2020 LTCMA volatility and correlation methodology
04-11-2019
Grace Koo
Xiao Xiao
Ivan Chan
Volatility and correlation methodology
This article describes the methodology we applied to formulate our Long-Term Capital Market Assumptions Volatility and Correlation forecasts. For the first time in the 2020 edition, we incorporate a statistical adjustment designed to ensure that our publicly released tables are numerically stable1, similar to what we have previously done internally. We are now making this positive semi-definite (PSD) adjusted output available more broadly.
Long-term asset class volatilities and correlations tend to exhibit stability when measured over multiple cycles. As such, we use the following process when estimating the long-term volatility and correlation assumptions for the majority of the asset classes with monthly data:
1. Start with monthly historical return data
- In our 2019 estimates, we used 12 years of historical data as the anchor, by adding 12 more monthly data points. For this 2020 edition, we continue to increase the data window, from 12 years to 13 years.
2. Filter data outliers
- Extreme data outliers could bias the volatility estimation so they are filtered to improve robustness. This is done by winsorizing2 the historical raw data.
- For extreme data points above 99.5% (or below 0.5%) significance level of normal distribution (or beyond a standard deviation of 2.58 from the mean), we adjust the return data by capping (or flooring) it at the 99.5% confidence (or 0.5% significance) level.
3. Construct anchor matrix
- We leverage the historical experience to help anchor our forward-looking expectations, focusing on:
- Simple historical return series (with each data point equally weighted)
- Historical return series with each data point weighted by “relevance,” based on forward looking frequency of economic regime
- We define stress periods based on National Bureau of Economic Research recession periods, assign them a long-run average probability of 15% and apply these weights on a global basis. Although one may argue that in the future the U.S. may not be the sole dominant economy, over the historical sample period there was no instance in which an economic contraction in the U.S. left the global financial markets unaffected. Analysis based on a market definition of stress produces a similar outcome.
- Once we have adjusted for periods of stress and calm, we incorporate further forward-looking adjustments, if needed.
- Simple historical return series (with each data point equally weighted)
- The variance-covariance matrix is calculated using the filtered dataset. We do this by:
- Demeaning filtered data. After filtering the data, we demean each data point by the average of the full sample
- Calculating the variance-covariance matrix
4. Adjust for key themes and structural changes
- Key themes and structural changes that are expected in the forecast horizon, such as those highlighted in this article, are reflected in the long-term risk forecast accordingly.
The above procedures describe the general process for liquid assets with monthly data. For alternative asset classes that are illiquid and hard to price, such as real estate, serial correlations can be prevalent. The difficulty in valuing the underlying assets at regular intervals is that fair prices are unobservable so we require an investment manager to estimate them. This is typically done by updating lagged prices with changes in the economic environment. Using previous prices as an input for the estimation artificially smooths the returns, biasing risk estimates downwards to the true economic risk, in our view.
We correct for this bias by adjusting the returns from these hard-to-price assets for first order serial correlation. We estimate the serial correlation coefficient using the same data window as for the liquid assets, applied to the returns of these assets before calculating their anchoring volatility and correlations. This adjustment helps to determine the true risk between these assets and the full investment opportunity set, preventing the underestimation of overall portfolio risk.
5. Ensure stable numerical property – symmetric positive semidefinite covariance matrix
For a covariance matrix to be used in optimizations, it needs to be symmetric and positive semi-definite (PSD). Due to inconsistency in underlying data sets’ periodicity (e.g. monthly returns for liquid assets and quarterly returns for illiquid assets), a covariance matrix including both liquid and illiquid assets may not guaranteed to be PSD. Therefore, we introduce a methodology by Higham3 to find the nearest symmetric positive semidefinite matrix in Frobenius norm to the original covariance matrix computed from raw returns. It is done through an iterative bisection algorithm on the upper and lower bounds on the distance of the original matrix to the nearest PSD matrix. Choleski decomposition is used as the criteria of the positive semidefiniteness test. Once we find the nearest matrix which passes the Choleski test, we then calculate the adjusted volatilities and correlations from the PSD covariance matrix as the numbers included in the official assumptions.
A few things to keep in mind
First, the standard deviation calculation is not subject to sequence risk. Thus, our assigned aggregate weighting of stress periods matters, but not the order of the data points or the continuity of the stress periods. Second, the weights are consistently applied to all the various currency matrices we publish. The forward-looking periods and the treatment of historical data are identical across regions and assets. Third, the volatility estimates capture the likely movement of the return around our central return forecasts. However, it does not incorporate distribution elements, such as the tail risk of the assets and other upper moments. It is particularly important for investors that hold assets known to have fat tails—such as high yield bonds, emerging market debt, convertible bonds, etc.—to account for risk aspects in addition to volatility.
Last but not least, the volatility and correlation forecasts are akin to expected long-run average experience. Under varying economic regimes, the variance-covariance matrix may clearly deviate from the long-run expectation.
Additional LTCMA references on risk assumptions
Daniel J. Scansaroli and Michael Feser, “Creating More Robust Forward-Looking Risk Statistics,” J.P. Morgan Asset Management, 2015 Long-Term Capital Market Return Assumptions (November, 2014).
Daniel J. Scansoroli, “Focusing on hedge fund volatility: Keeping alpha with the beta,” J.P. Morgan Asset Management. (November, 2016).
Grace Koo, Xiao Xiao and Ivan Chan, “Volatility and Correlation Assumptions: Long term risk forecast is stable, but watch for rough sailing ahead,” 2020 Long-Term Capital Market Assumptions (November, 2019).
2020 Long-Term Capital Market Assumptions
1Numerically stable calculations are those that can be proven not to magnify approximation errors.
2Winsorization applies a cap and a floor to extreme data values to remove the impact of potentially spurious outlier data on statistical results.
3Nicolas J. Higham, “Computing a Nearest Symmetric Positive Semidefinite Matrix.” Linear Algebra and Its Applications 103:103-118 (1988).
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