Perfect Correlation And Hedge Cases

Why Perfect Correlation Creates Special Cases

A portfolio combines assets with different expected returns and risks. Ordinarily, the correlation between their returns affects how much the assets’ movements offset one another. At the two extremes, correlation has a particularly direct effect: with perfect positive correlation, the assets’ return movements reinforce one another in the portfolio; with perfect negative correlation, they move in exactly opposite directions. That second case can allow a precise allocation to eliminate portfolio return variance.

The formulas in this tutorial assume a one-period portfolio with two risky assets, fixed beginning-of-period weights, and returns measured over the same period. Let \(w\) be the weight in asset A and \(1-w\) the weight in asset B. The weights sum to one; unless a restriction is stated, a weight may be negative, representing a short position. The examples specify long-only weights when both weights must be between zero and one.

Definition: Perfect positive correlation means the two assets’ returns have correlation \(\rho=+1\); perfect negative correlation means \(\rho=-1\). Assume both assets have positive standard deviations, \(\sigma_A\) and \(\sigma_B\), so their correlation is defined. The portfolio return is \(R_p=wR_A+(1-w)R_B\), and its expected return is \(E[R_p]=wE[R_A]+(1-w)E[R_B]\).

The portfolio variance formula, using \(\operatorname{Cov}(R_A,R_B)=\rho\sigma_A\sigma_B\), is:

\operatorname{Var}(R_p) =w^2\sigma_A^2+(1-w)^2\sigma_B^2 +2w(1-w)\rho\sigma_A\sigma_B

For perfect correlation, this expression becomes a square. The standard deviation is the nonnegative square root, so the absolute value in each standard-deviation formula is essential.

\rho=+1:\quad \sigma_p=\left|w\sigma_A+(1-w)\sigma_B\right|

\rho=-1:\quad \sigma_p=\left|w\sigma_A-(1-w)\sigma_B\right|

Why the Formulas Reduce to a Square

Substitute \(\rho=+1\) into the variance formula. The resulting three terms are exactly the expansion of a square:

\begin{aligned} \operatorname{Var}(R_p) &=w^2\sigma_A^2+(1-w)^2\sigma_B^2 +2w(1-w)\sigma_A\sigma_B\\ &=\left(w\sigma_A+(1-w)\sigma_B\right)^2. \end{aligned}

Taking the nonnegative square root gives the perfect-positive-correlation formula. For \(\rho=-1\), the covariance term has a minus sign instead. The terms then expand as a difference squared:

\begin{aligned} \operatorname{Var}(R_p) &=w^2\sigma_A^2+(1-w)^2\sigma_B^2 -2w(1-w)\sigma_A\sigma_B\\ &=\left(w\sigma_A-(1-w)\sigma_B\right)^2. \end{aligned}

These identities also explain the different portfolio outcomes. With long-only weights and \(\rho=+1\), portfolio standard deviation equals the weighted average of the two standard deviations. With \(\rho=-1\), the contributions to risk subtract, and the portfolio has zero variance if they exactly balance.

Perfect Positive Correlation: No Diversification Benefit

When \(0\leq w\leq1\) and \(\rho=+1\), both terms inside the absolute value are nonnegative. Therefore:

\sigma_p=w\sigma_A+(1-w)\sigma_B

This is the weighted average of the assets’ standalone standard deviations. Thus the portfolio has no risk reduction relative to that weighted average: the covariance does not create an offsetting effect. This statement does not mean the portfolio must be riskier than either asset. If one asset is less risky, moving some weight from a riskier asset to it can lower portfolio risk, but that reduction comes from the changed allocation, not from diversification between imperfectly aligned movements.

The long-only condition matters. If short positions are allowed, one of the weights can be negative, and the absolute-value formula still applies. In that case a zero-variance position can be possible even at \(\rho=+1\), but it requires the signed risk exposures to cancel and is not a conventional long-only diversification result.

