Beta

Builds onCorrelation from Covariance · Sample Standard Deviation — if this page feels steep, start there.

\beta_i = \frac{Cov(R_i, R_m)}{\sigma_m^2} = \rho_{im}\,\frac{\sigma_i}{\sigma_m}

$\beta_i$: asset i's sensitivity to the market: its average % move per 1% market move
$Cov(R_i, R_m)$: covariance — how the asset's returns move together with the market's
$\sigma_m^2$: the market's variance (its volatility, squared)
$\rho_{im}$: the correlation between the asset and the market (−1 to +1)
$\sigma_i / \sigma_m$: relative volatility — how much louder the asset swings than the market

Why it must be true

Diversification kills an asset's private wiggles — across a big portfolio they cancel. What survives is the part of its movement that tracks the whole market, because that part is shared by everything and cannot cancel. Beta measures exactly that surviving part: how many percent the asset moves, on average, when the market moves one percent.

The two forms say the same thing: covariance with the market, scaled by the market's own variance — or equivalently, "how correlated is it, and how much louder does it swing" ( $\rho \times \sigma_i/\sigma_m$ ). β = 1 moves with the market; β = 2 doubles every market move; β = 0.5 dampens it.

The derivation

Regress the asset's returns on the market's: $R_i = \alpha + \beta R_m + \varepsilon$ . The least-squares slope is covariance over variance:

\beta_i = \frac{Cov(R_i, R_m)}{\sigma_m^2}

Now substitute the covariance identity $Cov = \rho_{im}\sigma_i\sigma_m$ and one $\sigma_m$ cancels:

\beta_i = \frac{\rho_{im}\sigma_i\sigma_m}{\sigma_m^2} = \rho_{im}\frac{\sigma_i}{\sigma_m}

So beta is a correlation, amplified (or damped) by relative volatility — an asset can be wildly volatile yet low-beta if it ignores the market.

When to reach for it

Given an asset's correlation with the market (or covariance) plus volatilities, and asked for systematic risk / market sensitivity — usually as the input to CAPM one step later.

Listen for

“covariance with the market”“sensitivity to market movements”“systematic risk”“correlation with the market index is …”

Back-of-the-envelope

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

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The ρ·σi/σm form is the mental-math form: correlation 0.8, asset twice as volatile as the market → β = 0.8 × 2 = 1.6. No squaring needed.
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Bounds check: |β| can never exceed σi/σm (that is the ρ = ±1 extreme). And β > σi/σm in a choice set means someone divided by σm instead of σm².
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The market's beta with itself is exactly 1 — anything described as 'like the index' should land near 1.

Traps in applying it

✗Dividing covariance by σm instead of σm² — the denominator is the market's VARIANCE.
✗Confusing beta with correlation: a low-correlation but very volatile asset can still have β near 1.
✗Using the asset's variance in the denominator instead of the market's.

Limits & criticisms

Beta is estimated from a historical regression, so it wobbles with the sample window and frequency chosen — the same stock can screen at 0.9 or 1.4 from different providers. It assumes the future resembles the estimation period, captures only linear market sensitivity, and says nothing about company-specific blowups, which is precisely the risk diversification was supposed to handle.

Where it came from

Beta emerged from the CAPM work of William Sharpe, John Lintner and Jan Mossin (1962–66), building directly on Markowitz: if everyone diversifies, only undiversifiable market exposure should earn a premium — and beta is its measure. It became the most-quoted risk number in finance: brokerage screens list it beside every ticker, index-fund and hedging desks target it, and "high-beta" / "low-beta" are ordinary market vocabulary. Its 1990s-era wobbles (Fama–French showed beta alone underpredicts value and size effects) made it humbler but no less used.

One identity, 1 questions

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

Market sensitivity

\beta_i = \rho_{im}\,\frac{\sigma_i}{\sigma_m}

The one drillable face: correlation times relative volatility — the risk that diversification cannot remove.

Drill this face →

On the BA II Plus

Worked example: An analyst measures the covariance between a fund and the market index at 0.0184; the index's standard deviation is 17% (the fund's is 18%, correlation 0.6). What beta should the analyst report?

1.0.6 [×] 0.18 [÷] 0.17 [=]ρ × σi ÷ σm — one line

→ 0.6353

Where it leads

Master this and the following come almost for free:

CAPM (Security Market Line)Treynor Ratio

Drill this formula Back to the library