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

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  • 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

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