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typora-copy-images-to: Risk Management and Financial Institution

Risk Management and Financial Institution Chapter 12 —— VaR and Es

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• Value at risk (VaR) and expected shortfall (ES) are attempts to provide a single number that summarizes the total risk in a portfolio

12.1 Definition of VaR

• It is the loss level during a time period of length T that we are X% certain will not be exceeded

12.2 Examples (略)

12.3 Drawback of VaR

12.4 Expected Shortfall

• 又称作为conditional value at risk,expected tail loss

• ES is the expected loss during time T conditional on the loss being greater than the VaR

• ES has better properties than VaR in that it always recognizes the benefits of diversification , 分散化的收益可以被算入

• ES的缺陷：

  • 比VaR更加难以理解
  • 回测过程也更加难

12.5 Coherent Risk Measures

• 风险度量应该满足一定的性态：

  • monotonicity 单调性 收益小，意味着风险大
  • Translation Invariance 平移不变性 加入K的cash ，总风险减去cash
  • Homogeneity 同质性 保持组合中单个资产所占比例不变，调整组合大小，风险相应增减
  • Subadditivity 次可加性 两个交易组合组成的新交易组合风险度量小于等于单独的和

• VaR 并不满足次可加性

• ES 测量满足四种性态

12.5.1 Exponential Spectral Risk Measures

• A risk measure can be characterized by the weights it assigns to percentiles of the loss distribution

• VaR 给了百分之一百的权重在分界点上

• One idea is to make the weight assigned to the qth percentile proportional to where γ is a constant. This is referred to as the exponential spectral risk measure

• 

• 不同的γ值 ，不同的指数形态

12.6 VaR 和 ES 参数的选择

• 两个参数必须选择，时间区间以及置信区间

• 假设时间界限内，交易组合的变化服从正态分布，X 是置信区间
• ES 的估计公式如下，假设loss is normally distributed

• Y is the Xth percentile point of the standard normal distribution

• This shows that when μ is assumed to be zero, ES, like VaR, is proportional to σ

12.6.1 Time Horizon 展望期

• 流动性差的组合，展望期长
• market risks 展望期一般为1天

• 假设：when the changes in the value of the portfolio on successive days have independent identical normal distributions with mean zero

12.6.2 Impact of Autocorrelation 自相关的影响

• 现实中，组合价值的变化并不独立
• 如果天与天之间的组合价值变化并不独立，那么其实用平方根法则预估的VaR值偏低

12.6.3 Confidence Level 置信区间

• 选择置信区间的因素：
  • 与信用评级挂钩，AA级99.97%置信区间，一年的展望期

12.7 Marginal, Incremental and Component Measures 边际VaR，递增VaR及成分VaR

• The marginal value at risk for the ith subportfolio is the sensitivity of VaR to the amount invested in the ith subportfolio交易组合价格变化同某个组合成分变化的比率

• If an asset’s beta is high, its marginal VaR will tend to be high. If its beta is low, the marginal VaR tends to be low

• incremental value at risk 是指子交易组合对VaR的递增效应，是组合包含与不包含此子组合VaR的差

12.8 Euler’s Theorem 欧拉定理

• 欧拉定理与风险度量的第三性态相一致
• 当V 为 VaR时：

• 组合VaR 相当于Component VaR的相加
• ES的性质与VaR一致：

• 欧拉定理是用于 risk budgeting 的很好的工具

12.9 Aggregating VaRs and ESs VaR的聚合

• 不同部门的VaR聚合成total VaR

• ρ 代表着损失与损失之间的相关性
• 上式当损失与收益的预期为0时严格成立，其他时候也是很好的估计

12.10 Back- Testing 回测

• It is a test of how well the current procedure for calculating the measure would have worked in the past 历史的检验

• VaR 的回测要比ES来得容易，因此监管者在制定市场风险时转换为ES很勉强

• Back-testing involves looking at how often the loss in a day would have exceeded the one-day 99% VaR when the latter is calculated using the current procedure

• 实际损失超出VaR值叫做exceptions

• 在回测时，我们应该考虑展望期内交易组合本身的变化

  • 假定展望期内组合构成没有变化
  • 与交易组合真实变化进行比较

• In practice, risk managers usually compare VaR to both hypothetical portfolio changes and actual portfolio changes

• Kupiec test （双尾检验）与3.84做比较，大于3.84 则拒绝模型

• Generally speaking, the difficulty of back-testing a VaR model increases as the VaR confidence level increases

12.10.1 Bunching 聚束效应

• 使用Chiristofferson test 例外的发生是均匀分布的，还是聚束分布的