第八章 债券定价和风险管理
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 principles,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 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 attributes
Bond pricing
Appropriate yield to maturity
Intrinsic value
The relationship between the yield and price
Bond risk management
When the maturity and the payment time is not consistent
Interest risk
Reinvestment risk
The approach of risk management
Duration
Convexity
1,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 Value
Promised 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。
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例子：一个债券，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？
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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。
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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
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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 maturity
coupon rate
call provisions
tax status
marketability
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 provisions
call price
call 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
price
1200
1000
0 5% 10% 15% 20% interest rate
At high interest rates,the risk of call is negligible,
and the values of the straight and callable bonds
converge,At lower rates,however,the values of
bonds begin to diverge,with the difference
reflecting the value of the firm?s 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%
Y i e l d to ca l l Y i e l d to m a t ur i t y
Coupon pay m e nt 40 40
Number of
semia nnua l p e ri ods
20 60
Fina l pay m e nt 1 100 1000
pri c e 1 150 1 150
The higher the coupon rate of a 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 provisions
putable 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 default
investment grade
speculative 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 costs
Leverage ratio--debt to equity ratio
Liquidity ratios--current assets to current
liability ratio
Profitability ratios--return on assets or equity
Cash 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
Ex pe c t e d YTM Sta te d Y TM
Coupon pay m e nt $45 $45
Num be r of semian nua l
pe riods
20 pe riod 20 pe riod
Fina l pay m e nt $700 $1000
Price $750 $750
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。
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One particular manner in 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 commensurately
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 premium
yield
spread Risk premium
Default-free yield-
to-Maturity
12% promised yield-to-maturity
9% expected yield-to-maturity
8% yield-to-maturity on a default-free
bond of similar and coupon rate
0%
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，与股票市场比较，级别越低的债券，对股票市场的波动敏感性越大。
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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 smaller
the expected net cash flow is smaller
the 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 debt
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 assets
54321 99.06.03.34.12.1 XXXXXZ
1X
2X
3X
4X
5X
Z-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 的四种测度：
The extent to which the firm?s 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 a loss.
The ratio of the market value of the firm?s equity
to the par value of its debt.
The market value of the firm?s 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 accounted
for roughly 75% of the variation in the
bond?s yield spread
1.4,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,Interest 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 greater 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 these 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 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（） +1000（） =974.23
一年后公平价格为 =70（） +1000（） =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 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
receive only a fair rate of return.
Each 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,
Security returns all should be comparable on
an after-tax risk-adjusted basis,If they are
not,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 risk 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 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 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。
Today Maturity Date
Par Value
Price of a
premium bond
Price of a discount
bond
premium
discount
3,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=1000
Bond D,coupon rate=9%,yield=7%,
P=1082
when 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 Duration
The 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
Y ie ld to
m at urity
T= 1 y e ar T= 10 y e ars T= 20 y e ars
8% 1000 1000 1000
9 990.64 934.96 907.99
Change in
pric e
0.94% 6.5% 9.20%
Y ie ld to
m at uri t y
T =1 y ear T =1 0 y ears T =2 0 y ears
8% 924,56 456,39 208,29
9 915,73 414,64 171,93
Cha nge in
pri ce
0,96% 9,15% 17,46%
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
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,most of which come years before the
bond?s 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,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
bond?s 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 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。
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34.553
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964.540
.0358
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1.000
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1.7742
1.8852
Duration Calculation,Example using
Table 16.3
Duration
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
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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 risk
Duration 和股票价格变化之间的关系
这里 表示债券价格的变化
是债券的初始价格
是到期收益的变化
是初始的到期收益
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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
Maturity
Zero coupon bond
15% 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,duration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
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2.3 Convexity
As a measure of interest rate sensitivity,
duration clearly is a key tool in fixed
income portfolio management.
The duration rule for the impact of
interest rates on bond prices is only an
approximation.
Yield
Price
Duration
Pricing Error
from convexity
Duration and Convexity
The duration rule is a good approximation for
small changes in bond yield,but it is less
accurate for larger changes.
The duration approximation always
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 convexity as exhibiting
higher curvature in the price yield
relationship.
Convexity allows us to improve the
duration approximation for bond price
changes.
Correction for Convexity
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Ex,A 30-year maturity,an 8% 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 bond?s yield
increase from 8% to 10%,the bond price will
fall to $811.46,a decline of 18.85%.
The duration rule would predict
The duration-with-convexity rule
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Why do investors like convexity?
y_y?y
Bond A
Bond B
Bond A has greater price increases and
smaller price decreases when interest
rates fluctuate by larger amounts.
2.4 Passive bond management
Passive 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 funds
Create a portfolio that mirrors the composition of
an index that measures the broad market.
债券市场指标
Lehman Brothers,Merrill Lynch,Salomon Brothers
Number of issues
maturity of included bonds
excluded issues
weighting
reinvestment
Daily 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.
Sector
T-t-M
Treasury Agency Mortgage
-Backed
Industrial Finance Utility Yankee
<1 year 12.1%
1-3 years 5.4%
3-5 years 4.1%
5-7 years
7-10
years
0.1%
10-15
years
15-30
years
9.2% 3.4%
30+
years
Immunization
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 worth 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 1000000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%
121 ww
278.21 21 ww
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
28.1469308.110000 5
Terminal value of a bond portfolio after 5 years (all
proceeds reinvested)
A,rates remain at 8%
1 4 80 0
4
08.1? = 10 8 8,39
2 3 80 0
3
08.1? 10 0 7,77
3 2 80 0
2
08.1? 93 3,1 2
4 1 80 0
1
08.1? 86 4
5 0 80 0 80 0
Sa l e of b o nd 0 10 8 00/ 1,08 10 0 00
14 6 93,2 8
Terminal value of a bond portfolio after 5 years (all
proceeds reinvested)
B,rates fall to 7%
1 4 80 0
4
07.1?
