11.2 Measuring Risk: Types of Risk, Standard Deviation & Downside Measures

Return is only half the picture: two investments with the same expected return can carry very different risks. This section covers the main types of investment risk and how risk is measured, including variance and standard deviation, the normal distribution, and downside measures such as semi-variance, maximum drawdown and value at risk.

1. Measuring Total Risk: Variance and Standard Deviation

In modern financial economics, risk is defined as the uncertainty or dispersion of actual future outcomes around the expected return. A security whose annual returns fluctuate narrowly between 4% and 6% possesses low risk, whereas a junior mining exploration stock whose returns swing between -60% and +120% carries extreme risk.

The classical statistical metrics used to quantify total risk are variance ($\sigma^2$) and standard deviation ($\sigma$).

1. Probability-Weighted Variance (Ex-Ante Scenario Analysis)

When evaluating future probabilistic scenarios, variance measures the probability-weighted sum of squared deviations from the expected return:

$\sigma^2 = \sum_{i=1}^n p_i [R_i - E(R)]^2$

2. Standard Deviation ($\sigma$)

Because variance is expressed in "squared percent" (an unintuitive statistical unit), analysts take the positive square root of variance to calculate the standard deviation. Standard deviation is expressed in the identical units as the return itself (percentage points):

$\sigma = \sqrt{\sigma^2} = \sqrt{\sum_{i=1}^n p_i [R_i - E(R)]^2}$

Step-by-Step Numeric Calculation: Standard Deviation

Using the energy stock scenario data from earlier, where $E(R) = 8.50\%$ (0.085):

Now, calculate the standard deviation ($\sigma$):

$\sigma = \sqrt{0.019225} \approx 0.13865 = 13.87\%$

The energy conglomerate possesses an expected return of 8.50% with a total risk (standard deviation) of 13.87%.

Historical Sample Variance and Standard Deviation

When calculating standard deviation from historical time-series data (e.g., monthly returns over 5 years, $n = 60$), sample statistics are utilized. To eliminate sample bias, the sum of squared deviations is divided by $n - 1$ degrees of freedom:

$s^2 = \frac{1}{n - 1} \sum_{t=1}^n (R_t - \bar{R})^2$

$s = \sqrt{s^2} = \sqrt{\frac{1}{n - 1} \sum_{t=1}^n (R_t - \bar{R})^2}$

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Types of Investment Risk

Before measuring risk, identify its sources:

Some of these risks can be reduced by diversifying across many securities; others, such as the overall market's response to interest rates or inflation, affect almost all securities. That distinction between systematic and unsystematic risk is developed in the next section.

2. The Normal Distribution Framework in Finance

Many fundamental portfolio management theories assume that security returns conform approximately to a normal distribution—the classic symmetrical, bell-shaped probability density curve.

The Normal Distribution Bell Curve (Empirical Rule):

                 Mean (μ)
                   |
                 .-'-.
               .'  |  '.
             .'    |    '.
           .'      |      '.
        .-'        |        '-.
      .'           |           '.
    .'             |             '.
  .'               |               '.
--+--------+-------+-------+--------+--> Return (%)
 -3σ      -2σ     -1σ     +1σ      +2σ      +3σ

 [ <------- 68.26% -------> ]
 [ <-------------- 95.44% --------------> ]
 [ <--------------------- 99.74% ---------------------> ]

Key Mathematical Properties of the Normal Distribution

1. Perfect Symmetry: The distribution is completely symmetrical around its central location. The mean, median, and mode are all identical.
2. Parameter Specification: The entire distribution is fully defined by exactly two statistical parameters: the mean ($\mu$ or $\bar{R}$) and the standard deviation ($\sigma$).
3. Zero Skewness: There is no directional tilt toward positive or negative outliers ($Skewness = 0$).
4. Mesokurtic: The curve exhibits standard tail thickness ($Kurtosis = 3$).

The Empirical Rule (68–95–99.7% Rule)

In any normal distribution, the probability of an observed outcome falling within standardized bands around the mean is constant:

• $\mu \pm 1\sigma$: Encompasses approximately 68.26% of all observations.
• $\mu \pm 2\sigma$: Encompasses approximately 95.44% of all observations (commonly rounded to 95%).
• $\mu \pm 3\sigma$: Encompasses approximately 99.74% of all observations (nearly all outcomes).

