Glossary: M05 — Portfolio Mathematics and Probability Distributions

Module: M05 Formulas: Formula Sheet Concept pages: Portfolio Risk, Normal Distribution

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Portfolio Expected Return

The weighted average of expected returns of the assets in the portfolio, where weights are portfolio allocation proportions.

$E(R_p)={\sum }_{i=1}^n{w}_{i}E(R_i)$

LOS: 5.a | Related: Weighted Mean

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Covariance (Portfolio)

The expected value of the product of deviations of two asset returns from their respective expected values. Measures how two asset returns move together.

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

LOS: 5.a | Key: Negative covariance provides diversification benefits. | Related: Covariance

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Covariance Matrix

A square matrix displaying the covariances between all pairs of assets in a portfolio. Diagonal elements are variances; off-diagonal elements are covariances.

$\Sigma =\left(\begin{array}{ccc}{\sigma }_1^2& {\text{Cov}}_{12}& \cdots \\ {\text{Cov}}_{21}& {\sigma }_2^2& \cdots \\ ⋮& ⋮& \ddots \end{array}\right)$

LOS: 5.a | Note: A portfolio of $n$ assets has $n$ variances and $\frac{n(n-1)}{2}$ unique covariances.

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Correlation (Portfolio)

The standardized covariance between two asset returns, ranging from −1 to +1.

${\rho }_{ij}=\frac{\text{Cov}(R_i,R_j)}{{\sigma }_i{\sigma }_j}$

LOS: 5.a | Key: Diversification reduces risk most when correlation is low (ideally negative). | Related: Correlation

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Correlation Matrix

A square matrix displaying correlations between all pairs of assets. Diagonal elements equal 1; off-diagonal elements are correlations ${\rho }_{ij}$.

LOS: 5.a | Related: Covariance Matrix

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Portfolio Variance

The weighted sum of covariances among all pairs of assets in the portfolio. For a two-asset portfolio:

${\sigma }_p^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2\text{Cov}(R_1,R_2)$

General: ${\sigma }_p^2={\sum }_i{\sum }_j{w}_i{w}_j\text{Cov}(R_i,R_j)$

LOS: 5.a | Key: Portfolio variance depends on ALL pairwise covariances, not just individual asset variances.

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Portfolio Standard Deviation

The positive square root of Portfolio Variance. Represents the total risk of the portfolio.

${\sigma }_p=\sqrt{{\sigma }_p^2}$

LOS: 5.a | Related: Portfolio Risk Concept

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Diversification Benefit

The reduction in portfolio risk (variance/standard deviation) achieved by combining assets whose returns are not perfectly positively correlated. Diversification eliminates unsystematic (firm-specific) risk.

LOS: 5.a | Key: Maximum diversification benefit when $\rho =-1$. No benefit when $\rho =+1$. | Related: Portfolio Risk

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Probability Distribution

A function that assigns probabilities to all possible values of a random variable. Must satisfy: $P(X=x)\ge 0$ for all $x$ and the sum (or integral) of all probabilities equals 1.

LOS: 5.b | Types: Discrete distributions, Continuous distributions.

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Probability Function

For a discrete random variable, the function $p(x)=P(X=x)$ that gives the probability of each specific value. Also called the probability mass function (PMF).

LOS: 5.b | Contrast: For continuous random variables, a probability density function (PDF) is used.

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Cumulative Distribution Function (CDF)

The function $F(x)=P(X\le x)$ giving the probability that a random variable takes a value less than or equal to $x$. Ranges from 0 to 1 and is non-decreasing.

LOS: 5.b | Related: Cumulative Frequency

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Discrete Random Variable

A random variable that can take on only a countable number of distinct values (e.g., integers). Described by a probability mass function.

LOS: 5.b | Examples: Number of defaults in a portfolio; outcome of rolling a die. | Related: Binomial Distribution, Discrete Uniform Distribution

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Continuous Random Variable

A random variable that can take on any value in an interval. Described by a probability density function. The probability of any single point equals zero.

LOS: 5.b | Examples: Stock returns, interest rates. | Related: Normal Distribution, Continuous Uniform Distribution

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Discrete Uniform Distribution

A distribution where each of $n$ possible outcomes is equally likely. The simplest discrete distribution.

$P(X=x_i)=\frac{1}{n}\text{for each }i=1,2,\dots ,n$

LOS: 5.c | Example: Rolling a fair die — each face has probability 1/6.

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Continuous Uniform Distribution

A distribution where the random variable is equally likely to take any value between a lower bound $a$ and upper bound $b$.

$f(x)=\frac{1}{b-a}\text{for }a\le x\le b$

$P(x_1\le X\le x_2)=\frac{x_2-x_1}{b-a}$

LOS: 5.c

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Bernoulli Distribution

A distribution for a single trial with two possible outcomes: success (probability $p$) or failure (probability $1-p$).

$E(X)=p\text{Var}(X)=p(1-p)$

LOS: 5.d | Related: Binomial Distribution (sum of independent Bernoulli trials)

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Binomial Distribution

The probability distribution of the number of successes in $n$ independent Bernoulli trials, each with probability of success $p$.

$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$

$E(X)=np\text{Var}(X)=np(1-p)$

LOS: 5.d | Application: Probability of $k$ stocks rising out of $n$ stocks in a portfolio. | Related: Combination

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Normal Distribution

A symmetric, bell-shaped continuous probability distribution completely described by its mean $\mu$ and variance ${\sigma }^2$. The most important distribution in statistics.

$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}$

LOS: 5.e | Key properties: Symmetric; mean = median = mode; 68%/95%/99% rule. | See: Normal Distribution Concept

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Standard Normal Distribution

A normal distribution with mean $\mu =0$ and variance ${\sigma }^2=1$. Used with $z$-tables to compute probabilities for any normal distribution.

$Z\sim N(0,1)$

LOS: 5.e | Related: Z-Score, Normal Distribution

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Z-Score

The number of standard deviations by which an observation differs from the mean. Standardizes a normal random variable to the standard normal.

$Z=\frac{X-\mu }{\sigma }$

LOS: 5.e | Use: Look up $P(Z\le z)$ in standard normal table. | Related: Standard Normal Distribution

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Shortfall Risk

The probability that a portfolio’s return falls below a specified minimum acceptable return (threshold level or target). A downside risk measure.

LOS: 5.f | Related: Safety-First Ratio, Roy’s Safety-First Criterion

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Safety-First Ratio

A measure for comparing portfolios on the basis of downside risk. Higher is better.

$\text{SF Ratio}=\frac{E(R_p)-R_L}{{\sigma }_p}$

where $R_L$ = minimum acceptable (threshold) return.

LOS: 5.f | Note: Structurally similar to the Sharpe ratio, but uses $R_L$ instead of $R_f$. | Related: Roy’s Safety-First Criterion

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Roy’s Safety-First Criterion

An approach to portfolio selection that chooses the portfolio with the highest safety-first ratio — i.e., the portfolio that minimizes the probability of portfolio return falling below the threshold return $R_L$.

$\text{Optimal portfolio}:\max \left[\frac{E(R_p)-R_L}{{\sigma }_p}\right]$

LOS: 5.f | Related: Safety-First Ratio, Shortfall Risk