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

Canadian Securities Course CSC

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1 Introduction & Exam Overview

About the CSC Exams

2 Chapter 1: The Canadian Investment Marketplace

1.1 The Canadian Securities Industry & the Investment Dealer's Role 1.2 Financial Intermediaries: Banks, Trusts, Credit Unions, Insurers & Pension Funds 1.3 Financial Market Trends 1.4 Investment Capital, Financial Instruments & Financial Markets 1.5 Canadian Exchanges, ATSs, Dark Pools & the Order Protection Rule 1.6 Canadian Securities Regulators, the CSA & CIRO 1.7 Investor Protection: OSFI, CIPF and CDIC 1.8 Ethics, Standards of Conduct & the Client Focused Reforms 1.9 Conflicts of Interest, Complaint Handling & Remediation

3 Chapter 2: The Economy

2.1 Economics Foundations & Measuring Economic Growth 2.2 The Business Cycle & Economic Indicators 2.3 Inflation & Interest Rates 2.4 The Labour Market & Unemployment 2.5 The Bank of Canada & the Inflation-Control Target 2.6 Implementing Monetary Policy: The Operating Band, Market Operations & Transmission 2.7 Fiscal Policy, Deficits & the Challenges of Government Policy 2.8 International Finance & Trade: The Balance of Payments and the Canadian Dollar

4 Chapter 3: Features and Types of Fixed-Income Securities

3.1 The Fixed-Income Marketplace, Bond Terminology & Security 3.2 Protective Covenants, Retirement Provisions & Extendible/Retractable Features 3.3 Government of Canada Securities: T-Bills, Marketable Bonds & RRBs 3.4 Provincial & Municipal Debt and Strip Bonds 3.5 Corporate Bonds, Credit Spreads & Bond Ratings 3.6 Money Market Instruments, Floating-Rate Notes & International Bonds 3.7 Other Fixed-Income Securities & Reading Bond Quotes

5 Chapter 4: Pricing and Trading of Fixed-Income Securities

4.1 Bond Pricing, Price–Yield Relationships & Accrued Interest 4.2 Yield Calculations & Return Measures 4.3 The Term Structure of Interest Rates & Yield Curve Theories 4.4 Fundamental Bond Pricing Properties: Volatility Rules, Duration & Convexity 4.5 Bond Market Trading & Bond Indexes

6 Chapter 5: Common and Preferred Shares

5.1 Common Shares: Ownership Rights, Voting & Restricted Shares 5.2 Dividends, DRIPs, Stock Splits & Consolidations 5.3 Preferred Shares: Features, Cumulative Dividends & Taxation 5.4 Preferred Share Variations: Convertible, Callable, Retractable, Floating & Rate-Reset 5.5 Fundamental Equity Metrics & Ratios 5.6 Stock Indexes & Averages

7 Chapter 6: Equity Transactions

6.1 Order Types, Execution & Market Mechanics 6.2 Margin Trading: Long Accounts & Borrowing 6.3 Short Selling & Short-Margin Requirements 6.4 Trading & Settlement Procedures: T+1, CDS & Confirmations

8 Chapter 7: Derivatives

7.1 Derivatives: Role, Underlying Assets & Users 7.2 Options Fundamentals: Calls, Puts & Moneyness 7.3 Option Strategies & Risk Management 7.4 Forward & Futures Contracts 7.5 Rights & Warrants

9 Chapter 8: Corporations and their Financial Statements

8.1 Corporate Structures & Corporate Governance 8.2 The Annual Report, Accounting Standards & the Auditor's Report 8.3 The Balance Sheet & Income Statement 8.4 The Cash Flow Statement & Earnings Quality 8.5 Public Company Disclosure, Investor Rights, Take-Over Bids & Insider Trading

10 Chapter 9: Financing and Listing Securities

9.1 Government & Corporate Financing: Underwriting and Syndicates 9.2 The Prospectus Process, Stabilization & Purchaser Rights 9.3 The Exempt Market & Private Placements 9.4 The Listing Process: TSX, TSXV & Capital Pool Companies 9.5 Continuous Disclosure, Material Changes & Insider Reporting

