Understanding Portfolio Risk: Variance In Investments

Portfolio variance is a statistical measure of the volatility or risk of a portfolio of investments. It is calculated using the standard deviation of each security in the portfolio and the correlation between securities in the portfolio. The formula for portfolio variance takes into account the weights and variances of each asset in a portfolio, as well as their co-variances. A lower correlation between securities in a portfolio results in a lower portfolio variance. Portfolio variance is an important concept in modern investment theory, helping investors make informed decisions about diversification and risk management.

Portfolio variance is a measurement of risk

Portfolio variance is a statistical measure of risk. It assesses the degree of dispersion of a portfolio's returns, or how the aggregate actual returns of a set of securities in a portfolio fluctuate over time. In other words, it measures the total risk of a portfolio.

Portfolio variance is calculated using the standard deviation of each security in the portfolio and the correlation between securities in the portfolio. The formula for portfolio variance is:

Portfolio variance = w12σ12 + w22σ22 + 2w1w2Cov1,2

Where:

• W1 = the portfolio weight of the first asset
• W2 = the portfolio weight of the second asset
• Σ1 = the standard deviation of the first asset
• Σ2 = the standard deviation of the second asset
• Cov1,2 = the covariance of the two assets, which can be expressed as p(1,2)σ1σ2, where p(1,2) is the correlation coefficient between the two assets

The portfolio variance is equivalent to the portfolio standard deviation squared. A higher portfolio variance indicates greater volatility and risk, while a lower variance suggests a more stable portfolio.

Modern portfolio theory (MPT) states that portfolio variance can be reduced by selecting securities with low or negative correlations, such as stocks and bonds. This is because a lower correlation between securities in a portfolio results in a lower portfolio variance.

Calculating portfolio variance

The calculation of portfolio variance takes into account not only the riskiness of individual assets but also the correlation between each pair of assets in the portfolio. In other words, the statistical variance analyses how assets within a portfolio tend to move together. The general rule of portfolio diversification is to select assets with low or negative correlations with each other.

The formula for a two-asset portfolio is as follows:

Portfolio variance = w1^2 * σ1^2 + w2^2 * σ2^2 + 2 * w1 * w2 * Cov1,2

Where:

• W1 = the portfolio weight of the first asset
• W2 = the portfolio weight of the second asset
• Σ1 = the standard deviation of the first asset
• Σ2 = the standard deviation of the second asset
• Cov1,2 = the covariance of the two assets, which can be expressed as p(1,2)σ1σ2, where p(1,2) is the correlation coefficient between the two assets

As the number of assets in the portfolio increases, the terms in the formula for variance increase exponentially. For example, a three-asset portfolio has six terms in the variance calculation, while a five-asset portfolio has 15. Software like Excel can be used to simplify these calculations.

To illustrate the calculation, let's consider an example. Assume a portfolio consisting of two stocks, Stock A and Stock B. Stock A is worth $60,000 and has a standard deviation of 15%, while Stock B is worth $90,000 and has a standard deviation of 10%. The correlation between the two stocks is 0.85.

First, we need to determine the weight of each stock in the portfolio, which is calculated by dividing the value of each stock by the total portfolio value. So, the weight of Stock A is 40% (or 0.40), and the weight of Stock B is 60% (or 0.60).

Now, we can plug these values into the formula:

Variance = (0.40^2 * (0.15)^2) + (0.60^2 * (0.10)^2) + 2 * 0.85 * 0.40 * 0.60 * 0.15 * 0.10

Calculating these values gives us a variance of 1.33%.

While variance is a useful measure, it is often interpreted through its standard deviation, which is the square root of the variance. In this case, the standard deviation would be the square root of 1.33%, or approximately 11.53%.

Modern portfolio theory

MPT is based on the idea that rational investors want to maximise returns while minimising risk. It assumes that investors are risk-averse, meaning they prefer a less risky portfolio over a riskier one for the same level of return. The theory suggests that investors can achieve optimal results by choosing a mix of high-risk, high-return investments and low-risk, low-return investments, based on their individual tolerance for risk.

