[Paper Review] Quantum algorithm for calculating risk contributions in a credit portfolio

Koichi Miyamoto|arXiv (Cornell University)|Jan 27, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a quantum algorithm for efficiently calculating conditional value-at-risk (CVaR) contributions in a credit portfolio using simultaneous estimation of multiple expected values. By leveraging quantum amplitude estimation and optimized oracles, the method achieves $Ϫ{O}(√{N_{\text{gr}}}/\epsilon)$ query complexity, offering quantum quadratic speedup in accuracy $ε$ but limited scaling advantage over classical methods as the number of subgroups $N_{\text{gr}}$ increases.

ABSTRACT

Finance is one of the promising field for industrial application of quantum computing. In particular, quantum algorithms for calculation of risk measures such as the value at risk and the conditional value at risk of a credit portfolio have been proposed. In this paper, we focus on another problem in credit risk management, calculation of risk contributions, which quantify the concentration of the risk on subgroups in the portfolio. Based on the recent quantum algorithm for simultaneous estimation of multiple expected values, we propose the method for credit risk contribution calculation. We also evaluate the query complexity of the proposed method and see that it scales as $\widetilde{O}\left(\sqrt{N_{ m gr}}/ε ight)$ on the subgroup number $N_{ m gr}$ and the accuracy $ε$, in contrast with the classical method with $\widetilde{O}\left(\log(N_{ m gr})/ε^2 ight)$ complexity. This means that, for calculation of risk contributions of finely divided subgroups, the advantage of the quantum method is reduced compared with risk measure calculation for the entire portfolio. Nevertheless, the quantum method can be advantageous in high-accuracy calculation, and in fact yield less complexity than the classical method in some practically plausible setting.

Motivation & Objective

• To address the computational challenge of calculating risk contributions in large credit portfolios, which quantify concentration of risk across subgroups.
• To extend quantum Monte Carlo methods beyond aggregate risk measures (like VaR and CVaR) to conditional risk contributions.
• To leverage recent advances in quantum algorithms for simultaneous estimation of multiple expected values to improve efficiency.
• To evaluate the quantum advantage in practical settings where high-accuracy risk analysis is required.

Proposed method

• Utilizes the quantum algorithm for simultaneous estimation of multiple expected values (Cornelissen et al.) to compute CVaR contributions for multiple obligor groups in parallel.
• Constructs oracles for probability amplitude encoding, controlled rotations, and standard normal random variable state preparation using fixed-point quantum amplitude estimation.
• Employs quantum circuits to encode the joint default probability distribution and conditional loss values across subgroups.
• Reduces query complexity by estimating all risk contributions in a single quantum run, avoiding sequential application of single-expected-value algorithms.
• Applies quantum amplitude estimation with fixed-point iteration to achieve high-accuracy estimation of conditional expectations.
• Derives query complexity scaling as $Ϫ{O}(√{N_{\text{gr}}}/\epsilon)$, where $N_{\text{gr}}$ is the number of subgroups and $\epsilon$ is the error tolerance.

Experimental results

Research questions

• RQ1 Can quantum algorithms provide a speedup for calculating CVaR contributions across multiple subgroups in a credit portfolio?
• RQ2 How does the quantum query complexity scale with the number of subgroups $N_{\text{gr}}$ and accuracy $\epsilon$ compared to classical Monte Carlo?
• RQ3 In what practical settings does the quantum method outperform classical methods despite the lack of quadratic speedup in $N_{\text{gr}}$?
• RQ4 How can oracles for probability distribution encoding and conditional loss evaluation be efficiently constructed in a quantum framework?
• RQ5 What is the trade-off between quantum advantage and the granularity of risk subgrouping in credit risk analysis?

Key findings

• The proposed quantum algorithm achieves a query complexity of $Ϫ{O}(√{N_{\text{gr}}}/\epsilon)$, offering quantum quadratic speedup in the error tolerance $\epsilon$.
• Classical Monte Carlo complexity scales as $Ϫ{O}(\log(N_{\text{gr}})/\epsilon^2)$, making the quantum method more efficient for high-accuracy settings.
• Even when computing contributions for all individual obligors ($N_{\text{gr}} = N_{\text{obl}}$), the quantum method can outperform classical methods in some plausible scenarios.
• The quantum advantage diminishes with finer subgrouping, as the $\sqrt{N_{\text{gr}}}$ scaling is less favorable than the classical $\log(N_{\text{gr}})$ scaling.
• The method remains viable for $N_{\text{obl}} \ll 10^8$, suggesting practical applicability in real-world credit risk management.
• Theoretical analysis confirms that the variance of estimated risk contributions remains bounded, ensuring stability under quantum estimation.