Covariance and correlation from a joint probability table

What is a joint probability function?

A joint probability function gives the probability that two random variables take particular values at the same time. In finance it usually looks like a short scenario analysis : a handful of states of the world, each with a probability and a return for each asset. When you have forecasts rather than history, this table is the raw material for covariance and correlation.

Here is the running example. Two assets face three states of the economy.

State	Probability	R_A	R_B
Strong	0.25	18%	4%
Steady	0.45	6%	10%
Weak	0.30	−4%	0%

How do you find the covariance from the table?

Work in four steps: expected returns, deviations, probability-weighted products, then add.

$$E(R) = \sum_{i} P(s_i)\, R_i$$

$$\mathrm{Cov}(R_A, R_B) = \sum_{i} P(s_i)\,\big[R_{A,i} - E(R_A)\big]\big[R_{B,i} - E(R_B)\big]$$

Two shortcuts help here. A state in which either deviation is zero adds zero, and the identity below avoids deviations altogether.

$$\mathrm{Cov}(R_A, R_B) = E(R_A R_B) - E(R_A)\,E(R_B)$$

Change a return and watch where each state sits relative to the cross-hairs at the two expected returns. Top right and bottom left add to the covariance, the other two corners subtract from it, and a bigger probability means a bigger bubble and a bigger vote.

How do you turn the covariance into a correlation?

Covariance on its own tells you only the direction. Divide by both standard deviations to get a strength between −1 and +1. The standard deviations come from the same table: weight each squared deviation by its probability, with no n − 1.

$$\rho_{AB} = \frac{\mathrm{Cov}(R_A, R_B)}{\sigma_A\,\sigma_B}$$

Joint table: probability, R_A and R_B in each stateprobabilities add to 1; no n − 1

• 1. Expected returns Memorize E(R_A) = Σ P × R_A and E(R_B) = Σ P × R_B
• 2. Covariance Memorize Σ P × (R_A − E_A) × (R_B − E_B)
  • Shortcut E(R_A R_B) − E(R_A) E(R_B)
• 3. Standard deviations Use it √ of Σ P × (R − E)²
• 4. Correlation Memorize Cov ÷ (σ_A σ_B), between −1 and +1

Exam tip The positive covariance 12.0 looks large, yet the correlation is only 0.34. Always convert to correlation before judging strength.

What if the joint table is a grid?

Sometimes the joint probabilities come as a grid, with A’s possible returns down the side and B’s across the top. Each cell holds the probability that both happen together. The formula is unchanged: take every cell with a non-zero probability, treat it as a scenario, and sum.

P(R_A, R_B)	R_B = 12%	R_B = 5%	R_B = −2%
R_A = 18%	0.25	0.05	0
R_A = 7%	0.05	0.40	0
R_A = −4%	0	0.05	0.20

What changes when returns are independent?

Two random variables are independent random variables when the joint probability of any pair of outcomes is the product of the separate probabilities.

$$P(X, Y) = P(X)\,P(Y)$$

Suppose X has a 0.4 chance of “up” and Y has a 0.5 chance, and they are independent. Every joint cell is the product of its row and column totals.

P(X, Y)	Y up (0.5)	Y down (0.5)	Total
X up (0.4)	0.20	0.20	0.40
X down (0.6)	0.30	0.30	0.60
Total	0.50	0.50	1.00

With a table like this, the expected product splits too, and the covariance collapses to zero:

$$E(XY) = E(X)\,E(Y) \quad \text{if } X, Y \text{ are uncorrelated}$$

Many financial quantities are products of random variables. Revenue is price times quantity: if expected price is 20 and expected quantity is 300 and the two are uncorrelated, expected revenue is 20 × 300 = 6,000. The rule is for uncorrelated returns in the sense of zero covariance, and independent variables satisfy it automatically.

Two random returnscovariance only sees straight-line links

• Independent Use it P(X, Y) = P(X) P(Y); knowing one tells you nothing about the other
  • Implies covariance 0, correlation 0, E(XY) = E(X)E(Y)
• Uncorrelated Use it covariance 0 but a nonlinear link may remain
• Positive covariance Memorize returns tend to sit on the same side of their means
• Negative covariance Memorize one tends to be above while the other is below

Go deeper Zero covariance does not imply independence

Let X be −1, 0 or +1, each with probability 1/3, and let Y = X². Then Y is completely determined by X, so the two are certainly not independent. But E(X) = 0 and E(XY) = E(X³) = (−1 + 0 + 1) ÷ 3 = 0, so Cov(X, Y) = 0 − 0 × E(Y) = 0.

Covariance and correlation only detect straight-line association. Independence is the stronger property, which is why independent variables are always uncorrelated but uncorrelated variables are not always independent.

Go deeper Where scenario tables are used

An analyst builds a table like this by choosing a few states of the world, assigning probabilities and forecasting each asset’s return in each state. The same layout supports stress testing, where one state is made deliberately extreme. Because the probabilities are forecasts, the covariance is forward-looking, and there is no n − 1 anywhere. A sample covariance from historical data, in contrast, divides by n − 1.