The final result of the GDIC is then bounded between 1 and -1.

*Note that the threshold reference point for returns/drawdowns in this case is tied to portfolio data. For example if the CDaR is 90%, then to find the CDaR for Asset A and Asset B you would use the drawdowns referenced for each asset referenced to the date that drawdowns for Portfolio AB exceeded the threshold. For a triple reference you would average the results of using all 3 time series separately as a reference.

This is a very compact and simple framework that allows you to create a custom correlation matrix that reflects downside risk using joint time-series dynamics. The resulting matrix can be used for research and analysis as well as stress-testing. Here are several metrics worth considering:

CDaR or Condtional-Drawdown-at-Risk/ CVaR or Conditional-Value-at Risk:

Cheklov,Urysaev and Zabarankin, 2003: Portfolio Optimization with Drawdown Constraints

Krokhmal, Palmquist and Urysaev, 2001: Portfolio Optimization with Conditional Value-at-Risk Objective and Constraints

A former colleague Enn Kuutan wrote an excellent primer on CDaR and CVaR in his master’s thesis.

Michael Kapler of Systematic Investor previously wrote a post on using CDaR and CVaR tools for implementation in R on his excellent blog to implement these metrics in the context of portfolio management.

ERoD or Expected Regret of Drawdown:

Ding and Urysaev, 2021: Drawdown Beta and Portfolio Optimization

UI or Ulcer Index:

Martin and McCann, 1987: The Investor’s Guide to Fidelity Funds

All of these metrics are excellent candidates for the GDIC. But there is no need to make a single choice: you can also combine the downside implied correlations from multiple different metrics to form one cohesive matrix that reflects all of the loss dynamics you wish to capture. What is lost in precision is gained in flexibility, allowing the integration of diverse information into a cohesive and adaptable framework. In a subsequent post, I will compare and contrast these different metrics and their advantages/disadvantages.

A Simplified Downside Correlation Estimate

It is important to recognize that the GDIC is an approximation or a heuristic method to derive correlation. Unlike volatility there is no iron clad mathematical proof of a linkage between these nonlinear metrics and how they combine to form a correlation. However we showed the previous post using the Drawdown Implied Correlation that there is a clear mathematical link and that this approximation is still useful. While not mathematically equivalent to Pearson correlation, these implied correlations are interpretable and internally consistent for use in analysis or risk-based optimization. Extreme downside scenarios are often driven by nonlinear relationships (e.g., tail dependence) that traditional correlation fails to capture. Downside metric implied correlations can help approximate this tail dependence without requiring complex copula modeling, providing a simplified but effective representation of how assets behave under stress. As a result this heuristic provides a very useful and unique insight into joint risk dynamics.

Applications

In addition the generalized downside implied correlation matrix can be used along with the corresponding downside risk measure (ie if correlations are derived using CDaR, use CDaR in place of volatility) or matched with standard volatility to create a variance/covariance matrix. The inputs can then be used very conveniently in a “Pseudo-Optimization” framework (or approximation) using standard Markowitz/Mean-Variance mathematics and optimization methods. The benefit of this approach over standard linear programming is that it is much faster/more scaleable, can handle multiple objectives, and is potentially more robust because it is less overfit to past data. The process of deriving downside implied correlations and volatilities inherently smooths extreme risk patterns by distilling them into a quadratic framework. This smoothing can reduce overfitting and provide a more robust solution under different market conditions. In a sense, this is a more generalized solution that also preserves the integrity of downside metric data (the implied correlations use the original time series sequence of returns and baseline) without having to approximate it with a piecewise linear approach which preserves the relative rankings. You can then apply random matrices and resampling/Monte Carlo simulation to make the optimization more robust. One caveat is that for the purpose of optimization you still need to ensure the Generalized Downside Correlation Matrix is Positive Semi-Definite or PSD. In the interest of brevity I will present my algorithm for fast PSD adjustment in a part 3 of this series.

