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1、第2讲债券的收益率1.利差2.复利3.债券收益率的度量 1.债券收益率利差基准利率：国债收益率LIBOR央行对基准利率的影响影响yield spread的因素：信用品质 经济景气内含期权 Call or prepayment Put or conversion option流动性 发行规模税收其他5Copyright Chen,Rong&Zheng,Zhenlong,20112.复利普通复利计息频率或计息周期利率r的时间单位和期数N 的时间单位应该相同已发行债券的剩余期限不是计息期的倍数时计算要特别小心连续复利同一利率的不同表达6Copyright Chen,Rong&Zheng,Zhenlo
2、ng,2011连续复利的优点计息天数不整齐或是现金流时间间隔不规则计算多期收益率：算术平均符合正态分布假设取值范围多期收益率仍然符合正态分布不存在汇率收益率悖论缺点横截面组合收益率单个资产收益率加权平均7Copyright Chen,Rong&Zheng,Zhenlong,2011普通复利（Rm）和连续复利（Rc）的转换特别地，当m=1时，对数差分收益率年百分比收益率APR与年有效收益率AEY债券等价收益率BEY3.债券收益率的度量债券收益率的度量就传统的固定收益债券而言，投资收益包括：定期的利息支付 本金收回（资本利得）再投资收益 3.1 息票率3.2 当期收益率 current yield
3、 =Annual Interest/Market Price 3.3到期收益率Yield to Maturity 使债券未来承诺现金流的现值与其当前价格相等的贴现率 与债券价格包含同样信息 内在收益率 解下列关于y的方程:Yield to Maturity:Example10 yrs MaturityCoupon Rate=7%Price=$950 y=3.8635%YTM=3.86%x 2=7.72%bond equivalent yield(BEY)对价格为面值的债券而言,息票率=当期收益率=到期收益率对溢价债券而言,息票率当期收益率到期收益率对折价债券而言,到期收益率当期收益率息票率13
4、Copyright Chen,Rong&Zheng,Zhenlong,2011到期收益率的优缺点综合反应债券投资的三种未来现金收益与债券价格一一对应仅仅是特定条件下的承诺到期收益率，非预期收益率的精确指标到期收益率的隐含假设没有违约风险投资者持有到期每一期现金流都按照y进行再投资（忽略再投资风险）对复杂债券预期收益率的衡量精度尤其低YTC=6.64%YTM=6.82%3.4赎回收益率Yield to Callcall protectionyield to call for premium bonds yield to maturity for discount bonds3.53.6注意：YT
5、C、YTP、CFY与YTM同为内在收益率存在同样局限CFY有提前偿付率的假定3.7实现复利收益率Realized compound Yield 在a）的情况下，实现复利收益率为 1000(1+yrealized)2=1210 yrealized=10%在b）的情况下，实现复利收益率为 1000(1+yrealized)2=1208 yrealized=9.91%However,in an economy with future interest rate uncertainty,the rates at which interim coupons will be reinvested are
6、not yet known.This reduces much of the attraction of the realized yield measure.3.8 持有期回报率Holding-Period ReturnHPR=I+(Pn-P0 )/P0whereI=interest paymentP1=price in n periodsP0=purchase priceHolding-Period Return：ExampleCR=8%YTM=8%N=10 yearsSemiannual CompoundingP0=$1000In six months the rate falls to
7、 7%P1=$1068.55HPR=40+(1068.55-1000)/1000 HPR=10.85%(semiannual)When the yield to maturity is unchanged over the period,the rate of return on the bond will equal that yield.An increase in the bonds yield acts to reduce its price,which means that the holding period return will be less than the initial
8、 yield.Conversely,a decline in yield will result in a holding period return greater than the initial yield.Annualizing the holding period returnOver multiple yearsFrom quarterly holding period returns(1+HPR)=(1+HPR1)(1+HPR2)(1+HPR3)(1+HPR4)3.9 OID债券的税后收益 有些债券以较低的息票率发行，按面值打折出售（OID，original issue disc
9、ount）极端的例子为零息债券，没有息票利息，而以价格升值的形式提供全部收益。美国的国库券是短期零息债券 长期零息债券一般由中长期国债剥离而成（separate trading of registered interest and principal of securities,STRIPS）。例如，一张10年期国债、可以剥离成21张独立的零息债券，期限从6个月到10年。债券剥离bond stripping 债券重构bond reconstitutionFigure 14.7 The Price of a 30-Year Zero-Coupon Bond over Time at a Yiel
10、d to Maturity of 10%The tax authorities recognize that the“built-in”price appreciation on OID bonds such as zero-coupon bonds represents an implicit interest payment to the holder of the security.基于市场利率不变的价格升值计为应税利息收入；而因市场利率变化而导致的额外价格变动则被看作是资本利得。例1：如果当前市场利率为10，一个30年期零息票债券的发行价是 1000/(1.10)30=57.31元 第
11、二年，如果利率仍为10，则债券价格应为 1000/(1.10)29=63.04元 应税利息收入为 63.0457.31=5.73元 如果利率发生变化，如跌到9.9，那么债券价格将是 1000/(1.099)29=64.72元 如果卖出债券，则64.72元与63.04元之差就是资本利得并要按资本利得税率纳税；如果不卖出，那么这个价差就是未实现的资本利得，在当年不纳税。例2：30-year maturity bond，issued with a coupon rate of 4%（annually）and a yield to maturity of 8%P0=$549.69If the bond
