固定收益证券第二章

3.5 3.6
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+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%
年百分比收益率APR与年有效收益率AEY 债券等价收益率BEY
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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
P1=$553.66 Taxable interest income = $553.66 –$549.69 =$3.97
Suppose 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 investor’s tax rate on interest income is 36% and the tax rate on capital gains is 20%, what is the investor’s after-tax rate of return?
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
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？
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
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 内在收益率 解下列关于y的方程:
P
t 1
T
Ct t 1 y
Yield to Maturity: Example
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
Rm FV lim PV 1 m m
mn
PV e
R cn
PV lim
FV R cn FV e mn m R m 1 m
同一利率的不同表达
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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
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=63.04元 应税利息收入为 63.04－57.31=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。
对复杂债券预期收益率的衡量精度尤其低
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3.4 赎回收益率Yield to Call
YTC=6.64%
YTM=6.82%
call protection
yield to call for premium bonds yield to maturity for discount bonds
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 bond’s yield acts to reduce its price,
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% (semiannual)
1/ n
1
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。
which means that the holding period return will be less than the initial yield. return greater than the initial yield.
Conversely, a decline in yield will result in a holding period
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 authorities recognize that the “built-in” price appreciation on OID bonds such as zerocoupon bonds represents an implicit interest
yield measure.
3.8 持有期回报率Holding-Period Return
HPR = [ I + ( Pn- P0 )] / P0 where I = interest payment P1 = price in n periods P0 = purchase price
Holding-Period Return： Example
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 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）
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普通复利（Rm）和连续复利（Rc）的转换 Rm R c m ln 1 m c R m R m m e 1 特别地，当m=1时，对数差分收益率
P Rc ln 1 R1 ln 1 ln P1 ln P0 P0
ຫໍສະໝຸດ Baidu
Annualizing the holding period return
Over multiple years
annualized HPR ((I ( Pn P ) 0 )) / P 0 1
From quarterly holding period returns (1 + HPR)= (1 + HPR1)(1 + HPR2)(1 + HPR3)(1 + HPR4)
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. 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 remaining 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.
10 yrs Maturity
Price = $950
Coupon Rate = 7%
35 1000 950 T t t 1 ( 1 y ) (1 y )
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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,
例2： 30-year maturity bond， issued with a coupon
rate of 4% （annually）and a yield to maturity of 8% P0=$549.69 year,
If the bond’s yield to maturity remains at 8% in one
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