Perfect Correlation

The Zero Variance Portfolio

When the correlation between two risky assets $A$ and $B$ is either one or minus one, we say that the assets are perfectly correlated. In this case, it is possible to create a portfolio $P$ between the two assets with zero variance.

We saw in the previous note that the variance of a portfolio in which we invest $w_{A}$ in $A$ and $w_{B}$ in $B$ is \[ \sigma_{P}^{2} = w_{A}^{2} \sigma_{A}^{2} + w_{B}^{2} \sigma_{B}^{2} + 2 w_{A} w_{B} \sigma_{A, B}. \tag{1}\]

We start first considering the case where $\rho_{A, B} = 1.$ In this case, we have that $\sigma_{A, B} = \sigma_{A} \sigma_{B}$ and we can write Eq. 1 as \[ \begin{aligned} \sigma_{P}^{2} & = w_{A}^{2} \sigma_{A}^{2} + w_{B}^{2} \sigma_{B}^{2} + 2 w_{A} w_{B} \sigma_{A} \sigma_{B} \\ & = (w_{A} \sigma_{A} + w_{B} \sigma_{B})^{2}. \end{aligned} \tag{2}\]

Since $w_{A} = 1 - w_{B},$ Eq. 2 can be made equal to zero by using \[ w_{B} = \frac{\sigma_{A}}{\sigma_{A} - \sigma_{B}}. \]

Similarly, if $\rho_{A, B} = -1,$ Eq. 1 implies that \[ \sigma_{P}^{2} = (w_{A} \sigma_{A} - w_{B} \sigma_{B})^{2}. \tag{3}\] Subtituting $w_{A} = 1 - w_{B},$ the previous expression can be made equal to zero if we pick \[ w_{B} = \frac{\sigma_{A}}{\sigma_{A} + \sigma_{B}}. \] Therefore, we conclude that by picking \[ w_{B} = \begin{cases} \dfrac{\sigma_{A}}{\sigma_{A} - \sigma_{B}} && \text{if $\rho_{A, B} = 1$}, \\ \dfrac{\sigma_{A}}{\sigma_{A} + \sigma_{B}} && \text{if $\rho_{A, B} = -1$}, \end{cases} \tag{4}\] and $w_{A} = 1 - w_{B}$ we can make the variance of the portfolio equal to zero.

The portfolio characterized by $w_{A}$ and $w_{B}$ in Eq. 4 is the global minimum variance portfolio that achieves a variance equal to zero when the two assets are perfectly correlated. In probability, a random variable with zero variance must be constant. Thus, in the absence of arbitrage opportunities, the return of this portfolio must be equal to the risk-free rate. If not, you could borrow at a cheaper rate and invest at a higher rate without risk, generating arbitrarily large profits for free. Such a riskless profit would only last briefly in competitive financial markets.

Thus, we must have that \[ r_{f} = \ev(r_{P}) = w_{A} r_{A} + w_{B} r_{B}, \tag{5}\] where $w_{B}$ is determined by Eq. 4 and $w_{A} = 1 - w_{B}.$ Eq. 5 says that in the absence of arbitrage opportunities, it is possible to create your own risk-free asset if you can trade two perfectly correlated assets.

These assets typically do not exist as such in financial markets, but financial institutions can create them. For example, a call option is a contract that gives its purchaser the right but not the obligation to purchase an asset at a specific date in the future for a price agreed upon today. Call options exhibit a positive perfect correlation with their underlying asset. Thus, combining the underlying asset with a call option written on it can create a risk-free asset. This remarkable insight allowed Black and Scholes (1973) and Merton (1973) to derive a formula for pricing derivatives!

Black, Fischer, and Myron Scholes. 1973. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81 (3): 637–54.

Merton, Robert C. 1973. “Theory of Rational Option Pricing.” The Bell Journal of Economics and Management Science, 141–83.

A put option gives an example of an asset exhibiting a perfect negative correlation with an asset. A put gives its purchaser the right but not the obligation to sell an asset at a specific date in the future for a price agreed upon today. Again, combining the underlying asset with a put option written on it can create a risk-free asset. The idea of synthesizing a risk-free asset out of perfectly correlated risky assets has spawned a gigantic industry of derivatives products.

Example 1 (Perfect Positive Correlation) Suppose you have two risky assets $A$ and $B$ such that $\mu_{A} = 14\%,$ $\mu_{B} = 19\%,$ $\sigma_{A} = 20\%,$ $\sigma_{B} = 30\%,$ and $\rho_{A, B} = 1.$

We can use the expression for $w_{B}$ defined in Eq. 4 to compute the implied risk-free rate. Thus, the portfolio defined by \[ w_{B} = \frac{0.20}{0.20 - 0.30} = -2, \] and $w_{A} = 1 - (-2) = 3$ has zero variance. The implied risk-free rate is \[ r_{f} = 3 \times 0.14 - 2 \times 0.19 = 4\%. \] We can verify that the variance of the portfolio is indeed zero, \[ \sigma_{P} = 3 \times 0.2 -2 \times 0.3 = 0. \]

