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Consider the lognormal random variables $X_1$ and $X_2$ with $\log(X_1)\sim \mathcal{N}(0,1)$, and $\log(X_2)\sim \mathcal{N}(0,\sigma^2)$.

I'm trying to calculate $\rho_{\max}$ and $\rho_{\min}$ for $\rho (X_1,X_2)$. One step in the given solution I have is:

$\rho_{\max}=\rho (\exp(Z),\exp(\sigma Z))$ and $\rho_{\min}=\rho (\exp(Z),\exp(-\sigma Z))$,

but they've made some references to comonotonicity and countercomonotonicity. I was hoping someone help me understand how they're relevant. (I know how to get this from the general expression but want to know specifically what the comonotonicity parts were saying.)

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    $\begingroup$ Who are "they"? $\endgroup$
    – whuber
    Dec 7 '12 at 14:57
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I'll start by providing the definition of comonotonicity and countermonotonicity. Then, I'll mention why this is relevant to compute the minimum and maximum possible correlation coefficient between two random variables. And finally, I'll compute these bounds for the lognormal random variables $X_1$ and $X_2$.

Comonotonicity and countermonotonicity
The random variables $X_1, \ldots, X_d$ are said to be comonotonic if their copula is the Fréchet upper bound $M(u_1, \ldots, u_d) = \min(u_1, \ldots, u_d)$, which is the strongest type of "positive" dependence.
It can be shown that $X_1, \ldots, X_d$ are comonotonic if and only if $$ (X_1, \ldots, X_d) \stackrel{\mathrm{d}}{=} (h_1(Z), \ldots, h_d(Z)), $$ where $Z$ is some random variable, $h_1, \ldots, h_d$ are increasing functions, and $\stackrel{\mathrm{d}}{=}$ denotes equality in distribution. So, comonotonic random variables are only functions of a single random variable.

The random variables $X_1, X_2$ are said to be countermonotonic if their copula is the Fréchet lower bound $W(u_1, u_2) = \max(0, u_1 + u_2 - 1)$, which is the strongest type of "negative" dependence in the bivariate case. Countermonotonocity doesn't generalize to higher dimensions.
It can be shown that $X_1, X_2$ are countermonotonic if and only if $$ (X_1, X_2) \stackrel{\mathrm{d}}{=} (h_1(Z), h_2(Z)), $$ where $Z$ is some random variable, and $h_1$ and $h_2$ are respectively an increasing and a decreasing function, or vice versa.

Attainable correlation
Let $X_1$ and $X_2$ be two random variables with strictly positive and finite variances, and let $\rho_{\min}$ and $\rho_{\max}$ denote the minimum and maximum possible correlation coefficient between $X_1$ and $X_2$. Then, it can be shown that

  • ${\rm \rho}(X_1, X_2) = \rho_{\min}$ if and only if $X_1$ and $X_2$ are countermonotonic;
  • ${\rm \rho}(X_1, X_2) = \rho_{\max}$ if and only if $X_1$ and $X_2$ are comonotonic.

Attainable correlation for lognormal random variables
To obtain $\rho_{\max}$ we use the fact that the maximum correlation is attained if and only if $X_1$ and $X_2$ are comonotonic. The random variables $X_1 = e^{Z}$ and $X_2 = e^{\sigma Z}$ where $Z \sim {\rm N} (0, 1)$ are comonotonic since the exponential function is a (strictly) increasing function, and thus $\rho_{\max} = {\rm corr} \left (e^Z, e^{\sigma Z} \right )$.

