# Why do volatility and correlation increase in times of crisis?

can somebody please explain to me why volatility and correlation increase in times of crisis? It is connected somehow to the herding effect. But I cannot really explain it.

And also why are negative returns stronger correlated than positive ones?

• I sincerely doubt somebody can answer this concisely as this is broad and probably somehow opinion-based. Let's see. – SRKX Nov 22 '15 at 11:47

Extra market volatility alone will cause correlations and stock volatilities to spike as you describe, even when overall market structure remains unchanged.

There's a minor variation of the very simple CAPM model that captures precisely this behavior.

To be specific, let's say every security $S_A, S_B, \dots$ (or yield, if you want bonds in this) has a value linked by some constant beta $\beta_A, \beta_B, \dots$ between its return $r_A, r_B, \dots$ and some (potentially unobservable) overall market return $r$.

Assume also an unvarying idiosyncratic variability $\sigma_A, \sigma_B, \dots$

$$r_i = \sigma_i \epsilon_i + \beta_i r$$

with independent gaussian $\epsilon_i$ for $i=A,B,\dots$.

Furthermore, rather than assuming overall market return $r$ follows the usual gaussian process, let's assume market return has two regimes, namely a serene regime in which

$$r(t) \sim r_0 + N\left(\nu t, s \sqrt{t}\right)$$

and a crisis regime in which

$$r(t) \sim r_0 + N\left(\nu t, c \sqrt{t}\right)$$

with $c\gg s$. From now on, for convenience, we'll take $t=1, \nu=0$.

Increased Crisis Volatility

In serene regimes we will measure, for example, a relatively small volatility for $S_A$ of

$$s_A = \sqrt{\beta_A^2 s^2 + \sigma_A^2}$$

and in crisis regimes a much larger volatility of

$$c_A = \sqrt{\beta_A^2 c^2 + \sigma_A^2}$$

Where in particular we have that

$$c_A^2 = s_A^2 + \beta_A^2(c^2-s^2) \gg s_A^2.$$

Thus, the crisis regime has increased the volatility of $S_A$ even without any change to the idiosyncratic variability $\sigma_A$.

Increased Crisis Correlation

Using slightly more complicated math, we can compute the observed correlations between $S_A$ and $S_B$ in serene or crisis times, and show they increase. For convenience let's think in terms of $\rho^2$ and try to show it increases. Begin by noting the covariance

$$\text{Cov}(S_A,S_B) = s^2\beta_A \beta_B$$

will obviously increase when we start measuring it in crisis as $$\text{Cov}(S_A,S_B) = c^2\beta_A \beta_B.$$

Now we have

$$\rho_{AB}^2 = \frac{\text{Cov}^2(S_A,S_B)}{(s^2\beta_A^2+ \sigma_A^2)(s^2\beta_B^2+ \sigma_B^2)} = \frac{(s^2\beta_A \beta_B)^2}{(s^2\beta_A^2+ \sigma_A^2)(s^2\beta_B^2+ \sigma_B^2)} .$$

To see that this is increasing in market volatility $s$, we define

$$x = s^2\beta_A \beta_B \\ \eta = \sigma_A^2\sigma_B^2\\ \mu = \frac{\beta_A^2\sigma_B^2 + \beta_B^2\sigma_A^2}{\beta_A \beta_B}$$

and note that both constants $\eta,\mu > 0$ if the $\sigma_i$ and $\beta_i$ are strictly positive, and that $s \propto \sqrt{x}$.

This allows us to write

$$\rho_{AB}^2 = \frac{x^2}{x^2+\eta+\mu x}$$

Then, if we take the derivative with respect to $x$ we find

$$\frac{\partial \, \rho_{AB}^2}{\partial x} = \frac{2x(x^2+\eta+\mu x)-x^2(2x+\mu)}{(x^2+\eta+\mu x)^2} \\ = \frac{2\eta x}{(x^2+\eta+\mu x)^2}$$

which is always positive.

We conclude that if the betas are strictly positive, then we will observe a strictly larger correlation $\rho_{AB}$ in times of crisis than in serene times even without any change to betas.

• Can you point me to a source/paper for your model? – Phun Jan 7 '16 at 10:02

It is hard to give a definite answer to that question. Let me focus on the volatility for now (the answer for correlation is even harder).

Schwert (1989) tries to determine the Economic Determinants of Stock Market Volatility. He finds that only lagged ex-post volatilities have strong forecasting power.

• Macro volatility, Industrial production and Monetary growth do not explain volatility in times of crisis;
• Interest rate volatility, state of the economy and recession indicators also do not predict increase in stock market volatility
• The leverage effect (i.e. that when return are negative firms get more distress as their market value decreases in relation to book value) also does not predict stock market volatility spikes in crisis.

More recently Corradi, Distaso, and Melo (2013, JME) also study the determinants of stock market volatility. Again same conclusion as before: Volatility increases when stock prices fall but, no strong evidence of a tight link between stock return volatility and leverage ratios

If one defines a crisis as a major unexpected event (ie. increases volatility) and also has a broad impact (ie affects more than one company or asset class), then I think the answer to your first question should be obvious... As for why the downside is usually more demonstrative of this affect, I think there are probably a number of factors involved including; 1) the tendency for unexpected major news events to be negative for growth and/or inflation outlook, rather than positive, and 2) the reaction of investors to news events, whereby sharp market declines are more likely to trigger stop-loss sales, than are rises to trigger buying.