Diversification is a concept that is critical to most asset managers and traders. The foundation of this body of research is built upon the Pearson correlation coefficient, which is the most popular metric to determine whether adding an asset to a portfolio might enhance diversification. Despite its widespread use, most investment practitioners recognize its limitations. Some of the flawed assumptions include : 1) the assumption of linearity: correlations assume a simplistic linear relationship between assets, overlooking the complexity of real-world market dynamics 2) the assumption of normality: correlation assumes that returns are normally distributed despite the evidence that asset returns have “fat tails” 3) the failure to capture tail risk: correlations fail to adequately capture tail risk during times of financial crisis or market stress. 

But the greatest danger that the correlation metric presents is a false sense of security- leading investors to believe their portfolios are diversified (and thus insulated from systematic risk) when in reality they are not. For example, a negative correlation between two assets might suggest effective diversification, but this metric ignores the direction and magnitude of returns. Both assets can lose money simultaneously, even while maintaining a negative correlation, leaving the portfolio exposed to significant losses. This highlights a critical flaw in relying solely on correlation: it focuses on the relationship between returns rather than their actual behavior or contribution to portfolio risk, leading to decisions that fail to achieve real diversification.

After analyzing some basic asset class data, I noticed that 2022 was a very unusual year. Both stocks and bonds had simultaneous declines while inflation hit multi-decade highs. Traditional asset allocation failed, and for the most part classic tactical asset allocation also suffered losses. I noticed a chart posted by Callum Thomas of the Weekly S&P500 Chart Storm that showed just how unusual market conditions were:

Many people will look at this chart and think that this is just an anomaly, but to me this is yet another clear example of why you can’t just blindly assume that bonds will protect your equities in a crisis. It also shows that backtesting and model creation can be biased by certain data samples to provide a false sense of security (imagine only using data from 1970 to 2021 to create your model!). Here is a chart showing how the drawdowns in stocks vs bonds evolved during this period along with major Federal Reserve rates/cuts highlighted (note in 2018 there were 4 rate hikes but only the start is on the chart):

Note that most of the drawdowns for both assets seems to be driven by rising rates which makes sense. During COVID the Fed cut rates and bonds managed to provide diversification during the large stock market drawdown. This suggests that the diversification benefit of holding bonds to balance the risk of stocks has become conditional on expectations for future Fed policy and inflation. But in the absence of complex macroeconomic models we typically rely on a dynamic or rolling correlation to measure diversification. I took a look at the correlations to see if they signalled this diversification failure and to my surprise they were actually negative to barely positive through most of 2022!

So much for traditional Diversification!

The problem is obvious, but what is the alternative? The inspiration for this post is actually based on a conversation from nearly 12 years ago that I had with a former colleague- Dave Abrams- that used to work at CSSA. He once asked me if it was possible to derive a better correlation coefficient using drawdowns. The general idea was that what traders and investors are really interested in is whether the drawdowns from one asset are offset by gains in the other asset or vice versa. We didn’t end up creating anything based on this discussion but it was a great insight.

Drawdowns are definitely the key to a practical solution. They represent the true historical reality of investment risk rather than estimates from a normal distribution. Drawdowns focus specifically on the risk of capital erosion and the investor’s capacity to weather those losses. They provide crucial information that traditional measures often overlook: 1) the potential for catastrophic loss, 2) a quantifiable measure that captures both tail risk and the emotional toll of losing money 3) the “sequence of returns risk” or the difficulty of making a financial recovery after large losses. This makes drawdowns an essential component of risk analysis and portfolio management, particularly for long-term investors concerned with both the return and preservation of capital. In contrast to correlations and volatility, drawdowns are nonlinear and path-dependent making them complementary for risk analysis.

