第九章 風(fēng)險(xiǎn)價(jià)值度

1、Chapter 9Value at RiskRisk Management and Financial Institutions 3e, Chapter 919.1 Definition of VaR9.1.1 VaR backgroundl金融機(jī)構(gòu)的交易組合往往取決于成百上千個(gè)市場(chǎng)變量（例如，股指、利率或商品價(jià)格），因此，交易員每天要計(jì)算大量的Delta、Gamma和Vega，但是它們卻并不能為金融機(jī)構(gòu)的高管及金融機(jī)構(gòu)的監(jiān)管人員提供一個(gè)關(guān)于整體風(fēng)險(xiǎn)的完整圖像。l風(fēng)險(xiǎn)價(jià)值度試圖對(duì)金融機(jī)構(gòu)的資產(chǎn)組合提供一個(gè)單一風(fēng)險(xiǎn)度量，而這一度量恰恰能體現(xiàn)金融機(jī)構(gòu)的整體風(fēng)險(xiǎn)。Risk Management an

2、d Financial Institutions 3e, Chapter 92業(yè)界事例業(yè)界事例9-1 9-1 ： 有關(guān)有關(guān)VaRVaR的歷史回顧的歷史回顧lVaR在今天的廣泛應(yīng)用歸功于J.P.摩根。lJ.P.摩根總裁對(duì)每天收到冗長(zhǎng)的報(bào)告很不滿意（敏感度報(bào)告），這些報(bào)告對(duì)銀行整體風(fēng)險(xiǎn)管理意義不大。l希望收到更為簡(jiǎn)潔的報(bào)告，報(bào)告應(yīng)該闡明銀行的整體交易組合在今后的24小時(shí)所面臨的風(fēng)險(xiǎn)。Risk Management and Financial Institutions 3e, Chapter 93l首先是基于馬克維茨交易組合理論為基礎(chǔ)建立了風(fēng)險(xiǎn)價(jià)值度報(bào)告。l為了產(chǎn)生風(fēng)險(xiǎn)價(jià)值度報(bào)告，1990年完成系統(tǒng)

3、開發(fā)工作，這樣系統(tǒng)的好處是使得銀行高管對(duì)于銀行自身所面臨的風(fēng)險(xiǎn)有了清醒的認(rèn)識(shí)。l截止1993年，風(fēng)險(xiǎn)價(jià)值度已經(jīng)成了測(cè)定風(fēng)險(xiǎn)的一個(gè)重要工具。l巴塞爾委員會(huì)在1996年公布了基于風(fēng)險(xiǎn)價(jià)值度的協(xié)議修正案，這一修正案在1998年得到了執(zhí)行。Risk Management and Financial Institutions 3e, Chapter 94資料資料: J.P.Morgan&CompanylJ.P.摩根公司是在世界上享有盛譽(yù)的一摩根公司是在世界上享有盛譽(yù)的一家家綜合性金融公司綜合性金融公司，主要，主要提供商業(yè)銀行提供商業(yè)銀行、投資銀行和其他各種金融服務(wù)、投資銀行和其他各種金融服務(wù)。公

4、司。公司的的資產(chǎn)規(guī)模資產(chǎn)規(guī)模名列著名財(cái)經(jīng)雜志名列著名財(cái)經(jīng)雜志財(cái)富財(cái)富美國(guó)前美國(guó)前 500 500 家大企業(yè)的前家大企業(yè)的前2020位位，而且是，而且是全球金融機(jī)構(gòu)中信用評(píng)級(jí)最高的公司之全球金融機(jī)構(gòu)中信用評(píng)級(jí)最高的公司之一，一，J.P.摩根公司經(jīng)營(yíng)商業(yè)銀行業(yè)務(wù)的摩根公司經(jīng)營(yíng)商業(yè)銀行業(yè)務(wù)的子公司紐約摩根擔(dān)保信托公司是美國(guó)子公司紐約摩根擔(dān)保信托公司是美國(guó)惟惟一獲得一獲得AAAAAA信用評(píng)級(jí)的商業(yè)銀行信用評(píng)級(jí)的商業(yè)銀行。 l2000年J.P.摩根公司與大通銀行及富林明集團(tuán)完成合并成立摩根大通(JP Morgan Chase)。 John John PierpointPierpoint Morgan M

5、organ（1837-19121837-1912）華爾街之子華爾街之子 59.1.2 Definition of 9.1.2 Definition of VaRVaRlVaR（Value at Risk）：“風(fēng)險(xiǎn)價(jià)值”或“在險(xiǎn)價(jià)值”，指在一定的置信水平下，某一金融資產(chǎn)（或證券組合）在未來特定的一段時(shí)間內(nèi)的最大可能損失。 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年置信水平為95%的日VaR值為960萬美元。l含義:該公司可以以95%的把握保證，2014年某一特定時(shí)點(diǎn)上的金融資產(chǎn)在未來24小時(shí)內(nèi)，由于市場(chǎng)價(jià)格變動(dòng)帶來的損失不會(huì)超過960萬美元?；蛘哒f，只有5%的可能損失超過960萬美元。 Risk Management and Financial Institut

