第八章债券定价和风险管理(证券投资学-北大)

1、第八章债券定价和风险管理(证券投资学-北大)CAPM, APT: treat securities at a high level of abstraction, assuming implicitly that a prior, detailed analysis of each security already had been performed, and that its risk and return features had been assessed.Specific analyses of particular security markets: valuation princ

2、iples, determinants of risk and return, portfolio strategies commonly used within and across the various markets固定收益证券Promise either a fixed stream of income or a stream of income that is determined according to a specified formula.Have the advantage of being relative easy to understand because the

3、level of payments is fixed in advance.Risk consideration are minimal as long as the issuer of the security is sufficiently creditworthy.Bond analysis Bond attributesBond pricingAppropriate yield to maturityIntrinsic valueThe relationship between the yield and priceBond risk managementWhen the maturi

4、ty and the payment time is not consistentInterest riskReinvestment riskThe approach of risk managementDurationConvexity1. Bond analysis债券是最基本的固定收益证券。债券的定价债券的特性1.1 Capitalization of Income Method of ValuePromised yield-to-maturity如果 ，则定价过低(underpriced)。如果 ，则定价过高(overpriced)。 的确定依赖于债券的特征以及现时的市场条件。例子：一

5、个债券，现价为900元，面值为1000元，三年到期，息率为6%。得到如果通过分析，得到问题：定价如何？如何确定 ？Intrinsic value例子：两种程序所得结果的一致性，即价格与回报率之间的关系。为了利用Capitalization of Income Method of Value，必须决定 ， ， 的值。 和 容易确定 的确定依赖于投资者对债券的特征的主观看法，以及现实的市场条件。从而债券分析中，最关键的部分是确定1.2 Bond attributes在债券定价过程中，债券的六个重要的属性：length of time until maturitycoupon ratecall pr

6、ovisionstax statusmarketability likelihood of default在任何时间，这些性质不同的债券的市场价格结构，以到期收益来描述。整个结构也称为收益结构(yield structure)。期限结构(different maturities)：不同到期日风险结构(different default risk)：不同违约风险yield spread：两种债券之间的收益差。被考虑债券和具有相同的到期日和息率的无违约风险债券定价是一个相对的概念Coupon rate and length of time until maturity这两个性质决定了发行者承诺支付

7、给持有者的现金流的时间和规模。由这两个属性可以决定债券的到期收益率，再与基准的收益率作比较。通常以国库券的到期收益作为基准。例子：前面例子里的债券与下面的国库券比较：面值1000元，息率5%，价格910.61元。到期收益为8.5%，yield spread 为152个基点。Call and put provisionscall pricecall premium当收益率剧烈下降后，债券的发行者回购已经发行的债券具有财务上的优势，因为发行者能够用收益更低的债券来代替。例子：考虑10年期债券以面值（1000元）发行，息率为12%，上市5年以后，可以以回购价格1050元进行回购。5年后，类似的5年期

8、的债券的收益为8%。为什么会回购10年的到期收益（实际回报）为10.96%。Yield to call例子：面值为1000元，息率8%，30年到期的无回购协议的债券和面值为1000元，息率8%，30年到期，回购价格为1100元的债券price12001000 0 5% 10% 15% 20% interest rateAt high interest rates, the risk of call is negligible, and the values of the straight and callable bonds converge. At lower rates, however,

9、 the values of bonds begin to diverge, with the difference reflecting the value of the firms option to reclaim the callable bond at the call price. 两者之差反映了公司以1100元进行回购这样一个权利的价值。当有回购风险时，更加关注回购收益而不是到期收益回购收益的计算例子：假设息率为8%，30年到期的债券价格为1150元，10年后以1100元回购回购收益为6.64%，到期收益为6.82%The higher the coupon rate of a

10、callable bond, the greater is the likely divergence between actual and promised yields.一般来说，息率超过5%的债券都会回购回购可能性越大，到期收益应该越高，即，息率越高，或者回购酬金越低，到期收益应该越高当别的性质相同时，息率越高或者回购酬金越低的可回购债券，其内在价值(intrinsic value) 应该越低。Put provisionsputable bonds当利率上涨时，投资者采用该策略Tax status因为税收的延迟性，低息债券比高息债券有更高的内在价值。Marketability度量债券流动