The next question is obvious: how do we convert this valuable information into a correlation? Without pausing to cover the existing research and how this is unique (we will cover that in part 2) let’s first dive in and demonstrate how we can derive a simple equation that anyone can use and calculate the drawdown correlation from only a single reference or data point. But first let’s “correct” a standard volatility measure by reconciling empirical versus theoretical drawdowns. It is well understood that the theoretical/normal distribution estimate for drawdowns can be very different from the empirical or actual maximum drawdown. By capturing this difference we can get a different estimate for “true” volatility. A very basic model for expected maximum drawdowns (EMD) would use the mean or drift minus some multiplier times volatility times the square root of the time measurement. If we use a multiplier as 2, then we are saying that the loss we expect is a 2 standard deviation event which is quite rare. Regardless of the multiplier used, here is the formula from which we will “reverse imply” ( in this case use basic algebra) the volatility or “Drawdown Implied Volatility” (DIV) by solving for it by replacing the EMD with the empirical maximum drawdown.

This measurement can be useful on its own in the context of portfolio inputs or used within a trading indicator. But that isn’t the goal of this post, instead we will now use this volatility as the basis for solving for correlation by again “reverse-implying” it. To do that we will start with portfolio math since the Drawdown Implied Correlation (DIC) is in fact a “joint” or “portfolio” derived measurement which differentiates it from other metrics. After all it is common sense that in diversification we want to see how two assets combine in a portfolio context to see whether we reduce drawdowns above more than just the average of their individual drawdowns. This is in essence what the DIC is all about.

To do that we need to create a new portfolio time series (AB) that is the equal weight of two assets being compared (A and B). With this simplification we can now provide weights and use standard portfolio math. Let’s dive into the simple derivation by isolating the correlation from the portfolio volatility formula:

This formula is still relatively involved since it has nested formulas for DIV. But fortunately it can be simplified:

Before showing some examples using drawdowns let’s verify how this implied formula works by substituting daily historical volatility (HV) and comparing it to the daily correlation coefficient:

As you can see they are exactly the same. The only minor difference when calculating a dynamic DIC is that if you use drawdowns from all-time highs versus only drawdowns contained within some lookback window then it isn’t exactly the same because you are using a cumulative measure for drawdowns that contains information outside of the lookback window. You can certainly use drawdowns entirely within a lookback window to keep the measure mathematically consistent, but it is recommended that you use a much bigger window for calculation to avoid a lot of noise. Regardless using drawdowns from all-time highs will slighly change the final values in such a way that they can be more negative than -1 which is why you need to bound the DIC between 1 and -1 to provide a practical correlation measure.

Another detail to mention is that volatility is an average metric (average of squared deviations from the mean) and as such in its basic form it is calculated at the most recent date in the lookback window being measured. In contrast, a maximum drawdown is inherently tied to a “point-in-time” at which the maximum occurs (the trough) so for accuracy you need find the date when the maximum occurs within the lookback window as the point of reference. The goal of “point in time” is to align the measurement of diversification at the trough which is what investors actually experienced at the point of maximum stress rather than a period-based measure that looks at the maximum drawdown on the most recent date in the lookback window. The graph below depicts the point-in-time reference:

Now let’s elaborate on these details:

Point-in-Time Reference:

1. Step 1: Drawdown Calculation for Each Asset:
   • For Asset A and Asset B, calculate the drawdowns from their respective all-time highs over a rolling window (e.g., 60 days).
   • For the joint time series (AB), calculate the combined drawdown from all-time highs over the same 60-day window.
2. Step 2: Find Maximum Drawdown for AB:
   • Identify the maximum drawdown for the joint time series (AB) over the 60-day rolling window.
   • Retrieve the corresponding drawdown values for Asset A and Asset B on the same day that the maximum drawdown for AB occurs.
3. Step 3: Compute the DIC:
   • Calculate the implied correlation between the drawdowns of A, B, and AB on the specific day.
   • This gives the DIC for the pair of assets based on the maximum drawdown for the joint time series (AB).

The Key Point: Portfolio Drawdown vs. Individual Asset Drawdowns

The DIC uses portfolio drawdowns which capture the path dependent dynamics from combining two assets. When constructing a portfolio with multiple assets, the portfolio’s drawdown series (the peak-to-trough losses of the combined portfolio) behaves differently than the individual drawdown series of the constituent assets. This difference arises from how the assets interact in a portfolio.

