2.9 Dependence between assets · M02 Statistics of returns

Motivation

Diversification works because assets do not move together. It fails when they do, and they move together most in crises. LTCM (M03 · 3.10) lost its capital when trades it had modelled as independent all moved against it at once.

Correlation and diversification#

Setup

For returns $x$ , $y$ with finite variance, Pearson correlation $\rho = \operatorname{Cov}(x,y)/(\sigma_x\sigma_y)$ measures linear dependence. For an equal-weighted portfolio of $n$ assets, each with volatility $\sigma$ and pairwise correlation $\rho$ ,

\sigma_p^2 = \frac{\sigma^2}{n}\big(1 + (n-1)\rho\big) = \sigma^2\Big(\rho + \frac{1-\rho}{n}\Big) \xrightarrow[n\to\infty]{} \rho\,\sigma^2. \tag{2.17}

Derivation. $\operatorname{Var}(\frac1n\sum x_i) = \frac{1}{n^2}[n\sigma^2 + n(n-1)\rho\sigma^2]$ . ∎ Diversification removes only the idiosyncratic part; the floor $\rho\sigma^2$ is set by correlation, and if $\rho$ rises from 0.3 to 0.8 in a crisis the floor on volatility rises by a factor $\sqrt{0.8/0.3} = 1.63$ .

Correlation in turbulent markets#

Proposition 2.15Forbes–Rigobon conditioning bias

Let $y = a + \beta x + \varepsilon$ , with $\varepsilon$ independent of $x$ and of constant variance. Let $\rho$ be the correlation in a calm regime, and suppose that in a turbulent regime $\operatorname{Var}(x)$ rises by a factor $1 + \delta$ while $\beta$ and $\operatorname{Var}(\varepsilon)$ are unchanged. Then the turbulent-regime correlation is

\rho^* = \rho\sqrt{\frac{1+\delta}{1 + \delta\rho^2}} > \rho \quad (\delta > 0). \tag{2.18}

Proof

In any regime, $\rho^2 = \beta^2 V_x/(\beta^2 V_x + V_\varepsilon)$ . In the turbulent regime $V_x^* = (1+\delta)V_x$ , so

\rho^{*2} = \frac{\beta^2 V_x(1+\delta)}{\beta^2 V_x(1+\delta) + V_\varepsilon}.

Divide numerator and denominator by $\beta^2 V_x + V_\varepsilon$ and use $V_\varepsilon/(\beta^2V_x + V_\varepsilon) = 1 - \rho^2$ : $\rho^{*2} = \rho^2(1+\delta)/(\rho^2(1+\delta) + 1 - \rho^2) = \rho^2(1+\delta)/(1 + \delta\rho^2)$ .∎

So a measured rise in correlation during a high-volatility period can arise with no change in the underlying linear relation. With $\rho = 0.4$ and variance quadrupling ( $\delta = 3$ ), $\rho^* = 0.66$ . Forbes and Rigobon (2002) argued that much of what was called "contagion" in the 1987, 1994 and 1997 crises was this bias. Longin and Solnik (2001) used extreme-value theory to separate the effect and found that correlation between international equity markets rises in bear markets but not in bull markets, which the bias alone cannot explain because it is symmetric.

Worked example 2.9.1Nifty sample, weekly, Friday closes

The full-sample correlation of weekly log returns of Nifty and the S&P 500 was 0.53. The rolling 52-week correlation ranged from 0.17 (week of 6 Dec 2019) to 0.82 (week of 20 Mar 2020). In the weeks when the S&P 500 return was in its bottom decile, the correlation was 0.59; in its top-decile weeks, 0.38. Under a bivariate normal with constant correlation, these two conditional correlations would be equal by symmetry (and both below 0.53, because conditioning on a tail truncates $x$ and lowers $V_x$ ). The asymmetry is the Longin–Solnik pattern. Correlations of weekly Nifty and gold (COMEX futures, USD) returns were 0.11 over the sample: gold diversifies Indian equities, though not perfectly (0.35 in 2020).

Practitioner notes

Estimate correlations on returns of the same timestamps. Nifty closes at 15:30 IST, before New York opens; daily Nifty–S&P correlations computed from same-date closes are biased toward zero and show a spurious lead–lag. Weekly data reduce the problem.
Stress-test portfolios at correlation 1 for assets that share a funding source or a crowded investor base. The correlation that matters is the one in the state of the world where you need diversification.
Rank (Spearman, Kendall) correlations and tail-dependence coefficients describe dependence that Pearson correlation misses; copula models (beyond this course) formalise them.