11.3 Portfolio Risk & Return: Correlation and Diversification

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

11.3 Portfolio Risk & Return: Correlation and Diversification

Key Takeaways Explain This Topic AI

A portfolio's expected return is the weighted average of its holdings' expected returns.

Portfolio risk depends on weights, individual standard deviations and the correlation between holdings.

When correlation is below +1, the portfolio's standard deviation is less than the weighted average of the individual standard deviations.

Diversification removes unsystematic (company-specific) risk but not systematic (market) risk.

Last updated: October 2026

Prior to the 1950s, financial practitioners evaluated investments largely on an isolated, standalone basis—attempting to pick individual stocks or bonds with the most attractive individual fundamentals. In 1952, American economist Harry Markowitz revolutionized portfolio management by publishing Portfolio Selection , introducing what is now universally termed Modern Portfolio Theory (MPT) . Markowitz proved mathematically that an asset's standalone volatility matters far less than how that asset co-moves with other assets in a portfolio. Through proper portfolio diversification, investors can achieve a higher expected return for a given level of risk, or eliminate substantial risk without sacrificing expected return.

1. Portfolio Expected Return

The expected return of a multi-asset portfolio is straightforward: it is simply the linear, weighted average of the expected returns of the underlying component securities:

E ( R p ) = ∑ i = 1 n w i E ( R i ) = w 1 E ( R 1 ) + w 2 E ( R 2 ) + ⋯ + w n E ( R n ) E(R_p) = \sum_{i=1}^n w_i E(R_i) = w_1 E(R_1) + w_2 E(R_2) + \dots + w_n E(R_n) E ( R p ​ ) = ∑ i = 1 n ​ w i ​ E ( R i ​ ) = w 1 ​ E ( R 1 ​ ) + w 2 ​ E ( R 2 ​ ) + ⋯ + w n ​ E ( R n ​ )

Where:

w i w_i w i ​ = Portfolio weight allocated to asset i i i (expressed as a fraction of total portfolio value, such that ∑ w i = 1.0 \sum w_i = 1.0 ∑ w i ​ = 1.0 )

E ( R i ) E(R_i) E ( R i ​ ) = Expected return of asset i i i

For a two-asset portfolio containing Asset A and Asset B:

E ( R p ) = w A E ( R A ) + w B E ( R B ) E(R_p) = w_A E(R_A) + w_B E(R_B) E ( R p ​ ) = w A ​ E ( R A ​ ) + w B ​ E ( R B ​ )

Notice that portfolio return is strictly linear. Diversification cannot create a return higher than the highest returning individual asset, nor can it pull return below the lowest returning asset.

2. Portfolio Risk and the Role of Correlation

While portfolio return is a simple linear weighted average, portfolio risk (standard deviation) is not . The total risk of a portfolio depends fundamentally on three variables:

The weight assigned to each security ( w i w_i w i ​ ).

The individual volatility of each security ( σ i \sigma_i σ i ​ ).

The degree of co-movement between the securities, measured by covariance ( Cov A B \text{Cov}_{AB} Cov A B ​ ) or the correlation coefficient ( r A B r_{AB} r A B ​ ).

The Two-Asset Portfolio Variance Formula

For a portfolio composed of two assets, A and B, the portfolio variance ( σ p 2 \sigma_p^2 σ p 2 ​ ) is expressed as:

σ p 2 = w A 2 σ A 2 + w B 2 σ B 2 + 2 w A w B Cov ( A , B ) \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}(A,B) σ p 2 ​ = w A 2 ​ σ A 2 ​ + w B 2 ​ σ B 2 ​ + 2 w A ​ w B ​ Cov ( A , B )

Because covariance equals the product of the correlation coefficient and the two individual standard deviations ( Cov ( A , B ) = r A B σ A σ B \text{Cov}(A,B) = r_{AB} \sigma_A \sigma_B Cov ( A , B ) = r A B ​ σ A ​ σ B ​ ), the formula can be written as:

σ p 2 = w A 2 σ A 2 + w B 2 σ B 2 + 2 w A w B σ A σ B r A B \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B r_{AB} σ p 2 ​ = w A 2 ​ σ A 2 ​ + w B 2 ​ σ B 2 ​ + 2 w A ​ w B ​ σ A ​ σ B ​ r A B ​

Taking the square root gives the portfolio standard deviation ( σ p \sigma_p σ p ​ ):