7、ions 3e, Chapter 97Example 2:Example 2: 一個(gè)投資組合持有1天,置信水平95%, VaR等于45美元.其含義是:l 1.該組合在1天中只有5%的時(shí)間里損失超過45美元。l 2.給一天劃分無窮多個(gè)時(shí)段，損失大于45美元的時(shí)段只有5%。Risk Management and Financial Institutions 3e, Chapter 98VaRVaR的公式表示的公式表示l公式表示為： 其中，P資產(chǎn)價(jià)值損失小于可能損失上限的概率，即英文Probability。 L某一金融資產(chǎn)或組合在一定持有期T的價(jià)值損失額。 VaR給定置信水平X%下的在險(xiǎn)價(jià)值，即可能

8、的損失上限。 X%給定的置信水平。()%PL TVaRX Risk Management and Financial Institutions 3e, Chapter 99()()%PLTVaRPLTVaRX Risk Management and Financial Institutions 3e, Chapter 910圖9-1 由交易組合在時(shí)間T的收益概率分布來計(jì)算VaR()%PL TVaRX Risk Management and Financial Institutions 3e, Chapter 9圖9-2 由交易組合在時(shí)間T的損失概率分布來計(jì)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根據(jù)正態(tài)分布的性質(zhì)，置信區(qū)間為： 。l所以，最大損失為： 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：?jiǎn)握{(diào)性。lTranslation invariance：平移不變性。lHomogeneity：同質(zhì)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假定某交易組合在10天展望期上的價(jià)值變化服從正態(tài)分布，分布的期望值為0，標(biāo)準(zhǔn)差為2000萬美元，10天展望期的99%VaR為： 2000*N-1（0.99）=4650萬美元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資料：持有期間的選擇資料：持有期間的選擇l

26、即確定計(jì)算在哪一段時(shí)間內(nèi)的持有資產(chǎn)的最大損失值，也就是明確風(fēng)險(xiǎn)管理者關(guān)心資產(chǎn)在一天內(nèi)一周內(nèi)還是一個(gè)月內(nèi)的風(fēng)險(xiǎn)價(jià)值。 l持有期的選擇應(yīng)依據(jù)所持有資產(chǎn)的特點(diǎn)來確定比如對(duì)于一些流動(dòng)性很強(qiáng)的交易頭寸往往需以每日為周期計(jì)算風(fēng)險(xiǎn)收益和VaR值，如G30小組在1993年的衍生產(chǎn)品的實(shí)踐和規(guī)則中就建議對(duì)場(chǎng)外OTC衍生工具以每日為周期計(jì)算其VaR，而對(duì)一些期限較長(zhǎng)的頭寸如養(yǎng)老基金和其他投資基金則可以以每月為周期。 l從銀行總體的風(fēng)險(xiǎn)管理看持有期長(zhǎng)短的選擇取決于資產(chǎn)組合調(diào)整的頻度及進(jìn)行相應(yīng)頭寸清算的可能速率。巴塞爾委員會(huì)在這方面采取了比較保守和穩(wěn)健的姿態(tài)，要求銀行以兩周即10個(gè)營(yíng)業(yè)日為持有期限。 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一般來說對(duì)置信水平的選擇在一定程度上反映了金融機(jī)構(gòu)對(duì)風(fēng)險(xiǎn)的不同偏好。選擇較大的置信水平意味著其對(duì)風(fēng)險(xiǎn)比較厭惡，希望能得到把握性較大的預(yù)測(cè)結(jié)果，希望模型對(duì)于極端事件的預(yù)測(cè)準(zhǔn)確性較高。 l根據(jù)各自的風(fēng)險(xiǎn)偏好不同，選擇的置信區(qū)間也各不相同。比如J.P. Morgan與美洲銀行選擇95，花旗銀行選擇95.4，大通曼哈

29、頓選擇97.5，Bankers Trust選擇99。 l作為金融監(jiān)管部門的巴塞爾委員會(huì)則要求采用99的置信水平，這與其穩(wěn)健的風(fēng)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之間的關(guān)系之間的關(guān)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假定一交易組合一天的95%VaR為150萬美元，同時(shí)假定交易組合的價(jià)值變化服從正態(tài)分布，期望值為0，由式（9-4）得出，一天展望期的99%VaR為150*2.326/1.645=212萬美元。l如果我們假定交易組合的價(jià)值變化在每天相互獨(dú)立，因此10天的99%VaR=