Some properties of Euler capital allocation

ArXiv ID: 2405.00606 “View on arXiv”

Authors: Unknown

Abstract

The paper discusses capital allocation using the Euler formula and focuses on the risk measures Value-at-Risk (VaR) and Expected shortfall (ES). Some new results connected to this capital allocation is known. Two examples illustrate that capital allocation with VaR is not monotonous which may be surprising since VaR is monotonous. A third example illustrates why the same risk measure should be used in capital allocation as in the evaluation of the total portfolio. We show how simulation may be used in order to estimate the expected Return on risk adjusted capital in the commitment period of an asset. Finally, we show how Markov chain Monte Carlo may be used in the estimation of the capital allocation.

Keywords: Capital allocation, Value-at-Risk, Expected Shortfall, Risk measures, Markov chain Monte Carlo

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper involves advanced mathematical concepts like coherent risk measures, Euler’s theorem, and MCMC estimation, but lacks any empirical backtesting or dataset implementation, focusing instead on theoretical properties and illustrative examples.

  flowchart TD
    A["Research Goal: Investigate<br>Euler Capital Allocation<br>using VaR and ES"] --> B["Methodology: Theoretical Analysis<br>and Numerical Simulation"]
    
    B --> C["Key Analysis 1: Monotonicity<br>Test of VaR Allocation"]
    C --> D["Outcome 1: VaR capital allocation<br>can be non-monotonic<br>despite VaR being monotonic"]
    
    B --> E["Key Analysis 2: Consistency<br>of Risk Measures"]
    E --> F["Outcome 2: Allocation and evaluation<br>must use the same risk measure<br>for consistency"]
    
    B --> G["Key Analysis 3: Computational<br>Estimation via MCMC"]
    G --> H["Outcome 3: MCMC enables efficient<br>estimation of Expected Return on<br>Risk Adjusted Capital ERARC"]