CHAPTER 3 (PART 2): PORTFOLIO RETURN, RISK, AND OPTIMIZATION

3.5 EXPECTED RATE OF RETURN FOR A PORTFOLIO (EX-ANTE RETURN)

3.5.1 The Weighted Average Approach

While the risk of a portfolio is not a simple weighted average of the risks of its individual components, the expected rate of return of a portfolio is calculated using a straightforward weighted average. The expected rate of return for a portfolio of assets is the sum of the expected rates of return of the individual investments, with each investment's return weighted by the proportion of total wealth allocated to that specific asset.

3.5.2 The Mathematical Formula

Written in a simple, copy-ready single-line format:

Portfolio Expected Return: E(R_port) = Sum from i = 1 to n of [W_i * E(R_i)]

Where:

• E(R_port) = The expected (Ex-Ante) rate of return of the entire portfolio.
• W_i = The percentage or proportion of the total portfolio value invested in asset i.
• E(R_i) = The expected rate of return for individual asset i.
• n = The total number of individual assets held within the portfolio.

3.5.3 Worked Example: 4-Asset Portfolio

Suppose a Category III Alternative Investment Fund (AIF) constructs a portfolio comprising four distinct assets with varying allocations and expected returns. The step-by-step portfolio return calculation is outlined in Table 3.3 below:

Table 3.3: Portfolio Return Calculation (4-Asset Portfolio)

Step-by-Step Mathematical Formulation:

E(R_port) = (0.20 * 9%) + (0.10 * 12%) + (0.30 * 15%) + (0.40 * 18%) E(R_port) = 1.80% + 1.20% + 4.50% + 7.20% E(R_port) = 14.70% (or 0.147 as a decimal decimal fraction)

This demonstrates that the ex-ante portfolio return is 14.70%.

3.6 VARIANCE OF RETURN FOR A PORTFOLIO

3.6.1 The Foundations of Portfolio Risk

To compute the true risk of a portfolio, a portfolio manager cannot simply calculate the weighted average of individual standard deviations. Doing so completely ignores the interaction between assets. Harry Markowitz mathematically proved that portfolio risk is dictated by three distinct variables:

1. The weights of the individual investments.
2. The individual risk (standard deviation) of each investment.
3. The co-movement (covariance or correlation) between the investments.

3.6.2 Statistical Pillars: Covariance vs. Correlation

The interaction and co-movement of asset returns are measured using two fundamental statistical concepts: Covariance and Correlation.

1. Covariance

Covariance measures the degree and direction in which the returns of two assets move together relative to their respective individual mean values over time.

Written in a simple, copy-ready single-line format:

Covariance: Cov_ij = [Sum from t = 1 to n of ((R_it - E(R_i)) * (R_jt - E(R_j)))] / n

Where:

• Cov_ij = The covariance between the returns of asset i and asset j.
• R_it = The actual return of asset i in period t.
• E(R_i) = The expected return of asset i.
• R_jt = The actual return of asset j in period t.
• E(R_j) = The expected return of asset j.
• n = The number of historical observations or periods.

2. Correlation (Correlation Coefficient)

Because covariance is an unbounded measure (affected by the scale of the underlying asset returns), it is difficult to compare across different asset classes. To resolve this, covariance is standardized by dividing it by the product of the individual standard deviations of the two assets. This yields the correlation coefficient.

Written in a simple, copy-ready single-line format:

Correlation Coefficient: r_ij = Cov_ij / (sigma_i * sigma_j)

Where:

• r_ij = The correlation coefficient between the returns of asset i and asset j.
• Cov_ij = The covariance between the returns of asset i and asset j.
• sigma_i = The standard deviation of asset i.
• sigma_j = The standard deviation of asset j.

Key Properties of the Correlation Coefficient (r_ij):

• Bounded Range: The correlation coefficient is strictly bounded between -1.00 and +1.00.
• Perfect Positive Correlation (r_ij = +1.00): The returns of the two assets move together in a completely linear, parallel manner. Portfolio diversification in this scenario provides no risk-reduction benefits beyond a simple weighted average of risk.
• Perfect Negative Correlation (r_ij = -1.00): The returns of the two assets move in completely opposite directions. If constructed correctly, a portfolio with perfectly negatively correlated assets can eliminate risk entirely.
• No Correlation (r_ij = 0.00): The returns of the two assets move completely independently of one another. Significant diversification benefits are achieved.

