Instructor Micky Midha

Updated On - Describe how equity correlations and correlation volatilities behave throughout various economic states.
- Calculate a mean reversion rate using standard regression and calculate the corresponding autocorrelation.
- Identify the best-fit distribution for equity, bond, and default correlations.

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•Contrary to common beliefs, financial correlations display statistically significant and expected properties.

•Correlation levels as well as correlation volatility are generally higher in economic crises, which should be taken into consideration by traders and risk managers.

•Strong mean reversion can be found in correlations and the distribution of correlations is typically not normal or lognormal.

THE STUDY –

•In their study, the authors observed daily closing prices of the 30 stocks in the Dow Jones Industrial Average (Dow) from January 1972 to October 2012. This resulted in 10,303 daily observations of the Dow stocks and hence 10,303 × 30 = 309,090 closing prices. Monthly bins were built and 900 correlation values (30 × 30) were derived for each month, applying the Pearson correlation approach. Since there were 490 months in the study, all together 490 × 900 = 441,000 correlation values were derived. The unity correlation values on the diagonal of each correlation matrix were eliminated and 441,000 — (30 × 490) = 426,300 correlation values were obtained as inputs.

The composition of the Dow changes in time, with successful stocks being put into the Dow and unsuccessful stocks being removed. This study included the Dow stocks that represent the Dow at each particular point in time.

THE RESULT –

•The first figure shows the 490 monthly averaged correlation levels from 1972 to 2012 with respect to the state of the economy. The second figure shows the volatility of the averaged monthly correlations. The economy can be segregated into three states

- an expansionary period with gross domes- tic product (GDP) growth rates of 3.5% or higher,

2. a normal economic period with growth rates between 0% and 3.49%, and

3. a recession with two consecutive quarters of negative growth rates.

THE RESULT –

•From the previous figures, the erratic behaviour of Dow correlation levels and volatility is observed. However, the following table reveals some expected results. It can be observed that correlation levels are lowest in strong economic growth times. The reason may be that in strong growth periods equity prices react primarily to idiosyncratic, not macroeconomic factors. In recessions, correlation levels typically increase, as shown in this Table. It is also known that correlation levels increased sharply in the Great Recession from 2007 to 2009. In a recession, macroeconomic factors seem to dominate idiosyncratic factors, leading to a downturn of multiple stocks.

Correlation Level | Correlation Volatility | |
---|---|---|

Expansionary period | 27.46% | 71.17% |

Normal economic period | 32.73% | 83.40% |

Recession | 36.96% | 80.48% |

THE RESULT –

•A further expected result in the previous table is that correlation volatility is lowest in an economic expansion and highest in worse economic states. Although a higher correlation volatility could have been expected in a recession compared to a normal economic state, but it seems that high correlation levels in a recession remain high without much additional volatility. Generally, correlation volatility is high, as can also be observed from the somewhat erratic correlation function in the previous figures. Altogether, the higher correlation risk in bad economic times is displayed in the table, which traders and risk managers should consider in their trading and risk management.

•Overall, a generally positive relationship between correlation level and correlation volatility is observed.

•Mean reversion is the tendency of a variable to be pulled back to its long-term mean. In finance, many variables, such as bonds, interest rates, volatilities, credit spreads, and more, are assumed to exhibit mean reversion. Fixed coupon bonds, which do not default, exhibit strong mean reversion: A bond is typically issued at par, for example at $100. If the bond does not default, at maturity it will revert to exactly that price of $100, which is typically close to its long-term mean.

•Interest rates are also assumed to be mean reverting: In an economic expansion, typically demand for capital is high and interest rates rise. These high interest rates will eventually lead to a cooling off of the economy, possibly leading to a recession. In this process, capital demand decreases and interest rates decrease from their high levels towards their long-term mean, eventually falling below their long-term mean. Being in a recession, eventually economic activity increases again, often supported by monetary and fiscal policy. In this reviving economy, demand for capital increases, in turn increasing interest rates to their long-term means.

•Mean reversion is present if there is a negative relationship between the change of a variable, 𝑆_{t }— 𝑆_{t–1}, and the variable at 𝑡 — 1, 𝑆_{t–1}. Formally, mean reversion exists if

where

𝑆_{t}: price at time t

𝑆_{t–1}: price at the previous point in time 𝑡— 1

𝜕: partial derivative coefficient

The above equation tells us: If 𝑆_{t–1 }increases by a very small amount, 𝑆_{t }— 𝑆_{t–1}, will decrease by a certain amount, and vice versa. This is intuitive: If 𝑆_{t–1 }has decreased and is low at 𝑡 — 1 (compared to the mean of S, 𝜇_{S}), then at the next point in time t, mean reversion will pull up 𝑆_{t–1 }to 𝜇_{s }and therefore increase 𝑆_{t }— 𝑆_{t–1}. If 𝑆_{t–1 }has increased and is high in 𝑡 — 1 (compared to the mean of S,𝜇_{S}), then at the next point in time t, mean reversion will pull down 𝑆_{t–1}, to 𝜇_{s }and therefore decrease 𝑆_{t }— 𝑆_{t–1}. The degree of the pull is the degree of the mean reversion, also called mean reversion rate, mean reversion speed, or gravity.

