第八章风险价值度

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第八章光结局和夜结局推荐度：
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The VaR MeasureChapter 8

8.1 Definition of VaR

8.1.1 VaR background

金融风险

市场风险信用风险

流动性风险

操作风险其他风险

为什么用VaR来管理风险?

金融机构的交易组合往往取决于成百上千个市场变量(例如，股指、利率或商品价格),因此，交易员每天要计算大量的Delta、Gamma 和Vega,但是它们却幵不能为金融机构的高管及金融机构的监管人员提供一个关于整体风险的完整图像。风险价值度试图对金融机构的资产组合提供一个单一风险度量，而这一度量恰恰能体现金融机构的整体风险。

业界事例8-1

: 有关VaR的历史回顾

VaR在今天的广泛应用归功于J.P.摩根。 J.P.摩根总裁对每天收到冗长的报告很不满意(敏感度报告),这些报告对银行整体风险管理意义不大。

希望收到更为简洁的报告，报告应该阐明银行的整体交易组合在今后的24小时所面临的风险。

首先是基于马克维茨交易组合理论为基础建立了风险价值度报告。为了产生风险价值度报告，1990年完成系统开发工作，这样系统的好处是使得银行高管对于银行自身所面临的风险有了清醒的认识。截止1993年，风险价值度已经成了测定风险的一个重要工具。

巴塞尔委员会在1996年公布了基于风险价值度的协议修正案，这一修正案在1998年得到了执行。

测度风险:一个历史视角

资料：J.P.Morgan&CompanyJ.P.摩根公司是在世界上享有盛誉的一家综合性金融公司，主要提供商业银行、投资银行和其他各种金融服务。公司的资产规模名列著名财经杂志《财富》美国前 500 家大企业的前20位，而且是全球金融机构中信用评级最高的公司之一，J.P.摩根公司经营商业银行业务的子公司纽约摩根担保信托公司是美国惟一获得AAA信用评级的商业银行。 2000年J.P.摩根公司与大通银行及富林明集团完成合并成立摩根大通(JP Morgan Chase)。

John Pierpoint Morgan (1837-1912) 华尔街之子

8.1.2 Definition of VaR

VaR(Value at Risk):“风险价值”或“ 在险价值”,指在一定的置信水平下，某一金融资产(或证券组合)在未来特定的一段时间内的最大可能损失。 VaR is a function of two parameters: the time horizon,T and the confidence level, X percent. It is the loss level during a time period of length T that we are X% certain will not be exceeded.

Example 1:

假定J.P.摩根公司在2012年置信水平为 95%的日VaR值为960万美元，其含义指该公司可以以95%的把握保证，2012年某一特定时点上的金融资产在未来24小时内，由于市场价格变动带

来的损失不会超过 960万美元。或者说，只有5%的可能损失超过960万美元。

Example 2:一个投资组合持有1天,置信水平95%, VaR 等于45美元.其含义是: 1.该组合在1天中只有5%的时间里损失超过45美元。 2.给一天划分无穷多个时段，损失大于45 美元的时段只占有5%。

VaR的表示公式

用公式表示为：

P( L T VaR) X %其中， P—— 资产价值损失小于可能损失上限的概率，即英文Probability。 △ L—— 某一金融资产或组合在一定持有期△T的价值损失额。 VaR—— 给定置信水平 X% 下的在险价值，即可能的损失上限。 X%——给定的置信水平。

P( L T VaR) P( L T VaR) X %

P( L T VaR) X %

8.2 Examples of the calculation of VaR

Suppose that the gain from a portfolio during six months is normally distributed with a mean of $2 million and a standard deviation of $10 million. How to the VaR for the portfolio with a time horizon of six months and confidence level of 99%? 根据正态分布的性质，置信区间为：μ±Zασ。所以，最大损失为：2- Z0.99*10=2-2.33*10=2130美元(Z0.9901=2.33)

8.3 VaR vs. Expected ShortfallAdvantages of VaR It is easy to understand. Managers are very comfortable with the idea of compressing all the Greek letters for all the market variables underlying a portfolio into a single number. It captures an important aspect of risk

in a single number. It asks the simple question: “How bad can things get?”

Disadvantages of VaRHowever, when VaR is used in an attempt to limit the risks taken by a trader, it can lead to undesirable results. Suppose that a bank tells a trader that the one-day 99% VaR of the trader’s portfolio must be kept at less than $10 million. The trader can construct a portfolio where there is a 99.1% chance that the daily loss is less than $10 million and a 0.9% chance that it is $500 million. The trader is satisfying the risk limits imposed by the bank but is clearly taking unacceptable risks.16

Distributions with the Same VaR but Different Expected Shortfalls

VaR

Expected Shortfall

A measure that can produce better incentives for traders than VaR is expected shortfall. Expected shortfall is the expected loss (also called conditional VaR and Tail Loss). Expected shortfall, like VaR, is a function of two parameters:T(the time horizon) and X(the confidence level). Expected shortfall is more difficult to understand.18

8.4 VaR and Capital

VaR is used by regulators of financial institutions and by financial instituti

ons themselves to determine the amount of capital they should keep. Regulators calculate the capital required for market risk as a multiple of the VaR calculated using a ten-day time horizon and a 99% confidence level. They calculate capital for credit risk and operational risk as the VaR using a one-year time horizon and a 99.9% confidence level.

An example

Suppose that the VaR of a portfolio for a confidence level of 99.9% and a time horizon of one year is $50 million. This means that in extreme circumstances(once every thousand years)the financial institution is expected to lose more than $50 million in a year. It also means that if it keeps $50 million in capital it will have a 99.9% probability of not running out of capital in the course of one year.

Properties of risk measure

Monotonicity:单调性。

Translation invariance:平移不变性。Homogeneity:同质性。

Subadditivity:次可加性。

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