Two-Asset Portfolio Return & Risk

Builds onWeighted Mean (Portfolio Return) · Correlation from Covariance — if this page feels steep, start there.

\sigma_p = \sqrt{w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2 w_A w_B \rho_{AB}\sigma_A\sigma_B}

$w_A, w_B$: the portfolio weights — the fraction of money in each asset (they sum to 1)
$\sigma_A, \sigma_B$: each asset's own volatility (standard deviation of returns)
$\rho_{AB}$: the correlation between the two assets, from −1 (opposite) to +1 (lockstep)
$2 w_A w_B \rho\sigma_A\sigma_B$: the cross term — how much the assets' co-movement adds (or removes) risk
$\sigma_p$: the portfolio's volatility — the square root of the whole expression

Why it must be true

Portfolio return is boring: a plain weighted average — hold 60% of something returning 10% and 40% of something returning 6%, and you get exactly 8.4%. No surprises, no free lunch.

Portfolio risk is where the magic lives. Volatility is NOT a weighted average unless the two assets move in perfect lockstep ( $\rho = 1$ ). Any correlation below 1 means the assets sometimes zig while the other zags, offsetting each other — so the portfolio's volatility comes out below the weighted average of the pieces. That gap is the diversification benefit, and it is the only free lunch in finance: less risk without giving up any expected return.

The derivation

Return first — each dollar earns what its asset earns, so weights carry through:

R_p = w_A R_A + w_B R_B

Risk needs variance algebra. The variance of a sum $w_A R_A + w_B R_B$ expands like $(a+b)^2 = a^2 + 2ab + b^2$ :

\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2 w_A w_B \, Cov_{AB}

Substitute $Cov_{AB} = \rho_{AB}\sigma_A\sigma_B$ and take the square root. The cross term is the entire story: at $\rho = 1$ the expression collapses to $(w_A\sigma_A + w_B\sigma_B)^2$ — a perfect weighted average, zero benefit. Every notch of $\rho$ below 1 shrinks the cross term and pulls $\sigma_p$ below that line.

When to reach for it

Combining two assets (or a portfolio and a new position) and asked for the combined expected return or standard deviation, given weights and a correlation or covariance.

Listen for

“a portfolio of two assets / 60-40 allocation”“correlation between the assets is …”“portfolio standard deviation”“diversification benefit”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Bracket the answer before computing: σp can never exceed the weighted average of the two σs (that is the ρ=1 ceiling), and any ρ < 1 puts it strictly below. If a choice sits above the weighted average, it is wrong.
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Return is the easy half: a plain weighted mean you can do in your head. Compute it first and eliminate any answer set that gets it wrong.
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At ρ = 0 drop the cross term entirely: σp = √(w²σ² + w²σ²) — a Pythagorean shortcut.

Traps in applying it

✗Averaging the standard deviations — the weighted average of σs is only correct at ρ = +1; it ignores diversification entirely.
✗Forgetting the 2 in the cross term (the expansion of (a+b)² has 2ab).
✗Reporting the variance instead of taking the square root at the end.
✗Mixing units: if σ is in percent, σ² is in percent-squared — keep everything in decimals until the final answer.

Limits & criticisms

The inputs are the weak point: correlations are estimated from history, and they famously rise toward 1 in a crisis — diversification evaporates exactly when it is needed most (2008 taught this brutally). The formula also treats volatility as the whole of risk, ignoring skew and tail events, and assumes weights stay fixed while markets move them constantly.

Where it came from

Harry Markowitz published this algebra in his 1952 paper Portfolio Selection — legend has it the insight struck while reading in the library as a Chicago PhD student: investors obviously care about risk as well as return, yet no theory priced the interaction. The two-asset variance formula is the atom of Modern Portfolio Theory, earned him the 1990 Nobel Prize, and its n-asset generalization runs inside every robo-advisor, pension optimizer and risk system today. The efficient frontier is just this formula traced over every possible weight.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Portfolio expected return

R_p = w_A R_A + w_B R_B

The linear half: return diversifies away nothing — it is a plain weighted average of the parts.

Drill this face →

Portfolio risk (the diversification engine)

\sigma_p = \sqrt{w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2 w_A w_B \rho\sigma_A\sigma_B}

The nonlinear half: any correlation below +1 pulls σp under the weighted average of the σs — the free lunch, quantified.

Drill this face →

On the BA II Plus

Worked example: A portfolio is 60% asset A (σ = 25%) and 40% asset B (σ = 8%), with correlation 0.25 between them. What is the portfolio's standard deviation?

1.0.6 [x²] [×] 0.25 [x²] [=] [STO] 1w_A²σ_A²
2.0.4 [x²] [×] 0.08 [x²] [=] [STO] 2w_B²σ_B²
3.2 [×] 0.6 [×] 0.4 [×] 0.25 [×] 0.25 [×] 0.08 [=]the cross term
4.[+] [RCL] 1 [+] [RCL] 2 [=] [√x]sum the three, square-root → σp

→ 16.1%

Drill this formula Back to the library