# Modern Portfolio Theory: Efficient and Optimal Portfolios

A portfolio consists of a number of different securities or other assets selected for investment gains. However, a portfolio also has investment risks. The primary objective of portfolio theory or management is to maximize gains while reducing diversifiable risk. Diversifiable risk is so named because the risk can be reduced by diversifying assets. **Systemic risk**, on the other hand, cannot be reduced through diversification, since it is a risk that affects the entire economy and most investments. So even the most optimized portfolio will still be subject to systemic risk.

**Traditional portfolio management** is a nonquantitative approach to balancing a portfolio with different assets, such as stocks and bonds, from different companies and different sectors as a way of reducing the overall risk of the portfolio. The main objective is to select assets that have little or negative correlation with each other, so that the overall diversifiable risk is reduced.

**Modern portfolio theory** (**MPT**) reduces portfolio risk by selecting and balancing assets based on statistical techniques that quantify the amount of diversification by calculating expected returns, standard deviations of individual securities to assess their risk, and by calculating the actual coefficients of correlation between assets, or by using a good proxy, such as the single-index model, allowing a better choice of assets that have negative or no correlation with other assets in the portfolio. Modern portfolio management differs from the traditional approach by the use of quantitative methods to reduce risk. The main objective of modern portfolio theory is to have an **efficient portfolio**, which is a portfolio that yields the highest return for a specific risk, or, stated in another way, the lowest risk for a given return. Profits can be maximized by selecting an efficient portfolio that is also an **optimal portfolio**, which is one that provides the most satisfaction — the greatest return — for an investor based on his tolerance for risk.

## Efficient Frontier

Because portfolios can consist of any number of assets with differing proportions of each asset, there is a wide range of risk-return ratios. If the universe of these risk-return possibilities—the **investment opportunity set**—were plotted as an area of a graph with the expected portfolio return on the vertical axis and portfolio risk on the horizontal axis, the entire area would consist of all feasible portfolios—those that are actually attainable. In this set of attainable portfolios, there would be some which have the greatest return for each risk level, or, for each risk level, there would be portfolios that have the greatest return. The **efficient frontier** consists of the set of all **efficient portfolios** that yield the highest return for each level of risk. The efficient frontier can be combined with an investor's utility function to find the investor's optimal portfolio, the portfolio with the greatest return for the risk that the investor is willing to accept.

On the efficient frontier, there is a portfolio with the minimum risk, as measured by the variance of its returns — hence, it is called the **minimum variance portfolio** — that also has a minimum return, and a **maximum return portfolio** with a concomitant maximum risk. Portfolios below the efficient frontier offer lower returns for the same risk, so a wise investor would not choose such portfolios.

## Utility Values and Risk Aversion

Most investors will assume a greater risk for a greater return. However, investors differ in the amount of risk they are willing to take for a given return. Investors who are **risk averse** require a greater return for a given amount of risk than a **risk lover**. A **risk-neutral** investor is only concerned with the magnitude of the return. However, most investors are risk averse to varying degrees.

Although investors differ in their risk tolerance, they should be consistent in their selection of any portfolio in terms of the risk-return trade-off. Because risk can be quantified as the sum of the variance of the returns over time, it is possible to assign a **utility score** (aka **utility value**, **utility function**) to any portfolio by subtracting its variance from its expected return to yield a number that would be commensurate with an investor's tolerance for risk, or a measure of their satisfaction with the investment. Because risk aversion is not an objectively measurably quantity, there is no unique equation that would yield such a quantity, but an equation can be selected, not for its absolute measure, but for its comparative measure of risk tolerance. One such equation is the following **utility formula**:

Utility Score = Expected Return – 0.005σ^{2} × Risk Aversion Coefficient

The risk aversion coefficient is a number proportionate to the amount of risk aversion of the investor and is usually set to integer values less than 6, and 0.005 is a normalizing factor to reduce the size of the variance, σ^{2}, which is the square of the standard deviation (σ), a measure of the volatility of the investment and therefore its risk. This equation is normalized so that the result is a yield percentage that can be compared to investment returns, which allows the utility score to be directly compared to other investment returns, such as the return of risk-free T-bills. For example, if a risk-free T-bill pays 4%, and XYZ stock has a return of 12% and a standard deviation of 25%, the utility score of XYZ stock is equal to:

Utility Score = 12 – 0.005 × 25^{2} × 2= 12 – 6.25 = 5.75%

In the above example, we let the risk aversion coefficient be equal to 2. If someone were more risk averse, we might use 3 instead of 2 to indicate the investor's greater aversion to risk. In this case, the above equation yields:

Utility Score = 12 – 0.005 × 25^{2} × 3 = 12 – 9.375 = 2.625%

Since 2.625% is less than the 4% yield of risk-free T-bills, this risk-averse investor will reject XYZ stock in favor of T-bills while the other investor will invest in XYZ stock since he assigns a utility score of 5.75% to the investment, which is higher than the T-bill yield.