11、性的一个方式是Bid-Ask spread交易活跃的债券比交易不活跃的债券具有更低的bid-ask spread.交易活跃的债券比交易不活跃的债券具有更低的到期收益和更高的内在价值。Likelihood of defaultinvestment gradespeculative grade对公司债券而言，好的级别一般与下列条件有关：低的财务杠杆大的和稳定的利润大的公司规模大的现金流从属债务少级别评估公司以发行公司的财务比的水平和未来趋势为基础，通过建立模型对公司债券级别进行评估：Coverage ratios-ratio of company earnings to fixed costsLe

12、verage ratio-debt to equity ratioLiquidity ratios-current assets to current liability ratioProfitability ratios-return on assets or equityCash flow-to-debt ratio-total cash flow to outstanding debt债券有违约风险，计算期望到期收益率(expected yield-to-maturity)只要有违约或者推迟支付的可能，期望收益就会小于承诺收益(promised yield)一般来说，违约的风险越大，违约

13、时损失的数量越大，在收益上的差别越大。Promised yield to maturity and expected yield to maturity.例子：在1993年8月，Wang Laboratories, Inc.即将破产，它的到期日为2009年的债券发行时以面值的35%折价发行，使得Promised yield to maturity 超过26%。投资者并不真正期望获得26%的回报率，他们预期不可能得到所有承诺支付，以期望现金流为基础的收益远远小于以承诺现金流为基础的收益。例子：某公司20年前发行的债券还有10年到期，息率为9%，现在公司有财务困难，投资者预期利息将不受影响，但在到

14、期日，公司将破产，投资者只能得到面值的70%，债券现在的价格为750元。具体情况见下表承诺到期收益为13.7%.期望到期收益为11.6%Default premium: the difference between the promised yield on a corporate bond and the expected yield.违约的概率越大，违约酬金越高。一个关于违约酬金大小的模型：如果每期违约的概率为 ，违约时支付的数量为前一年市价的 ，则当债券公平定价时承诺的到期收益 为承诺到期收益和期望到期收益 之间的差为债券的违约酬金。例子：One particular manner in

15、 which yield spreads seem to very over time is related to the business cycle. Yield spreads tend to be wider when the economy is in a recession. Apparently, investors perceive a higher probability of bankruptcy when the economy is faltering, even holding bond ratings constant. They require a commens

16、urately higher default premium. Risk premium对风险债券而言，它的期望到期收益和具有同样到期日、息率的无风险债券到期收益之间的差称为风险酬金。每种具有违约风险的债券都有违约酬金。但风险酬金是另一种酬金。例子：考虑一组公司，都有破产的可能性，但破产的原因各不相同，由这些公司的债券组成的证券组合的实际回报率近似等于其期望回报率，系统风险为0。每种债券的风险酬金为0，但违约酬金显然大于0。一种债券，它的持有期收益率可能和别的债券以及股票的收益率相关。最重要的，在某种程度上，它 的持有期收益率可能和风险分散化的市场证券组合的收益率相关。这部分风险称为债券的系统

17、风险，使得债券具有风险酬金，以它的期望收益率与无违约风险利率的差表示。例子： Default premiumyieldspread Risk premiumDefault-free yield-to-Maturity12% promised yield-to-maturity9% expected yield-to-maturity8% yield-to-maturity on a default-freebond of similar and coupon rate0%债券的违约性越大，它对市场潜在的敏感度越大。实证结果表明，债券的级别越低，平均回报率越高，标准差越大。实证结果表明，与股票市

18、场比较，级别越低的债券，对股票市场的波动敏感性越大。预测违约的财务比(financial ratio)对一个公司而言，当下面情况发生时，违约的概率变大：the existing cash balance is smallerthe expected net cash flow is smallerthe net cash flow is more variable单变量方法多变量方法单变量方法the ratio of net cash flow（income before depreciation, depletion, and amortization charges) to total d

19、ebt 多变量方法Z-score =(current assets-current liabilities)/total assets =retained earnings/total assets =earnings before interest and taxes/total assets =market value of equity/book value of total debt =sales/total assetsZ-score小于1.8公司作为违约对象，并且Z-score越小，违约的可能性越大。1.3. Yield spread 的确定评估 yield spread 的四种测

20、度：The extent to which the firms net income had varied over the preceding nine years(measured by the coefficient of variation of earnings-that is, the ratio of standard deviation of earnings to average earnings)The length of time that the firm had operated without forcing any of its creditors to take