3.6.3 The General Portfolio Variance Formula

For a portfolio containing multiple assets, the general mathematical formula for portfolio risk (standard deviation) incorporates the individual variances of each asset and the pairwise covariances between all unique combinations of assets:

Written in a simple, copy-ready single-line format:

Portfolio Risk: sigma_port = Square Root of (Sum from i = 1 to n of [W_i^2 * sigma_i^2] + Sum from i = 1 to n of Sum from j = 1 to n of [W_i * W_j * Cov_ij]) where i is not equal to j

3.6.4 Standard Two-Asset Portfolio Risk Formula

For a simplified portfolio consisting of exactly two assets (Asset A and Asset B), the formulas for portfolio variance and standard deviation are written as follows:

Written in simple, copy-ready single-line formats:

Portfolio Variance (2 Assets): sigma_port^2 = (W_A^2 * sigma_A^2) + (W_B^2 * sigma_B^2) + (2 * W_A * W_B * r_AB * sigma_A * sigma_B)

Portfolio Standard Deviation (2 Assets): sigma_port = Square Root of Portfolio Variance (sigma_port^2)

Where:

• sigma_port^2 = The total variance of the two-asset portfolio.
• sigma_port = The standard deviation of the two-asset portfolio.
• W_A, W_B = The portfolio weights of Asset A and Asset B.
• sigma_A, sigma_B = The standard deviations (individual risk) of Asset A and Asset B.
• r_AB = The correlation coefficient between the returns of Asset A and Asset B.
• Cov_AB (which equals r_AB * sigma_A * sigma_B) = The covariance between Asset A and Asset B.

3.6.5 Step-by-Step Worked Mathematical Example

Let us compute the total portfolio variance and standard deviation for a two-security portfolio using the parameters outlined in Table 3.4:

Table 3.4: Portfolio Risk Parameters (Two Securities)

Step-by-Step Risk Calculation Walkthrough:

Step 1: Calculate the Weighted Variance of Security A Term 1 = W_A^2 * sigma_A^2 Term 1 = (0.50^2) * (0.05^2) Term 1 = 0.25 * 0.0025 = 0.000625

Step 2: Calculate the Weighted Variance of Security B Term 2 = W_B^2 * sigma_B^2 Term 2 = (0.50^2) * (0.05^2) Term 2 = 0.25 * 0.0025 = 0.000625

Step 3: Calculate the Covariance Contribution (Co-movement Term) Term 3 = 2 * W_A * W_B * r_AB * sigma_A * sigma_B Term 3 = 2 * 0.50 * 0.50 * 0.50 * 0.05 * 0.05 Term 3 = 0.25 * 0.0025 Term 3 = 0.000625

Step 4: Sum the terms to find Portfolio Variance (sigma_port^2) sigma_port^2 = Term 1 + Term 2 + Term 3 sigma_port^2 = 0.000625 + 0.000625 + 0.000625 sigma_port^2 = 0.001875 (Variance)

Step 5: Calculate Portfolio Standard Deviation (sigma_port) sigma_port = Square Root of 0.001875 sigma_port = 0.0433 (or 4.33%)

Key Insight for Exam Prep:

Note that the standard deviation of each stock in isolation is 5.00%. However, because they are not perfectly correlated (r_AB = 0.50), combining them in equal weights reduces the overall portfolio standard deviation to 4.33%. This is the mathematical proof of diversification reducing total portfolio volatility.

3.7 THE EFFICIENT FRONTIER

3.7.1 Weight Combinations and Curve Plotting

When we take two risky securities and plot their expected return and risk (standard deviation) across every possible weight combination, they map out a curve rather than a straight line, provided the correlation coefficient between them is less than perfect (+1.00).

When we expand this process to include hundreds of assets in the investment universe, we can combine them in an infinite number of weight combinations. Plotting all of these potential combinations on a graph yields a cloud of feasible portfolios.

3.7.2 Defining the Efficient Frontier (The "Umbrella" Curve)

The upper boundary of this feasible cloud forms a distinct, umbrella-shaped curve known as the Efficient Frontier.

The Efficient Frontier is defined as the set of optimal portfolios that offer:

1. The maximum expected rate of return for a given level of risk.
2. The minimum level of risk for a given level of expected return.

Exhibit 3.2: Feasible Cloud and Efficient Frontier (Conceptual Plot)

3.7.3 The Dominance Principle: Portfolio Analysis

We can evaluate the efficiency of specific portfolios using the Dominance Principle, comparing portfolios lying on the frontier against those lying inside the cloud:

• Portfolio A vs. Portfolio C (Same Return, Different Risk): Portfolio A lies directly on the Efficient Frontier, whereas Portfolio C lies deep inside the feasible cloud. Both portfolios offer the exact same expected rate of return. However, Portfolio C carries a much higher level of risk than Portfolio A. Therefore, a rational, risk-averse investor will always choose Portfolio A, and Portfolio A is said to dominate Portfolio C.

• Portfolio B vs. Portfolio C (Different Return, Same Risk): Portfolio B lies directly on the Efficient Frontier, while Portfolio C lies inside the feasible cloud. Both portfolios carry the exact same level of risk (standard deviation). However, Portfolio B offers a significantly higher expected return than Portfolio C. Consequently, a rational investor will always select Portfolio B, and Portfolio B dominates Portfolio C.