Practical Application: Forecasting Canadian Equity Ranges

Suppose the S&P/TSX Composite Index has an expected annual return of 8.0% and an annual standard deviation of 16.0%. Assuming returns are normally distributed:

• There is a 68.3% probability that next year's return will fall between:
  $8.0\% \pm 16.0\% \implies -8.0\% \text{ to } +24.0\%$
• There is a 95.4% probability that next year's return will fall between:
  $8.0\% \pm 2(16.0\%) = 8.0\% \pm 32.0\% \implies -24.0\% \text{ to } +40.0\%$
• There is only approximately a 2.3% probability (the lower tail beyond $-2\sigma$) that the market will suffer a catastrophic loss exceeding -24.0% in any single year: $\frac{100\% - 95.44\%}{2} = 2.28\%$

Limitations of Normal Distribution in Real Financial Markets

While mathematically elegant, real-world equity returns deviate from pure normality in two critical ways:

• Fat Tails (Leptokurtosis): Extreme market crashes and surges occur far more frequently in reality than predicted by a normal distribution bell curve. Events beyond 3 or 4 standard deviations (such as the 1987 crash or the March 2020 liquidity shock) occur with higher frequency than the 0.26% theoretical probability.
• Negative Skewness: Financial markets typically exhibit longer, fatter negative tails—stock prices drop faster and more violently during panic sell-offs than they rise during bull markets.

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3. Downside Risk Measures

A primary limitation of standard deviation is that it treats upside volatility (windfall gains) identically to downside volatility (capital losses). To retail and institutional investors, unexpected outsized gains are welcomed, whereas severe drops represent actual economic harm. To address this asymmetry, portfolio analysts utilize specialized downside risk metrics.

Specialized Downside Risk Metrics:
1. Semi-Variance: Measures dispersion only for returns falling below the mean or a hurdle rate.
2. Maximum Drawdown (MDD): Measures the largest peak-to-trough cumulative drop in portfolio equity.
3. Value at Risk (VaR): Quantifies the minimum dollar or percentage loss expected at a given confidence level over a specific timeframe.

1. Semi-Variance and Semi-Deviation

Semi-variance calculates the variance of returns that fall strictly below a designated benchmark—most commonly the mean return or a minimum acceptable return (MAR, such as 0% or the risk-free rate):

$\text{Semi-Variance} = \frac{1}{k} \sum_{R_t < \text{MAR}} (R_t - \text{MAR})^2$

Where $k$ is the number of observations falling below the target. Taking the square root yields the semi-deviation (or downside deviation). Assets with identical standard deviations can have very different semi-deviations if one has a positively skewed return profile (desirable) and the other has severe downside spikes.

2. Maximum Drawdown (MDD)

Maximum Drawdown evaluates the worst cumulative peak-to-trough decline experienced by a portfolio before a new equity peak is achieved. It measures capital impairment and psychological stress during market crises:

$\text{Maximum Drawdown} = \frac{\text{Trough Value} - \text{Peak Value}}{\text{Peak Value}}$

If a Canadian balanced portfolio reaches a peak value of $1,200,000, drops to a trough of $900,000 during a market contraction, and subsequently recovers to $1,300,000, the Maximum Drawdown is:

$\text{MDD} = \frac{\$900,000 - \$1,200,000}{\$1,200,000} = \frac{-\$300,000}{\$1,200,000} = -25.00\%$

3. Value at Risk (VaR)

Value at Risk (VaR) is the standard institutional metric used by Canadian banks, pension funds, and asset managers to summarize potential downside loss. VaR answers a precise question: What is the minimum loss expected over a specified time horizon at a given statistical confidence level?

VaR requires three core parameters:

1. Time Horizon: e.g., 1 day, 10 days, 1 month, or 1 year.
2. Confidence Level: typically 95% or 99%.
3. Loss Amount: expressed in currency ($) or percentage (%).

Interpreting VaR Statements

If a $10,000,000 institutional equity portfolio has a 1-month 95% VaR of $500,000 (5.0%), this indicates:

• Over any given month, the portfolio manager can be 95% confident that losses will not exceed $500,000.
• Conversely, there is a 5% probability (roughly 1 month out of every 20) that the portfolio will suffer a loss of at least $500,000 or greater.

Comprehensive Summary of Risk and Return Metrics