: credit (default) risk, foreign exchange risk, interest rate risk, political risk, market risk

12. Diversification Risk management technique Mixes a wide variety of investments within a portfolio Objective is to minimize the impact that any one security will have on overall performance of the portfolio For the best diversification, portfolio should be spread among many different assets; cash, stocks, bonds securities should vary in risk and securities should vary by industry to minimize unsystematic risk to different companies

13. Portfolio Risk For n securities, portfolio risk – variance of the portfolio return Expected value of squared variable- how much rate of return of portfolio tends to vary from its mean

14. Example: Variance of Portfolio The variance and covariance are calculated as

15. Example: Variance of Portfolio The portfolio risk is For equally weighted portfolio

16. Short Sale It is possible to sell an asset that you do not own through the short selling (shorting) the asset. In order to implement short selling borrow the asset from the owner such as brokerage firm sell the borrowed asset to someone else, receiving an amount x 0 at a later date purchase the asset for x 1 , and return the asset to lender The short selling is profitable if the asset price declines Risky since the potential for loss is unlimited –if asset value increases Although prohibited within certain financial institutions, not universally forbidden

17. Example:Short-sale Assume that company X has a poor outlook next month. The stock is now trading at £65, but you see it trading much lower than this price in the future . You decide to take risk and trade on this stock. Two things can happen; stock price can go up or down Make Money Lose Money RISKY: no guarantee that the price of a short stock will drop

18. Optimal Portfolio Portfolio theory effects of investor decisions on security prices relationship that should exist between the returns and risk. possible to have different portfolios varying levels of risk &return decide how much risk you can handle and allocate (or diversify) portfolio according to this decision Harry Markowitz: 1990 Nobel Prize winner in Economic Sciences published in 1952 Journal of Finance titled “Portfolio Selection ” formalized an integrated theory of diversification, portfolio risks, efficient & inefficient portfolios Maximize (expected return of portfolio) (weighted sum of individual expected rates of return) Minimize (portfolio risk) (expected value of squared variable- how much rate of return of portfolio tends to vary from its mean)

19. The Markowitz Model Construct portfolio under uncertainty Suppose that there are n assets with random rates of return with mean values The portfolio consists of n assets are the portfolio weights such that

20. Maximum Expected Wealth Problem No risk –maximize expected portfolio return –risk-neutral approach

21. Minimum Variance Portfolio Take risk into account

22. Mean-Variance Optimisation Problem trade-off between expected rate of return &variance of rate of return in a portfolio naturally leads to diversification by taking risk attitudes in to account

23. Efficient Frontier Varying the desired level of return and repeatedly solving QP problems identifies the minimum variance portfolio for each value of : efficient portfolios