MPT takes into account the diversification of a portfolio, arguing that an asset's risk and return should be evaluated in the context of its contribution to the overall portfolio's risk and return. It uses statistical measures such as variance and correlation to assess risk and return. The variance of return, or its transformation, the standard deviation, is used as a measure of risk.

MPT helps investors model different portfolios with varying levels of risk and expected returns. Investors can then construct an efficient frontier, which is a graph representing the most efficient or risk-optimised portfolio allocation for different anticipated returns. The efficient frontier allows investors to identify the optimal portfolio that offers the highest return for a specified level of risk.

While MPT is a widely used tool, it has been criticised for its reliance on historical data, which may not reflect current or future market conditions. Additionally, MPT's measure of risk does not account for downside risk, which can make it difficult to distinguish between portfolios with the same level of risk but different risk characteristics.

Portfolio standard deviation

The formula for portfolio variance with two assets is as follows:

Portfolio variance = w12σ12 + w22σ22 + 2w1w2Cov1,2

Where:

• W1 = the portfolio weight of the first asset
• W2 = the portfolio weight of the second asset
• Σ1 = the standard deviation of the first asset
• Σ2 = the standard deviation of the second asset
• Cov1,2 = the co-variance of the two assets, which can be expressed as p(1,2)σ1σ2, where p(1,2) is the correlation coefficient between the two assets

The formula for portfolio variance with three assets is as follows:

Portfolio variance = w12σ12 + w22σ22 + w32σ32 + 2w1w2Cov1,2 + 2w1w3Cov1,3 + 2w2w3Cov2,3

Where:

• W1 = the portfolio weight of the first asset
• W2 = the portfolio weight of the second asset
• W3 = the portfolio weight of the third asset
• Σ1 = the standard deviation of the first asset
• Σ2 = the standard deviation of the second asset
• Σ3 = the standard deviation of the third asset
• Cov1,2 = the co-variance of the first and second assets
• Cov1,3 = the co-variance of the first and third assets
• Cov2,3 = the co-variance of the second and third assets

The portfolio standard deviation for a portfolio with two assets, A and B, is given by:

Standard deviation = sqrt(WA^2 * σA^2 + WB^2 * σB^2 + 2 * WA * WB * CovAB)

Where:

• WA = Weight of asset A in the portfolio
• WB = Weight of asset B in the portfolio
• ΣA = Standard deviation of returns on asset A
• ΣB = Standard deviation of returns on asset B
• CovAB = Covariance of returns on assets A and B

The portfolio standard deviation for a portfolio with three assets, A, B, and C, is given by:

Standard deviation = sqrt(WA^2 * σA^2 + WB^2 * σB^2 + WC^2 * σC^2 + 2 * WA * WB * CovAB + 2 * WA * WC * CovAC + 2 * WB * WC * CovBC)

Where:

• WA = Weight of asset A in the portfolio
• WB = Weight of asset B in the portfolio
• WC = Weight of asset C in the portfolio
• ΣA = Standard deviation of returns on asset A
• ΣB = Standard deviation of returns on asset B
• ΣC = Standard deviation of returns on asset C
• CovAB = Covariance of returns on assets A and B
• CovAC = Covariance of returns on assets A and C
• CovBC = Covariance of returns on assets B and C

Covariance and correlation

Covariance is a statistical measure of the relationship between the movement of two asset prices or returns. It helps investors understand the relationship between two stocks and can be used to predict how they might perform relative to each other in the future.

A positive covariance means that asset prices or returns are moving in the same general direction, while a negative covariance means they are moving in opposite directions. Investors using modern portfolio theory (MPT) seek to include assets with a negative covariance in their portfolio to optimise returns and reduce risk.

Covariance is closely related to correlation, which is a better measure of the strength of the relationship between assets. The correlation coefficient ranges from 0 indicating no linear relationship between the random variables. The correlation coefficient is calculated by taking the covariance and dividing it by the product of each variable's standard deviation.

By including assets with a negative covariance and a low correlation coefficient in a portfolio, investors can reduce the overall risk. This is because a lower correlation between securities in a portfolio results in a lower portfolio variance, which is a measure of a portfolio's overall risk.

In summary, covariance and correlation are important tools for investors to measure the relationship between assets and construct portfolios that optimise returns while minimising risk.

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