第八章 利率风险

,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,Risk Management and Financial Institutions, 2e, Chapter 7, Copyright John C. Hull 2009,4.,*,Interest Rate Risk,Chapter 8,Risk Management and Financial Institutions,Copyright John C. Hull,1,8.1 Management of Net Interest Income,Net Interest Income=Interest received-Interest paid,Consider a simple situation where a bank offers consumers a one-year and a five-year deposit rate as well as a one-year and five-year mortgage rate. The rates are shown in Table 8.1:,Maturity (yrs),Deposit Rate,Mortgage Rate,1,3%,6%,5,3%,6%,Risk Management and Financial Institutions,Copyright John C. Hull,2,Table 8.1 Example of rates offered by a bank to its customers,Two question:,（,1,）,What would happen if a bank posted the rates in Table 8.1?,（,2,）,How can the bank manage its risks?,Assumption:,market participants expect the one-year interest rate for future time periods to equal the one-year rates prevailing in the market today,.,Suppose you have money to deposit. Would you choose to deposit your money for one year at 3% per annum or for five year at 3% per annum?,If deposit one year:,(1+3%)(1+3%)(1+3%)(1+3%)(1+3%)=(1+3%),5,If deposit five year,:(1+3%),5,So,most customers would choose one year,because this gives them more financial flexibility. It ties up their funds for a shorter period of time.,（,流动性偏好,）,Now suppose that you want a mortgage. Would you choose a one-year mortgage at 6% or a five-year mortgage at 6%?,One-year mortgage:,(1+6%)(1+6%)(1+6%)(1+6%)(1+6%)=(1+6%),5,Five-year mortgage,:(1+6%),5,So most would choose a five-year mortgage because it fixes your borrowing rate for the next five years and subjects you to less refinancing risk.,由于多数客户会选择,1,年期存款，,5,年期贷款，所以，导致银行的资产与负债的不匹配（,短借长放现象,），从而对净利息收入产生风险冲击。,若利率下降,，贷款利率,6%,，存款利率低于,3%,，,利息收入增加,。,若利率上升,，贷款利率,6%,，存款利率高于,3%,，,利息收入减少,。,总结,解决方案：,实现资产负债匹配,。,Maturity (yrs),Deposit Rate,Mortgage Rate,1,3%,6%,5,4%,7%,表,8-2,提高,5,年期利率以达到资产负债的匹配,8.2 LIBOR Rates and Swap Rates,LIBOR rates are,1-, 3-, 6-, and 12-month,borrowing rates,for companies that have a,AA-rating,Swap Rates are the fixed rates exchanged for floating in an interest rate swap agreement,Risk Management and Financial Institutions,Copyright John C. Hull,9,资料：,LIBOR rates and SHIBOR rates,LIBOR rates are provided by,British Bankers Association,(BBA).,The BBA is the leading trade association for the UK banking and financial services sector. We speak for over,200 member banks from 60 countries,on the full range of UK and international banking issues.,Understanding BBA LIBOR,LIBOR rates closely reflect the real rates of interest being used by the,worlds largest financial institutions,.,Whereas central banks (such as the Bank of England, the US Federal Reserve and the European Central Bank) fix,official base rates monthly, LIBOR reflects the rates at which these prime banks borrow money from each other each day, in the worlds 10 major currencies and for 15 borrowing periods ranging from overnight loans to 12 month.,Once calculated, the LIBOR figures are then published by Thomson Reuters:,they appear on more than one million screens around the world and are widely reported in