Portfolio Theory

PORTFOLIO THEORY It proposes how rational investors will use diversification to optimize their portfolios, and how a risky asset should be priced. MPT models an asset's return as a random variable, and models a portfolio as a weighted combination of assets so that the return of a portfolio is the weighted combination of the assets' returns. Moreover, a portfolio's return is a random variable, and consequently has an expected value and a variance. Risk, in this model, is the standard deviation of return.

Assumptions Investors are assumed to seek out maximizing their returns while minimizing the risk involved in generating those returns. The model recognizes that different investors have different risk tolerance and return requirements. As such, the model defines investor utility functions that can be used to graph indifference curves. These indifference curves plot the various

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• The model then defines a universe of potential risky securities for investment. • Each risky security in the universe requires an estimate of:− expected returns,− standard deviation,− correlation between each pair of risky securities. • When these sets of securities are combined into a portfolio in such a way as to minimize risk for any desired level of return (or

The Markowitz Model • The Markowitz model describes a set of rigorous statistical procedures used to select the optimal portfolio for wealth maximizing/risk-averse investors. • The model is described under the framework of a risk-return tradeoff graph. • Investors are assumed to seek out maximizing their returns while

Contd.. • The model recognizes that different investors have different risk tolerance and return requirements. As such, the model defines investor utility functions that can be used to graph indifference curves. These indifference curves plot the various risk levels that give an investor the same level of satisfaction (utility) on a riskreturn trade-off graph.

Risk Free Assets • The introduction of a risk free security to the Markowitz model changes the efficient frontier from a curved line to a straight line called the Capital Market Line (CML). This CML represents the allocation of capital between risk free securities and risky securities for all investors combined. • The optimal portfolio for an investor is the point where the new CML is tangent to the old efficient frontier when only risky securities were graphed. This

The risk-free asset • The risk-free asset is the (hypothetical) asset which pays a risk-free rate. It is usually provided by an investment in short-dated Government securities. The risk-free asset has zero variance in returns (hence is risk-free); it is also uncorrelated with any other asset (by definition: since its variance is zero). As a result, when it is combined with any other asset, or portfolio of

The market portfolio

• The efficient frontier is a collection of portfolios, each one optimal for a given amount of risk. A quantity known as the Sharpe ratio represents a measure of the amount of additional return (above the risk-free rate) a portfolio provides compared to the risk it carries. The portfolio on the efficient frontier with the highest Sharpe Ratio is known as the market portfolio, or sometimes the superefficient portfolio; it is the tangencyportfolio in the above diagram. This portfolio has the property that any combination of it and the risk-free asset

Capital market line • When the market portfolio is combined with the risk-free asset, the result is the Capital Market Line. All points along the CML have superior risk-return profiles to any portfolio on the efficient frontier. Just the special case of the market portfolio with zero cash weighting is on the efficient frontier. Additions of cash or leverage with the risk-free

Systematic Risk vs. NonSystematic Risk

• Non-systematic risk is the risk that disappears in the portfolio construction process when you diversify among assets that are not correlated. • Systematic risk is the risk that remains after constructing the market portfolio, which presumably contains all risky assets. It is the risks that cannot be diversified away. • Total Risk = Systematic Risk + Non-

Systematic risk and specific risk • Specific risk is the risk associated with individual assets - within a portfolio these risks can be reduced through diversification (specific risks "cancel out"). Specific risk is also called diversifiable, unique, unsystematic, or idiosyncratic risk.

Contd.. • Systematic risk (a.k.a. portfolio risk or market risk) refers to the risk common to all securities - except for selling short as noted below, systematic risk cannot be diversified away (within one market). Within the market portfolio, asset specific risk will be diversified away to the extent possible. Systematic risk is therefore equated with the risk (standard

Risk and return The model assumes that investors are risk averse, meaning that given two assets that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher returns must accept more risk. A rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-return profile – i.e., if for that level of risk an alternative portfolio

The theory uses a historical parameter, volatility, as a proxy for risk, while return is an expectation on the future. The basic assumption here is that the best forecast for tomorrow is the price of today. “The investor is indifferent to other characteristics" seems not to be true.

Diversification • An investor can reduce portfolio risk simply by holding combinations of instruments which are not perfectly positively correlated (correlation coefficient -1<(r)<1)). In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification will allow for the same portfolio return with reduced risk.

If all the assets of a portfolio have a correlation of +1, i.e., perfect positive correlation, the portfolio volatility (standard deviation) will be equal to the weighted sum of the individual asset volatilities. Hence the portfolio variance will be equal to the square of the total weighted sum of the individual asset volatilities.

If all the assets have a correlation of 0, i.e., perfectly uncorrelated, the portfolio variance is the sum of the individual asset weights squared times the individual asset variance (and volatility is the square root of this sum).

If correlation coefficient is less than zero (r=0), i.e., the assets are inversely correlated, the portfolio variance and hence volatility will be less than if the correlation coefficient is 0.

First Principles n Invest in projects that yield a return greater than the minimum acceptable hurdle rate. • The hurdle rate should be higher for riskier projects and reflect the financing mix used - owners’ funds (equity) or borrowed money (debt) • Returns on projects should be measured based on cash flows generated and the timing of these cash flows; they should also consider both positive and negative side effects of these projects. n Choose a financing mix that minimizes the hurdle rate and matches the assets being financed. n If there are not enough investments that earn the hurdle rate, return the cash to stockholders. • The form of returns - dividends and stock buybacks - will depend upon the stockholders’ characteristics. Objective: Maximize the Value of the Firm