σ p = w A 2 σ A 2 + w B 2 σ B 2 + 2 w A w B σ A σ B r A B \sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B r_{AB}} σ p ​ = w A 2 ​ σ A 2 ​ + w B 2 ​ σ B 2 ​ + 2 w A ​ w B ​ σ A ​ σ B ​ r A B ​ ​ Decomposition of Two-Asset Portfolio Risk:

• [ Portfolio Variance σ_p^2 ]
• |
• +---------------+---------------+
• | |
• [ Independent Risk Components ] [ Interactive Co-Movement Term ]
• w_A^2 σ_A^2 + w_B^2 σ_B^2 + 2 w_A w_B σ_A σ_B r_AB
• (Weighted asset variances) (Diversification benefit driven by r_AB)
• Analyzing the Correlation Coefficient ( r A B r_{AB} r A B ​ )

The correlation coefficient ( r r r or ρ \rho ρ ) is a standardized statistical measure that quantifies the strength and direction of the linear relationship between the returns of two securities. It is strictly bounded between -1.0 and +1.0 :

− 1.0 ≤ r A B ≤ + 1.0 -1.0 \le r_{AB} \le +1.0 − 1.0 ≤ r A B ​ ≤ + 1.0

Correlation Value Relationship Impact on Portfolio Risk ( σ p \sigma_p σ p ​ )

r = + 1.0 r = +1.0 r = + 1.0 Perfect Positive Correlation Assets move in lockstep. The formula simplifies to σ p = w A σ A + w B σ B \sigma_p = w_A \sigma_A + w_B \sigma_B σ p ​ = w A ​ σ A ​ + w B ​ σ B ​ . Zero diversification benefit ; portfolio risk is simply the weighted average of individual risks.

0.0 < r < + 1.0 0.0 < r < +1.0 0.0 < r < + 1.0 Imperfect Positive Correlation Assets generally move in the same direction, but not identically. Portfolio risk is strictly less than the weighted average of individual risks. Meaningful diversification benefits occur .

r = 0.0 r = 0.0 r = 0.0 Uncorrelated No linear relationship. The interactive covariance term vanishes ( 2 w A w B σ A σ B ( 0 ) = 0 2 w_A w_B \sigma_A \sigma_B (0) = 0 2 w A ​ w B ​ σ A ​ σ B ​ ( 0 ) = 0 ). Substantial diversification benefits occur .

− 1.0 < r < 0.0 -1.0 < r < 0.0 − 1.0 < r < 0.0 Negative Correlation Assets tend to move in opposite directions. When one suffers losses, the other generates offsetting gains. Dramatic risk reduction .

r = − 1.0 r = -1.0 r = − 1.0 Perfect Negative Correlation Assets move in exact opposite directions. By setting weights inversely proportional to standard deviations ( w A = σ B σ A + σ B w_A = \frac{\sigma_B}{\sigma_A + \sigma_B} w A ​ = σ A ​ + σ B ​ σ B ​ ​ ), total portfolio risk can theoretically be reduced to zero ( σ p = 0 \sigma_p = 0 σ p ​ = 0 ) .

The Golden Rule of Diversification

Exam Key Concept: Whenever the correlation coefficient between two assets is strictly less than +1.0 ( r < 1.0 r < 1.0 r < 1.0 ) , the standard deviation of the portfolio is strictly less than the weighted average of the individual standard deviations:

σ p < w A σ A + w B σ B \sigma_p < w_A \sigma_A + w_B \sigma_B σ p ​ < w A ​ σ A ​ + w B ​ σ B ​

In real-world Canadian financial markets, finding assets with perfect negative correlation ( r = − 1.0 r = -1.0 r = − 1.0 ) is virtually impossible. However, most financial assets exhibit imperfect correlations between +0.10 and +0.65 (e.g., Canadian equities vs. Government of Canada bonds, or Canadian banks vs. gold mining equities). Because r < 1.0 r < 1.0 r < 1.0 , combining these assets consistently lowers total portfolio volatility.

3. Comprehensive Worked Numeric Calculation: Two-Asset Diversification

An advisor constructs a two-asset portfolio allocating 50% ( w A = 0.50 w_A = 0.50 w A ​ = 0.50 ) to a Canadian chartered bank (Stock A) and 50% ( w B = 0.50 w_B = 0.50 w B ​ = 0.50 ) to a gold mining exploration firm (Stock B).

