[论文解读] Beyond cash-additive capital requirements: when changing the numeraire fails

Walter Farkas, Pablo Koch‐Medina+1 more|arXiv (Cornell University)|Jun 3, 2012

Risk and Portfolio Optimization被引用 4

一句话总结

本文通过分析当合格资产为违约债券时的资本要求，挑战了传统现金加性风险度量的使用，表明在此类情况下贴现方法失效。本文引入并表征了现金次可加性风险度量，证明当合格资产具有风险时，此类度量并非普遍现象，而是例外；并确立了在非线性定价规则下风险度量保持拟凸性的条件。

ABSTRACT

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numeraire. However, discounting does not work in all financially relevant situations, typically when the eligible asset is a defaultable bond. In this paper we fill this gap allowing for general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on Value-at-Risk and Tail Value-at-Risk on $L^p$ spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that, when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules.

研究动机与目标

• 解决当合格资产为违约债券时，传统现金加性风险度量的局限性，此时标准的计价单位变换方法失效。
• 建立基于任意合格资产（包括违约债券）的一般风险度量框架。
• 表征现金次可加性成立的条件，特别是在违约资产背景下的情况。
• 研究非线性、 illiquid 定价规则对风险度量拟凸性的影响。
• 明确现金次可加性在 illiquid 市场中与连续定价规则兼容的条件。

提出的方法

• 将风险度量形式化为：为实现充分资本化，必须投资于预设合格资产的最低资本要求。
• 通过允许使用一般合格资产（包括违约债券）扩展风险度量理论，超越现金加性框架。
• 将理论应用于具体风险度量：在 $L^p$ 空间上的风险价值（Value-at-Risk）、尾部风险价值（Tail Value-at-Risk），以及在 Orlicz 空间上的短路风险。
• 引入并分析现金次可加性这一属性，作为在存在违约资产时对现金加性的替代。
• 确立在一般合格资产下风险度量的有限性与连续性条件。
• 研究当定价规则为非线性时（尤其在 illiquid 市场中），风险度量的拟凸性。

实验结果

研究问题

• RQ1 当合格资产为违约债券时，标准计价单位变换在何种条件下会失效？
• RQ2 基于违约债券的风险度量在何种条件下满足现金次可加性？
• RQ3 风险度量的有限性与连续性属性如何依赖于合格资产的选择？
• RQ4 在非线性、illiquid 定价规则下，何种条件可确保风险度量的拟凸性？
• RQ5 现金次可加性是否与 illiquid 市场中的连续定价规则兼容？

主要发现

• 当合格资产为违约债券时，通过计价单位变换将一般合格资产简化为现金加性度量的标准方法会失效。
• 当合格资产为违约债券时，现金次可加性并非普遍现象；相反，它属于例外而非常规。
• 对于基于 $L^p$ 空间上的风险价值与尾部风险价值的风险度量，在适当的可积性条件下可建立其有限性与连续性。
• 在 Orlicz 空间上的短路风险度量，若损失函数满足适当的增长条件，同样满足有限性与连续性。
• 现金次可加性仅在定价机制为线性或足够规则时，才与连续定价规则兼容。
• 在非线性定价规则下，风险度量的拟凸性仅在定价规则满足特定连续性与单调性假设时才能保证。