北大光华证券投资学第8章10年

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1、第八章 债券定价与风险管理2主要内容n利率风险n久期n凸性n消极的债券组合管理3n这里所谓的风险管理，是针对债券的利率风险控制，而债券本身的风险（例如，违约风险）不在讨论范围之内。4n积极策略（Active strategy）： attempts to achieve returns greater than to those commensurate with the risk borne.nTrade on interest rate predictionsnTrade on market inefficienciesn消极策略（Passive strategy）：takes market

2、 prices of securities as fairly set. nControl risknBalance risk and return管理固定收益证券的基本策略：1、利率风险6利率风险n当利率上涨和下降时，债券持有者就会面临资金损失和收益。这些损失或者收益使得在债券投资中，即使利息和本金支付能够保证得到（例如国债），投资者也面临风险。7利率风险n为什么利率变动时，债券价格会变动？n在一个完全竞争市场中，所有的证券提供的都是公平合理的期望回报率（fair expected rates of return）。8利率风险n例子：一种债券，息率为8% 。如果市场的竞争收益率为8% ，则它

3、的价格为面值。n如果市场竞争收益率上升为9%，则债券价格将下降，以使得总期望回报率为9%。n如果市场的竞争收益率下降为7%，则债券价格将上升，以使得总期望回报率为9%。 9利率风险：折价债券nWhen bond prices are set according to the present value formula, any discount from par value provides an anticipated capital gain that will augment a below-market coupon rate just sufficiently to provide

4、a fair total rate of return. 10利率风险：溢价债券nIf the coupon rate exceeds the market interest rate, the interest income by itself is greater than available elsewhere in the market. The price is greater than the par value, the resulting capital losses offset the large coupon payments so that the investor r

5、eceive only a fair rate of return.11利率风险：公平合理的期望回报率nEach bond offers investors the fair total rate of return. Although the capital gain versus income components differ, the price of each bond is set to provide competitive rates, as we should expect in well-functioning capital markets. 12利率风险n债券价格受市场

6、的影响，市场利率波动是固定收入证券市场的主要风险根源。n价格和收益率之间的反向关系：The inverse relationship between price and yield is a central feature of fixed-income securities. n决定价格对利率波动敏感度一个关键因素是到期日。Interest rate fluctuations represent the main source of risk in the fixed-income market, and one key factor that determines that sensiti

7、vity is the maturity of the bond. A general rule in evaluating bond price risk is that, keeping all other factors the same, the longer the maturity of the bond, the greater the sensitivity of price to fluctuations in the interest rate. This is why short-term Treasury securities such as T-bills are c

8、onsidered to be the safest. They are free not only of default risk, but also largely free of price risk attributable to interest rate volatility.n到期日是唯一因素吗？13债券定价定理：定性描述利率风险n债券定价定理：说明市场收益变化和价格变动之间的关系（定性描述）。假设每年支付一次利息，以到期收益为研究对象：14债券定价定理：定性描述利率风险n1. 如果债券的市场价格上升，则收益下降；反过来，如果债券价格下降，则收益上升。15债券定价定理：定性描述利

9、率风险n2. 如果债券的收益在到期日之前不变，则它的折价或者酬金的规模将随着到期日的接近而下降。TodayMaturity DatePar ValuePrice of a premium bondPrice of a discount bondpremiumdiscount16债券定价定理：定性描述利率风险n3. 如果债券的收益在到期日之前不变，则它的折价或者酬金的规模变化速度随着到期日的靠近加快。n4. 当债券的收益上升和下降相同的数量时，收益上升导致价格下降的规模，小于收益下降导致价格上升的规模。（凸性）17Change in Bond Price as a Function of Cha