Using the properties of lognormal random variables, we have ${\rm E}(e^Z) = e^{1/2}$, ${\rm E}(e^{\sigma Z}) = e^{\sigma^2/2}$, ${\rm var}(e^Z) = e(e - 1)$, ${\rm var}(e^{\sigma Z}) = e^{\sigma^2}(e^{\sigma^2} - 1)$, and the covariance is \begin{align} {\rm cov}\left (e^Z, e^{\sigma Z}\right ) &= {\rm E}\left (e^{(\sigma + 1) Z}\right ) - {\rm E}\left (e^{\sigma Z}\right ){\rm E}\left (e^Z\right ) \\ &= e^{(\sigma + 1)^2/2} - e^{(\sigma^2 + 1)/2} \\ &= e^{(\sigma^2 + 1)/2} ( e^{\sigma} -1 ). \end{align} Thus, \begin{align} \rho_{\max} & = \frac{ e^{(\sigma^2 + 1)/2} ( e^{\sigma} -1 ) } { \sqrt{ e(e - 1) e^{\sigma^2}(e^{\sigma^2} - 1) } } \\ & = \frac{ ( e^{\sigma} -1 ) } { \sqrt{ (e - 1) (e^{\sigma^2} - 1) } }. \end{align}

Similar computations with $X_2 = e^{-\sigma Z}$ yield \begin{align} \rho_{\min} & = \frac{ ( e^{-\sigma} -1 ) } { \sqrt{ (e - 1) (e^{\sigma^2} - 1) } }. \end{align}

Comment
This example shows that it is possible to have a pair of random variable that are strongly dependent — comonotonicity and countermonotonicity are the strongest kind of dependence — but that have a very low correlation. The following chart shows these bounds as a function of $\sigma$.

enter image description here

This is the R code I used to produce the above chart.

curve((exp(x)-1)/sqrt((exp(1) - 1)*(exp(x^2) - 1)), from = 0, to = 5,
      ylim = c(-1, 1), col = 2, lwd = 2, main = "Lognormal attainable correlation",
      xlab = expression(sigma), ylab = "Correlation", cex.lab = 1.2)
curve((exp(-x)-1)/sqrt((exp(1) - 1)*(exp(x^2) - 1)), col = 4, lwd = 2, add = TRUE)
legend(x = "bottomright", col = c(2, 4), lwd = c(2, 2), inset = 0.02,
       legend = c("Correlation upper bound", "Correlation lower bound"))
abline(h = 0, lty = 2)
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    $\begingroup$ (+6) Nice thorough exposition and well illustrated. It is interesting that attempts to confirm your chart through simulation will be doomed when $\sigma$ is much larger than $3$ because the sample correlation coefficient is extremely variable (due to the chance of getting one extremely high value of $X_2$, which will have high leverage). That places a higher value than usual on a solid theoretical analysis. $\endgroup$
    – whuber
    Jul 17 '13 at 21:20
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    $\begingroup$ This exposition is an adaptation of Example 2.1 (pg. 23) of M. Denuit and J. Dhaene (2003), Simple characterizations of comonotonicity and countermonotonicity by extremal correlations, Belgian Actuarial Bulletin, vol. 3, 22-27. $\endgroup$
    – cardinal
    Jul 17 '13 at 22:34
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    $\begingroup$ @cardinal I wasn't aware of this article, thanks. Other potential references include ebooks.cambridge.org/… or McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton: Princeton University Press. $\endgroup$
    – QuantIbex
    Jul 17 '13 at 22:45
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    $\begingroup$ The example goes back to at least R. D. De Veaux (1976), Tight upper and lower bounds for correlation of bivariate distributions arising in air pollution models, Tech. Report 5, Dept. of Statistics, Stanford University. See Section 3 starting on page 6. The underlying tools were known to Hoeffding. $\endgroup$
    – cardinal
    Jul 17 '13 at 23:03
  • $\begingroup$ @QuantIbex in your proof there's something unclear to me. You first claim that $X_1$ and $X_2$ are comonotonic if and only if their joint distribution is equal to $(h_1(Z), h_2(Z))$, for $h_1, h_2$ increasing, etc., but when you apply this result to the lognormal random variables, you say that this implies that the random variables themselves are such that $X_1=e^Z$ and $X_1=e^{\sigma Z}$, i.e., it seems you apply the claim to the random variables themselves, not just their distributions. How is it? $\endgroup$
    – RandomGuy
    Dec 25 '16 at 17:10

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