Efficient Portfolio and Introduction to Capital Market Line Benninga Chapter 9

Efficient Portfolio and Introduction to Capital Market Line Benninga Chapter 9

Optimal Investment with Risky Assets There are N risky assets, named 1, 2,, N, but no risk-free asset. With fixed total dollar amount of investment, you choose the proportion of investment in each asset, i.e., portfolio choice. Since you are risk averse, you want to maximize the return of your investment, given a certain level of risk. Alternatively, you want to minimize the risk of your investment for a given level of return. 2

A review

Optimization The optimal investment can be summarized as: choosing portfolio x, which solves the following minimisation problem Min{Var(r x )=x T Sx} st. E(r x ) = x T E(r) = r g, å = N i 1 xi = 1 r is the return vector of the N assets, S is their covariance matrix, r g (or r target ) is the expected return (target). Note, x i is the proportion of investment in asset i, it can be negative in this topic. 4

Minimum-Variance Portfolios When E(r), S and r target are known, Excel Solver can solve the optimal portfolio. The next example provides a case of 4 risk assets and it solves three optimal portfolios for r target = 10%, 12% and 7%. Excel Solver can be found in Data/Analysis (the very right end of your menu ribbon) 5

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Envelope and Global Minimum-Variance Portfolio (GMV) Portfolios that have minimum variance for a given return are called envelope portfolios Consider a new problem: you want the risk of your investment as low as possible, don t mind the return of your investment. This means your are looking for a global minimum variance portfolio (GMV) Mathematically, this can be solved by: min Var(r x )=x T Sx st. å = N i 1 xi = 1 Note we here drop the constraint of E(r x ) = r g in the previous minimization problem (we care about risk only). 8

Feasible Portfolios 11% 10% Infeasible portfolio Portfolio mean return 9% 8% 7% GMV Efficient and envelope Feasible, not efficient 6% 5% Envelope, not efficient 4% 10% 20% 30% 40% 50% 60% 70% 80% 90% Portfolio standard deviation FM3: Chapter 9 Efficient portfolio theorems 9

GMV and Efficient Frontier The global minimum variance portfolio (GMV) provides lowest risk for all feasible investment choices. GMV separate the frontier to two parts: 1. The upper segment: efficient frontier and portfolios on this segment are efficient portfolios. An efficient portfolio maximizes return for a given variance level. 2. The Lower part: of course consists inefficient portfolios 2. Similarly, we can use Solver to find the GMV portfolio (See excel example) 10

Envelope portfolios We can generate minimum-variance portfolios or envelope portfolios by changing the required (expected) rate of return r target to obtain many portfolios. A curve linking all these portfolios in a mean-standard deviation space is called minimum-variance frontier or the envelope of the feasible investment set All envelope portfolios can be found by repeating the previous exercise using Solver for different r target 11

Find Envelope Portfolios: alternative methods The disadvantage of previous method: Have to use Solver many times to find the envelope/efficient frontier. The alternative is using concept/theorem about Black (1972) 2-fund proposition: Use Solver directly or find tangency portfolio Data Table 12

Black proposition: The convex combination of any two envelope portfolios is also an envelope portfolio. Thus: if x and y are envelope portfolios, so is lx ( 1 l) ì lx1+ - y1 ü ï ï + ( 1- l) y =í L ý ïlxn + ( 1-l) y ï î Nþ FM3: Chapter 9 Efficient portfolio theorems 13

Use proposition one To use proposition one and data table to generate the envelope, we need at least 2 efficient portfolios. How to find? (2 ways) Solver directly apply to mean and variance function Tangency portfolio Use Solver

Second way to find efficient portfolios: Tangency portfolio r c s x E(r x ) - c Tencency porfolios for a given c s Except for the GMV, any envelope portfolio or minimum variance portfolio is a tangency portfolio of a constant c. Given c, at least one tangency portfolio exists. 16

If C=r f, the tangent portfolio is on the CML line (more about this later ) Efficient Frontier with CML Capital market line, Portfolio mean return Risk-free rate, r f Market Portfolio standard deviation

Find the Tangency portfolio using constant and Solver Mathematically, we can obtain tangency portfolio x for a given c by solving the following problem Max (or Min) [E(r x ) c]/s x st. å = N i 1 xi = 1 The maximization à a portfolio on the upper part of the envelope (efficient frontier) The minimization à a portfolio on the lower segment (not efficient). 18

Capital Market Line: Idea about investment with a risk-free Asset Let the risk-free rate is r f, so you will not consider any investment with a return lower than r f as long as you are risk averse. For any target rate of expected return greater than r f, we can still determine the optimal investment proportions to N risky assets + 1 risk free asset by minimizing the portfolio s risk. 20

Efficient Portfolios with a Risk-Free Asset and CML Similarly, we can vary the target return to obtain all efficient portfolios and the efficient frontier. The efficient frontier (with the risk-free asset) is a straight line in the standard deviations-mean space. This straight line passes through the risk-free asset and the tangency portfolio of efficient frontier of risky assets. If the portfolio has N assets that include all risky assets in the market, the portfolio is the market portfolio (MP) The straight line is called Capital Market Line (CML) 21

Market portfolio (MP) and Capital Market Line (CML) CML is a straight line from the risk-free rate through the market portfolio in an s-r plane. r Capital market line Market portforlio rf Envelope s 22

Market portfolio Most important implication of CAPM All investors hold the same optimal portfolio of risky assets The optimal portfolio is at the highest point of tangency between r f and the efficient frontier The portfolio of all risky assets is the optimal risky portfolio Called market portfolio

Characteristics of Market portfolio All risky assets must be in portfolio, so it is completely diversified Includes only systematic risk All securities included in proportion to their market value Unobservable but proxied by some market index, e.g., all ordinaries, S&P500

CAPM s prediction CAPM prediction: Efficient portfolio = invest a proportion in the risk-free asset and (1 a) in the MP The expected return and risk: E(r p ) = ar f +(1 a)e(r M ) s p =(1 a)s M Note the particularly simple form of the standard deviation here. What are the return and risk on CML when a=0, a=1, a<0, 0<a<1? 25

How to Find the CML? Two steps of finding the CML 1. Following tangency approach, obtain the MP by setting c = r f. 2. Use Data Table: Change the weights of risk-free asset and the market portfolio to create all expected returns and standard deviations on the CML. 26

Find a Portfolio with Desired Return or Risk Since all efficient portfolios are on the CML, an investor with a particular target of mean return or risk in mind can choose an investment portfolio on the CML. But with Excel, we can find the portfolio even without drawing the CML. The tool is Goal Seek or Solver, See example next 28