10、nge in Yield to Maturity18债券定价定理：定性描述利率风险n5.长期债券的价格对利率变化的敏感度大于短期债券的敏感度。即，长期债券有更大的利率风险。n6.债券的息率越高，由收益变化导致的价格变化的百分比越小。19例子nBond G: coupon rate=7%, yield=7%, P=1000nBond H: coupon rate=9%, yield=7%, P=1082nwhen yield change to be 8% nbond G: price 1000 960.03, 3.993%nbond H: price 1082 1039.93 3.889%20

11、债券定价定理：定性描述利率风险n7. 债券发行时的初始到期收益越低，则它对收益变化的敏感度越大。n债券价格对市场利率变化的敏感度受三个关键因素的影响：到期日，息率，到期收益2、Duration22仅仅只用到期日描述利率风险是不够的n债券定价定理说明，到期日是决定利率风险的主要因素，但是，仅仅只有到期日不能完全度量债券价格对利率的敏感度。例如债券B、C23例子：息率8%的债券（每年支付两次）与零息债券Yield tomaturityT=1 yearT=10 yearsT=20 years8%1000100010009990.64934.96907.99Change inprice0.94%6.5

12、%9.20%Yield tomaturityT=1 yearT=10 yearsT=20 years8%924.56456.39208.299915.73414.64171.93Change inprice0.96%9.15%17.46%24例子说明n这里的到期日并不是债券长期或者短期的完美度量。n有效到期日：Because we know that long term bonds are more sensitive to interest rate movements than are short term bonds, in some sense a zero coupon bond r

13、epresents a longer-term bond than an equal-time-maturity coupon bond. This is the insight about effective maturity.25例子说明:有效到期日n比较20年到期的零息债券和带息债券(8% coupon rate)。nThe 20-year 8% bond makes many coupon payments, most of which come years before the bonds maturity date. Each of these payments may be co

14、nsidered to have its own “maturity date”, and the effective maturity of the bond is therefore some sort of average of the maturities of all the cash flows paid out by the bond.nThe zero-coupon bond, by contrast, makes only one payment at maturity. Its time to maturity is a well defined concept. 26例子

15、说明:有效到期日nTo deal with the ambiguity of the maturity of a bond making many payments, we need a measure of the average maturity of the bonds promised cash flows to serve as a useful summary statistic of the effective maturity of the bond. We would like also to use the measure as a guide to the sensiti

16、vity of a bond to interest rate changes.27Durationn这里 表示在时间 接受的现金流的现值，利用债券的到期收益作为折现率得到。n 表示债券现在的市场价格。n 表示债券剩下的距到期日的时间。01PtCPVDTtttCPVt0PT28Cash flows paid by 9% coupon, annual payment bond with 8-year maturity and 10 y-t-m298%BondTimeyearsPaymentPV of CF(5% per period)WeightC1 XC40.54038.095.0395.01

17、9714036.281.0376.03761.52.0401040sum34.553855.611964.540.0358.88711.000.05371.77421.8852Duration Calculation: Example30Durationn当到期收益保持不变时，证券组合duration 是单个债券duration的加权和TttCPVP10TtttPCPVD1031DurationnDuration 在固定收益投资组合管理中的作用n测量证券组合有效平均到期日的统计量n度量证券组合对利率的敏感度（定量刻画） nan essential tool in immunizing port

18、folios from interest rate risk32Duration 和股票价格变化之间的关系n这里 表示债券价格的变化n 是债券的初始价格n 是到期收益的变化n 是初始的到期收益yyDPP1PPyy33例子nBond : coupon rate 8%, yield to maturity 8%, par value 1000, price 1000, duration 10 when yield to maturity 8% 9% 34What determines duration?35Rule for durationn1.零息债券的duration等于其到期日n2.到期日保

19、持不变，息率越低， duration越高n3.息率不变，到期日越大， duration一般越大。对等价或者溢价发行的债券，上述关系总是成立n4.别的因素不变，到期收益越低，带息债券的duration越高。36Rules for Duration (contd)n5.永久性现金流的duration为n到期日与duration的差别n当到期日越来越大时， duration接近于相应永久性现金流的durationn注意支付时间单位与利率之间的一致性yy137Rules for Duration (contd)Rule 6 The duration of a level annuity is equa