11 Chapter 10: Investment Analysis

10.1 Methods of Equity Analysis & Macroeconomic Analysis 10.2 Fundamental Industry Analysis 10.3 Company Analysis: Competitive Advantages, Governance & the Dividend Discount Model 10.4 Discounted Cash Flow & Relative Valuation Multiples 10.5 Financial Statement Ratio Analysis 10.6 Assessing Preferred Share Investment Quality 10.7 Technical Analysis: Charts, Trends & Patterns 10.8 Technical Indicators, Volume, Breadth & Sentiment

12 Chapter 11: Portfolio Analysis

11.1 Measuring Investment Return 11.2 Measuring Risk: Types of Risk, Standard Deviation & Downside Measures 11.3 Portfolio Risk & Return: Correlation and Diversification 11.4 Beta, the Capital Asset Pricing Model & the Security Market Line 11.5 Portfolio Manager Styles: Equity & Fixed Income 11.6 The Portfolio Management Process: Objectives, Constraints & the IPS 11.7 Asset Mix, Monitoring, Performance Evaluation & Rebalancing

13 Chapter 12: Alternative Investments, Other Managed, and Structured Products

12.1 Alternative Investments: Structures, Fees & Liquid Alternatives 12.2 Alternative Strategies, Performance Measurement & Due Diligence 12.3 Segregated Funds, Labour-Sponsored Funds & Listed Private Equity 12.4 Income Trusts, REITs & Closed-End Funds 12.5 Structured Products: PPNs & Market-Linked GICs 12.6 Split Shares & Asset-Backed Securities

14 Chapter 13: Mutual Funds

13.1 Managed Products & Mutual Fund Structure 13.2 Pricing Mutual Fund Units: NAVPS & Forward Pricing 13.3 Mutual Fund Fees, Sales Charges & Unit Series 13.4 Mutual Fund Regulation, Fund Facts & Account Opening 13.5 Types of Mutual Funds & Fund Management Styles 13.6 Redemptions, Withdrawal Plans & Measuring Fund Performance

15 Chapter 14: Exchange-Traded Funds (ETFs)

14.1 ETF Origins, Structure, Tax Efficiency & ETF Facts 14.2 ETF Creation, Redemption & Arbitrage 14.3 Types of ETFs & Advanced Strategies 14.4 Comparing ETFs, Mutual Funds & Individual Securities

16 Chapter 15: Working with the Institutional Client

15.1 The Institutional Marketplace & Institutional Clients 15.2 Institutional Investment Guidelines, Styles & Restrictions 15.3 Buy Side & Sell Side: Trading Desks, Revenue, Clearing & Best Execution 15.4 Block Trading, Dark Pools & Algorithmic Trading

17 Chapter 16: Fee-Based Accounts and Working with the Retail Client

16.1 Fee-Based Accounts: Managed & Non-Managed Models 16.2 Fee Transparency, CRM2 & Total Cost Reporting, and MWRR vs. TWRR 16.3 The Financial Planning Approach & the Life Cycle 16.4 Behavioral Finance, Sequence-of-Returns Risk & Client Coaching 16.5 Estate Planning Basics

18 Chapter 17: Canadian Taxation

17.1 The Canadian Taxation System 17.2 Taxation of Investment Income 17.3 Capital Gains, Losses & The Superficial Loss Rule 17.4 Registered vs Non-Registered Tax-Advantaged Accounts 17.5 Tax Planning Strategies, Pension Plans & the RDSP

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

Key Takeaways Explain This Topic AI

Main risk types include inflation, business, political, liquidity, interest rate, foreign exchange and default risk.

Standard deviation measures how widely returns vary around the mean.

In a normal distribution, about 68% of outcomes fall within one standard deviation of the mean and about 95% within two.

Value at risk states a loss that should not be exceeded at a given confidence level over a given period, but it is not a maximum possible loss.

Last updated: October 2026

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 ( σ 2 \sigma^2 σ 2 ) and standard deviation ( σ \sigma σ ).

