Employing variance and standard deviation to quantify financial risk can be misleading. An asset that consistently exhibits an upward trend may still display significant volatility. Consequently, depending on the frequency at which variance is estimated, an asset with a persistent positive trajectory may receive a “high risk score” due to its volatility, despite its overall upward movement.
In finance, risk and uncertainty are fundamental concepts that, while related, have distinct meanings.
Risk refers to situations where the probabilities of different outcomes are known or can be estimated. It involves measurable uncertainties where past data and statistical methods can be used to predict future outcomes with some level of confidence.
Uncertainty, on the other hand, involves situations where the probabilities of outcomes are unknown or cannot be quantified. It represents unmeasurable uncertainties where it’s impossible to assign exact probabilities to future events due to a lack of sufficient information or inherent unpredictability.
Understanding this distinction is crucial for investors:
With risk, investors can use statistical tools and models (like variance and standard deviation) to assess potential outcomes and make informed decisions.
With uncertainty, investors face unknown probabilities, making it challenging to predict outcomes and requiring judgment, experience, and qualitative assessments.
In the context of financial returns, risk is commonly associated with the variability or volatility of returns. This includes the possibility that an investment’s actual return will differ from its expected return, potentially resulting in financial loss. This uncertainty arises from various factors such as market fluctuations, economic changes, and company-specific events. Investors are generally concerned with the possibility of losing money or not achieving their financial goals.
Variance and standard deviation are statistical tools used to measure the dispersion of returns around the mean (average) return. They quantify how much the returns on an asset fluctuate over time.
Variance (\(\sigma^2\)) calculates the average of the squared deviations from the mean:
\[\sigma^2 = \frac{1}{N}\sum_{i=1}^{N} (R_i - \mu)^2\]Standard deviation (\(\sigma\)) is the square root of the variance:
\[\sigma = \sqrt{\sigma^2} = \sqrt{\frac{1}{N}\sum_{i=1}^{N} (R_i - \mu)^2}\]Where:
These measures are popular because they provide a single number that reflects the volatility of an asset’s returns. In many financial models, especially those rooted in Modern Portfolio Theory (MPT), volatility is equated with risk under the assumption that higher volatility means a less predictable and, therefore, riskier investment.
The relation between risk and variance comes from the fact that the central limit theorem tells us that expected financial returns are asymptotically normal, and therefore symmetrical around the mean.
While variance and standard deviation are useful, they have notable limitations:
Variance and standard deviation treat all deviations from the mean equally, whether they’re above or below it. This means that unusually high returns increase the measured risk just as much as unusually low returns. Investors, however, are typically more concerned with downside risk—the possibility of returns falling below a certain threshold.
These measures assume that returns are normally distributed. In reality, financial returns often exhibit skewness (asymmetry) and kurtosis (fat tails), leading to underestimation or overestimation of risk. For instance, extreme events (like market crashes) are more common than a normal distribution would predict.
The frequency of data (daily, monthly, yearly) affects the calculated variance and standard deviation. High-frequency data may show more volatility due to short-term fluctuations, potentially overstating risk for long-term investors. Conversely, using long-term data might smooth out important short-term risks.
To address these limitations, finance people use additional or alternative risk measures that focus more on the downside or provide a different perspective on risk.
Semi-variance focuses only on returns below the mean or a target return, measuring the average squared deviations of returns that fall below a certain threshold.
\[\text{Semi-Variance} = \frac{1}{N}\sum_{R_i < \mu} (R_i - \mu)^2\]This provides a measure of downside risk, aligning more closely with investors’ concerns.
Value at Risk estimates the maximum expected loss over a specific time frame at a given confidence level.
VaR focuses on tail risk but does not provide information about the magnitude of losses beyond the threshold.
Also known as Expected Shortfall, CVaR measures the average loss exceeding the VaR threshold.
\[\text{CVaR} = E[L \mid L > \text{VaR}]\]Where \(L\) represents losses. CVaR provides insight into extreme losses and is considered a more coherent risk measure than VaR.
The Sortino Ratio adjusts the Sharpe Ratio by using downside deviation in the denominator, penalizing only undesirable volatility.
\[\text{Sortino Ratio} = \frac{R_p - R_f}{\sigma_d}\]Where:
In the Capital Asset Pricing Model (CAPM), beta (\(\beta\)) measures an asset’s volatility relative to the overall market.
\[\beta = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2}\]Where:
Beta indicates systematic risk that cannot simply be diversified away, helping investors understand how an asset might respond to market movements.
The interpretation of risk measures depends on the investor’s objectives, time horizon, and risk tolerance. Short-term traders might be more concerned with daily volatility, while long-term investors focus on fundamental value drivers and are less sensitive to short-term fluctuations.
Professionals often use a combination of risk metrics to capture different dimensions of risk. This includes:
Many investors prioritize the potential for loss over variability per se. Measures that emphasize downside risk, like semi-variance and CVaR, provide a more accurate reflection of their concerns and help in making investment decisions that align with their risk tolerance.
Variance and standard deviation are foundational tools in risk assessment due to their mathematical properties and ease of use. However, they are not without shortcomings. They serve as starting points in understanding an asset’s volatility but should be complemented with other measures that account for the asymmetry of returns and focus on downside risk.
Understanding the nuances of these risk measures helps investors make more informed decisions that align with their specific goals and risk appetites. A holistic approach to risk management, utilizing a variety of metrics, provides a more comprehensive view of potential risks and rewards.