20、l to:1)1 (1TyTyy38The modified durationyDPP*yDD1*3、Convexity40仅仅只需Duration就够了吗？nAs a measure of interest rate sensitivity, duration clearly is a key tool in fixed income portfolio management.nThe duration rule for the impact of interest rates on bond prices is only an approximation.41YieldPriceDurat

21、ionPricing Error from convexityDuration and Convexity42nThe duration rule is a good approximation for small changes in bond yield, but it is less accurate for larger changes.nThe duration approximation always understates the value of the bond, it underestimates the increase in bond price when the yi

22、eld falls, and it over estimates the decline in price when the yield rises.nThe curvature of the price yield curve is called the convexity of the bond.43nAs a practical rule, we can view bonds with higher convexity as exhibiting higher curvature in the price yield relationship.nConvexity allows us t

23、o improve the duration approximation for bond price changes.44Correction for ConvexityntttttyCFyPConvexity122)()1 ()1 (1Correction for Convexity:)(212yConveixityyDPP45nThe convexity is more important as a practical matter when potential interest rate changes are large.46例子：nA 30-year maturity, an 8%

24、 coupon, and sells at an initial yield to maturity of 8%. The bond sells at par value, $1000. The modified duration is 11.26years. If the bonds yield increase from 8% to 10%, the bond price will fall to $811.46, a decline of 18.85%.nThe duration rule would predictnThe duration-with-convexity rule%52

25、.2202. 026.11*yDPP%27.1802. 04 .2125 . 002. 026.11)(2122yConveixityyDPP47nIf the change in yield were smaller ,0.1%, convexity would matter less.the price of the bond actually would fall to 988.85, a decline 1.115% .%126. 1001. 026.11*yDPP%115. 1001. 04 .2125 . 002. 026.11)(2122yConveixityyDPP48Why

26、do investors like convexity?y_yyn Bond ABond B49nBond A has greater price increases and smaller price decreases when interest rates fluctuate by larger amounts.4、Passive bond management51nPassive methodsn假设债券市场时半强有效的。证券选择(security selection)和决定交易时间(market timing)都是无用的，不会带来超平均的收益。nActive methodsn假设债券

27、市场不是非常有效的。通过准确预测利率来辨别误定价的债券或者制定交易时间，从而能够获得超额收益。52Passive methodsn消极债券管理认为债券的价格是公平的，只能控制固定收入证券组合的风险n主要策略：n指标化策略:复制给定债券指标的行为nImmunization 策略：shield the overall financial status of the institution from exposure to interest rate fluctuations.53n两者认为市场价是公平的n两者的区别n债券-指标证券组合和债券市场指标具有相同的风险-收益回报nImmunization

28、建立了零风险的证券组合，利率的波动对公司的价值没有影响。54nBond-index fundsnCreate a portfolio that mirrors the composition of an index that measures the broad market.n债券市场指标nLehman Brothers, Merrill Lynch, Salomon BrothersnNumber of issuesnmaturity of included bondsnexcluded issuesnweightingnreinvestmentnDaily availability 55

29、n构造反映债券市场指标的证券组合n问题：n债券种类过多，难以一一购买n交易少，很难以公平市价买到指标中包含的所有债券n指标不断更换到期日少于1年的债券n不断调整，利息收入重投资n方法：精确复制债券指标不可行，采用cellular方法n把债券市场分成几类n指标中债券在各类中占的比例n按这一比例构造债券组合56nIn these way, the characteristics of the portfolio in terms of maturity, coupon rate, credit risk, industrial representation, and so on, will mat