Worked Example: Portfolio Risk With Perfect Positive Correlation

A hypothetical portfolio invests 40% in asset A and 60% in asset B. Their expected returns are 8% and 11%, their standard deviations are 12% and 20%, and their return correlation is exactly \(+1\). Calculate the portfolio expected return and standard deviation, and interpret the risk result. Assume long-only weights.

Method. Use the weighted-average formula for expected return and the perfect-positive-correlation formula for standard deviation. The weights are \(w=0.40\) and \(1-w=0.60\).

Expected return.

E[R_p]=(0.40)(0.08)+(0.60)(0.11) =0.032+0.066=0.098=9.8\%

Standard deviation.

\sigma_p=(0.40)(0.12)+(0.60)(0.20) =0.048+0.120=0.168=16.8\%

As a check, the variance terms are \(0.40^2(0.12^2)=0.002304\), \(0.60^2(0.20^2)=0.014400\), and \(2(0.40)(0.60)(0.12)(0.20)=0.011520\). Their sum is \(0.028224\), whose square root is \(0.168\), or 16.8%.

The portfolio’s 16.8% standard deviation is below asset B’s 20% because the portfolio holds some of the less risky asset A. It equals the weighted average, \(0.40(12\%)+0.60(20\%)=16.8\%\), so there is no diversification benefit relative to weighted-average standalone risk.

Perfect Negative Correlation: The Hedge Allocation

With long-only weights and \(\rho=-1\), risk depends on the difference between the two weighted standard deviations. A zero-variance allocation requires:

w\sigma_A-(1-w)\sigma_B=0

Rearranging gives \(w\sigma_A=\sigma_B-w\sigma_B\), so \(w(\sigma_A+\sigma_B)=\sigma_B\). When both standard deviations are positive, the unique solution is:

w=\frac{\sigma_B}{\sigma_A+\sigma_B}, \qquad 1-w=\frac{\sigma_A}{\sigma_A+\sigma_B}

The weight in each asset is proportional to the other asset’s standard deviation. This may seem reversed, but it follows from balancing the weighted risks: the higher-risk asset needs the smaller weight. The weights are both between zero and one, so this hedge is feasible with long-only holdings when both standard deviations are positive.

Zero variance means the portfolio return has no variation under the exact model; it does not mean its expected return is zero. Its expected return is still the weighted average of the expected returns. A portfolio can therefore have a deterministic return that is positive, negative, or zero.

Worked Example: Finding a Zero-Variance Allocation

A hypothetical investor considers asset A, with expected return 9% and standard deviation 15%, and asset B, with expected return 5% and standard deviation 10%. The returns have correlation exactly \(-1\). Find the long-only weights that eliminate variance and calculate the resulting expected return.

Method. Set the two weighted risk contributions equal, then use the weights in the expected-return formula.

The weight in A is:

w=\frac{\sigma_B}{\sigma_A+\sigma_B} =\frac{0.10}{0.15+0.10} =\frac{0.10}{0.25}=0.40

The weight in B is \(1-w=0.60\). Check the hedge condition: \(0.40(15\%)=6\%\) and \(0.60(10\%)=6\%\). The weighted risk contributions match, so:

\sigma_p=|6\%-6\%|=0\%

The portfolio expected return is:

E[R_p]=(0.40)(9\%)+(0.60)(5\%) =3.6\%+3.0\%=6.6\%

The portfolio’s variance is \(0^2=0\), while its expected return is 6.6%. The check confirms that variance elimination concerns return variability, not the level of the expected return.

Allocations Away From the Hedge

If the weights do not satisfy the hedge equation, perfect negative correlation still reduces risk relative to adding the two weighted risk contributions. The remaining standard deviation is the absolute difference. The absolute value tells which asset’s weighted risk contribution is larger; it does not change the amount of risk into a negative number.

Worked Example: Risk and Return Away From the Hedge

A hypothetical fund invests 25% in asset A and 75% in asset B. Asset A has expected return 4% and standard deviation 10%; asset B has expected return 12% and standard deviation 6%. Their correlation is exactly \(-1\). Calculate the portfolio’s expected return and standard deviation, and compare its risk with the hedge allocation.

Method. Apply the expected-return formula and take the absolute difference between weighted standard deviations.

