Question

Based on Modern Portfolio Theory, explain what the variance and covariance terms are capturing. Which is a more relevant measure in a well-diversified portfolio? Briefly explain. If a portfolio has 5 individual stocks in it, how many total variance and covariance terms are there?

Answer 1

Modern portfolio theory (MPT)

is a theory on how risk-averse investors can construct portfolios to optimize or maximize expected return based on a given level of market risk, emphasizing that risk is an inherent part of higher reward. According to the theory, it's possible to construct an "efficient frontier" of optimal portfolios offering the maximum possible expected return for a given level of risk. This theory was pioneered by Harry Markowitz in his paper "Portfolio Selection," published in 1952 by the Journal of Finance. He was later awarded a Nobel prize for developing the MPT.

Modern portfolio theory argues that an investment's risk and return characteristics should not be viewed alone, but should be evaluated by how the investment affects the overall portfolio's risk and return.

MPT shows that an investor can construct a portfolio of multiple assets that will maximize returns for a given level of risk. Likewise, given a desired level of expected return, an investor can construct a portfolio with the lowest possible risk. Based on statistical measures such as variance and correlation, an individual investment's return is less important than how the investment behaves in the context of the entire portfolio.

MPT makes the assumption that investors are risk-averse, meaning they prefer a less risky portfolio to a riskier one for a given level of return. This implies that an investor will take on more risk only if he or she is expecting more reward.

The expected return of the portfolio is calculated as a weighted sum of the individual assets' returns. If a portfolio contained four equally-weighted assets with expected returns of 4, 6, 10, and 14%, the portfolio's expected return would be:

(4% x 25%) + (6% x 25%) + (10% x 25%) + (14% x 25%) = 8.5%

The portfolio's risk is a complicated function of the variances of each asset and the correlations of each pair of assets. To calculate the risk of a four-asset portfolio, an investor needs each of the four assets' variances and six correlation values, since there are six possible two-asset combinations with four assets. Because of the asset correlations, the total portfolio risk, or standard deviation, is lower than what would be calculated by a weighted sum.

EFFICIENT FRONTEIR

Every possible combination of assets that exists can be plotted on a graph, with the portfolio's risk on the X-axis and the expected return on the Y-axis. This plot reveals the most desirable portfolios. For example, assume Portfolio A has an expected return of 8.5% and a standard deviation of 8%, and that Portfolio B has an expected return of 8.5% and a standard deviation of 9.5%. Portfolio A would be deemed more "efficient" because it has the same expected return but lower risk. It is possible to draw an upward sloping hyperbola to connect all of the most efficient portfolios, and this is known as the efficient frontier. Investing in any portfolio not on this curve is not desirable.

CO VARIANCE - UNDER PORT FOLIO THEORY

Covariance is used in portfolio theory to determine what assets to include in the portfolio. Covariance is a statistical measure of the directional relationship between two asset prices. Modern portfolio theory uses this statistical measurement to reduce the overall risk for a portfolio.

Covariance is used in portfolio theory to determine what assets to include in the portfolio. Covariance is a statistical measure of the directional relationship between two asset prices. Modern portfolio theory uses this statistical measurement to reduce the overall risk for a portfolio. A positive covariance means that assets generally move in the same direction. Negative covariance means assets generally move in opposite directions. Here we'll discuss how covariance is used to reduce investment risk and provide portfolio diversification.

VARIANCE UNDER PORT FOLIO THEORY

Portfolio variance is a measurement of risk, of how the aggregate actual returns of a set of securities making up a portfolio fluctuate over time. This portfolio variance statistic is calculated using the standard deviations of each security in the portfolio as well as the correlations of each security pair in the portfolio.

Portfolio variance looks at the covariance or correlation coefficients for the securities in the portfolio. Generally, a lower correlation between securities in a portfolio results in a lower portfolio variance.

Portfolio variance is calculated by multiplying the squared weight of each security by its corresponding variance and adding twice the weighted average weight multiplied by the covariance of all individual security pairs.

Modern portfolio theory says that portfolio variance can be reduced by choosing asset classes with a low or negative correlation, such as stocks and bonds, where the variance (or standard deviation) of the portfolio is the x-axis of the efficient frontier.

Modern Portfolio Theory is a framework for constructing an investment portfolio. MPT takes as its central premise the idea that rational investors want to maximize returns while also minimizing risk, sometimes measured using volatility. Investors seek what is called an efficient frontier, or the lowest level or risk and volatility at which a target return can be achieved.

Risk is lowered in MPT portfolios by investing in non-correlated assets. Assets that might be risky on their own can actually lower the overall risk of a portfolio by introducing an investment that will rise when other investments fall. This reduced correlation can reduce the variance of a theoretical portfolio. In this sense, an individual investment's return is less important that its overall contribution to the portfolio, in terms of risk, return and diversification.

The level of risk in a portfolio is often measured using standard deviation, which is calculated as the square root of the variance. If data points are far away from the mean, the variance is high, and the overall level of risk in the portfolio is high, as well. Standard deviation is a key measure of risk used by portfolio managers, financial advisors and institutional investors. Asset managers routinely include standard deviation in their performance reports.

FOR DIVERSIFIED PORTFOLIO CO-VARIANCE IS MORE IMPORTANT MEASURE

Covariance can maximize diversification in a portfolio of assets. Adding assets with a negative covariance to a portfolio reduces the overall risk. At first, this risk drops off quickly; as additional assets are added, it drops off slowly. Diversifiable risk cannot significantly be reduced beyond including 25 different stocks in a portfolio. However, including more assets with negative covariance means that the risk drops more quickly.

Covariance has some limitations. While covariance can show the direction between two assets, it cannot be used to calculate the strength of the relationship between the prices. Determining the correlation coefficient between the assets is a better way to measure the strength of the relationship.

An additional drawback to the use of covariance is that the measurement is subject to being skewed by the presence of outliers in the underlying data. Thus, large single-period price movements may skew the overall volatility of the price series and provide an unreliable statistical measurement of the nature of the direction between the assets.

Modern portfolio theory (MPT) uses covariance as an important element in the construction of portfolios. MPT assumes investors are risk averse yet still seek the best return possible. MPT thus attempts to determine an efficient frontier for a mix of assets in a portfolio, or an optimal point at which the relationship between risk and return is most beneficial. The efficient frontier calculates the maximum return for a portfolio versus the amount of risk for the combination of the underlying assets. The goal is to create a group of assets with an overall standard deviation that is less than that of the individual securities. The graph of the efficient frontier is curved, demonstrating how higher-volatility assets may be mixed with lower-volatility assets to maximize return but reduce the impact of large price fluctuations. By diversifying the assets in a portfolio, investors can reduce risk while obtaining returns on their investments.