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:

σ 2 = ∑ i = 1 n p i [ R i − E ( R ) ] 2 \sigma^2 = \sum_{i=1}^n p_i [R_i - E(R)]^2 σ 2 = ∑ i = 1 n ​ p i ​ [ R i ​ − E ( R ) ] 2

2. Standard Deviation ( σ \sigma σ )

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):

σ = σ 2 = ∑ i = 1 n p i [ R i − E ( R ) ] 2 \sigma = \sqrt{\sigma^2} = \sqrt{\sum_{i=1}^n p_i [R_i - E(R)]^2} σ = σ 2 ​ = ∑ 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 % E(R) = 8.50\% E ( R ) = 8.50% (0.085):

Scenario p i p_i p i ​ R i R_i R i ​ Deviation: [ R i − E ( R ) ] [R_i - E(R)] [ R i ​ − E ( R )] Squared Deviation: [ R i − E ( R ) ] 2 [R_i - E(R)]^2 [ R i ​ − E ( R ) ] 2 Weighted Squared Deviation: p i [ R i − E ( R ) ] 2 p_i [R_i - E(R)]^2 p i ​ [ R i ​ − E ( R ) ] 2

Recession 0.20 -0.150 − 0.150 − 0.085 = − 0.235 -0.150 - 0.085 = -0.235 − 0.150 − 0.085 = − 0.235 ( − 0.235 ) 2 = 0.055225 (-0.235)^2 = 0.055225 ( − 0.235 ) 2 = 0.055225 0.20 × 0.055225 = 0.011045 0.20 \times 0.055225 = 0.011045 0.20 × 0.055225 = 0.011045

Normal 0.50 +0.080 + 0.080 − 0.085 = − 0.005 +0.080 - 0.085 = -0.005 + 0.080 − 0.085 = − 0.005 ( − 0.005 ) 2 = 0.000025 (-0.005)^2 = 0.000025 ( − 0.005 ) 2 = 0.000025 0.50 × 0.000025 = 0.0000125 0.50 \times 0.000025 = 0.0000125 0.50 × 0.000025 = 0.0000125

Boom 0.30 +0.250 + 0.250 − 0.085 = + 0.165 +0.250 - 0.085 = +0.165 + 0.250 − 0.085 = + 0.165 ( + 0.165 ) 2 = 0.027225 (+0.165)^2 = 0.027225 ( + 0.165 ) 2 = 0.027225 0.30 × 0.027225 = 0.0081675 0.30 \times 0.027225 = 0.0081675 0.30 × 0.027225 = 0.0081675

Totals 1.00 — — — σ 2 = 0.019225 \sigma^2 = 0.019225 σ 2 = 0.019225

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

σ = 0.019225 ≈ 0.13865 = 13.87 % \sigma = \sqrt{0.019225} \approx 0.13865 = 13.87\% σ = 0.019225 ​ ≈ 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 n = 60 n = 60 ), sample statistics are utilized. To eliminate sample bias, the sum of squared deviations is divided by n − 1 n - 1 n − 1 degrees of freedom:

s 2 = 1 n − 1 ∑ t = 1 n ( R t − R ˉ ) 2 s^2 = \frac{1}{n - 1} \sum_{t=1}^n (R_t - \bar{R})^2 s 2 = n − 1 1 ​ ∑ t = 1 n ​ ( R t ​ − R ˉ ) 2

s = s 2 = 1 n − 1 ∑ t = 1 n ( R t − R ˉ ) 2 s = \sqrt{s^2} = \sqrt{\frac{1}{n - 1} \sum_{t=1}^n (R_t - \bar{R})^2} s = s 2 ​ = n − 1 1 ​ ∑ t = 1 n ​ ( R t ​ − R ˉ ) 2 ​

Types of Investment Risk

Before measuring risk, identify its sources:

Risk Description Example

Inflation (purchasing power) risk Returns fail to keep pace with rising prices A GIC earning less than inflation

Business risk A company's earnings fall because of competition, poor management or other problems A retailer losing market share

• Political risk Government actions hurt an investment New royalties, sanctions or expropriation
• Liquidity risk The asset cannot be sold quickly without a price concession A thinly traded small-cap stock

Interest rate risk Rising rates lower the prices of bonds and rate-sensitive stocks A long-term bond after a rate hike

Foreign exchange risk Currency moves change the value of foreign investments U.S. stocks when the Canadian dollar strengthens

Default (credit) risk The issuer fails to pay interest or principal A high-yield bond issuer going bankrupt

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

Perfect Symmetry: The distribution is completely symmetrical around its central location. The mean, median, and mode are all identical.