21、 a loss.The ratio of the market value of the firms equity to the par value of its debt.The market value of the firms outstanding debt.Yield spread=1.987 +0.307(earnings variability) -0.253(time without default) -0.537(equity/debt ratio) -0.275(market value of debt)This form of the relationship accou

22、nted for roughly 75% of the variation in the bonds yield spread1.4. 债券定价给定合理的利率，给债券公平定价何为合理利率？合理的利率（或者折现率）是由市场唯一确定的，包括：实利率通货膨胀率yield spread一级市场：以面值发行息率近似为市场收益率二级市场：债券价格受市场的影响，市场利率波动是固定收入证券市场的主要风险根源。The inverse relationship between price and yield is a central feature of fixed-income securities. Inte

23、rest rate fluctuations represent the main source of risk in the fixed-income market, and one key factor that determines that sensitivity 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 gre

24、ater the sensitivity of price to fluctuations in the interest rate. This is why short-term Treasury securities such as T-bills are considered to be the safest. They are free not only of default risk, but also largely free of price risk attributable to interest rate volatility.当息率等于市场利率时，价格等于面值In the

25、se circumstances, the investor receives fair compensation for the time value of money in the form of the recurring interest payments. No further capital gain is necessary to provide fair compensation.当息率小于市场利率时，价格小于面值The coupon payments alone will not provide investors as high a return as they could

26、 earn elsewhere in the market. To receive a fair return on such an investment, investors also need to earn price appreciation on their bonds. The bond would have to sell below par value to provide a “built-in” capital gain on the investment.例子：面值1000元，息率7%，公平利率8%，三年到期现在公平价格=70讲义+1000讲义=974.23一年后公平价格

27、为=70讲义+1000讲义=982.17过去的一年的回报率为70+7.94/973.23=8%When 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 a fair total rate of return. If the coupon rate e

28、xceeds 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 receive only a fair rate of return.Each bond offers investors

29、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. Security returns all should be comparable on an after-tax risk-adjusted basis. If they are no

30、t, investors will try to sell low-return securities, thereby deriving down the prices until the total return at the now lower price is competitive with other securities. Prices should continue to adjust until all securities are fairly priced in that expected returns are appropriate(given necessary r

31、isk and tax adjustments)2. Fixed-income portfolio management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。2.1 利率风险As interest rates rise and fall, bondholders experience capital losses and gains. These gains or losses make fixed-income investments risky, even if the coupon and principal payments

32、are guaranteed, as in the case of Treasury obligations.Why do bond prices respond to interest rate fluctuations?In a competitive market all securities must offer investors fair expected rates of return. The sensitivity of bond prices to changes in market interest rates is obviously of great concern

33、to investors.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：1. 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。2. 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 bondpremiumdiscount3. 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。4. 当债券的收益上升和下降

34、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。5. 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。例子Bond C: coupon rate=7%, yield=7%, P=1000Bond D: coupon rate=9%, yield=7%, P=1082when yield change to be 8% bond C: price 1000 960.03, 3.993%bond D: price 1082 1039.93 3.889%6. 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。7. 债券发行时的初始到期

35、收益越低，则它对收益变化的敏感度越大。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2.2 DurationThe propositions confirm that maturity is a major determinant of interest rate risk. However, the also show that maturity alone is not sufficient to measure interest rate sensitivity.例子：息率8%的债券Because we know that long term bonds ar

36、e more sensitive to interest rate movements than are short term bonds, in some sense a zero coupon bond represents a longer-term bond than an equal-time-maturity coupon bond. This is the insight about effective maturity.比较20年到期的零息债券和带息债券(8% coupon rate)。The 20-year 8% bond makes many coupon payments

37、, most of which come years before the bonds maturity date. Each of these payments may be considered 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.The zero-coupon bond, by contrast

38、, makes only one payment at maturity. Its time to maturity is a well defined concept. To 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

39、 of the bond. We would like also to use the measure as a guide to the sensitivity of a bond to interest rate changes.Duration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。 表示债券现在的市场价格。 表示债券剩下的距到期日的时间。8%BondTimeyearsPaymentPV of CF(10%)WeightC1 XC4.54038.095.0395.019714036.281.0376.03761.52.0401040sum34.553855

40、.611964.540.0358.88711.000.05371.77421.8852Duration Calculation: Exle using Table 16.3Duration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Duration 在固定收益投资组合管理中的作用测量证券组合有效平均到期日的统计量度量证券组合对利率的敏感度（定量刻画） an essential tool in immunizing portfolios from interest rate riskDuration 和股票价格变化之间的关系这里 表示债券价格的变化 是债券的