•The degree of mean reversion can be calculated by starting with the discrete Vasicek process,

where

𝑆_{t }: price at time t

𝑆_{t–1}: price at the previous point in time 𝑡 — 1

𝑎: degree of mean reversion, also called mean reversion rate or gravity, 0 ≤ 𝑎 ≤ 1

𝜇_{s}: long-term mean of 𝑆

𝜎_{s}: volatility of 𝑆

𝜀: random drawing from a standardized normal distribution at time 𝑡, ε(𝑡): 𝑛 ~ (0,1)

To deal with mean reversion, the stochasticity part in the above equation (i.e. 𝜎_{s}𝜀 Δ𝑡) can be ignored. If Δ𝑡 = 1, then a mean reversion parameter of a = 1 will pull 𝑆_{t–1 }to the long-term mean 𝜇_{s }completely at every time step. For example, if is 80 and 𝜇_{s }is 100, then a × (𝜇_{s }— 𝑆_{t–1}) = 1 × (100 — 80) = 20, so the 𝑆_{t–1 }of 80 is mean reverted up to its long- term mean of 100. Naturally, a mean reversion parameter a of 0.5 will lead to a mean reversion of 50% at each time step, and a mean reversion parameter a of 0 will result in no mean reversion.

•To quantify mean reversion. Setting Δ𝑡 to 1, and ignoring stochasticity, we get

To find the mean reversion rate 𝑎, a standard regression analysis can be done of the form

𝑌 = 𝛼 + 𝛽𝑋

Hence, 𝑆_{t }— 𝑆_{t–1}_{ }is being regressed with respect to 𝑆_{t–1}:

It can be observed that the regression coefficient 𝛽 is equal to the negative mean reversion parameter 𝑎.

•After this the author runs a regression to find the empirical mean reversion of the earlier obtained correlation data. Hence 𝑆 represents the 30 × 30 Dow stock monthly average correlations from 1972 to 2012. The regression analysis is displayed in this figure.

•The regression function in this figure displays a strong mean reversion of 77.51%. This means that on average in every month, a deviation from the long-term correlation mean (34.83% in our study) is pulled back to that long-term mean by 77.51%. This strong mean reversion can also be observed by looking at the first figure in this chapter. An upward spike in correlation is typically followed by a sharp decline in the next time period, and vice versa.

•Autocorrelation is the degree to which a variable is correlated to its past values. Autocorrelation can be quantified with the Nobel Prize-winning autoregressive conditional heteroscedasticity (ARCH) model or its extension, generalized autoregressive conditional heteroscedasticity (GARCH). However, we can also regress the time series of a variable to its past time series values to derive autocorrelation. This is the approach discussed here.

•In finance, positive autocorrelation is also termed persistence. In mutual fund or hedge fund performance analysis, an investor typically wants to know if an above-market performance of a fund has persisted for some time (i.e., is positively correlated to its past strong performance).

•The question whether autocorrelation exists is an easy one. Autocorrelation is the “reverse property” to mean reversion: The stronger the mean reversion (i.e., the more strongly a variable is pulled back to its mean), the lower the autocorrelation (i.e., the less it is correlated to its past values), and vice versa.

•The autocorrelation can be derived as

where

𝐴𝐶: autocorrelation

𝜌_{t}: correlation values for time period 𝑡

𝜌_{t–1}: correlation values for time period 𝑡 — 1

𝐶𝑜𝑣: covariance;

This equation is algebraically identical with the Pearson correlation coefficient equation. The autocorrelation just uses the correlation values of time period 𝑡 and time period 𝑡 — 1 as inputs.

•It was found that the one-period lag autocorrelation of the correlation values from 1972 to 2012 was 22.49%. As mentioned earlier, autocorrelation is the opposite property of mean reversion. Therefore, not surprisingly, the autocorrelation of 22.49% and the mean reversion in our study of 77.51% add up to 1. This Figure shows the autocorrelation with respect to different time lags.

•It can be observed that time lag 2 autocorrelation is highest, so autocorrelation with respect to two months prior produces the highest autocorrelation. Altogether we observe the expected decay in autocorrelation with respect to time lags of earlier periods.

•The input data of the distribution tests were daily correlation values between all 30 Dow stocks from 1972 to 2012. This resulted in 426,300 correlation values. The distribution is shown in this figure. It can be observed that correlations between the stocks in the Dow are mostly positive . In fact, 77.23% of all 426,300 correlation values were positive.

•The distributions were tested for fitting the histogram, applying three standard fitting tests:

(1)Kolmogorov-Smirnov,

(2)Anderson-Darling

(3)Chi-squared.

The versatile Johnson SB distribution with four parameters, 𝛾 and 𝛿 for the shape, μ for location, and 𝜎 for scale, provided the best fit.

Standard distributions such as normal distribution, lognormal distribution, or beta distribution provided a poor fit.

•Preliminary studies of 7,645 bond correlations and 4,655 default probability correlations display properties similar to those of equity correlations. Correlation levels were higher for bonds (41.67%) and slightly lower for default probabilities (30.43%) compared to equity correlation levels (34.83%). Correlation volatility was lower for bonds (63.74%) and slightly higher for default probabilities (87.74%) compared to equity correlation volatility (79.73%).

•Mean reversion was present in bond correlations (25.79%) and in default probability correlations (29.97%). These levels were lower than the very high equity correlation mean reversion of 77.51%.

•The default probability correlation distribution is similar to the equity correlation distribution and can be replicated best with the Johnson SB distribution. However, the bond correlation distribution shows a more normal shape and can be best fitted with the generalized extreme value distribution and quite well with the normal distribution. The bond correlation and default probabilities results are currently being verified with a larger sample database.

•The following table summarizes the information:

Correlation Type | Average Correlation | Correlation Volatility | Reversion Rate | Best Fit Distribution |
---|---|---|---|---|

Equity | 34.83% | 79.73% | 77.51% | Johnson SB |

Bond | 41.67% | 63.74% | 25.79% | Generalized Extreme Value |

Default Probability | 30.43% | 87.74% | 29.97% | Johnson SB |