第九章 风险价值度

1、Chapter 9Value at RiskRisk Management and Financial Institutions 3e, Chapter 919.1 Definition of VaR9.1.1 VaR backgroundl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，但是它们却并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。l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。Risk Management an

2、d Financial Institutions 3e, Chapter 92业界事例业界事例9-1 9-1 ： 有关有关VaRVaR的历史回顾的历史回顾lVaR在今天的广泛应用归功于J.P.摩根。lJ.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。l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。Risk Management and Financial Institutions 3e, Chapter 93l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。l为了产生风险价值度报告，1990年完成系统

3、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。l截止1993年，风险价值度已经成了测定风险的一个重要工具。l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。Risk Management and Financial Institutions 3e, Chapter 94资料资料: J.P.Morgan&CompanylJ.P.摩根公司是在世界上享有盛誉的一摩根公司是在世界上享有盛誉的一家家综合性金融公司综合性金融公司，主要，主要提供商业银行提供商业银行、投资银行和其他各种金融服务、投资银行和其他各种金融服务。公

4、司。公司的的资产规模资产规模名列著名财经杂志名列著名财经杂志财富财富美国前美国前 500 500 家大企业的前家大企业的前2020位位，而且是，而且是全球金融机构中信用评级最高的公司之全球金融机构中信用评级最高的公司之一，一，J.P.摩根公司经营商业银行业务的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子公司纽约摩根担保信托公司是美国惟惟一获得一获得AAAAAA信用评级的商业银行信用评级的商业银行。 l2000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 John PierpointPierpoint Morgan M

5、organ（1837-19121837-1912）华尔街之子华尔街之子 59.1.2 Definition of 9.1.2 Definition of VaRVaRlVaR（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。 lVaR 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

6、length T that we are X% certain will not be exceeded. Risk Management and Financial Institutions 3e, Chapter 96Example 1:Example 1:l假定J.P.摩根公司在2014年置信水平为95%的日VaR值为960万美元。l含义:该公司可以以95%的把握保证，2014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。 Risk Management and Financial Institut

7、ions 3e, Chapter 97Example 2: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:l 1.该组合在1天中只有5%的时间里损失超过45美元。l 2.给一天划分无穷多个时段，损失大于45美元的时段只有5%。Risk Management and Financial Institutions 3e, Chapter 98VaRVaR的公式表示的公式表示l公式表示为： 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

8、的损失上限。 X%给定的置信水平。()%PL TVaRX Risk Management and Financial Institutions 3e, Chapter 99()()%PLTVaRPLTVaRX Risk Management and Financial Institutions 3e, Chapter 910图9-1 由交易组合在时间T的收益概率分布来计算VaR()%PL TVaRX Risk Management and Financial Institutions 3e, Chapter 9图9-2 由交易组合在时间T的损失概率分布来计算VaR119.2 Examples

9、of the calculation of VaRlSuppose 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%?l根据正态分布的性质，置信区间为： 。l所以，最大损失为： 2

10、- N-1（0.99）*10= 2-2.33*10=-2130美元 （N-1（0.9901）=2.33）Risk Management and Financial Institutions 3e, Chapter 91()XN 129.3 VaR vs. Expected ShortfallAdvantages of VaRlIt is easy to understand. Managers are very comfortable with the idea of compressing all the Greek letters for all the market variables

11、underlying a portfolio into a single number.l It captures an important aspect of riskin a single number.lIt asks the simple question: “How bad can things How bad can things get?”get?”Risk Management and Financial Institutions 3e, Chapter 913Disadvantages of VaR However, when VaR is used in an attemp

12、t to limit the risks taken by a trader, it can lead to undesirable results.lSuppose that a bank tells a trader that the one-day 99% VaR of the traders portfolio must be kept at less than $10 million.lThe trader can construct a portfolio where there is a 99.1% chance that the daily loss is less than

13、$10 million and a 0.9% chance that it is $500 million.lThe trader is satisfying the risk limits imposed by the bank but is clearly taking unacceptable risks.Risk Management and Financial Institutions 3e, Chapter 914Distributions with the Same VaR but Different Expected Shortfalls VaRVaRRisk Manageme

14、nt and Financial Institutions 3e, Chapter 915Expected Shortfall lA measure that can produce better incentives for traders than VaR is expected shortfall.lExpected shortfall is the expected lossexpected loss (also called conditional VaR and Tail Loss).lExpected shortfall, like VaR, is a function of t

15、wo parameters:T(the time horizon) and X(the confidence level).lExpected shortfall is more difficult to understand.Risk Management and Financial Institutions 3e, Chapter 9169.4 VaR and CapitallVaR is used by regulators of financial institutions and by financial institutions themselves to determine th

16、e amount of capital they should keep.lRegulators 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.lThey calculate capital for credit risk and operational risk as the VaR using a one-year time horizon and a 99.9