41、初始价格 是到期收益的变化 是初始的到期收益例子Bond : coupon rate 8%, yield to maturity 8%, par value 1000, price 1000, duration 10 when yield to maturity 8% 9% What determines duration?Duration MaturityZero coupon bond15% coupon YTM=6%3% coupon YTM=15%15% coupon YTM=15%Rule for duration零息债券的duration等于其到期日到期日保持不变，息率越低， du

42、ration越高息率不变，到期日越大， duration一般越大。对等价或者溢价发行的债券，上述关系总是成立别的因素不变，到期收益越低，带息债券的duration越高。永久性现金流的duration为到期日与duration的差别当到期日越来越大时， duration接近于相应永久性现金流的duration注意支付时间单位与利率之间的一致性2.3 ConvexityAs a measure of interest rate sensitivity, duration clearly is a key tool in fixed income portfolio management.The d

43、uration rule for the impact of interest rates on bond prices is only an approximation.YieldPriceDurationPricing Error from convexityDuration and ConvexityThe duration rule is a good approximation for small changes in bond yield, but it is less accurate for larger changes.The duration approximation a

44、lways understates the value of the bond, it underestimates the increase in bond price when the yield falls, and it over estimates the decline in price when the yield rises.The curvature of the price yield curve is called the convexity of the bond.As a practical rule, we can view bonds with higher co

45、nvexity as exhibiting higher curvature in the price yield relationship.Convexity allows us to improve the duration approximation for bond price changes.Correction for ConvexityCorrection for Convexity:Ex. A 30-year maturity, an 8% coupon, and sells at an initial yield to maturity of 8%. The bond sel

46、ls 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%.The duration rule would predictThe duration-with-convexity ruleyWhy do investors like convexity? Bond ABond BBond A has greater price incr

47、eases and smaller price decreases when interest rates fluctuate by larger amounts.2.4 Passive bond managementPassive methods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。Active methods假设债券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。Passive methods消极债券管理认为债券的价格是公平的，只能

48、控制固定收入证券组合的风险主要策略：指标化策略:复制给定债券指标的行为Immunization 策略：shield the overall financial status of the institution from exposure to interest rate fluctuations.两者认为市场价是公平的两者的区别债券-指标证券组合和债券市场指标具有相同的风险-收益回报Immunization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。Bond-index fundsCreate a portfolio that mirrors the composition of

49、 an index that measures the broad market.债券市场指标Lehman Brothers, Merrill Lynch, Salomon BrothersNumber of issuesmaturity of included bondsexcluded issuesweightingreinvestmentDaily availability 构造反映债券市场指标的证券组合问题：债券种类过多，难以一一购买交易少，很难以公平市价买到指标中包含的所有债券指标不断更换到期日少于1年的债券不断调整，利息收入重投资方法：精确复制债券指标不可行，采用cellular方

50、法把债券市场分成几类指标中债券在各类中占的比例按这一比例构造债券组合In these way, the characteristics of the portfolio in terms of maturity, coupon rate, credit risk, industrial representation, and so on, will match the characteristics of the index, and the performance of the portfolio likewise should match the index. SectorT-t-MTre

51、asuryAgencyMortgage-BackedIndustrialFinanceUtilityYankee1 year12.1%1-3 years5.4%3-5 years4.1%5-7 years7-10 years0.1%10-15 years 15-30 years9.2%3.4%30+ yearsImmunization两种不同的看待利率风险的方式银行，使得资产净现值不受利率波动的影响养老金，使得资产将来的值不受风险的影响What is common to the bank and the pension fund is interest rate risk. The net w

52、orth of the firm or the ability to meet future obligations fluctuates with interest rates.通过适当调整证券组合的到期日结构，规避利率风险Immunization techniques refer to strategies used by such investors to shed their overall financial status from exposure to interest rate fluctuations.Net worth immunization例子：承诺在两年后支付1000

53、000元，有两种债券可供选择：债券 1年 2年 3年 yield 1 80 80 1080 10% 2 1070 10% 例子：保险公司以价格10000元发行一种guaranteed investment contract(GIC)，5年到期，保证利率为8%。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。 价格风险重投资风险Terminal value of a bond portfolio after 5 years (all proceeds reinvested)A. rates remain at 8%Terminal value of a bond portfolio after 5 years (all proc