the press, the wire services and online.,Thomson Reuters undertakes this work for the British Bankers Association., How is it calculated?,Each day at 11:00 hrs London time the banks which contribute to the LIBOR-setting process send their,interbank borrowing rates,confidentially,to Thomson Reuters.,Thomson Reuters,discards the highest and lowest contributions,(the top and bottom quartiles),and then uses the middle two quartiles to calculate an average.,The,Australian Dollar,Danish,Krone,New Zealand Dollar,and,Swedish,Krone,panels have eight banks,The Canadian Dollar,and,Swiss Franc,panels have 12 banks. The,Sterling,Yen,and,Euro,panels have 16 members and the,US Dollar,panel has 19 members. Each follows the same procedure of discarding the upper and lower quartiles and averaging the centre quartiles to create a rate., How did it become so important?,LIBOR was,first developed in the 1980s,as demand grew for an accurate measure of the rate at which banks would lend money to each other. This became increasingly important as Londons status grew as an international financial centre. More than 20 per cent of all international bank lending and more than 30 per cent of all foreign exchange transactions now take place in London.,LIBOR rates are the,basis for a range of financial instruments: derivatives based on the LIBOR rates are now traded on exchanges such as LIFFE and the Chicago Mercantile Exchange (CME) as well as over-the-counter.,The rates are also used as the basis for many types of lending, from syndicated and commercial lending, to residential mortgages., SHIBOR rates,Shibor,全称是,“,上海银行间同业拆放利率,”,（,Shanghai Interbank Offered Rate,，,SHIBOR,），,被称为中国的,LIBOR,（,London Interbank Offered Rate,，伦敦同业拆放利率），自,2007,年,1,月,4,日正式运行。,Shibor,是由信用等级较高的银行组成报价团自主报出的人民币同业拆出利率计算确定的,算术平均利率，是单利、无担保、批发性利率。,目前，对社会公布的,Shibor,品种包括,隔夜、,1,周、,2,周、,1,个月、,3,个月、,6,个月、,9,个月及,1,年,。,上海首批,16,家报价行分别为：,工商银行，农业银行，中国银行，建设银行，交通银行，兴业银行，浦发银行，北京银行，上海银行，招商银行，光大银行，中信银行，南京商行，德意志上海，汇丰上海，渣打上海。,2010,年,5,月，广发银行也成为,SHIBOR,基准利率互换业务报价行。,Why Swap Rates Are an Average of LIBOR Forward Rates,A bank can,Lend to a series AA-rated borrowers for ten successive six month periods,Swap the LIBOR interest received to the five-year swap rate,It can,Lend to a certain principal for six months to a AA borrower and relend it for nine successive six-month periods to AA borrowers; and,Enter into a swap to,exchange the LIBOR for the five-year swap rate,.,Risk Management and Financial Institutions,Copyright John C. Hull,15,LIBOR VS Treasury Rate,The risk-free rate is important in the pricing of financial contracts.,The usual practice among financial institutions is to assume that the,LIBOR/swap yield curve provides the risk-free rate,.,The,Treasury curve is about 50 basis points below the LIBOR/swap zero curve,Risk Management and Financial Institutions,Copyright John C. Hull,16,Treasury rates are considered to be artificially low for a variety of regulatory and tax reasons,8.3 Duration ,度量利率风险,的常用指标是,麦考利,（,Frederick Macaulay,）在,1938,年提出的,久期,。,假设债券收益率为,y,，债券价格为,B,，债券久期的定义为：,或等价于：,(8-1),式中，,y,为债券收益率的一个小的变化，,B,为相应债券价格的变化。,Risk Management and Financial Institutions,Copyright John C. Hull,18,久期的理解：,久期,就是,债券价格,相对于,利率水平,正常变动的,敏感度，,它可以精确地,量化利率变动给债券价格造成的影响,。