Optimizing Portfolio Returns through Risk Management

Investment Risks

Every form of investment carries a certain degree of uncertainty. Systematic Risk refers to risks that affect the entire market or economy and cannot be eliminated through diversification. Examples include market risk, interest rate risk, inflation risk, and geopolitical events. Unsystematic Risk is specific to a particular company or industry, such as management decisions, product failures, or industry competition. This type of risk can be reduced or eliminated through diversification.

The main types of risks include:

  • Market Risk: The risk of loss due to overall market fluctuations.
  • Credit Risk: The chance that a company may default on its debt.
  • Liquidity Risk: The risk of not being able to buy or sell stocks quickly without significantly affecting their price.
  • Inflation Risk: The possibility that inflation will erode the purchasing power of future returns.
  • Interest Rate Risk: The risk of financial losses due to changes in interest rates.
  • Political/Regulatory Risk: Involves the potential for financial losses due to changes in government policies, regulations, or political instability.
  • Company-Specific Risk: Refers to factors unique to a particular company, such as management decisions, product recalls, or competitive pressures.

These and other risks may result in partial or complete loss of the invested capital.

Identifying and Managing Risks

The primary task of investors is to identify the risks associated with their portfolio, measure them, and determine the strategy they will follow regarding these risks (elimination, reduction, or acceptance).

While diversification can mitigate unsystematic risk, systematic risks cannot be eliminated.

Variance and standard deviation are the most common measures of total risk (including both systematic and unsystematic risk).

Variance

Variance is a key measure of risk, indicating the volatility of an investment’s returns. It assesses the degree to which individual return observations are dispersed around the mean return. A higher variance indicates a higher risk associated with the financial instrument.

The formula for sample variance:

\[
s^2 = \frac{\sum (X – \hat{X})^2}{n – 1}
\]

Where:

  • s^2: Sample variance
  • X: Each value in the data set
  • X̂: Mean of all values in the data set
  • n: Number of values in the data set​

The numerator represents the sum of the squared differences between each observation and the sample mean. When working with a sample, the denominator is the number of observations minus one (n−1). If working with a population, the denominator is simply the number of observations (n).

Let’s use one-year monthly data for S&P 500 as an example:

DateS&P 500Returns(X-X̂)^2
12/1/20223,839.50
1/1/20234,076.606.18%0.18%
2/1/20233,970.20-2.61%0.20%
3/1/20234,109.303.50%0.03%
4/1/20234,169.501.46%0.00%
5/1/20234,179.800.25%0.03%
6/1/20234,450.406.47%0.21%
7/1/20234,589.003.11%0.01%
8/1/20234,507.70-1.77%0.14%
9/1/20234,288.10-4.87%0.46%
10/1/20234,193.80-2.20%0.17%
11/1/20234,567.808.92%0.49%
12/1/20234,769.804.42%0.06%

Table 1: S&P 500 monthly returns for one year. Source: tradepretty.com

By using the information in the table we can calculate:

  • Average Monthly Return (1.91%): The sum of Column 3 divided by the number of observations (12).
  • Sample Variance (0.18%):  The sum of Column 4 divided by the number of observations minus one (12-1=11).
  • Standard Deviation (4.25%): Square root of the variance.

The sum of squared differences between the returns for each specific month and the mean return can be calculated from Column 4. To find the sample variance, this sum is divided by the number of observations minus one (12 – 1 = 11). As a result, the monthly sample variance is 0.18%.

Two financial instruments may share the same average monthly return, but one could exhibit much larger fluctuations over time. For instance, consider a hypothetical stock XYZ, which averages a 1.91% monthly return, similar to the S&P 500. However, if XYZ’s individual returns are more widely spread around the mean, its volatility is higher. As a result, it reaches the same average return but at a greater level of risk.

The variance is not particularly intuitive for interpretation. The above value of 0.18% for S&P 500 do not allow for substantial conclusions about the risk of investment in the index.

Standard Deviation

Standard deviation is the square root of the variance. It is the most popular measure of volatility and is widely used in many risk models.

In a normal distribution, 68% of the observations will be within 1 standard deviation from the mean, 95% will be within 2 standard deviations, and 99.7% will be within 3 standard deviations from the mean.

Using the example of the S&P 500, where the average monthly return is 1.91% and the standard deviation is 4.25%, we can expect that in 68% of cases, the monthly changes will be within 1.91% +/- 4.25% (-2.34% to 6.16%).

The assumption that observations are normally distributed is crucial. This will be discussed further below.

Variance and standard deviation can be used to assess the volatility of a specific financial instrument (such as the stock of a particular company) or an entire portfolio (stocks, bonds, commodities).