Stock A (Bank): E ( R A ) = 10.0 % E(R_A) = 10.0\% E ( R A ​ ) = 10.0% , σ A = 16.0 % \sigma_A = 16.0\% σ A ​ = 16.0%

Stock B (Gold): E ( R B ) = 6.0 % E(R_B) = 6.0\% E ( R B ​ ) = 6.0% , σ B = 24.0 % \sigma_B = 24.0\% σ B ​ = 24.0%

Step 1: Calculate Portfolio Expected Return

E ( R p ) = ( 0.50 × 10.0 % ) + ( 0.50 × 6.0 % ) = 5.0 % + 3.0 % = 8.00 % E(R_p) = (0.50 \times 10.0\%) + (0.50 \times 6.0\%) = 5.0\% + 3.0\% = 8.00\% E ( R p ​ ) = ( 0.50 × 10.0% ) + ( 0.50 × 6.0% ) = 5.0% + 3.0% = 8.00% Notice that the expected return remains 8.00% regardless of the correlation coefficient between the two stocks.

Step 2: Calculate Weighted Variance Components

w A 2 σ A 2 = ( 0.50 ) 2 × ( 16.0 ) 2 = 0.25 × 256 = 64.0 w_A^2 \sigma_A^2 = (0.50)^2 \times (16.0)^2 = 0.25 \times 256 = 64.0 w A 2 ​ σ A 2 ​ = ( 0.50 ) 2 × ( 16.0 ) 2 = 0.25 × 256 = 64.0

w B 2 σ B 2 = ( 0.50 ) 2 × ( 24.0 ) 2 = 0.25 × 576 = 144.0 w_B^2 \sigma_B^2 = (0.50)^2 \times (24.0)^2 = 0.25 \times 576 = 144.0 w B 2 ​ σ B 2 ​ = ( 0.50 ) 2 × ( 24.0 ) 2 = 0.25 × 576 = 144.0

Sum of weighted variance terms: 64.0 + 144.0 = 208.0 64.0 + 144.0 = 208.0 64.0 + 144.0 = 208.0

Interactive term multiplier: 2 w A w B σ A σ B = 2 × 0.50 × 0.50 × 16.0 × 24.0 = 192.0 2 w_A w_B \sigma_A \sigma_B = 2 \times 0.50 \times 0.50 \times 16.0 \times 24.0 = 192.0 2 w A ​ w B ​ σ A ​ σ B ​ = 2 × 0.50 × 0.50 × 16.0 × 24.0 = 192.0

Now, observe how portfolio standard deviation changes across four distinct correlation scenarios:

Case 1: Perfect Positive Correlation ( r A B = + 1.0 r_{AB} = +1.0 r A B ​ = + 1.0 )

σ p 2 = 208.0 + ( 192.0 × 1.0 ) = 400.0 \sigma_p^2 = 208.0 + (192.0 \times 1.0) = 400.0 σ p 2 ​ = 208.0 + ( 192.0 × 1.0 ) = 400.0 σ p = 400.0 = 20.00 % \sigma_p = \sqrt{400.0} = 20.00\% σ p ​ = 400.0 ​ = 20.00% Note: This equals the weighted average of individual risks: ( 0.50 × 16.0 % ) + ( 0.50 × 24.0 % ) = 8.0 % + 12.0 % = 20.00 % (0.50 \times 16.0\%) + (0.50 \times 24.0\%) = 8.0\% + 12.0\% = 20.00\% ( 0.50 × 16.0% ) + ( 0.50 × 24.0% ) = 8.0% + 12.0% = 20.00% . There is zero diversification benefit.

Case 2: Realistic Moderate Correlation ( r A B = + 0.20 r_{AB} = +0.20 r A B ​ = + 0.20 )

σ p 2 = 208.0 + ( 192.0 × 0.20 ) = 208.0 + 38.4 = 246.4 \sigma_p^2 = 208.0 + (192.0 \times 0.20) = 208.0 + 38.4 = 246.4 σ p 2 ​ = 208.0 + ( 192.0 × 0.20 ) = 208.0 + 38.4 = 246.4 σ p = 246.4 ≈ 15.70 % \sigma_p = \sqrt{246.4} \approx 15.70\% σ p ​ = 246.4 ​ ≈ 15.70% Result: By combining the two assets, portfolio volatility drops from 20.00% to 15.70% (a 4.30 percentage point risk reduction), while the expected return remains steady at 8.00%. Remarkably, the portfolio risk (15.70%) is lower than Stock A's risk (16.00%) alone!

