Two correlated assets · RiskAtoms Academy

By the end you will be able to

1Calculate the volatility of any two-asset mix from the weights, volatilities and correlation.
2Draw the curve of two-asset mixes on a risk-return chart and explain why it bends to the left.
3Describe the two extremes: a straight line at ρ = +1 and a perfect hedge at ρ = −1.
4Say when adding a riskier asset lowers the risk of a calmer one, using the threshold ρ < σ₂/σ₁.

Why it matters

Put 25% of the group's money in equities, three times as volatile as government bonds, and the portfolio becomes less risky than bonds alone: 5.71% against 6%. Whether that happens depends on one number, the correlation.

Core explanation

One curve for every correlation

L2.5.03: Correlation and diversification calculated the volatility of one mix, the 60/40 fund, at a few correlations. L3.1.02: Two independent assets looked at the special case ρ = 0. Now we look at every mix at once, for every correlation, and ask what shapes appear.

The tool is the risk-return chart: volatility across, expected return up. Global equities sit at (16%, 7%), government bonds at (6%, 3.5%). As the weight in equities runs from 0 to 1, the mix traces a path from the bond point to the equity point: the two-asset curve. Its vertical position is easy: the expected return is the weighted average, so it rises evenly with the weight. Its horizontal position, the volatility, depends on the correlation.

ρ = 1.0ρ = 0.5ρ = 0.0ρ = −0.5ρ = −1.0

Same two assets, five correlations. Every curve passes through both assets; the lower the correlation, the further the curve bulges to the left.

ρ = +1: a straight line. The two always move in proportion, so nothing cancels. The volatility of any mix is the weighted average of the two volatilities, and the mixes lie on the straight line between the assets. No diversification at all.
ρ between −1 and +1: a curve bulging left. Some of the swings cancel, so every mix is less volatile than the straight line. The lower the correlation, the deeper the bulge. The leftmost point, the nose of the curve, is the lowest-risk mix (L3.1.04: The minimum-variance portfolio).
ρ = −1: two straight lines meeting at zero. The two always move in exactly opposite proportion. Choose the weights so that the swings are equal and opposite, and they cancel completely: a perfect hedge. For our pair that is 27.3% in equities, with an expected return of 4.45% and no risk at all.

When does a riskier asset reduce risk?

Start from all bonds and add a little equity. If the correlation is high, the equity's swings mostly add to the bonds' and risk rises at once. If it is low enough, the new swings partly cancel the old, and risk falls before it rises. The dividing line is

\rho = \sigma_2/\sigma_1

, here 6% ÷ 16% = 0.375. Our universe's correlation, −0.10, is well below it, which is why the 25/75 mix in the hook beats bonds alone.

Sort itWhich correlation?

Put each statement with the correlation it describes. Some hold for every correlation.

The mixes lie on a straight line between the two assets.
Some mix of the two, without short sales, has zero risk.
The curve passes through both assets' own points.
The mix's volatility equals the weighted average of the two volatilities.
The curve is two straight pieces meeting at the vertical axis.
The mix's expected return is the weighted average of the two expected returns.

The formal version

The full two-asset formula, from L3.1.01: Portfolio return and risk, with $w_1 = w$ and $w_2 = 1 - w$ :

\sigma_p^2 = w^2\sigma_1^2 + (1-w)^2\sigma_2^2 + 2\,w(1-w)\,\rho\,\sigma_1\sigma_2

The two extremes. At $\rho = \pm 1$ the right-hand side is a perfect square, so the square root comes out cleanly:

\begin{aligned} \rho = +1: \quad \sigma_p^2 &= w^2\sigma_1^2 + (1-w)^2\sigma_2^2 + 2\,w(1-w)\,\sigma_1\sigma_2 = \big(w\sigma_1 + (1-w)\sigma_2\big)^2 \\ \sigma_p &= \big|\,w\sigma_1 + (1-w)\sigma_2\,\big| \\[4pt] \rho = -1: \quad \sigma_p^2 &= w^2\sigma_1^2 + (1-w)^2\sigma_2^2 - 2\,w(1-w)\,\sigma_1\sigma_2 = \big(w\sigma_1 - (1-w)\sigma_2\big)^2 \\ \sigma_p &= \big|\,w\sigma_1 - (1-w)\sigma_2\,\big| = 0 \quad\text{when}\quad w = \frac{\sigma_2}{\sigma_1 + \sigma_2} \end{aligned}

For weights between 0 and 1 the $\rho = +1$ case is the weighted average: a straight line. The $\rho = -1$ case is a V: two straight lines that meet at zero volatility. With short sales the $\rho = +1$ line also reaches zero, at $w = -\sigma_2/(\sigma_1 - \sigma_2)$ : for our pair, short 60% equities, for an expected return of 1.4%.