Covariance

So far, we have discussed how to calculate variance and standard deviation for a single financial instrument (S&P 500). Reducing risk is one of the main goals of portfolio management. This is achieved by combining two or more assets into a basket. When building a portfolio, it is important to determine the relationships between its constituent securities. Covariance and correlation are coefficients that measure exactly this.

The formula for covariance between two assets is:

\[
\text{Cov}(X,Y) = \frac{\sum (X – \hat{X})(Y – \hat{Y})}{n}
\]

Where:

  • Cov(X,Y): The covariance between X and Y,
  • X and Y:  Represent the data points,
  • X^ and Y^: The means of X and ,
  • The number of observations.

Let’s try to establish the relationship between the S&P 500 and Nasdaq 100. We can prepare our data and make preliminary calculations:

DateS&P 500Returns(X-X̂)Nasdaq 100Returns(Y-Ŷ )(X- X̂)(Y- Ŷ)
12/1/20223,839.5010,939.76
1/1/20234,076.606.18%4.27%12,101.9310.62%6.84%0.29%
2/1/20233,970.20-2.61%-4.52%12,042.12-0.49%-4.28%0.19%
3/1/20234,109.303.50%1.60%13,181.359.46%5.68%0.09%
4/1/20234,169.501.46%-0.44%13,245.990.49%-3.29%0.01%
5/1/20234,179.800.25%-1.66%14,254.097.61%3.83%-0.06%
6/1/20234,450.406.47%4.57%15,179.216.49%2.71%0.12%
7/1/20234,589.003.11%1.21%15,757.003.81%0.02%0.00%
8/1/20234,507.70-1.77%-3.68%15,501.07-1.62%-5.41%0.20%
9/1/20234,288.10-4.87%-6.78%14,715.24-5.07%-8.85%0.60%
10/1/20234,193.80-2.20%-4.10%14,409.78-2.08%-5.86%0.24%
11/1/20234,567.808.92%7.01%15,947.8710.67%6.89%0.48%
12/1/20234,769.804.42%2.52%16,825.935.51%1.72%0.04%

Table 2: S&P 500 and Nasdaq 100 data for sample covariance. Source: tradepretty.com

Table 2 provides information about:

  • X and Y: Monthly returns for S&P 500 and Nasdaq 100,
  • X^ and Y^: Means of X and
  • n: number of observations

Based on the data from the table we can make the following calculations:

  • Average Monthly Return S&P 500 (1.91%): The average from Column 3 (Returns).
  • Average Monthly Return Nasdaq 100 (3.78%): The average from Column 6 (Returns).
  • Variance S&P 500 (0.18%): The sum of squared values from Column 4 (X-X̂) divided by the number of observations minus one (12-1=11).
  • Variance Nasdaq 100 (0.29%): The sum of squared values from Column 7 (Y-Ŷ ) divided by the number of observations minus one (12-1=11).
  • Standard Deviation S&P 500 (4.25%): Square root of the variance of S&P 500.
  • Standard Deviation Nasdaq 100 (5.41%): Square root of the variance of Nasdaq 100.
  • Sample covariance between S&P 500 and Nasdaq 100 (0.2%): Sum of Column 8 (X- X̂)(Y- Ŷ) divided by the number of observations minus one (12-1=11).

Based on monthly data, we calculate a sample covariance between the S&P 500 and Nasdaq 100 of 0.2%.

Covariance is difficult to interpret because its result is not standardized and can take on any positive or negative value. For this reason, correlation is more commonly used in practice, as it is normalized and always ranges between +1 and -1.

Correlation

The correlation coefficient between two financial instruments can be found with the following formula:

\[
p(1,2) = \frac{\text{Cov}_{1,2}}{\sigma_1 \cdot \sigma_2}
\]

Where:

  • Cov1,2: The co-variance of the two assets
  • p(1,2): The correlation coefficient between the two assets
  • σ1: The standard deviation of the first asset
  • σ2: The standard deviation of the second asset

A correlation of +1 means that the two instruments are perfectly positively correlated – both assets increase or decrease in value in the same way. A correlation of -1 means that the two assets are perfectly negatively correlated – an increase in one is accompanied by a decrease in the other. A correlation of 0 means that a relation between the two assets can not be identified.

It is rare for two financial instruments to be perfectly positively (+1) or negatively (-1) correlated. However, a coefficient of 0.8 or -0.8 indicates a significant positive or negative relationship.

The correlation between the S&P 500 and Nasdaq 100 from the above example is 0.88.

\[
p(1,2) = \frac{0.2}{4.25\% \cdot 5.41\%} \approx 0.88
\]
Importance of the Correlation Coefficient

The correlation plays a key role in portfolio management. This is because it helps reduce the volatility (risk) of the portfolio (unless the correlation between assets is +1).