The expected return is:

E[R_p]=(0.25)(4\%)+(0.75)(12\%) =1\%+9\%=10\%

The weighted risk contributions are \(0.25(10\%)=2.5\%\) and \(0.75(6\%)=4.5\%\). Thus:

\sigma_p=|2.5\%-4.5\%|=2.0\%

The hedge weight in A for these standard deviations would be \(6\%/(10\%+6\%)=0.375\), with 0.625 in B. The actual 25% weight in A is below that hedge weight, leaving a 2.0% risk exposure from the larger weighted contribution of asset B. As a check, the portfolio variance is \(0.02^2=0.0004\), consistent with a 2.0% standard deviation.

Common Exam Trap: Dropping the Absolute Value

A frequent error is to calculate the signed difference in the perfect-negative-correlation formula and report it directly as the standard deviation. Standard deviation cannot be negative. For example, with \(\sigma_A=10\%\), \(\sigma_B=6\%\), and \(w=0.80\), the weighted contributions are \(8.0\%\) and \(1.2\%\). The signed difference is \(8.0\%-1.2\%=6.8\%\), which happens to be positive. But if \(w=0.20\), the contributions are \(2.0\%\) and \(4.8\%\), so their signed difference is \(-2.8\%\). Reporting \(-2.8\%\) as portfolio standard deviation is wrong; the correct value is \(|-2.8\%|=2.8\%\).

Another common mistake is to claim that perfect positive correlation means every mix is as risky as the riskier asset. That is not the formula: for long-only weights, portfolio risk is the weighted average of the two standard deviations. It may be below the riskier asset’s standard deviation while still providing no reduction relative to that weighted average.

Key takeaway: For long-only weights, perfect positive correlation gives \(\sigma_p=w\sigma_A+(1-w)\sigma_B\), while perfect negative correlation gives \(\sigma_p=|w\sigma_A-(1-w)\sigma_B|\). Under perfect negative correlation, the long-only zero-variance allocation is \(w=\sigma_B/(\sigma_A+\sigma_B)\). These conclusions require the stated correlation to be exactly \(+1\) or \(-1\); they do not establish the result for other correlations.

These special cases clarify the role of covariance in portfolio risk and are aligned with the SOA VEE Accounting and Finance syllabus. In practice, an estimated correlation may not remain exactly at an extreme value, so a theoretical zero-variance hedge should not be treated as a guarantee of risk-free performance.

Check Your Understanding

For each problem, state the method you use and check that the reported standard deviation is nonnegative.

Assets A and B have standard deviations of 5% and 11%, respectively, and correlation \(+1\). A long-only portfolio places 30% in A. Calculate its standard deviation and compare it with the weighted average of standalone standard deviations. (Answer: 9.2%; equal to the weighted average, so no diversification benefit relative to that average.)
A long-only portfolio has 40% in A and 60% in B. Their expected returns are 7% and 10%, their standard deviations are 12% and 8%, and correlation is \(-1\). Calculate expected return and standard deviation. (Answer: Expected return 8.8%; standard deviation 0.0%.)
Two assets have standard deviations of 9% and 15% and correlation \(-1\). Find the long-only weight in A and the weight in B that eliminate variance. Verify the two weighted risk contributions. (Answer: A weight 62.5%, B weight 37.5%; each weighted risk contribution is 5.625%, giving zero standard deviation.)
Two assets have standard deviations of 7% and 13% and correlation \(+1\). A long-only portfolio invests equally in them. Calculate portfolio standard deviation and determine whether that risk is below the riskier asset’s standalone risk. (Answer: 10.0%; yes, it is below 13%, but it equals the weighted-average risk and does not reflect a diversification benefit.)
Assets A and B have standard deviations of 4% and 10% and correlation \(-1\). A portfolio places 50% in each. Calculate the standard deviation, identify which asset has the larger weighted risk contribution, and find the weight in A that would eliminate variance. (Answer: Standard deviation 3.0%; B has the larger weighted contribution; the zero-variance weight in A is 71.4286% and in B is 28.5714%.)

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