Why Portfolio Drawdown Is Different:

The drawdown of the portfolio is not simply the sum or average of individual asset drawdowns; instead, it reflects the combined behavior of the assets as they interact over time. Two or more assets in the portfolio may experience drawdowns at different times or to different extents, and their drawdown implied correlations will directly influence how the portfolio’s total drawdown evolves.

For example:

• If two assets experience drawdowns simultaneously, their joint drawdown will be greater than what you would expect from either asset alone, this will lead to a measurement of high correlation.
• If the portfolio drawdown is moderate to low compared to the individual assets’ drawdowns this will lead to a measurement of low correlation.

Therefore, the drawdown of the portfolio can reflect behavior and interactions between assets that individual asset drawdowns and returns cannot capture. This is the key reason correlating individual asset drawdowns will not fully explain the portfolio drawdowns.

This can be demonstrated using a simple example. Using the method described above we will compute the DIC using large drawdowns over the past 5 years to demonstrate the calculated values versus the inputs from Step 1 and Step 2. The only difference is that we will use the market (S&P500) instead of the portfolio as the reference point and isolate significant drawdowns (>10%) over the past 5 years. Note that using a threshold is one of many different ways to compute the DIC. The standard approach is to use a rolling lookback period similar to the Pearson correlation requires additional calculations that I will discuss shortly. In this example it is interesting to compare using the Pearson correlation between drawdowns vs the DIC. The table below shows the difference between the Pearson correlation of drawdowns for stocks and bonds vs an average of the DIC values over the same 3 periods. The DIC is a joint measurement and reflects the daily compounding and sequence of returns of the portfolio of both assets, the DIC is calculated for each date and averaged. Both measure different relationships:

• Regular correlation: Pure A-to-B relationship.
• Drawdown Implied correlation: Portfolio-centric A-to-B dynamics.

Both are valuable, but their interpretations diverge. Using both metrics together often provides richer insights into asset behavior.

Notice that the correlation of drawdowns is much more negative than the Drawdown Implied Correlation, suggesting strong diversification benefits that don’t match actual investor experience over the same time period. The DIC (the average of the DIC values on each date) shows that in the last two market drawdowns that bonds have been positively correlated with stocks and only provided diversification to equity investors during the COVID drawdown in 2020 when the Fed cut rates. Clearly using the DIC can help provide alternative or complementary analysis to computing standard correlations. Another interesting advantage is that the DIC can be calculated using only one drawdown point while you need a minimum of 3 data points to compute a correlation between drawdowns.

Next, lets compute the “standard” version of the DIC which uses the max drawdown over some window and compare it to the rolling Pearson correlation of returns over the same window length. Note you can certainly use the top % or drawdowns above a threshold as well. But because we are only looking at maximum drawdowns with this variation, in order to create a rolling daily measurement I suggest a slight modification to the original calculation by using a “triple point” reference. This means we are going to look at three reference points which represent the maximum drawdown for each asset and the portfolio. The purpose of a triple reference is to get as much information as possible from a shorter window and increase accuracy while reducing indicator volatility. The graph below depicts a visual of the triple point reference:

The triple point reference introduces a layer of robustness by also calculating DIC at the point of maximum drawdown for A and B, in addition to the joint series AB and averaging the three results. This helps capture correlations more comprehensively and reduces bias by considering all assets’ drawdown behavior. If only the joint series AB is considered, the correlation will reflect the behavior of both assets combined during the maximum drawdown period. However, this doesn’t capture how A and B behave individually during drawdowns. By referencing the max drawdowns of A and B separately, you gain insights into how each asset is contributing to the joint drawdown and whether they are truly behaving as diversifiers or correlated assets during stress periods. By considering the drawdowns of A and B individually in addition to AB, you reduce the bias that might be introduced when only looking at the joint series. The maximum drawdown for AB could be influenced by an unusually strong movement in one asset, which might not reflect the risk dynamics between A and B themselves. By averaging the DIC from the three scenarios (max drawdown of AB, A, and B), you smooth out this potential bias and get a more robust measure of correlation.