Example 2 (Perfect Negative Correlation) Suppose you have two risky assets $A$ and $B$ such that $\mu_{A} = 13\%,$ $\mu_{B} = -2\%,$ $\sigma_{A} = 20\%,$ $\sigma_{B} = 10\%,$ and $\rho_{A, B} = -1.$

We can use the expression for $w_{B}$ defined in Eq. 4 to compute the implied risk-free rate. The portfolio characterized by \[ w_{B} = \frac{0.20}{0.20 + 0.10} = 2/3, \] and $w_{A} = 1/3,$ has zero variance. The implied risk-free rate is \[ r_{f} = 1/3 \times 0.13 + 2/3 \times (-0.02) = 3\%. \] The variance of the portfolio is indeed zero since \[ \sigma_{P} = 1/3 \times 0.2 - 2/3 \times 0.1 = 0. \]

Eq. 5 can be written as $w_{A} R_{A} + w_{B} R_{B} = 0,$ or \[ R_{B} = \pm \frac{\sigma_{B}}{\sigma_{A}} R_{A}, \tag{6}\] where capital letters denote excess returns over the risk-free rate. The sign in Eq. 6 is the same as the correlation coefficient betwee the two assets. Thus, if two risky assets are perfectly correlated, their excess returns over the risk free rate must be proportional.

Creating Perfectly Correlated Assets

The converse of the previous statement is also true. Suppose you start with a risky asseet $A$ and you create a new asset $B$ with proportions $w$ in $A$ and $1 - w$ in the risk-free asset. The returns of $B$ are described by \[ r_{B} = (1 - w) r_{F} + w r_{A} = r_{F} + w (r_{A} - r_{F}), \] or \[ R_{B} = w R_{A}. \tag{7}\]

We will see now that the excess returns of $A$ and $B$ are perfectly correlated with correlation $1$ or $-1,$ the sign of the correlation depending on the sign of $w.$¹

¹ Remember that a negative $w$ means that you short $A$ to invest more in the risk-free asset.

To see this, we can compute \[ \begin{aligned} \cov(R_{A}, R_{B}) & = \cov(R_{A}, w R{A}) \\ & = w \sigma_{A}^{2}. \end{aligned} \] Thus, the correlation between $A$ and $B$ is \[ \rho_{A, B} = \frac{\cov(R_{A}, R_{B})}{\sigma_{A} \sigma_{B}} = \frac{w \sigma_{A}^{2}}{\sigma_{A} |w| \sigma_{A}} = \begin{cases} \phantom{-} 1 && \text{if $w > 0$}, \\ -1 && \text{if $w < 0$}. \end{cases} \] By combining any risky asset with the risk-free rate we obtain a whole family of portfolios that are perfectly correlated, either positively or negatively, with each other.

The Investment Opportunity Set

We just saw that we can make perfectly correlated assets out of a risky asset and the risk-free rate. Take a risky asset $A$ with expected return equal to $\mu_{A}$ and standard deviation equal to $\sigma_{A}.$ \[ \begin{aligned} \mu_{P} & = (1 - w) rf + w \mu_{A}, \\ \sigma_{P} & = |w| \sigma_{A}. \end{aligned} \] Combining the previous two expressions, we find that he investment opportunity set of combining $A$ and the risk-free asset is given by: \[ \mu_{P} = \begin{cases} rf + \left(\dfrac{\mu_{A} - rf}{\sigma_{A}}\right) \sigma_{P} && \text{if $w > 0$}, \\ rf - \left(\dfrac{\mu_{A} - rf}{\sigma_{A}}\right) \sigma_{P} && \text{if $w < 0$}. \end{cases} \tag{8}\] The previous expression describes two lines that have the same intercept equal to the risk-free rate. The slope coefficient of the lines is equal to plus/minus the Sharpe ratio of asset $A,$ depending on whether you invest or borrow asset $A.$ The figure below plots the investment opportunity set generated by asset $A$ and the risk-free asset.

In the figure above, the point $B$ denotes a portfolio between $A$ and the risk-free asset where $R_{B} = w R_{A}$ and $w > 1.$ Thus, assets $A$ and $B$ are perfectly positively correlated. If there was no risk-free asset, both risky assets could be combined together to create a risk-free asset.

Example 3 Suppose you have two risky assets $A$ and $B$ such that $\mu_{A} = 15\%,$ $\mu_{B} = 25\%,$ $\sigma_{A} = 25\%,$ $\sigma_{B} = 50\%,$ and $\rho_{A, B} = 1.$

This would be the situation described in Figure 1. The slope coefficient of the line created by $A$ and $B$ is \[ \mathit{SR} = \frac{\mu_{B} - \mu_{A}}{\sigma_{B} - \sigma_{A}} = \frac{0.25 - 0.15}{0.50 - 0.25} = 0.40. \] The line between $A$ and $B$ is described by \[ \mu = r_{f} + 0.40 \sigma, \] where $r_{f}$ is the risk-free rate and represents the line intercept with the y-axis. The equation should be valid for both $A$ and $B$, so we can pick either to compute the implied risk-free rate. If we use asset $A$ we have that \[ 0.15 = r_{f} + 0.40 \times 0.25, \] or \[ r_{f} = 0.15 - 0.40 \times 0.25 = 5\%. \]