Every correlation in between. Collect the variance by powers of $w$ . Write $\sigma_{12} = \rho\,\sigma_1\sigma_2$ :

\begin{aligned} \sigma_p^2 &= w^2\sigma_1^2 + (1 - 2w + w^2)\,\sigma_2^2 + 2w\,\sigma_{12} - 2w^2\sigma_{12} \\ &= w^2\big(\sigma_1^2 + \sigma_2^2 - 2\sigma_{12}\big) - 2w\big(\sigma_2^2 - \sigma_{12}\big) + \sigma_2^2 \\ &= K\,w^2 - 2L\,w + \sigma_2^2, \qquad K = \sigma_1^2 + \sigma_2^2 - 2\sigma_{12}, \quad L = \sigma_2^2 - \sigma_{12} \end{aligned}

$K$ is the variance of $R_1 - R_2$ , so it is positive unless the two assets are identical. The variance is therefore a U-shaped parabola in the weight. Its slope at $w = 0$ (all bonds) is $-2L$ : negative, so a little equity lowers risk, exactly when $\sigma_2^2 > \rho\,\sigma_1\sigma_2$ , that is $\rho < \sigma_2/\sigma_1$ . Now switch from the weight to the expected return to see the shape on the chart.

Conventions

Annual expected returns and volatilities, simple returns, weights fixed over the year. The curve needs the two expected returns to differ; if they were equal, every mix would sit at the same height and only the volatility would change.

Try it

Start at the universe's correlation, −0.10, and press Play: the curve bends further left as ρ falls and folds into a V at −1. Then drag ρ up past 0.375 and watch the nose slide into the bond point: equities stop reducing risk. Last, allow short sales and set ρ to +1: the straight line runs on until it hits zero volatility.

Try itThe two-asset curve: drag the correlation

Correlation ρ−0.10

Weight in global equities60% / 40%

Your mix

9.66%

Expected return5.60%

Weights60% / 40%

Lowest-risk mix5.41%

… weight in equities14.7%

Bonds alone6.00%

ρ = −0.10ρ = −1, 0, +1your mix

Ring: the lowest-risk mix on the curve. Expected returns stay at 7% and 3.5%, volatilities at 16% and 6%: only the correlation moves.

Worked example

Mixes of equities and bonds across correlations

Global equities ( $\sigma_1 = 16\%, \ \mu_1 = 7\%$ ) and government bonds ( $\sigma_2 = 6\%, \ \mu_2 = 3.5\%$ ).

Four points worth knowing

Step 1 of 4

1
The 25/75 mix at the universe's ρ = −0.10
$\begin{aligned} \sigma_p^2 &= 0.25^2 \times 0.0256 + 0.75^2 \times 0.0036 + 2 \times 0.25 \times 0.75 \times (-0.10) \times 0.16 \times 0.06 \\ &= 0.0016 + 0.002025 -0.00036 = 0.003265 \\ \sigma_p &= 5.71\%, \qquad \mu_p = 0.25 \times 7\% + 0.75 \times 3.5\% = 4.375\% \end{aligned}$

Volatility of each mix (weight in equities across, correlation down):

Correlation	0% eq.	25% eq.	50% eq.	75% eq.	100% eq.
Expected return	3.50%	4.38%	5.25%	6.13%	7.00%
−1.00	6.00%	0.50%	5.00%	10.50%	16.00%
−0.50	6.00%	4.27%	7.00%	11.32%	16.00%
−0.10 (universe)	6.00%	5.71%	8.26%	11.94%	16.00%
0.00	6.00%	6.02%	8.54%	12.09%	16.00%
0.50	6.00%	7.37%	9.85%	12.82%	16.00%
1.00	6.00%	8.50%	11.00%	13.50%	16.00%

The 60/40 mix runs from 12.00% at ρ = +1 down to 7.20% at ρ = −1. The same portfolio can be half as risky or twice as risky depending on a number that is hard to estimate and changes over time.