Triple Point Reference:

1. Step 1: Find the point of Max Drawdown for Asset A:
   • Now, repeat the process but for Asset A as the reference. Find the maximum drawdown for A over the same 60-day rolling window.
   • Retrieve the corresponding drawdown values for Asset B and AB on the same day that the maximum drawdown for A occurs. Calculate the DIC using the exact same formula.
2. Step 2: Find the point of Max Drawdown for Asset B:
   • Similarly, find the maximum drawdown for Asset B over the same 60-day window.
   • Retrieve the corresponding drawdown values for A and AB on the same day as the maximum drawdown for B. Calculate the DIC using the same formula.
3. Step 3: Calculate and Average DICs:
   • You now have three DICs: one from the maximum drawdown for AB, one from the maximum drawdown for A, and one from the maximum drawdown for B.
   • The final DIC is the average of these three DICs, providing a comprehensive view of the correlation during drawdown periods for both individual assets and their joint performance.

This methodology ensures that DIC reflects correlations during significant drawdown events at the point that they happen during adverse market conditions, which is crucial for portfolio risk management.

Now let’s take a look at what the 60-day DIC looks like versus the 60-day Pearson Correlation for Stocks and Bonds in 2022 to see if we notice an improvement:

What we notice is that the DIC is positive for most of 2022 and rises much faster than the traditional correlation which provides an early warning signal. This is what we want to see in the context of diversification and risk management.

The initial results were encouraging, so I decided to substitute the DIC values manually into a correlation matrix including Stocks, Bonds and Commodities to see if it would assist with improving the minimizing variance portfolios during 2022. For volatility I used the standard volatility metric in both cases, however you could use Drawdown Implied Volatility (DIV) with the DIC in a more drawdown centric optimization. To stabilize the correlation matrix, I used Ledoit-Wolf Shrinkage to shrink the DIC correlations to the identity matrix ensuring that the diagonals remained equal to 1. I verified that the eigenvalues were positive using the Cholesky Decomposition. In Part 2, I will present a very fast algorithm to ensure PSD. Here is how the drawdown implied correlation matrix improved performance over using Pearson correlations during a challenging year:

This limited example shows that the first half of the year year when correlations were negative and the DIC was positive this provided a substantial advantage, while later in the year when both correlations converged the portfolios showed very similar performance. It is important to keep in mind that the DIC can only help if there is an asset in the universe that is a true diversifier which in this case was commodities. This highlights the timeless wisdom that universe selection is extremely important. In 2022, bonds and stocks both had deep drawdowns so in a simple two asset case there is nowhere to go regardless of what the correlation is indicating. Portfolio math for two-assets shows that a more positive correlation will increase the allocation to the lower risk asset (100% allocation as the correlation approaches 1) while a more negative correlation will balance risk between both (risk parity with a -1 correlation), but if both assets have steep drawdowns that are similar and you identify a positive correlation it mathematically doesn’t matter very much for portfolio outcomes.

In terms of practical use, the combination of the Drawdown Implied Correlation matrix and the regular correlation matrix can be valuable as it can provide more information than either metric can provide individually. Using more drawdowns (the top “n” drawdowns or top % or drawdowns that exceed a percentile threshold) and a longer time period for the DIC as well as combining multiple windows for both drawdowns (maximum drawdowns over the past “n” days vs all time highs, or only considering drawdowns within the window) and their measurement period could improve the robustness and quality of the information that it provides. In part 2 we will discuss the existing research and where DIC fits, and also show a fast method to create DIC matrices that are PSD along with some new applications. In the meantime, I wanted to thank all of my readers for your support and valuable contributions over the years. I wish you all a very Happy Holidays and an awesome New Year in 2025!