# Chapter 12 Choosing an Investment Portfolio. Objectives To understand the process of personal portfolio selection in theory and in practice To build.

Post on 18-Jan-2016

214 views

TRANSCRIPT

Chapter 12Choosing 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

ContentsThe Process of Personal Portfolio Selection

The Trade-Off between Expected Return and Risk

Efficient Diversification with Many Risky Assets

Portfolio SelectionA 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 CycleIn portfolio selection the best strategy depends on an individual s personal circumstances:

Family statusOccupationIncomeWealth

Time HorizonsPlanning Horizon: The total length of time for which one plansDecision 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 ToleranceThe 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 AssetA 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 investors decision horizon

Combining a Riskless Asset and a Single Risky Asset

Riskless asset:

Risky asset:

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

Combining the Riskless Asset and a Single Risky AssetThe 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 AssetWe 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)1/2sp = |W1| * s1

- Combining the Riskless Asset and a Single Risky AssetIn summarysp = |W1| * s1, And:mp = W1*m1 + (1- W1)*rf , So:If W1
To obtain a 20% ReturnYou settle on a 20% return, and decide not to pursue on the computational issueRecall: 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% ReturnAssume that you manage a $50,000,000 portfolioA W1 of 1.5 or 150% means you invest (go long) $75,000,000, and borrow (short) $25,000,000 to finance the difference

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

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

Equations for Two SharesThe 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 trueThe expected return on the portfolio is the sum of its weighted expectations

Equations for Two SharesIdeally, 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 StockThe next slide shows statistics of two common stock with these statistics:mean return 1 = 0.15mean return 2 = 0.10standard deviation 1 = 0.20standard deviation 2 = 0.25correlation of returns = 0.90initial price 1 = $57.25Initial price 2 = $72.625

Formulae for Minimum Variance Portfolio

Formulae for Tangent Portfolio

Example: Whats the Best Return given a 10% SD?

Achieving the Target Expected Return (2): WeightsAssume 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

*********************

Recommended