Let’s continue with the above example: we have equally distributed our assets in the S&P 500 and Nasdaq 100. The S&P 500 achieves an average monthly return of 1.91% with a standard deviation of 4.25%, while the Nasdaq 100 achieves an average monthly return of 3.78% with a standard deviation of 5.41%.

The expected monthly return when investing in this portfolio is:

\[
\text{Return}_{\text{portfolio}} = 0.5 \times 1.91\% + 0.5 \times 3.78\% = 2.85\%
\]

The variance for a portfolio of two assets can be calculated using the following formula:

\[
\text{Portfolio Variance} = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \text{Cov}_{1,2}
\]

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 thus be expressed as p(1,2)σ1σ2, where p(1,2) is the correlation coefficient between the two assets.

Substituting in the equation, we find that that the portfolio variance is:

\[
\text{Portfolio Variance} = (0.5^2 \times (4.25\%)^2) + (0.5^2 \times (5.41\%)^2) + 2 \times 0.5 \times 0.5 \times 0.2\% = 0.22\%
\]

The standard deviation is the square root of the variance, therefore 4.68%.

If the correlation between the two assets were +1, it would not help reduce the risk, and the portfolio’s standard deviation would be:

\[
0.5 \times 4.25\% + 0.5 \times 5.41\% = 4.83\%
\]

However, the correlation between S&P 500 and Nasdaq 100 from the example is 0.88. As a result, the monthly standard deviation for this portfolio drops to 4.68%, indicating a slight reduction in risk while maintaining the expected return.

This is the most significant benefit of portfolio diversification – optimizing the return-to-risk ratio. The above example demonstrates the effect of diversification with two assets, but the same logic applies to a basket of more financial securities.

It is important to note that the sample size in the above example is statistically too small. It is used solely for illustrative purposes.

Other Investment Characteristics

Distribution of Returns

The distribution of returns refers to how the returns of an investment or portfolio are spread out over a period. It can be visualized through various statistical measures and charts, such as histograms or probability distributions.

For example, a normal distribution of returns suggests that most returns will cluster around the mean, while a skewed or fat-tailed distribution indicates a higher likelihood of extreme outcomes.

Graph 1: Standard Normal Distribution. Mean = 0, Standard Deviation = 1. Source: tradepretty.com.

Graph 1: Standard Normal Distribution. Mean = 0, Standard Deviation = 1. Source: tradepretty.com.

However, returns are not always normally distributed. Sometimes the skewness of the observations is positive (positive skew) or negative (negative skew).

Skewness

Skewness is a measure of the asymmetry of a probability distribution or dataset around its mean. In simple terms, it tells us whether the data is skewed to the left or the right.

  • Positive skewness (right skewed): The right tail of the distribution is longer or fatter than the left tail. This means there are a larger number of low values with a few high values stretching the tail on the right.
  • Negative skewness (left skewed): The left tail is longer or fatter than the right tail, indicating that there are more high values and fewer low values stretching the tail on the left.
  • Zero skewness: The distribution is symmetric around the mean (normal).
Graph 2: Positive and Negative Skewness of Observations. Source: tradepretty.com.

Graph 2: Positive and Negative Skewness. Source: tradepretty.com.

Skewness provides insights into the potential for outliers and can help in understanding the underlying dynamics of the data, influencing decisions on risk assessment and statistical modeling.

Most stocks are negatively skewed – meaning the risk of significant declines is high.

Kurtosis

Kurtosis measures the extent to which data points deviate from the mean and how heavy or light the tails are compared to a normal distribution.

Graph 3: Leptokurtic and Platykurtic Distributions compared to Normal Distribution. Source: tradepretty.com.

Graph 3: Leptokurtic and Platykurtic Distributions compared to Normal Distribution. Source: tradepretty.com.

There are three main types of kurtosis:

  1. Leptokurtic: This distribution has heavy tails and a sharp peak, meaning there are more extreme values (outliers) compared to a normal distribution. The kurtosis value is greater than 3.
  2. Platykurtic: This distribution has light tails and a flatter peak, indicating fewer outliers and less extreme values compared to a normal distribution. The kurtosis value is less than 3.
  3. Mesokurtic: This distribution has tails and a peak similar to a normal distribution. The kurtosis value of a normal distribution is equal to 3.

Kurtosis helps in understanding the probability of extreme events or outliers in the dataset.

Conclusion

In conclusion, effective portfolio risk management is essential for navigating the complexities of financial markets and achieving long-term investment goals. By employing a range of strategies—such as diversification, asset allocation, and regular risk assessment—investors can mitigate potential losses and enhance the stability of their portfolios. Understanding the concepts presented in this chapter provides deeper insights into the return and risk characteristics of assets, as well as the relationships between them.

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