,例：如果一只短期债券的久期是,2,，那么利率每变化,1,个百分点，该债券价格将上升或下降,2%,。一只长期债券的久期是,12.0,，那么利率每变化,1,个百分点，其价格将上升或下降,12%,。,久期越大，债券价格对利率的变动就越敏感，,利率上升,所引起的,债券价格下降幅度,就,越大,，,而,利率下降,所引起的,债券价格上升幅度,也,越大,。,久期特征给债券投资提供了参照。,当前的利率水平存在上升可能，就可以集中投资于短期品种、缩短债券久期,；而,当前的利率水平有可能下降，则拉长债券久期、加大长期债券的投资,，这就可以帮助我们在债市的上涨中获得更高的溢价。,Duration Continued,债券久期用于检测,债券价格,对,收益率,的敏感度。利用微积分的符号，有：,(8-2),假定一个债券在,t,1,t,2, ,t,n,时刻给债券持有人提供的现金流为,c,1, c,2,c,n,(,现金流包括债息和本金,),，债券收益率,y,是使得债券理论价格等于市场价格的贴现率，如果收益率为连续复利，债券价格与收益率的关系式为：,Duration Continued,求导,所以,(8-3),Duration Continued,债券久期：,上式中括号中的项为,t,i,时刻,债券支付的现金流的现值与债券价格的比率,，债券价格等于未来所有支付的现金本息贴现值的总和，因此久期是付款时间,t,i,的加权平均。即,久期是投资者收到所有现金流所要等待的平均时间。,(8-3),Calculation of Duration for a 3-year bond paying a coupon 10%. Bond yield=12%.,(Table 8.3),Risk Management and Financial Institutions,Copyright John C. Hull,23,Time (yrs),Cash Flow ($),PV ($),Weight,Time,Weight,0.5,5,4.709,0.050,0.025,1.0,5,4.435,0.047,0.047,1.5,5,4.176,0.044,0.066,2.0,5,3.933,0.042,0.083,2.5,5,3.704,0.039,0.098,3.0,105,73.256,0.778,2.333,Total,130,94.213,1.000,2.653,例,8-1:,由表,8-3,描述的债券价格为,94.213,，久期为,2.653,，根据公式（,8-1,）：,B=-B*D*y=-94.213*2.653y=-249.95y,当收益率增加了,10,个基点（,0.1%,），即,y=0.001,，久期公式预计,B,为：,B=-249.95*0.001=-0.25,久期公式预期债券价格会下降到,94.213-0.25=93.963,。,为了检验这个预测的准确性，我们计算当收益率增加,10,个基点到,12.1%,时的债券价格，其数量为：,5e,-0.121*0.5,+5e,-0.121*1,+5e,-0.121*1.5,+5e,-0.121*2,+5e,-0.121*2.5,+105e,-0.121*3,=93.963,这一数值同久期公式预计的变化相同。,8.3.1 Modified Duration,When the yield,y,is expressed with compounding,m,times per year,The expression,is referred to as the “modified duration”,Risk Management and Financial Institutions, Copyright John C. Hull,25,例,8-2,由表,8-3,描述的债券价格为,94.213,，久期为,2.653,，每年复利两次的收益率为,12.3673%,，修正久期为：,=2.653/(1+0.123673/2),=2.4985,8.3.2 Dollar Duration,绝对额久期等于修正久期与债券价格的乘积：,采用微分的记号为：,8.4 Convexity,对于收益曲线上,较小的平移,，,久期,可以检测,组合价值的变化,。,相同久期的债券，导数相同,（,图,8-1,）。这意味着当收益率有较小的变动时，两个债券同收益率变化的百分比相同。,所引出问题：,当利率变化较大时，如何来衡量它们的价值变化呢？,图,8-1,两个具备同样久期的交易组合,B/B,y,曲率,变量用于检验曲线的,凸凹变化程度,，它可以,改善近似式的准确性,。,债券的曲率,被定义为：,其中，,y,对应于债券的收益率。,涵义,：,债券曲率,是,将来收到现金流的时间平方的平均值,。,可得关于债券价格的两阶近似式（附录,G,）：,由此：,（,8-4,）,注意：和久期公式对比！,例,8-3,由表,8-3,描述的债券价格为,94.213,，久期为,2.653,，曲率为：,0.05*0.5,2,+0.047*1,2,+0.044*1.5,2,+0.042*2,2,+0.039*2.5,2,+0.779*3,2,=7.57,由式（,8-4,）得出：,B/B=-2.653,y+0.5*7.57*(y),2,假设债券收益率由,12%,变为,14%,，久期公式预期债券价值变化将是：,-94.213*2.653*0.02=-4.999,，,曲率关系式预计变化为：,-94.213*2.653*0.02+0.5*94.213*7.57*0.022=-4.856,而债券价格的实际变化为,-4.859.,这个例子说明，当债券收益率变化较大时，曲率公式比久期公式更精确。,Dollar Convexity,绝对额曲率等于曲率与债券价格的乘积：,绝对额久期、绝对额曲率对应的,Delta,度量和,Gamma,度量。,7.5 Generalization,Duration and convexity can be defined similarly for,portfolios of bonds,and other,interest-rate dependent securities,。,The,duration of a portfolio,is,the weighted average of the durations of the components of the portfolio,. Similarly for convexity.,Portfolio Immunization,A portfolio consisting of long and short positions in interest-rate-dependent assets can be,protected against relatively small parallel shifts,in the yield curve by ensuring that,its duration is zero,.,I,t can be,protected against relatively large parallel shifts,in the yield curve by ensuring that,its duration and convexity are both zero or close to zero,.,The end,