30、ch the characteristics of the index, and the performance of the portfolio likewise should match the index.5730+ years3.4%9.2%15-30 years10-15 years 0.1%7-10 years5-7 years4.1%3-5 years5.4%1-3 years12.1%1 yearYankeeUtilityFinanceIndustrialMortgage-BackedAgencyTreasury SectorT-t-M58Immunizationn两种不同的看

31、待利率风险的方式n银行，使得资产净现值不受利率波动的影响n养老金，使得资产将来的值不受利率风险的影响nWhat is common to the bank and the pension fund is interest rate risk. The net worth of the firm or the ability to meet future obligations fluctuates with interest rates.n通过适当调整证券组合的到期日结构，规避利率风险nImmunization techniques refer to strategies used by su

32、ch investors to shed their overall financial status from exposure to interest rate fluctuations.59nNet worth immunizationnF. M. Redington60例子：n承诺在两年后支付1000000元，有两种债券可供选择：n债券 1年 2年 3年 yieldn 1 80 80 1080 10%n 2 1070 10%n 121 ww278. 2121ww61例子：n保险公司以价格10000元发行一种guaranteed investment contract(GIC)，5年到期

33、，保证利率为8%。n假设公司选择息率8%，6年到期、价格为10000元的带息债券为债务提供基金。 n价格风险n重投资风险28.1469308. 110000562nTerminal value of a bond portfolio after 5 years (all proceeds reinvested)nA. rates remain at 8%14800408. 1=1088.3923800308. 11007.7732800208. 1933.1241800108. 186450800800Sale of bond010800/1.081000014693.2863nTermina

34、l value of a bond portfolio after 5 years (all proceeds reinvested)nB. rates fall to 7%14800407. 123800307. 132800207. 141800107. 150800Sale of bond010800/1.0714694.0564nTerminal value of a bond portfolio after 5 years (all proceeds reinvested)nC. rates increase to 9%14800409. 123800309. 132800209.

35、141800109. 150800Sale of bond010800/1.0914696.0265nFor a horizon equal to the portfolios duration, price risk and reinvestment risk exactly cancel out.66nAccumulated value of invested fundsnfundsn0 t* D t67Figure Immunization68n在8%，资产和债务的现值相等；当利率变化幅度不大时，资产和债务的值的变化量相等；当利率变化幅度很大时，资产和债务值的变化量不再相等。69Reba

36、lancing immunized portfolion当资产的收益变化时，其久期也发生了变化，这时，资产和债务的久期不再相匹配。n即使利率不变，当时间变化时，久期也会发生变化70Rebalancing immunized portfolionEven if an obligation is immunized at the outset, as time passes the durations of the asset and liability will fall at different rates. Without portfolio rebalancing, durations w

37、ill become unmatched and the goals of immunization will not be realized.nImmunization is a passive strategy only in the sense that it does not involve attempts to identify undervalued securities. Immunization mangers still actively update and monitor their positions. 71n例子：假设证券组合经理在7年后有一笔19487元的债务，以

38、现在10%的市场利率计算现值为10000元。经理希望通过3年的零息债券和永久性现金流来immunize这笔债务。n对永久性现金流，在利率10%之下的久期是11年。设 为零息债券的权711)1 (35 . 072n第二年，债务的现值为11000元，资金也变为11000元：零息债券5500元，永久性500+5000元n第2年，即使利率不变，经理也需要调整策略n投资者总共投资 在零息债券，即500元息加上再卖掉111.11永久性债券，投在零息债券上。611)1 (29511.6111110009573nAn appropriate compromise must be established bet

39、ween the desire for perfect immunization and the need to control trading costs.74Immunization 在实际中的问题nThe notion of duration is strictly valid only for a flat yield curve. nmultiple nonparallel shifts in a nonhorizontal yield curven通货膨胀 75nOn this note, it is worth pointing out that immunization is a goal that may well be inappropriate for many investors who would find a zero-risk portfolio strategy unduly conservative. Full immunization is a fairly extreme position for a portfolio manager to pursue.