17、% confidence level.Risk Management and Financial Institutions 3e, Chapter 917An examplelSuppose that the VaR of a portfolio for a confidence level of 99.9% and a time horizon of one year is $50 million. lThis means that in extreme circumstances(once every thousand years)the financial institution is

18、expected to lose more than $50 million in a year.l It also means that if it keeps $50 million in capitalif 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.Risk Management and Financial Institutions 3e, Chapter 918Properties of

19、risk measurelMonotonicity：单调性。lTranslation invariance：平移不变性。lHomogeneity：同质性。lSubadditivity：次可加性。Risk Management and Financial Institutions 3e, Chapter 919VaR vs Expected ShortfalllVaR satisfies the first three conditions but not the fourth onelExpected shortfall satisfies all four conditions.Risk M

20、anagement and Financial Institutions 3e, Chapter 9209.5 Coherent Risk Measures lRisk measures satisfying all four conditions given above are referred to as coherent.Risk Management and Financial Institutions 3e, Chapter 9219.6 Choice of parameters for VaRlTo calculate VaR, the user must choose two p

21、arameters: the time horizon and the confidence level. lA common assumption is that the change in the portfolio value over the time horizon is normally distributed.lThe mean change in the portfolio value is usually assumed to be zero.Risk Management and Financial Institutions 3e, Chapter 922lThese as

22、sumptions are convenient because they lead to a simple formula for a simple formula for VaRVaR: where X is the confidence level, is the standard deviation of the portfolio change over the time horizon, and N-1() is the inverse cumulative normal distribution. this equation shows that, regardless of t

23、he time horizon, VaR for a particular confidence level is proportional to .1()VaRXNRisk Management and Financial Institutions 3e, Chapter 923例例9-9l假定某交易组合在10天展望期上的价值变化服从正态分布，分布的期望值为0，标准差为2000万美元，10天展望期的99%VaR为： 2000*N-1（0.99）=4650万美元Risk Management and Financial Institutions 3e, Chapter 924The time

24、horizonlBanksBanks calculate the profit and loss daily, when their positions are fairly liquid and actively managed, it therefore makes sense to calculate a calculate a VaRVaR over a time horizon of one trading day over a time horizon of one trading day.lFor an investment portfolio held by a pension

25、 pension fundfund, a time horizon of one month is often time horizon of one month is often chosen.chosen. this is because the portfolio is traded less actively and some of the instruments in the portfolio are less liquid.Risk Management and Financial Institutions 3e, Chapter 925资料：持有期间的选择资料：持有期间的选择l

26、即确定计算在哪一段时间内的持有资产的最大损失值，也就是明确风险管理者关心资产在一天内一周内还是一个月内的风险价值。 l持有期的选择应依据所持有资产的特点来确定比如对于一些流动性很强的交易头寸往往需以每日为周期计算风险收益和VaR值，如G30小组在1993年的衍生产品的实践和规则中就建议对场外OTC衍生工具以每日为周期计算其VaR，而对一些期限较长的头寸如养老基金和其他投资基金则可以以每月为周期。 l从银行总体的风险管理看持有期长短的选择取决于资产组合调整的频度及进行相应头寸清算的可能速率。巴塞尔委员会在这方面采取了比较保守和稳健的姿态，要求银行以两周即10个营业日为持有期限。 Risk Man

27、agement and Financial Institutions 3e, Chapter 926N-day VaRlWhatever the application, when market risks are being considered, analysts almost invariably start by calculating VaR for a time horizon of one day.lThe N-day VaR equals times the one-day VaRlChanges in the value have independent identical

28、normal distributions with mean zero.1*NdayVaRdayVaRN NRisk Management and Financial Institutions 3e, Chapter 927Confidence levelConfidence levell一般来说对置信水平的选择在一定程度上反映了金融机构对风险的不同偏好。选择较大的置信水平意味着其对风险比较厌恶，希望能得到把握性较大的预测结果，希望模型对于极端事件的预测准确性较高。 l根据各自的风险偏好不同，选择的置信区间也各不相同。比如J.P. Morgan与美洲银行选择95，花旗银行选择95.4，大通曼哈

29、顿选择97.5，Bankers Trust选择99。 l作为金融监管部门的巴塞尔委员会则要求采用99的置信水平，这与其稳健的风格是一致的。 Risk Management and Financial Institutions 3e, Chapter 928不同置信水平不同置信水平VaRVaR之间的关系之间的关系lFrom equation(9.1):lA VaR with a confidence level of X can be calculated from a VaR with a lower confidence level of X using:11()()()*()NXVaRVaR XXXN1()()VaR XXN1()()VaRNXXRisk Management and Financial Institutions 3e, Chapter 929例例9-11l假定一交易组合一天的95%VaR为150万美元，同时假定交易组合的价值变化服从正态分布，期望值为0，由式（9-4）得出，一天展望期的99%VaR为150*2.326/1.645=212万美元。l如果我们假定交易组合的价值变化在每天相互独立，因此10天的99%VaR=