12、s yield to maturity remains at 8%in one year,P1=$553.66 Taxable interest income=$553.66$549.69=$3.97Suppose that the YTM actually falls to 7%by the end of the first year,and that the investor sells the bond after the first year.If the investors tax rate on interest income is 36%and the tax rate on c
13、apital gains is 20%,what is the investors after-tax rate of return?3.10到期收益率与违约风险 违约溢价default premium:公司债券的承诺收益率与类似期限的无违约风险政府债券收益率之差.违约风险越大的债券，提供的违约溢价越高，因而到期收益率也就越高。这一关系被称为“利率的风险结构”.违约溢价随经济周期变动.Figure 14.11 Yields on Long-Term Bonds,1954 2006承诺到期收益率promised YTM:公司履行全部本息支付期望到期收益率expected YTM:考虑公司违约可能
14、 例 假设某公司20年前发行了一种息票利率为9的债券，到目前为止还有10年到期。由于公司面临财务困境，投资者虽然相信公司有能力按期支付利息，但预期在债券到期日公司将被迫破产，届时债券持有人将只能收回面值的70。债券当前的市场价格为750元。该债券的期望到期收益率和承诺到期收益率分别是多少？承诺到期收益率=13.7期望到期收益率=11.61.Bonds of Zello Corporation with a par value of$1,000 sell for$960,mature in five years,and have a 7%annual coupon rate paid semia
15、nnually.a.Calculate the:i.Current yield.ii.Yield to maturity(to the nearest whole percent,i.e.,3%,4%,5%,etc.).iii.Realized compound yield for an investor with a three-year holding period and a reinvestment rate of 6%over the period.At the end of three years the 7%coupon bonds with two years remainin
16、g will sell to yield 7%.b.Cite one major shortcoming for each of the following fixed-income yield measures:i.Current yield.ii.Yield to maturity.iii.Realized compound yield.2.A newly issued bond pays its coupons once annually.Its coupon rate is 5%,its maturity is 20 years,and its yield to maturity is
17、 8%.a.Find the holding period return for a one-year investment period if the bond is selling at a yield to maturity of 7%by the end of the year.b.If you sell the bond after one year,what taxes will you owe if the tax rate on interest income is 40%and the tax rate on capital gains income is 30%?The b
18、ond is subject to original-issue discount tax treatment.c.What is the after-tax holding period return on the bond?d.Find the realized compound yield before taxes for a two-year holding period,assuming that(1)you sell the bond after two years,(2)the bond yield is 7%at the end of the second year,and(3
19、)the coupon can be reinvested for one year at a 3%interest rate.e.Use the tax rates in(b)above to compute the after-tax two-year realized compound yield.Remember to take account of OID tax rules.3.Suppose that a 10%15-year bond has the following call structure:Not callable for the next 5 years;First
20、 callable in 5 years at$105;First par call date is in 10 years;The price of the bond is$127.5880.A.Is the yield to maturity for this bond 7%,7.4%,or 7.8%?B.Is the yield to first call for this bond 4.65%,4.55%,or 4.85%?C.Is the yield to first par call for this bond 6.25%,6.55%,or 6.75%?4.A bond with
21、a coupon rate of 7%makes semiannual coupon payments on January 15 and July 15 of each year.The Wall Street Journal reports the ask price for the bond on January 30 at 100:02.What is the invoice price of the bond?The coupon period has 182 days.5.Suppose that an amortizing security pays interest monthly.Based on the projected principal payments and interest,suppose that the monthly interest rate that makes the present value of the cash flows equal to the price of the security is 0.41%.What is the cash flow yield on a bond-equivalent basis?