Perfect Correlation
Fall 2024
Perfectly Correlated Assets
- 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}
Perfect Positive Correlation
- 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 (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}, (2) can be made equal to zero by using w_{B} = \frac{\sigma_{A}}{\sigma_{A} - \sigma_{B}}.
Perfect Negative Correlation
- Similarly, if \rho_{A, B} = -1, (1) implies that \sigma_{P}^{2} = (w_{A} \sigma_{A} - w_{B} \sigma_{B})^{2}. \tag{3}
- Substituting 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}}.
The Zero Variance Portfolio
- 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.
Implications of Zero Variance
- The portfolio characterized by w_{A} and w_{B} in (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.
The Implicit Risk-Free Rate
- Thus, we must have that r_{f} = \operatorname{E}(r_{P}) = w_{A} r_{A} + w_{B} r_{B}, \tag{5} where w_{B} is determined by (4) and w_{A} = 1 - w_{B}.
- Equation (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.
Synthetic Assets
- Perfectly correlated 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!
Put Options and Perfect Negative Correlation
- 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 (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 (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.
Perfect Correlation Implies Returns Are Proportional
- Equation (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 (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.
Combining a Risky Asset with a Risk-Free Asset
- The converse of the previous statement is also true.
- Suppose you start with a risky asset 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}
Creating Perfectly Correlated Assets
- 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.
- To see this, we can compute \operatorname{Cov}(R_{A}, R_{B}) = \operatorname{Cov}(R_{A}, w R{A}) = w \sigma_{A}^{2}.
- Thus, the correlation between A and B is \rho_{A, B} = \frac{\operatorname{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, the 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}
Plotting the Investment Opportunity Set
- The figure below plots the investment opportunity set generated by asset A and the risk-free asset.
Figure 1: The figure shows the investment opportunity set of combining a risky asset A and the risk-free asset.
Implications
- 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.
- The point C is a portfolio between A and the risk-free asset such that R_{C} = w R_{A} and w < 0.
- Thus, assets A and C are perfectly negatively correlated.
- If there was no risk-free asset, any two 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. Using A, 0.15 = r_{f} + 0.40 \times 0.25 \Rightarrow r_{f} = 5\%.