Choosing an Investment Portfolio Chapter 12 Choosing an Investment Portfolio

Objectives To understand the process of personal portfolio selection in theory and in practice To build a quantitative model of the trade-off between risk and reward

Contents The Process of Personal Portfolio Selection The Trade-Off between Expected Return and Risk Efficient Diversification with Many Risky Assets

Portfolio Selection A process of trading off risk and expected return to find the best portfolio of assets and liabilities

Portfolio Selection The Life Cycle Time Horizons Risk Tolerance

The Life Cycle In portfolio selection the best strategy depends on an individual ‘s personal circumstances: Family status Occupation Income Wealth

Time Horizons Planning Horizon: The total length of time for which one plans Decision Horizon: The length of time between decisions to revise the portfolio Trading Horizon: The minimum time interval over which investors can revise their portfolios.

Risk Tolerance The characteristic of a person who is more willing than the average person to take on additional risk to achieve a higher expected return

The Trade-Off between Expected Return and Risk Objective: To find the portfolio that offers investors the highest expected rate of return for any degree of risk they are willing to tolerate

Portfolio Optimization Find the optimal combination of risky assets Mix this optimal risky-asset portfolio with the riskless asset.

Riskless Asset A security that offers a perfectly predictable rate of return in terms of the unit of account selected for the analysis and the length of the investor’s decision horizon

Combining a Riskless Asset and a Single Risky Asset

Combining the Riskless Asset and a Single Risky Asset The expected return of the portfolio is the weighted average of the component returns mp = W1*m1 + W2*m2 mp = W1*m1 + (1- W1)*m2

Combining the Riskless Asset and a Single Risky Asset The volatility of the portfolio is not quite as simple: sp = ((W1* s1)2 + 2 W1* s1* W2* s2 + (W2* s2)2)1/2

Combining the Riskless Asset and a Single Risky Asset We know something special about the portfolio, namely that security 2 is riskless, so s2 = 0, and sp becomes: sp = ((W1* s1)2 + 2W1* s1* W2* 0 + (W2* 0)2)1/2 sp = |W1| * s1

Combining the Riskless Asset and a Single Risky Asset In summary sp = |W1| * s1, And: mp = W1*m1 + (1- W1)*rf , So: If W1<0, mp = [(rf -m1)/ s1]*sp + rf , Else mp = [(m1-rf )/ s1]*sp + rf

100% Risky 100% Risk-less Long both risky and risk-free Long risky and short risk-free 100% Risky Long both risky and risk-free 100% Risk-less

To obtain a 20% Return You settle on a 20% return, and decide not to pursue on the computational issue Recall: mp = W1*m1 + (1- W1)*rf Your portfolio: s = 20%, m = 15%, rf = 5% So: W1 = (mp - rf)/(m1 - rf) = (0.20 - 0.05)/(0.15 - 0.05) = 150%

To obtain a 20% Return Assume that you manage a $50,000,000 portfolio A W1 of 1.5 or 150% means you invest (go long) $75,000,000, and borrow (short) $25,000,000 to finance the difference

sp = |W1| * s1 = 1.5 * 0.20 = 0.30 To obtain a 20% Return How risky is this strategy? sp = |W1| * s1 = 1.5 * 0.20 = 0.30 The portfolio has a volatility of 30%

Portfolio of Two Risky Assets Recall from statistics, that two random variables, such as two security returns, may be combined to form a new random variable A reasonable assumption for returns on different securities is the linear model:

Equations for Two Shares The sum of the weights w1 and w2 being 1 is not necessary for the validity of the following equations, for portfolios it happens to be true The expected return on the portfolio is the sum of its weighted expectations

Equations for Two Shares Ideally, we would like to have a similar result for risk Later we discover a measure of risk with this property, but for standard deviation:

Correlated Common Stock The next slide shows statistics of two common stock with these statistics: mean return 1 = 0.15 mean return 2 = 0.10 standard deviation 1 = 0.20 standard deviation 2 = 0.25 correlation of returns = 0.90 initial price 1 = $57.25 Initial price 2 = $72.625

Formulae for Minimum Variance Portfolio

Formulae for Tangent Portfolio

Example: What’s the Best Return given a 10% SD?

Achieving the Target Expected Return (2): Weights Assume that the investment criterion is to generate a 30% return This is the weight of the risky portfolio on the CML

Achieving the Target Expected Return (2):Volatility Now determine the volatility associated with this portfolio This is the volatility of the portfolio we seek