3.8 PORTFOLIO OPTIMIZATION PROCESS

3.8.1 Defining the Optimal Portfolio

An optimum portfolio is a specific combination of investments designed to maximize expected return for a given level of risk, or minimize risk for a specified return target, based on the investor's unique utility function and a given set of constraints.

3.8.2 The Three Critical Statistical Inputs

To execute the Modern Portfolio Theory framework and perform portfolio optimization, a portfolio manager must estimate three critical categories of statistical inputs for the entire investment universe:

1. The expected rate of return for every asset class, security, and investment opportunity.
2. The standard deviation (volatility) of each individual asset's expected returns.
3. The correlation coefficients between every unique pair of assets in the investment universe, taken two at a time.

3.8.3 The Structural Optimization Flowchart

The systematic process of constructing and selecting an optimal portfolio can be conceptualized in the following linear flow:

3.9 ESTIMATION ISSUES AND ESTIMATION RISK

3.9.1 The Challenge of Input Scale

While the Markowitz model is mathematically elegant, applying it in practice presents a major computational hurdle. As the number of securities in an investment universe grows, the number of required correlation estimates increases exponentially.

3.9.2 The Correlation Estimation Formula

To find the exact number of correlation estimates required to construct a portfolio, use the following simple line formula:

Correlation Estimates Required = (n^2 - n) / 2

Where:

• n = The total number of individual securities in the investment universe.

Example Calculation:

For a moderate portfolio consisting of 50 securities (n = 50), let us calculate the number of unique correlation estimates a portfolio manager must generate: Correlation Estimates = (50^2 - 50) / 2 Correlation Estimates = (2500 - 50) / 2 Correlation Estimates = 2450 / 2 Correlation Estimates = 1,225 estimates

This means that for just 50 stocks, the manager must calculate and maintain 1,225 unique correlation coefficients. For 100 stocks, this number surges to 4,950 estimates.

3.9.3 Understanding Estimation Risk

The outputs of any portfolio optimization and allocation model are highly sensitive to the accuracy of the statistical inputs. If a manager makes slight errors in estimating expected returns, standard deviations, or correlation coefficients, the resulting "optimized" portfolio may perform poorly. This risk is formally known as Estimation Risk.

3.9.4 Managing Portfolios with the Variance-Covariance Matrix

For portfolios containing more than two securities, portfolio risk calculations require the construction of a Variance-Covariance Matrix. This matrix acts as a grid displaying the variances of individual assets along the diagonal and the covariances between all asset pairs in the remaining cells. Today, these matrices are calculated programmatically using spreadsheets or financial modeling software to automate optimization.

3.10 KEY TERMINOLOGY AND DEFINITIONS (PART 2)

• Expected Return of a Portfolio: The weighted average of the expected rates of return of the individual assets making up the portfolio.
• Covariance: A statistical measure indicating how the returns of two assets move together relative to their respective averages.
• Correlation Coefficient: A standardized measure of covariance ranging from -1.00 to +1.00, indicating the linear strength and direction of co-movement between two assets.
• Efficient Frontier: An umbrella-shaped curve displaying the set of all optimal portfolios that offer the highest possible return for a given level of risk.
• Dominance Principle: An investment rule stating that a portfolio is preferred if it offers a higher return for the same level of risk, or lower risk for the same return.
• Optimal Portfolio: The portfolio that best matches an investor's specific risk-return preferences and constraints.
• Estimation Risk: The potential for portfolio optimization models to produce inaccurate or sub-optimal allocations due to errors in estimating returns, standard deviations, and correlations.
• Variance-Covariance Matrix: A mathematical matrix displaying the individual variances and pairwise covariances of all assets in an investment universe.

3.11 EXAM-FOCUSED KEY TAKEAWAYS (PART 2)

1. Portfolio Return is Weighted: Unlike risk, portfolio expected return is always a simple weighted average of the expected returns of its component assets.
2. Risk is Non-Linear: Portfolio risk (standard deviation) is not a simple weighted average of individual standard deviations. It depends heavily on the co-movement (covariance and correlation) between asset returns.
3. The Diversification Benefit: As long as the correlation coefficient between assets is less than perfect positive (+1.00), combining them will reduce the overall portfolio standard deviation below the weighted average of individual standard deviations.
4. Efficient Frontier Rule: Every portfolio lying on the Efficient Frontier dominates any portfolio lying inside the cloud because it offers higher return for the same risk, or lower risk for the same return.
5. Optimization Input Trio: The three essential inputs required for MPT portfolio optimization are: (1) Expected returns, (2) Standard deviations, and (3) Pairwise correlation coefficients.
6. Estimation Scale Formula: The number of correlation estimates required for a portfolio of size n is calculated as (n^2 - n) / 2. Understand this formula, as it is a common target for computational exam questions.