14.6 Correlation in crises

Core about 10 min

Motivation

Diversification is a promise that losses will not all arrive together. In crises that promise is weakest.

Evidence Longin & Solnik ("Extreme Correlation of International Equity Markets", Journal of Finance 56(2), 2001) use extreme-value theory to show that correlation between international equity markets rises in bear markets but not in bull markets. In the 2008 crisis and in March 2020, most risky assets fell together.

A statistical trap Measuring correlation only on "crisis days" (high volatility) biases it upward even if the true correlation never changes (Forbes & Rigobon, "No Contagion, Only Interdependence", Journal of Finance 57(5), 2002).

Proposition 14.11Conditioning bias

Let $y = \beta x + u$ with $u$ independent of $x$ , and let unconditional correlation be $\rho$ . Condition on an event $A$ depending only on $x$ , and let $k = \operatorname{Var}(x\mid A)/\operatorname{Var}(x)$ . Then

\rho_A = \frac{\rho\sqrt k}{\sqrt{1 + \rho^2(k - 1)}},

which exceeds $|\rho|$ in magnitude when $k > 1$ .

Proof

$\operatorname{Cov}(x, y\mid A) = \beta\operatorname{Var}(x\mid A) = \beta k\sigma_x^2$ . $\operatorname{Var}(y\mid A) = \beta^2k\sigma_x^2 + \sigma_u^2$ (since $u$ is independent of $x$ , hence of $A$ ). Using $\rho^2 = \beta^2\sigma_x^2/(\beta^2\sigma_x^2 + \sigma_u^2)$ , so $\sigma_u^2 = \beta^2\sigma_x^2(1-\rho^2)/\rho^2$ :

\rho_A = \frac{\beta k\sigma_x^2}{\sqrt{k\sigma_x^2}\sqrt{\beta^2k\sigma_x^2 + \sigma_u^2}} = \frac{\sqrt k}{\sqrt{k + (1-\rho^2)/\rho^2}}\operatorname{sgn}\beta = \frac{\rho\sqrt k}{\sqrt{\rho^2k + 1 - \rho^2}}.\ ∎

With $\rho = 0.3$ , selecting days where $|x| > 1.5\sigma$ gives $k \approx 3.9$ and a measured correlation of 0.53 (simulation: 0.528; formula: 0.528), with no change in the data-generating process.

What to do Both effects are real: genuine correlation increases (forced selling, common funding) and the conditioning artefact. For risk limits the distinction does not matter. Stress-test the portfolio with all risky-asset correlations set to 0.8–1 and ask whether the loss is survivable. For a book of 10 positions each with unit risk, portfolio standard deviation is $\sqrt{10(1 + 9\rho)}$ : 4.36 at $\rho = 0.1$ and 9.06 at $\rho = 0.8$ , a factor of 2.1. If leverage was set at $\rho = 0.1$ , a shift to $\rho = 0.8$ doubles risk without any position changing. This is the mechanism of the LTCM case (14.12.1).

Data lab Open on OI Charts (opens in a new tab)

Correlation in crises: Nifty, gold in rupees and US yields

Weekly log returns of the Nifty 50 and of gold in rupees (COMEX gold futures converted at USD/INR), and weekly changes in the US 10-year Treasury yield. The chart shows rolling correlations of the Nifty with each; the shaded bands are three stress windows chosen for this lab (from Lehman's failure to March 2009, the February–April 2020 sell-off, and the months after 24 February 2022).

What to look at First, how much the rolling correlations wander in calm periods: a 26-week window has a standard error of roughly $1/\sqrt{26} \approx 0.2$ around zero. Second, the table: full-sample correlations, correlations inside each stress window, and the correlation measured only on high-volatility Nifty weeks next to the value Proposition 14.11 predicts from the full-sample correlation alone.

What it tests 14.6 says diversification is weakest in crises, and that part of the measured increase is a conditioning artefact (Forbes–Rigobon). If the high-volatility correlation is close to the Proposition 14.11 prediction, the data are consistent with no change in the underlying relation.

Questions

Did gold in rupees diversify the Nifty in each of the three windows? How much of that comes from the rupee rather than from gold in dollars?
Compare the high-volatility-week correlations with the predicted values. Which pair shows a change beyond the conditioning effect?
Change the window from 13 to 104 weeks. Which features of the chart survive, and which are noise?
Using the worst-window correlations, recompute the volatility of a 50/50 Nifty–gold (INR) portfolio. How much larger is it than with full-sample correlations?