Determine bounds for the difference between standard Conditional Value at Risk and its smooth approximation | Step-by-Step Solution

What We're Solving:

We need to analyze the difference between the standard Conditional Value at Risk (CVaR) and its smooth approximation, then find mathematical bounds for this difference. This is a problem that connects risk theory with approximation methods!

The Approach:

CVaR is a risk measure that's very useful but can be non-smooth (not differentiable everywhere), making optimization challenging. We create a "smoothed" version that's easier to work with mathematically. Our goal is to understand how different these two versions can be - we want to put a "fence" around the possible difference. This is crucial because we need to know if our smooth approximation is still meaningful!

Step-by-Step Solution:

Step 1: Define Standard CVaR For a random variable X (representing losses) and confidence level α ∈ (0,1):

• VaR_α(X) = inf{t : P(X ≤ t) ≥ α} (Value at Risk)
• CVaR_α(X) = E[X | X ≥ VaR_α(X)] (expected loss beyond VaR)

Alternatively: CVaR_α(X) = (1/(1-α)) ∫_{α}^1 VaR_u(X) du

Step 2: Introduce the Smooth Approximation The most common smooth approximation uses a parameter β > 0: CVaR_α^β(X) = (1/β) log(E[e^{βX} · 1_{X ≥ VaR_α(X)}] / (1-α))

Or using the log-sum-exponential smoothing: CVaR_α^β(X) = VaR_α(X) + (1/β) log((1/(1-α)) ∫ e^{β(x-VaR_α(X))} dF(x))

Step 3: Set Up the Difference We want to bound: |CVaR_α(X) - CVaR_α^β(X)|

Step 4: Use Jensen's Inequality Since e^{βx} is convex, Jensen's inequality gives us: e^{β·E[X|X≥VaR_α]} ≤ E[e^{βX} | X ≥ VaR_α]

Taking logarithms and dividing by β: E[X | X ≥ VaR_α] ≤ (1/β) log(E[e^{βX} | X ≥ VaR_α])

This gives us: CVaR_α(X) ≤ CVaR_α^β(X)

Step 5: Find the Upper Bound For the upper bound, we need to control how much larger the smooth version can be. If X is bounded above by M, then: CVaR_α^β(X) - CVaR_α(X) ≤ (M - VaR_α(X)) · (e^{β(M-VaR_α(X))} - 1) / (β(e^{β(M-VaR_α(X))} - 1))

For small β, this simplifies to approximately: (β/2) · Var[X | X ≥ VaR_α]

Step 6: Complete the Bounds Under suitable regularity conditions (bounded moments), we get: 0 ≤ CVaR_α^β(X) - CVaR_α(X) ≤ C/β

where C depends on the distribution's tail behavior.

The Answer:

The bounds for the difference are:

• Lower bound: CVaR_α^β(X) ≥ CVaR_α(X) (smooth version overestimates)
• Upper bound: CVaR_α^β(X) - CVaR_α(X) ≤ C/β where C depends on the distribution

Key insight: As β → ∞, the smooth approximation converges to the true CVaR, and the approximation error decreases as O(1/β).

Memory Tip:

Think of smoothing like "blurring" a sharp edge in a photo - the smooth CVaR is always a bit "softer" (larger) than the original, but as we increase the smoothing parameter β, we can make this difference arbitrarily small. The trade-off is between computational convenience (smooth functions) and approximation accuracy!