Parameter Specification: The entire distribution is fully defined by exactly two statistical parameters: the mean ( μ \mu μ or R ˉ \bar{R} R ˉ ) and the standard deviation ( σ \sigma σ ).

Zero Skewness: There is no directional tilt toward positive or negative outliers ( S k e w n e s s = 0 Skewness = 0 S k e w n ess = 0 ).

Mesokurtic: The curve exhibits standard tail thickness ( K u r t o s i s = 3 Kurtosis = 3 K u r t os i s = 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:

μ ± 1 σ \mu \pm 1\sigma μ ± 1 σ : Encompasses approximately 68.26% of all observations.

μ ± 2 σ \mu \pm 2\sigma μ ± 2 σ : Encompasses approximately 95.44% of all observations (commonly rounded to 95%).

μ ± 3 σ \mu \pm 3\sigma μ ± 3 σ : 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 % ± 16.0 % ⟹ − 8.0 % to + 24.0 % 8.0\% \pm 16.0\% \implies -8.0\% \text{ to } +24.0\% 8.0% ± 16.0% ⟹ − 8.0% to + 24.0%

There is a 95.4% probability that next year's return will fall between:

8.0 % ± 2 ( 16.0 % ) = 8.0 % ± 32.0 % ⟹ − 24.0 % to + 40.0 % 8.0\% \pm 2(16.0\%) = 8.0\% \pm 32.0\% \implies -24.0\% \text{ to } +40.0\% 8.0% ± 2 ( 16.0% ) = 8.0% ± 32.0% ⟹ − 24.0% to + 40.0%

There is only approximately a 2.3% probability (the lower tail beyond − 2 σ -2\sigma − 2 σ ) that the market will suffer a catastrophic loss exceeding -24.0% in any single year: 100 % − 95.44 % 2 = 2.28 % \frac{100\% - 95.44\%}{2} = 2.28\% 2 100% − 95.44% ​ = 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.

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):

Semi-Variance = 1 k ∑ R t < MAR ( R t − MAR ) 2 \text{Semi-Variance} = \frac{1}{k} \sum_{R_t < \text{MAR}} (R_t - \text{MAR})^2 Semi-Variance = k 1 ​ ∑ R t ​ < MAR ​ ( R t ​ − MAR ) 2

Where k k 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:

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

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

Time Horizon: e.g., 1 day, 10 days, 1 month, or 1 year.

Confidence Level: typically 95% or 99%.

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

Metric Classification Formula / Definition Strategic Value in Portfolio Analysis

Total Return ( R R R ) Performance ( P 1 − P 0 ) + D P 0 \frac{(P_1 - P_0) + D}{P_0} P 0 ​ ( P 1 ​ − P 0 ​ ) + D ​ Measures aggregate wealth change from capital gains and cash income.

Expected Return ( E ( R ) E(R) E ( R ) ) Performance Forecast ∑ p i R i \sum p_i R_i ∑ p i ​ R i ​ Probability-weighted projection across prospective economic states.

Geometric Mean ( R G R_G R G ​ ) Compounded Return [ ∏ ( 1 + R t ) ] 1 / n − 1 [\prod(1 + R_t)]^{1/n} - 1 [ ∏ ( 1 + R t ​ ) ] 1/ n − 1 Measures true historical compound annual wealth accumulation rate.

Standard Deviation ( σ \sigma σ ) Total Risk ∑ p i [ R i − E ( R ) ] 2 \sqrt{\sum p_i [R_i - E(R)]^2} ∑ p i ​ [ R i ​ − E ( R ) ] 2 ​ Quantifies overall volatility and dispersion around expected value.

Semi-Deviation Downside Risk 1 k ∑ R < MAR ( R − MAR ) 2 \sqrt{\frac{1}{k}\sum_{R < \text{MAR}}(R - \text{MAR})^2} k 1 ​ ∑ R < MAR ​ ( R − MAR ) 2 ​ Isolates harmful downside volatility below target hurdle rate.

Maximum Drawdown (MDD) Capital Preservation Trough − Peak Peak \frac{\text{Trough} - \text{Peak}}{\text{Peak}} Peak Trough − Peak ​ Measures historical worst peak-to-trough capital impairment.

Value at Risk (VaR) Tail Risk Exposure Minimum loss at X % X\% X % confidence over period T T T Establishes capital adequacy and regulatory risk reserves for institutions.