Library/CFA (Chartered Financial Analyst)/JuiceNotes™ 2024: Quantitative Methods - CFA® Level I/Portfolio Mathematics

Portfolio Mathematics

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Portfolio Return and Risk Measures10 min

The fundamental building blocks of portfolio mathematics are the expected return and the standard deviation (risk). The expected return is a linear combination of asset returns weighted by their allocation. However, portfolio risk is non-linear due to the interaction between assets, quantified by covariance and correlation. The variance of a two-asset portfolio is derived from the square of individual weighted risks plus a cross-term: 2 Weight_A Weight_B * Covariance_AB. The text highlights that correlation is a standardized measure of this interaction. As the number of assets increases (e.g., to three), the number of covariance terms grows, emphasizing the impact of inter-asset relationships on total portfolio risk.

Key Points

Portfolio Expected Return is a weighted average of individual returns.
Portfolio Variance depends on weights, individual variances, and covariance.
Covariance equals Correlation times the product of individual Standard Deviations.
The variance formula expands with the number of assets, adding covariance terms for every pair.

Joint Probability and Covariance Calculation10 min

Covariance can be calculated directly from a joint probability distribution of asset returns. This process involves constructing a table where probabilities are assigned to joint outcomes of returns for Asset A and Asset B. The process requires three steps: 1) Calculate the expected return for A and B individually. 2) Calculate the expected value of the product of returns, E(AB), by summing the product of Prob Return_A Return_B for all scenarios. 3) Compute covariance as E(AB) minus the product of individual expected returns E(A)E(B). This method provides a granular view of how assets co-move across different economic scenarios.

Key Points

Joint probability tables map probabilities to pairs of asset returns.
E(AB) is the probability-weighted sum of the product of returns.
Covariance formula: Cov(A,B) = E(AB) - [E(A) * E(B)].
This method links scenario analysis directly to risk metrics.

Mean-Variance Analysis and Safety First Rule10 min

Mean-Variance Analysis (MVA) is the standard framework for weighing risk against reward, relying on the assumption that investors are risk-averse and that returns follow a normal distribution or utility is quadratic. To address downside risk specifically, Roy's Safety First Rule is used. It focuses on the likelihood of returns dropping below a critical threshold (Minimum Acceptable Return). The Safety First Ratio (SFR) measures how many standard deviations the expected return is above this threshold. A higher SFR indicates a 'safer' portfolio. The shortfall risk is then determined by finding the probability associated with the negative SFR value in a standard normal distribution table.

Key Points

MVA assumes risk aversion and normal distributions (or quadratic utility).
Safety First Ratio (SFR) = (E(Rp) - MAR) / Sigma_p.
Investors should maximize the SFR to minimize Shortfall Risk.
Shortfall Risk is the probability of returns < Minimum Acceptable Return.

Value at Risk and Stress Testing5 min

Financial risk management employs metrics like Value at Risk (VaR) to quantify potential losses. VaR estimates the minimum loss expected over a specific time frame with a certain confidence level (e.g., 5 percent or 1 percent). While VaR handles standard market conditions, Stress Testing and Scenario Analysis are used to assess resilience against extreme, often non-statistical, market shocks. These methods simulate unfavorable combinations of events to ensure the portfolio can withstand rare but catastrophic scenarios.

Key Points

VaR is a money measure of minimum expected loss at a given probability.
VaR is defined by a time period and a confidence level (probability).
Stress testing evaluates losses in extremely unfavorable scenarios.
Scenario analysis complements VaR by addressing tail risks.

Questions

Question 1

How is the expected return of a portfolio calculated?

Portfolio Mathematics

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