Tools Portfolio 4
Econ 333 Spring 08
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Portfolio Tools 4
Our goals: Understand the importance of mean-standard deviation diagram and know how to locate within the diagram the efficient frontier of risky assets, the capital market line, the minimum variance portfolio, and the tangency portfolio. Compute and use both the tangency portfolio and the efficient frontier of risky assets. Understand the linkage between mean-variance efficiency and risk-expected return equations. Describe how to compute the beat of a portfolio given the betas of individual assets in the portfolio and their respective weights. Understand what the market portfolio is, what assumptions are needed for the market portfolio to be the tangency portfolio, i.e. for the Capital Asset pricing Model (CAPM) to hold.
Econ 333 Spring 08
2
Portfolio Tools 4
The issue is that of analyzing the portfolio selection of an investor seeking to maximize the expected return given the variance of its future returns. As tools for illustrating how to achieve higher average returns with lower risk, mean-variance analysis and the CAPM are routinely applied by brokers, pension fund managers, and consultants when formulating investment strategies and giving financial advices. A firm grasp of mean-variance analysis and the CAPM also is becoming increasingly important for corporate managers. The need to understand the determinants of share value, and what actions to take to increase this value in response to the pressures of stockholders and directors is higher than ever before. They can apply these tools to hedge their risks optimally and diversify their portfolio. One of the lessons of the CAPM is that while diversifying investments can reduce the variance of a firm's stock price, it does not reduce the firm's cost of capital (weighted average of the expected rates of return required by the financial markets for a firm's debt and equity financing). As a result, diversification can create value only if it increases the expected returns of real assets. Corporate also use these tools to evaluate their capital expenditure decisions.
Econ 333 Spring 08 3
Portfolio Tools 4
The assumptions of mean-variance analysis: When making investment decisions today, investors care only about the means and the variances of the returns of their portfolios over a particular period: They are risk-averse and have preference for higher means and lower variances. Higher mean return because on average they'll be wealthier. Lower variance because it implies less dispersion in the possible wealth outcomes. Notice that when using the variance as a measure of risk / dispersion it is assumed that returns are normally distributed. Financial markets are frictionless: This assumption simplifies the computation of the feasible set. It implies that all investments are tradable at any price and in any quantity, both positive or negative , there is no transaction costs, regulations, or taxes. Although these assumptions may not hold in some cases, the general results are perceived to remain: portfolio dominates individual assets; covariances are more important than variances for risk-return equations; and optimal portfolios (in the meanvariance sense) can generally be found if one knows the inputs.
Econ 333 Spring 08 4
Portfolio Tools 4
Given the previous assumptions, an investor would always like to be on the efficient frontier. This frontier contains many portfolios and which of these investors select depends on their personal trade-off between mean and variance. Because investors who treat variance as the sole measure of risk want to select meanvariance efficient portfolio, it is important to know how to construct them. Result (two-fund separation): All portfolios on the mean-average efficient frontier can be formed as a weighted average of any two portfolios on the efficient frontier. Illustration and example: (In class)
Econ 333 Spring 08
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Portfolio Tools 4
So far (portfolio tools 4) we have only consider optimal portfolios selection when there is no risk-free asset. Accounting for a risk-free asset sharply simplifies the analysis. Because of the two-fund separation principle, when there is a risk-free asset only "one key" portfolio need to be found. If there is a risk-free asset, it is necessarily the minimum variance one and thus must be on the efficient frontier. As a result, we need to be concerned with only one efficient risky portfolio (once combined with the efficient risky portfolio, the risk-free generate asset the entire boundary of the feasible set). This optimal risky portfolio will be called the tangency portfolio: it is the unique optimal portfolio that contains no investment in the risk-free asset. The line from rf and tangent to the efficiency frontier is called the capital market line CML. Result: Under the assumptions of mean-variance analysis, and assuming the existence of a risk-free asset, all investors will select portfolios on the CML.
Econ 333 Spring 08
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Portfolio Tools 4
The result above suggests that the CML is the key to optimal investment for every investor interested in maximizing expected return for a given amount of variance risk. Investors who are extremely risk-averse will select a portfolio close to the risk-free asset, achieving a low expected return but also a low variance for their future wealth. Investors who are only slightly risk averse will select a portfolio high up on the CML, possibly above T if they choose to sell short the risk-free asset. They would achieve a higher expected return than more risk-averse investors, but the larger variance also makes the possibility of realizing large losses quite high.
CML line equation: Mean return CML T
R p = rf +
RT - r f
T
p
rf
Trade-off between risk and return: larger implies the financial markets offer greater expected return improvements for given increases in risk.
Standard Econ 333 Spring 08 deviation
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Portfolio Tools 4
General lesson from the mean-standard deviation diagram: there is no necessary relation between mean and standard deviation (portfolios with the same standard deviation can have different means) Implication: it pays to identify efficient portfolios which maximize mean return for a given standard deviation. The tangency portfolio serves double duty: useful for identifying proper investment weights, but also useful for addressing what determines mean returns. But, how to identify the tangency portfolio? Result: The ratio of the risk premium of every stock and the portfolio to its covariance with the tangency portfolio is constant. In clear, is constant. ri - r f Illustration: (In class)
~ cov( ~i , R T ) r
The result holds for the tangent portfolio as well (of course!)
Econ 333 Spring 08
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Portfolio Tools 4
Finding the efficient frontier of risky assets: Because of two-fund separation, the identification of any two portfolios on the boundary is enough to construct the entire set of risky portfolios that minimize variance for a given mean return. One can use the minimum variance portfolio of the risky assets as of the two portfolios since computing its weights is so easy. For the other portfolio, it is also possible to draw a tangent line from every point on the vertical axis of the mean-standard deviation diagram to the hyperbolic boundary (except for minimum variance point V). (i) Select any return that is less than the expected return of the minimum-variance portfolio. (ii) Compute the hypothetical tangency portfolio by pretending that the return in step i is the risk-free asset, even if a risk-free asset does not exist. (iii) Take the weighted averages of the minimum variance portfolio and the hypothetical tangency portfolio found in step ii to generate the entire set of mean-variance efficient portfolios. The weights on the minimum variance portfolio must be less than 1 to be the top half of the hyperbolic boundary. Illustration: (in class)
Econ 333 Spring 08
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Portfolio Tools 4
The relation between risk and expected return: A secondary benefit of identifying a mean-variance efficient portfolio is that it generates an equation that relates the risk of an asset to its expected return. If historical data provide unreliable estimates of the true expected rates of return of the stocks of individual corporations, how to obtain such estimates? Solution: When a risk-free asset exists, the relation between the relevant risk of an investment and its expected return is given by:
~ cov(~, RT ) ri ri - rf = ~ ( RT - rf ) var(RT )
Beta Illustration: (In class)
Econ 333 Spring 08
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Portfolio Tools 4
Betas In a regression, give the slope of the coefficient
ri - rf = ( RT - rf )
Betas and covariances are essentially the same measure of marginal variance Securities market line versus mean-standard deviation diagram
The securities market line is the graphical representation of mean versus beta. The tangency portfolio has a beta of 1 The risk free asset has a beta of 0 Illustration: In class
Econ 333 Spring 08
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