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Extreme Value Theory Applied to US Treasury Tail Risks
In traditional financial modeling, daily market movements are often assumed to follow a normal (Gaussian) distribution. However, history proves that markets are fundamentally asymmetric and prone to "black swan" events.
To illustrate this, we examine the 10-Year US Treasury Yield. Looking at the raw yield over the last decades,we see prolonged periods of rising and falling yields :
Because interest rates in level are non-stationary, quantitative risk models must focus on First Differences (the daily yield shocks in basis points). When we plot these daily shocks, a classic stylized fact of financial time series emerges: volatility clustering. Quiet periods are violently interrupted by clusters of extreme stress.
The histogram of these daily changes reveals the fundamental flaw of normal distributions: the presence of "fat tails". While most days see minimal movement, the extremes stretch far beyond what standard variance models predict.
To properly quantify these tail risks — specifically the extreme negative shocks (yield drops) associated with rumors of recession — we turn to Extreme Value Theory (EVT).
The first method in EVT is the Block Maxima approach. Instead of looking at all daily variations, we divide the history into equal blocks (e.g., one year) and extract only the single most extreme event of that period. For this study, we focus on the left tail (the deepest daily rate drop of the year, mathematically inverted to fit maximum-likelihood algorithms).
By isolating these extreme annual shocks, we fit the Generalized Extreme Value (GEV) distribution, whose cumulative distribution function is given by:
The critical parameter here is the Shape (ξ), which dictates the weight of the tail, while μ represents the location and σ the scale. Our Maximum Likelihood Estimation (MLE) yielded ξ=0.2612. Because ξ>0, we mathematically confirm that the tail follows a Fréchet distribution (a heavy-tailed distribution, meaning variance can theoretically explode).
The mathematical fit is highly accurate. Plotting the Empirical CDF against the Theoretical GEV CDF shows that the model captures the curvature of the extremes perfectly:
Using the calibrated GEV parameters, we can project T(x), the expected waiting time (Return Period) in years for a catastrophe of magnitude x to occur. Mathematically, this is the inverse of the exceedance probability:
The Return Period Curve below illustrates this non-linear relationship between the magnitude of a market shock and its historical rarity. For instance, a 75 bps single-day drop (like the one seen in 1987) is modeled as a "1-in-100 years" event.
While Block Maxima is robust, it wastes data. If a highly volatile year has five massive shocks, Block Maxima discards four of them. To solve this, modern risk management uses the Peaks-Over-Threshold (POT) method, modeled by the Generalized Pareto Distribution (GPD), given by the following cumulative distribution:
First, we must define the threshold (u). We use a Mean Excess Plot. According to EVT, if the tail follows a Pareto distribution, the mean excess over the threshold should be a straight, upward-sloping line.
The linearity clearly begins at u=10 bps. By setting our threshold there, we isolate 639 true "stress events" from the historical noise. Fitting the GPD to these exceedances refines our tail shape parameter to ξ = 0.0979, providing a more stable heavy-tail estimation.
Once again, the model's goodness-of-fit is exceptional, accurately capturing the cumulative probabilities of the excesses
With the GPD properly calibrated, we can accurately calculate the boundaries of extreme risk at a 99% confidence level:
Value at Risk (VaR 99%): The threshold where the worst 1% of days begin.
Conditional VaR (CVaR 99% / Expected Shortfall): The mathematical expectation of the shock given that the VaR has been breached.
As seen in the probability density plot above, the heavy tail pulls the CVaR significantly to the right of the VaR. While a normal distribution assumes things don't get much worse once the VaR is breached, EVT proves that in the US Treasury market, the "unforeseen" averages a brutal 28 bps daily shock when panic truly sets in.
Financial markets do not behave normally during crises. By applying Extreme Value Theory to the 10-Year US Treasury Yield, we mathematically validate that interest rate shocks possess heavy tails (ξ>0).
Both the Block Maxima (GEV) and Peaks-Over-Threshold (GPD) approaches provide a rigorous framework for quantifying events that standard models dismiss as "impossible." The substantial gap between VaR and Expected Shortfall highlights a crucial lesson for risk managers: surviving the market means preparing not just for the edge of the distribution, but for the depth of the tail. Meeting the unforeseen requires accepting that extreme events are not anomalies—they are structural features of the system.
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