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 prepaymentPut or conversion option 流动性发行规模 税收 其他,2. 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,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
2、 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,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BEY,3. 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=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
3、率解下列关于y的方程:,Yield to Maturity: Example,10 yrs Maturity Coupon 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,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
4、 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 Call,call protectionyield to call for premium bonds yield to maturity for discount bonds,3.53.6,注意： YTC、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+yrealize
5、d)2=1210yrealized=10%在b）的情况下，实现复利收益率为1000(1+yrealized)2=1208yrealized=9.91%,However, in an economy with future interest rate uncertainty, the rates at which interim coupons will be reinvested are not yet known.This reduces much of the attraction of the realized yield measure.,3.8 持有期回报率Holding-Perio
6、d ReturnHPR = I + ( Pn- P0 ) / P0 where I = interest payment P1 = price in n periods P0 = purchase price,Holding-Period Return： Example,CR = 8% YTM = 8%N=10 years Semiannual Compounding P0 = $1000 In six months the rate falls to 7%P1 = $1068.55 HPR = 40 + ( 1068.55 - 1000) / 1000 HPR = 10.85% (semia
7、nnual),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 yield.Conversely, a decline in yield will result in a
8、 holding period return greater than the initial yield.,Annualizing the holding period return,Over multiple years,From 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 discount ）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
9、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 reconstitution,Figure 14.7 The Price of a 30-Year Zero-Coupon Bond over Time at a Yield to Maturity of 10%,The tax authori
10、ties 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元第二年，如果利率仍为10，则债券价格应为1000/(1.10)29=
11、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 bonds yield to maturity remains at 8% i
12、n one year, P1=$553.66Taxable 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 capital gains is 20%, what
13、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,例 假设某公司20年前发行了一种息票利率
14、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=11.6,1. 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 semiannually. a.
15、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 rem
16、aining 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 yi
17、eld to maturity is 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
18、gains income is 30%? The bond is subjectto 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%
19、at the end of the second year, and (3) 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 str
20、ucture: Not callable for the next 5 years; First 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 p
21、ar call for this bond 6.25%,6.55%,or 6.75%?,4.A bond with 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
22、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?,
