11.3 Portfolio Risk & Return: Correlation and Diversification

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.

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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:

$E(R_p) = \sum_{i=1}^n w_i E(R_i) = w_1 E(R_1) + w_2 E(R_2) + \dots + w_n E(R_n)$

Where:

• $w_i$ = Portfolio weight allocated to asset $i$ (expressed as a fraction of total portfolio value, such that $\sum w_i = 1.0$)
• $E(R_i)$ = Expected return of asset $i$

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

$E(R_p) = w_A E(R_A) + w_B E(R_B)$

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.

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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:

1. The weight assigned to each security ($w_i$).
2. The individual volatility of each security ($\sigma_i$).
3. The degree of co-movement between the securities, measured by covariance ($\text{Cov}_{AB}$) or the correlation coefficient ($r_{AB}$).

The Two-Asset Portfolio Variance Formula

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

$\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}(A,B)$

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

$\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B r_{AB}$

Taking the square root gives the portfolio standard deviation ($\sigma_p$):

$\sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B r_{AB}}$

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 ($r_{AB}$)

The correlation coefficient ($r$ or $\rho$) 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:

$-1.0 \le r_{AB} \le +1.0$

The Golden Rule of Diversification

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

In real-world Canadian financial markets, finding assets with perfect negative correlation ($r = -1.0$) 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 $r < 1.0$, combining these assets consistently lowers total portfolio volatility.

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3. Comprehensive Worked Numeric Calculation: Two-Asset Diversification

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

• Stock A (Bank): $E(R_A) = 10.0\%$, $\sigma_A = 16.0\%$
• Stock B (Gold): $E(R_B) = 6.0\%$, $\sigma_B = 24.0\%$

Step 1: Calculate Portfolio Expected Return

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

Step 2: Calculate Weighted Variance Components

• $w_A^2 \sigma_A^2 = (0.50)^2 \times (16.0)^2 = 0.25 \times 256 = 64.0$
• $w_B^2 \sigma_B^2 = (0.50)^2 \times (24.0)^2 = 0.25 \times 576 = 144.0$
• Sum of weighted variance terms: $64.0 + 144.0 = 208.0$
• Interactive term multiplier: $2 w_A w_B \sigma_A \sigma_B = 2 \times 0.50 \times 0.50 \times 16.0 \times 24.0 = 192.0$

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

Case 1: Perfect Positive Correlation ($r_{AB} = +1.0$)

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

Case 2: Realistic Moderate Correlation ($r_{AB} = +0.20$)

$\sigma_p^2 = 208.0 + (192.0 \times 0.20) = 208.0 + 38.4 = 246.4$ $\sigma_p = \sqrt{246.4} \approx 15.70\%$ 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 ($r_{AB} = 0.0$)

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

Case 4: Moderate Negative Correlation ($r_{AB} = -0.40$)

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

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4. Systematic vs. Unsystematic Risk

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

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

$\sigma_{\text{total}}^2 = \sigma_{\text{systematic}}^2 + \sigma_{\text{unsystematic}}^2$

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