Covariance and correlation from a joint probability table

What is a joint probability table?

A joint probability is the chance that two random variables take particular values at the same time. A joint probability table (also called a joint distribution) lists those chances. 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 the unconditional covariance : the covariance computed over all scenarios, before you know which one happens. Its symbol is σ_XY = Cov(X, Y); for two assets A and B we write σ_AB or Cov(R_A, R_B).

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 three steps: the expected returns, the expected product of the two returns, then subtract. This shortcut is the main route, because it needs no deviations.

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

$$\sigma_{AB} = \mathrm{Cov}(R_A, R_B) = E[R_A R_B] - E[R_A]\,E[R_B]$$

Why the shortcut works: the deviation form

Covariance is the probability-weighted average of the product of the two deviations from the means. That is what the shortcut computes without taking the deviations, and it is the form that shows what the sign means.

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

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. The Independent returns view adds row and column totals, the marginal probabilities.

How do you turn the covariance into a correlation?

Covariance on its own tells you only the direction. A large covariance does not by itself mean a strong relationship: covariance grows with the volatility of both variables. Strength is the correlation’s job. Divide by both standard deviations to get a number between −1 and +1. The standard deviations come from the same table: weight each squared deviation by its probability, with no n − 1.

$$\rho_{XY} = \dfrac{\sigma_{XY}}{\sigma_X\,\sigma_Y} = \dfrac{\mathrm{Cov}(X, Y)}{\sigma_X\,\sigma_Y}$$

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. Unconditional covariance Memorize E[R_A R_B] − E[R_A] E[R_B]
  • Deviation form Σ P × (R_A − E_A) × (R_B − E_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%	0.05	0.20	0
R_A = 7%	0.25	0.05	0
R_A = −4%	0	0.20	0.25

What are joint and marginal distributions?

The joint distribution function gives the probability that both variables are at or below chosen values at the same time. It is built from the joint probabilities.

$$F_{12}(x_1, x_2) = P(X_1 \le x_1,\; X_2 \le x_2)$$

In the grid above, P(R_A ≤ 7% and R_B ≤ 5%) adds the cells (7, 5), (7, −2), (−4, 5) and (−4, −2): 0.05 + 0 + 0.20 + 0.25 = 0.50.

The marginal probability is the distribution of one variable on its own. For a grid it is the row or column total, as in the expected-return step above. For continuous variables you find it by integrating the joint density over the other variable; that is for understanding only.

A contingency table shows how often combinations of categories occur, and works with counts. Here are 500 corporate bonds by rating and by whether they were downgraded within a year.

	Downgraded	Not downgraded	Total
Investment grade	38	287	325
High yield	42	133	175
Total	80	420	500

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 0.80 chance of “up” and Y has a 0.70 chance, and they are independent. Every joint cell is the product of its row and column totals.

P(X, Y)	Y up (0.70)	Y down (0.30)	Total
X up (0.80)	0.56	0.24	0.80
X down (0.20)	0.14	0.06	0.20
Total	0.70	0.30	1.00

For continuous variables the same idea holds: the joint density is the product of the marginal densities, and the joint cumulative function is the product of the marginal cumulative functions. 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 (Learning Module 5), in contrast, divides by n − 1. Learning Module 8 uses the same scenario table to get a portfolio’s variance directly.