第八章-风险价值度

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1、Chapter 8Risk Management and Financial Institutions 2e,Chapter 8,Copyright John C.Hull 2009The VaR Measure18.1 Definition of VaRl8.1.1 VaR background金融风险市场风险流动性风险操作风险信用风险其他风险为什么用为什么用VaRVaR来管理风险来管理风险?l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

2、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。业界事例业界事例8-18-1 ：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。l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

3、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。测度风险测度风险:一个历史视角一个历史视角l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。公司。公司的的资产规模名列著名财经杂志资产规模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

4、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 Pierpoint MorganJohn Pierpoint Morgan（1837-19121837-1912）华尔街之子华尔街之子 8.1.2 Definition of VaR8.1.2 Definition of

5、 VaRlVaR（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 length T that we are X%certain will not be exceeded.Example 1:Example 1:l假定J.P.摩根公司

6、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: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美元的时段只占有5%。VaRVaR的表示公式的表示公式l用公式表示为：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

7、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。()%PL TVaRX()()%PL TVaRPL TVaRX ()%PL TVaRX 8.2 Examples 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 f

8、or 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：Z。l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 VaRlIt is easy to understand.Managers are very comfortable with the idea of compressing all the Greek lette

9、rs for all the market variables 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?”Disadvantages of VaR However,when VaR is used in an attempt to limit the risks taken b

10、y 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$10 million and a 0.9%chance that

11、 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 2e,Chapter 8,Copyright John C.Hull 200916Distributions with the Same VaR but Different Expected Shortfalls Risk Management and Financia

12、l Institutions 2e,Chapter 8,Copyright John C.Hull 2009VaRVaR17Expected 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

13、 function of two parameters:T(the time horizon)and X(the confidence level).lExpected shortfall is more difficult to understand.Risk Management and Financial Institutions 2e,Chapter 8,Copyright John C.Hull 2009188.4 VaR and CapitallVaR is used by regulators of financial institutions and by financial

14、institutions themselves to determine the 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 us

15、ing a one-year time horizon and a 99.9%confidence level.An 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 expected to lose more th

16、an$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.Properties of risk measurelMonotonicity:单调性。lTranslation invariance:平移不变性。lHomogeneity:同质性。lSubaddit

17、ivity:次可加性。VaR vs Expected ShortfalllVaR satisfies the first three conditions but not the fourth onelExpected shortfall satisfies all four conditions.Risk Management and Financial Institutions 2e,Chapter 8,Copyright John C.Hull 2009228.5 Coherent Risk Measures(page 120)lRisk measures satisfying all

18、four conditions given above are referred to as coherent.Risk Management and Financial Institutions 2e,Chapter 8,Copyright John C.Hull 2009238.6 Choice of parameters for VaRlTo calculate VaR,the user must choose two parameters:the time horizon and the confidence level.lA common assumption is that the

19、 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.lThese assumptions are convenient because they lead to a simple formula for VaRa simple formula for VaR:where X is the confidence level,is the standard d

20、eviation of the portfolio change over the time horizon,and N-1()is the inverse cumulative normal distribution.this equation shows that,regardless of the time horizon,VaR for a particular confidence level is proportional to.1()VaRXN例例8-9l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：

21、2000*N-1（0.99）=4650万美元The time 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 VaR over a time horizon of one trading dayVaR over a time horizon of one trading day.lFor an investmen

22、t portfolio held by a pension 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.资料：持有期间的选择资料：持有期间的选择l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

23、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。T-day VaRlWhatever the application,when ma

24、rket 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 normal distributions with mean zero.Risk Management and Financial Institutions 2e,Chapter 8,Co

25、pyright John C.Hull 2009291*NdayVaRdayVaRN NImpact of autocorrelation(了解了解)lWhen daily changes in a portfolio are identically distributed and independent the variance over T days is T times the variance over one daylWhen there is autocorrelation equal to r the multiplier is increased from T to 1322)

26、3(2)2(2)1(2rrrrTTTTTRisk Management and Financial Institutions 2e,Chapter 8,Copyright John C.Hull 200930Impact of Autocorrelation:Ratio of N-day VaR to 1-day VaR(Table 8.1,page 204)Risk Management and Financial Institutions 2e,Chapter 8,Copyright John C.Hull 2009T=1T=2T=5T=10T=50T=250r=01.01.412.243

27、.167.0715.81r=0.05 1.01.452.333.317.4316.62r=0.11.01.482.423.467.8017.47r=0.21.01.552.623.798.6219.3531Confidence 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，大通曼哈顿选择97.5，Ban

28、kers 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。不同置信水平不同置信水平VaR之间的关系之间的关系lFrom equation(8.1):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()()VaRNXX例例8-11l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

29、从正态分布，期望值为0，由式（8-4）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=*212=671万美元，250天的99%VaR=*212=3354万美元。102508.7 Marginal VaR,Incremental VaR and Component VaR(page 123-124)lMarginal VaR:Marginal VaR:consider a portfolio with a number of subportfolios where the investment

30、 in the ith subportfolio is xi.The marginal VaR is the sensitivity of VaR to the size of the ith subportfolio.it is:ixVaRRisk Management and Financial Institutions 2e,Chapter 8,Copyright John C.Hull 200935lIncremental VaR:Incremental effect on VaR of the ith subportfolio(what is the difference betwe

31、en VaR with and without the subportfolio).lIncremental VaR of the ith subportfolio:*iiV a RxxlComponent VaR:式中，N为成分的个数，其中第i个子交易组合成分VaR被定义为：1*niiiVaRVaRxx*iiiVaRxxC8.8 Back-testing(page 124-125)l计算出的VaR必须同现实进行比较，这一测试被称为回顾测试。例：假定一天99%的VaR模型，在回顾测试中，需要找出交易组合每天的损失有多少次超出了一天的99%的VaR，实际损失超出VaR的情形称为例外，如果例外的天数大约占整体天数的1%，欣慰；远大于1%，VaR估计偏低。Risk Management and Financial Institutions 2e,Chapter 8,Copyright John C.Hull 200938The end