Portfolio Mathematics - Quantitative Methods | CFA® Level 1 Guide

Portfolio Expected Return and Variance of Return

Learning Outcome Statement:

calculate and interpret the expected value, variance, standard deviation, covariances, and correlations of portfolio returns

Summary:

This LOS focuses on the calculation and interpretation of key statistical measures used in portfolio management, including expected value, variance, standard deviation, covariance, and correlation of portfolio returns. These measures are essential for assessing the performance and risk of a portfolio, and they play a crucial role in portfolio construction and optimization.

Key Concepts:

Expected Return of Portfolio

The expected return on a portfolio is calculated as a weighted average of the expected returns of the individual assets in the portfolio, where the weights are the proportions of the total portfolio value that each asset represents.

Portfolio Variance

Portfolio variance is a measure of the dispersion of returns from the expected return. It incorporates both the variances of individual asset returns and the covariances between pairs of assets.

Covariance

Covariance measures the directional relationship between the returns on two assets. A positive covariance indicates that asset returns move together, while a negative covariance indicates that they move inversely.

Correlation

Correlation is a standardized measure of the strength of the linear relationship between two variables, ranging from -1 to 1. It is derived from covariance and the standard deviations of the variables involved.

Formulas:

Expected Return of Portfolio

E(R_p) = \sum_{i=1}^{n} w_i E(R_i)

This formula calculates the expected return of a portfolio by summing the products of the weights and expected returns of each asset.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$w_i$ :: Weight of asset i in the portfolio
$E(R_i)$ :: Expected return of asset i
$n$ :: Total number of assets in the portfolio

Units: percentage or decimal

Portfolio Variance

\sigma^2(R_p) = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \text{Cov}(R_i, R_j)

This formula calculates the variance of the portfolio return, considering both the variances of individual asset returns and the covariances between different assets.

Variables:

$\sigma^2(R_p)$ :: Variance of the portfolio return
$w_i, w_j$ :: Weights of assets i and j in the portfolio
$\text{Cov}(R_i, R_j)$ :: Covariance between returns of assets i and j
$n$ :: Total number of assets in the portfolio

Units: percentage squared or decimal squared

Covariance

\text{Cov}(R_i, R_j) = E[(R_i - E(R_i))(R_j - E(R_j))]

Covariance measures the extent to which the returns on two assets move in tandem. A positive value indicates that they tend to move in the same direction, while a negative value indicates they tend to move in opposite directions.

Variables:

$\text{Cov}(R_i, R_j)$ :: Covariance between returns of assets i and j
$R_i, R_j$ :: Returns of assets i and j
$E(R_i), E(R_j)$ :: Expected returns of assets i and j

Units: percentage squared or decimal squared

Correlation

\rho(R_i, R_j) = \frac{\text{Cov}(R_i, R_j)}{\sigma(R_i) \sigma(R_j)}

Correlation is a normalized measure of the strength of the linear relationship between two variables, scaled to be between -1 and 1. It is derived from the covariance of the variables divided by the product of their standard deviations.

Variables:

$\rho(R_i, R_j)$ :: Correlation between returns of assets i and j
$\text{Cov}(R_i, R_j)$ :: Covariance between returns of assets i and j
$\sigma(R_i), \sigma(R_j)$ :: Standard deviations of returns of assets i and j

Units: unitless

Portfolio Risk Measures: Applications of the Normal Distribution

Learning Outcome Statement:

define shortfall risk, calculate the safety-first ratio, and identify an optimal portfolio using Roy’s safety-first criterion

Summary:

This LOS explores the application of the normal distribution in portfolio risk measures, specifically through the concept of shortfall risk and Roy's safety-first criterion. It discusses how mean-variance analysis can be applied under the assumption of normally distributed returns or quadratic utility functions. The safety-first criterion focuses on minimizing the probability that portfolio returns fall below a certain threshold, using the safety-first ratio (SFRatio) to compare portfolios.

Key Concepts:

Shortfall Risk

Shortfall risk refers to the risk that portfolio returns will fall below a minimum acceptable level over a specified time horizon. It is particularly relevant in contexts like pension funds where meeting future liabilities is crucial.

Safety-First Ratio (SFRatio)

The safety-first ratio is calculated as the difference between the expected portfolio return and the threshold level, divided by the standard deviation of the portfolio. It measures how many standard deviations the threshold is away from the expected return.

Roy's Safety-First Criterion

This criterion is used to select the optimal portfolio by maximizing the safety-first ratio. The optimal portfolio under this criterion has the highest safety-first ratio, thus minimizing the probability of the return falling below the threshold level.

Normal Distribution Assumption

The assumption that portfolio returns are normally distributed simplifies the calculation of the probability of achieving returns above or below a certain threshold, using the properties of the normal distribution.

Formulas:

Safety-First Ratio (SFRatio)

SFRatio = \frac{E(RP) - RL}{\sigma_P}

This formula calculates the safety-first ratio, which is used to determine the optimal portfolio under Roy's safety-first criterion. It measures the distance of the threshold from the mean return in units of standard deviation.

Variables:

$E(RP)$ :: Expected return of the portfolio
$RL$ :: Return threshold level (shortfall level)
$\sigma_P$ :: Standard deviation of the portfolio returns

Units: dimensionless (ratio)