Giddy/NYU Foundations of Finance Course

Professor Ian Giddy's Foundations of Finance

Portfolio Optimization: Highlights

The Portfolio Risk-Return Trade-Off

Investors face a trade-off between risk and expected return. Shifting funds from the risky portfolio to the risk-free asset is the simplest way to reduce risk. Another method involves diversification of the risky portfolio. This is the focus here.
A risky investment portfolio (referred to here as the risky asset) can be characterized by its reward-to-variability ratio. This ratio is the slope of the capital allocation line (CAL), the line that goes from the risk-free asset through the risky asset. All combinations of the risky and risk-free assets lie on this line. Investors would prefer a steeper sloping CAL, because that means higher expected returns for any level of risk. If the borrowing rate is greater than the lending rate, the CAL will be "kinked" at the point corresponding to investment of 100% of the complete portfolio in the risky asset.
An investor's preferred choice among the portfolios on the capital allocation line will depend on risk aversion. Risk-averse investors will weight their complete portfolios more heavily toward Treasury bills. Risk-tolerant investors will hold higher proportions of their complete portfolios in the risky asset.
The capital market line is the capital allocation line that results from using a passive investment strategy that treats a market index portfolio such as the Standard & Poor's 500 as the risky asset. Passive strategies are low-cost ways of obtaining well-diversified portfolios with performance close to that of the market as a whole.

Optimal Portfolio Diversification

The variance of a portfolio is a sum of the contributions of the component-security variances plus terms involving the correlation among assets.
Even if correlations are positive, the portfolio standard deviation will be less than the weighted average of the component standard deviations, as long as the assets are not perfectly positively correlated. Thus, portfolio diversification is of value as long as assets are less than perfectly correlated.
The contribution of an asset to portfolio variance depends on its correlation with the other assets in the portfolio, as well as on its own variance. An asset that is perfectly negatively correlated with a portfolio can be used to reduce the portfolio variance to zero. Thus, it can serve as a perfect hedge.
The efficient frontier of risky assets is the graphical representation of the set of portfolios that maximizes portfolio expected return for a given level of portfolio standard deviation. Rational investors will choose a portfolio on the efficient frontier.
A portfolio manager identifies the efficient frontier by first establishing estimates for the expected returns and standard deviations, and the correlations among them. The input data are then fed into an optimization program that produces the investment proportions, expected returns, and standard deviations of the portfolios on the efficient frontier.
In general, portfolio managers will identify different efficient portfolios because of differences in the methods and quality of security analysis. Managers compete on the quality of their security analysis relative to their management fees.
If a risk-free asset is available and input data are identical, all investors will choose the same portfolio on the efficient frontier, the one that is tangent to the CAL. All investors with identical input data will hold the identical risky portfolio, differing only in how much each allocates to this optimal portfolio and to the risk-free asset. This result is characterized as the separation principle of portfolio selection.
The single-index representation of a single-factor security market expresses the excess rate of return on a security as a function of the market excess return: Ri = (xi + PiRm + ei) This equation also can be interpreted as a regression of the security excess return on the market index excess return. The regression line has intercept cLi and slope Pi, and is called the security characteristic line.
In a single-index model, the variance of the rate of return on a security or portfolio can be decomposed into systematic and firm-specific risk. The systematic component of variance equals Beta times the variance of the market excess return. The firm-specific component is the variance of the residual term in the index model equation.
The beta of a portfolio is the weighted average of the betas of the component securities. A security with negative beta reduces the portfolio beta, thereby reducing exposure to market risk.
The unsystematic risk of a portfolio approaches zero as the portfolio becomes more diversified.

Optimizing Fixed-Income Portfolios

Duration measures the percentage price change of a bond for a 1% change in yield. Convexity measures the degree to which duration changes as the yield to maturity changes.
Duration is additive, so the duration of a portfolio of bonds is the weighted sum of the duration of the individual bonds. Because duration and convexity measure price risk, they can be helpful in bond portfolio management.
Immunization strategies are characteristic of passive fixed-income portfolio management. Such strategies attempt to render the individual or firm immune from movements in interest rates. This may take the form of immunizing net worth or, instead, immunizing the future accumulated value of a fixed-income portfolio. Immunization of a fully funded plan is accomplished by matching the durations of assets and liabilities.
When the weighted duration of assets does not equal that of the liabilities, derivatives (such as futures or swaps) can be employed to match asset with liability duration.
Duration changes as time passes and as yields change. Therefore to maintain an immunized position, the portfolio must be periodically rebalanced.
A more precise form of immunization is dedication or cash flow matching. If a portfolio is perfectly matched in cash flow with projected liabilities, rebalancing will be unnecessary.

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