Case 3: Zero Correlation ( r A B = 0.0 r_{AB} = 0.0 r A B ​ = 0.0 )

σ p 2 = 208.0 + ( 192.0 × 0.0 ) = 208.0 \sigma_p^2 = 208.0 + (192.0 \times 0.0) = 208.0 σ p 2 ​ = 208.0 + ( 192.0 × 0.0 ) = 208.0 σ p = 208.0 ≈ 14.42 % \sigma_p = \sqrt{208.0} \approx 14.42\% σ p ​ = 208.0 ​ ≈ 14.42% Result: Risk drops further to 14.42% .

Case 4: Moderate Negative Correlation ( r A B = − 0.40 r_{AB} = -0.40 r A B ​ = − 0.40 )

σ p 2 = 208.0 + ( 192.0 × − 0.40 ) = 208.0 − 76.8 = 131.2 \sigma_p^2 = 208.0 + (192.0 \times -0.40) = 208.0 - 76.8 = 131.2 σ p 2 ​ = 208.0 + ( 192.0 × − 0.40 ) = 208.0 − 76.8 = 131.2 σ p = 131.2 ≈ 11.45 % \sigma_p = \sqrt{131.2} \approx 11.45\% σ p ​ = 131.2 ​ ≈ 11.45% Result: Portfolio risk plummets to 11.45% —nearly half the volatility of Stock B alone.

4. Systematic vs. Unsystematic Risk

Modern Portfolio Theory reveals that the total risk of any individual security consists of two fundamentally distinct parts:

Total Risk = Systematic Risk + Unsystematic Risk \text{Total Risk} = \text{Systematic Risk} + \text{Unsystematic Risk} Total Risk = Systematic Risk + Unsystematic Risk

σ total 2 = σ systematic 2 + σ unsystematic 2 \sigma_{\text{total}}^2 = \sigma_{\text{systematic}}^2 + \sigma_{\text{unsystematic}}^2 σ total 2 ​ = σ systematic 2 ​ + σ unsystematic 2 ​ Reduction of Portfolio Risk Through Diversification:

• Portfolio Risk (σ_p)
• ^
• |
• High | * Total Risk
• | \
• | \ [ Unsystematic Risk ] <--- Eliminated by diversification
• | ' - . _
• | '- . _
• | - - - - - - - - - - -'=========> [ Systematic Market Risk ]
• Low | (Cannot be diversified away)
• +---------------------------------------> Number of Stocks
• 0 10 20 30 40+
• 1. Unsystematic Risk (Specific, Diversifiable, or Idiosyncratic Risk)

Unsystematic risk represents the hazards unique to a single company, industry, or sector. Examples include:

An oil spill or pipeline shutdown affecting a specific energy producer.

A regulatory rejection of a drug developed by a Canadian biotechnology firm.

An unexpected executive departure, labor union strike, or forensic accounting scandal.

Because these firm-specific events occur independently across corporations, positive surprises in some companies cancel out negative surprises in others. Empirical research confirms that holding a portfolio of approximately 30 to 40 broadly diversified common stocks across diverse economic sectors virtually eliminates unsystematic risk.

2. Systematic Risk (Market, Non-Diversifiable, or Macroeconomic Risk)

Systematic risk reflects pervasive macroeconomic forces that impact the entire capital market simultaneously. Examples include:

Sudden shifts in the Bank of Canada overnight policy rate.

Nationwide economic recessions or spikes in domestic unemployment.

Major currency swings (CAD/USD exchange rate).

Global geopolitical conflicts or pandemics.

Because systematic forces affect all market participants to varying degrees, systematic risk cannot be eliminated through diversification . An investor holding all 220+ stocks in the S&P/TSX Composite Index remains fully exposed to systematic market downturns.

Direct Comparison: Systematic vs. Unsystematic Risk

• Feature Systematic Risk Unsystematic Risk
• Alternative Names Market risk, Non-diversifiable risk Specific risk, Diversifiable risk, Idiosyncratic risk
• Scope of Impact Entire financial market / macroeconomy Specific company, narrow industry, or issuer
• Can it be Diversified? No ; persists across all portfolios Yes ; eliminated with 30–40 diversified stocks

Key Examples Interest rate shocks, inflation, recession, wars CEO resignation, industrial strike, product recall, default

Metric Used Beta ( β \beta β ) Residual standard deviation (standard error)

Market Pricing Compensated by the market with a risk premium Not compensated ; market pays zero premium for avoidable risk