Regulatory lens

The two-asset formula appears word for word in insurance rules. Solvency II's standard formula charges type 1 equities (listed on regulated markets in EEA or OECD countries) and type 2 equities (everything else) separately, then combines the two charges with a correlation of 0.75, exactly as a two-asset volatility: the square root of $\text{SCR}_1^2 + 2 \times 0.75 \times \text{SCR}_1\text{SCR}_2 + \text{SCR}_2^2$ (Commission Delegated Regulation (EU) 2015/35, the equity risk sub-module). The regulator fixes the correlation so that firms cannot pick a flattering one. More in M4.6: Solvency II and insurance regulation.

Spark Atoms

Spark AtomPerfect hedges do not survive

A correlation of exactly −1 between two assets with different expected returns would hand out a riskless profit: hold the hedged mix, borrow at the risk-free rate, keep the difference. That is an arbitrage, and traders would buy until prices moved and the gap closed. So in a working market a perfect hedge earns the risk-free rate. Real hedges leave some basis risk.

Spark AtomThe stock-bond correlation changes sign

The 60/40 portfolio leans on bonds rising when equities fall. That has not always held. In the United States, stock and government bond returns tended to move together for much of the 1970s, 1980s and 1990s, when inflation was the main worry; they moved mostly in opposite directions from about 2000 until around 2020; and in 2022 both fell sharply as inflation returned. The curve you dragged above is not fixed.

Key takeaways

On a risk-return chart, the mixes of two assets trace a curve through both asset points. Expected return is linear in the weight; volatility is not.
ρ = +1: a straight line, no diversification. ρ = −1: two lines meeting at zero, a perfect hedge at w = σ₂/(σ₁ + σ₂).
In between, the curve bulges left; the lower the correlation, the deeper the bulge. Mathematically it is a hyperbola: variance is a quadratic in expected return.
Adding a riskier asset lowers the risk of a calmer one only if ρ < σ₂/σ₁ (here 0.375).
At the universe's ρ = −0.10, the 25/75 mix has 5.71% volatility, below bonds alone, and a higher expected return.

Reaction

Check what stuck

Question 1 · objective 1

Half in an asset with 20% volatility, half in one with 10%, correlation −0.5. What is the portfolio's volatility, in per cent, to two decimals?

%

Question 2 · objective 3

Two assets with volatilities of 30% and 10% have a correlation of exactly −1. What weight in the 30% asset removes all risk, in per cent?

%

Question 3 · objective 2

On the risk-return chart, what does the curve of two-asset mixes look like when ρ = +1 and there are no short sales?

AA straight line between the two assetsBTwo straight lines meeting at zero volatilityCA curve bulging to the leftDA single point

Question 4 · objective 4

A bond fund has 10% volatility and an equity fund 25%. Below what correlation does adding a little of the equity fund lower the risk of an all-bond portfolio? Give the correlation as a decimal.

Question 5 · objective 3

Two assets have a correlation of exactly −1, and their zero-risk mix is expected to earn 4.5% while Treasury bills pay 3%. What should happen?

ANothing: correlations of −1 are commonBInvestors borrow at 3% to buy the riskless 4.5% mix until prices move and the gap closesCThe bills' rate must fall to zeroDThe two assets' volatilities must rise

Glossary

Risk-return chart: A chart with volatility across and expected return up. Each portfolio is one point on it.
Two-asset curve: The points that mixes of two assets reach as the weight runs from all in one to all in the other.
Perfect hedge: A mix whose risk cancels completely. With two assets it needs a correlation of exactly −1.
Nose of the curve: The leftmost point of the curve: the mix with the lowest volatility.
Arbitrage: A riskless profit: earning more than the risk-free rate with no risk, or something for nothing.

Deep Dives & Origins

Deep Dives

E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, Wiley, the chapter on combining two risky assets. Intermediate.
J. Y. Campbell, A. Sunderam and L. M. Viceira, “Inflation Bets or Deflation Hedges? The Changing Risks of Nominal Bonds”, Critical Finance Review, 6(2), 2017. Advanced.

Origins

H. Markowitz, “Portfolio Selection”, Journal of Finance, 7(1), 1952, pp. 77–91.
Commission Delegated Regulation (EU) 2015/35 (Solvency II), the equity risk sub-module, Official Journal of the European Union, L 12, 17 January 2015.
J. Y. Campbell, A. Sunderam and L. M. Viceira, “Inflation Bets or Deflation Hedges? The Changing Risks of Nominal Bonds”, Critical Finance Review, 6(2), 2017.

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