Unit VII - Portfolio Theory

Portfolio Theories

Markowitz risk-return analysis

Sharpe Portfolio Optimization

Other theories

Markowitz Portfolio Selection Model

Till 18th century – Bernoulli and Cramer – Mean, Rule of thumb, Intuition

Dr Harry M Markowitz – Mean-Variance Model (Rand Corporation, 1952)

Based on concept of Efficient Market and Portfolios.

Efficient market & portfolios - implies

All investors have common (homogeneous) expectations regarding the expected returns, variances and correlation of returns among all securities;All investors have the same information about securities;There are no restrictions on investments;There are no taxes;There are no transaction costs; and No single investor can affect the market price significantly.

Minimize Portfolio Risk / Maximize Portfolio Return

Risk & Return for a 2- asset portfolio

Efficient Frontier

Efficient Frontier with Investor’s Indifference Curve

Corner Portfolios

In which a new security is added to a previous efficient portfolio or in which a security is dropped from a previously efficient portfolio.

Investor’s Risk Frontier

Modifications to the Efficient Frontier

A. Short Selling – 2 effects – a security sold short produces a positive return when a security has a large decrease in price and a negative return when its price increases.If it pays to s-s, the efficient frontier shifts up and to the left – as it allows disinvest in poor investments (hence gain if they do poorly)Extension of the efficient frontier to the right – arises as s-s increases the risk and return on the portfolio, as s-s can involve huge losses.

Efficient Frontier with and without Short Sales

B. Leveraged Portfolios – Markowitz did not allow for borrowing and lending opportunities, though it recognizes the existence of both systematic and unsystematic risk.

(i) Risk free asset – investment in risk free asset is referred to risk free lending. Standard deviation of risk free asset = 0. Hence, covariance = 0.

(ii) Investing in both risk free and risky asset –

Reward-to-risk-ratio = (Return-risk free return) / standard deviation

Reward-to-Risk Ratio

By combining the securities in portfolio T with risk free securities at rf, the investor would actually reduce risk more than the reduction in return. The reduction of risk makes the combination of securities more attractive at point P than an all equity portfolio at U.Rf – T = lending portfolioT – Z = borrowing portfolio

Real life lending and borrowing curves

Critique of the ModelWhy assume all rational investors are risk averse?Why is variance most appropriate measure of risk? – for a long term investor price volatility is not a real risk.Investment managers are vary of mathematics.Managers more comfortable with individual returns rather than covariance.The entire portfolio needs to be revaluated even if 1 security changes.

Sharpe’s Single Index Model

The Security Characteristic Line - Dimensions

SCL – Different Dimensions

SCL - Variances

Portfolio Characteristic Line

PCL – Portfolio Beta

Constructing the optimal portfolio

To determine which securities are to be included in the optimum portfolio, the following steps are necessary:

1. Calculate the excess return to Beta ratio for each security under review and rank them.

2. The optimum portfolio consists of investing in all securities for which (Ri-T) / Beta(i,m) is > a particular cut-off point.

Other Portfolio theories

Capital Market theory & CAPM

Security Market Line

Arbitrage Pricing Theory