Another way to measure the risk averseness of an investor is by comparing the desirability of a risky investment to a risk-free investment. The **certainty equivalent rate** is the rate of return of a risk-free investment that would be equally attractive as a risky investment. Since the utility score of a risk-free investment is simply its rate of return (in other words, the variance of a risk-free investment is considered zero, hence the 2^{nd} term of the utility score formula is zero), the certainty equivalent rate would equal the utility score of the risky investment. So for the 1^{st} investor above, a risk-free yield of 5.75% would be equally attractive as XYZ stock yielding a risky 12%, while the 2^{nd} investor would only consider XYZ stock if the risk-free rate were only 2.625%. In other words, each investor would be **indifferent** to either investment if the risk-free rate were equal to their certainty equivalent rate.

The set of all portfolios with the same utility score plots as a **risk-indifference curve**. An investor will accept any portfolio with a utility score on her risk-indifference curve as being equally acceptable.

However, there are many possible portfolios on many risk-indifference curves that do not yield the highest return for a given risk. All of these portfolios lie below the efficient frontier. The **optimal portfolio** is a portfolio on the efficient frontier that would yield the best combination of return and risk for a given investor, which would give that investor the most satisfaction.

In the graph below, risk-indifference curves are plotted along with the investment opportunity set of attainable portfolios. Data points outside of the investment opportunity set designate portfolios that are not attainable, while those portfolios that lie along the northwest boundary of the investment opportunity set is the efficient frontier. All portfolios that lie below the efficient frontier have a risk-return trade-off that is inferior to those that lie on the efficient frontier. If a utility curve intersects the efficient frontier at 2 points, there are a number of portfolios on the same curve that lie below the efficient frontier; hence they are not optimal. Remember that all points on a risk-indifference curve are equally attractive to the investor; therefore, if any points on the indifference curve lie below the efficient frontier, then no point on that curve can be an optimum portfolio for the investor. If a utility curve lies wholly above the efficient frontier, then there is no attainable portfolio on that utility curve.

However, there is a utility curve such that it intersects the efficient frontier at a single point—this is the **optimum portfolio**. The only attainable portfolio is on the efficient frontier, and thus, provides the greatest satisfaction to the investor. The optimum portfolio will yield the highest return for the amount of risk that the investor is willing to take.

## Portfolio Betas: A Measure of the Systematic Risk of Portfolios

By selecting the right assets in the right proportions, it may be possible to reduce diversifiable risk to near zero, but the portfolio would still have systematic risk, which also affects the general market. Portfolios, like stocks, have betas which measure the systematic risk of the portfolio compared to that of the market. The **portfolio beta** is equal to the sum of the beta of the weighted average of each security's value over the value of the portfolio.

Portfolio Beta | = | n ∑ k=1 | Dollar Amount of Asset k Dollar Amount of Portfolio | × | Beta for Asset k |

### Example—Calculating the Portfolio Beta

You can find the beta of most stocks at websites that give stock quotes. The financial data for Microsoft and Google is provided below. What is the beta of a portfolio consisting of 100 shares of each stock?

Financial Data (accessed 11/6/2008):

- Microsoft
- Share Price: $20.88
- Beta: 0.99

- Google
- Share Price: $331.22
- Beta: 1.55

- Total Portfolio Value = $20.88 × 100 + $331.22 × 100 = $2,088 + $33,122 = $35,210

**Portfolio Beta** = $2,088/$35,210 × 0.99 + $33,122/$35,210 × 1.55 = 0.06 + 1.46 = **1.52**

Hence, this portfolio would be 1.5 times as volatile as the S&P 500 index, which is considered to have a beta of 1. When the S&P rises by 1%, this portfolio will tend to increase by 1.5%; when the S&P is down 1%, this portfolio will tend to be down 1.5%. Remember that the beta is a statistical value, and therefore, the actual ratios will deviate frequently from these values.

The portfolio beta is interpreted in the same way that it is for stocks. A portfolio with a beta of 1 has the volatility of the stock market—the value of the portfolio moves 1%, up or down, for each 1% move in the stock market; a portfolio beta value of 0.5 would have half the volatility of the market and a beta of 2 would have twice the volatility.

Hence, the portfolio beta allows an investor to choose securities or other assets that would limit the investor's exposure to systematic risk to her risk tolerance.