2 3 80 0
3
07.1?
3 2 80 0
2
07.1?
4 1 80 0
1
07.1?
5 0 80 0
Sa l e of b o nd 0 10 8 00/ 1,07
14 6 94,0 5
Terminal value of a bond portfolio after 5 years (all
proceeds reinvested)
C,rates increase to 9%
1 4 80 0
4
09.1?
2 3 80 0
3
09.1?
3 2 80 0
2
09.1?
4 1 80 0
1
09.1?
5 0 80 0
Sa l e of b o nd 0 10 8 00/ 1,09
14 6 96,0 2
For a horizon equal to the portfolio?s
duration,price risk and reinvestment
risk exactly cancel out.
Accumulated value of invested funds
funds
0 t* D t
value
10000
6000
8% interest rate
Coupon
bond
Single
payment
obligation
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。
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。
即使利率不变，当时间变化时，久期也会发生变化
Rebalancing immunized portfolio
Even 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 will become
unmatched and the goals of immunization will
not be realized.
Immunization 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,
例子：假设证券组合经理在 7年后有一笔
19487元的债务，以现在 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。
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(3
5.0
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
第 2年，即使利率不变，经理也需要调整策略
611)1(2
95
An appropriate compromise must be
established between the desire for perfect
immunization and the need to control
trading costs.
Immunization 在实际中 的问题
The notion of duration is strictly valid only for a
flat yield curve,
multiple nonparallel shifts in a nonhorizontal
yield curve
通货膨胀
On 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.
2.5 Active bond management
Interest rate forecasting
identification of relative mispricing
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 principles,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 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 attributes
Bond pricing
Appropriate yield to maturity
Intrinsic value
The relationship between the yield and price
Bond risk management
When the maturity and the payment time is not consistent
Interest risk
Reinvestment risk
The approach of risk management
Duration
Convexity
1,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 Value
Promised 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。
n
t
t
t
y
CP
1 1
*yy?
*yy?
*y
例子：一个债券，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？
%02.10?y
%00.9*?y
*y
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。
n
t
t
t
y
CV
1
*1
06.924?V
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
P *y
tC
*y
tC
P
*y
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 maturity
coupon rate
call provisions
tax status
marketability
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 provisions
call price
call 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
price
1200
1000
0 5% 10% 15% 20% interest rate
At high interest rates,the risk of call is negligible,
and the values of the straight and callable bonds
converge,At lower rates,however,the values of
bonds begin to diverge,with the difference
reflecting the value of the firm?s 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%
Y i e l d to ca l l Y i e l d to m a t ur i t y
Coupon pay m e nt 40 40
Number of
semia nnua l p e ri ods
20 60
Fina l pay m e nt 1 100 1000
pri c e 1 150 1 150
The higher the coupon rate of a 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 provisions
putable 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 default
investment grade
speculative 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 costs
Leverage ratio--debt to equity ratio
Liquidity ratios--current assets to current
liability ratio
Profitability ratios--return on assets or equity
Cash 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
Ex pe c t e d YTM Sta te d Y TM
Coupon pay m e nt $45 $45
Num be r of semian nua l
pe riods
20 pe riod 20 pe riod
Fina l pay m e nt $700 $1000
Price $750 $750
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。
dp
1
d
d
p
pyy
1
y
y
yppyyyd
d
d
1
例子：
06.0?dp 6.01 09.0?y
0 3 1 3.009.006.01 06.04.009.0d
One particular manner in 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 commensurately
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 premium
yield
spread Risk premium
Default-free yield-
to-Maturity
12% promised yield-to-maturity
9% expected yield-to-maturity
8% yield-to-maturity on a default-free
bond of similar and coupon rate
0%
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，与股票市场比较，级别越低的债券，对股票市场的波动敏感性越大。
y
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 smaller
the expected net cash flow is smaller
the 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 debt
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 assets
54321 99.06.03.34.12.1 XXXXXZ
1X
2X
3X
4X
5X
Z-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 的四种测度：
The extent to which the firm?s 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 a loss.
The ratio of the market value of the firm?s equity
to the par value of its debt.
The market value of the firm?s 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 accounted
for roughly 75% of the variation in the
bond?s yield spread
1.4,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,Interest 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 greater 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 these 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 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（） +1000（） =974.23
一年后公平价格为 =70（） +1000（） =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 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
receive only a fair rate of return.
Each 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,
Security returns all should be comparable on
an after-tax risk-adjusted basis,If they are
not,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 risk 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 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 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。
Today Maturity Date
Par Value
Price of a
premium bond
Price of a discount
bond
premium
discount
3,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=1000
Bond D,coupon rate=9%,yield=7%,
P=1082
when 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 Duration
The 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
Y ie ld to
m at urity
T= 1 y e ar T= 10 y e ars T= 20 y e ars
8% 1000 1000 1000
9 990.64 934.96 907.99
Change in
pric e
0.94% 6.5% 9.20%
Y ie ld to
m at uri t y
T =1 y ear T =1 0 y ears T =2 0 y ears
8% 924,56 456,39 208,29
9 915,73 414,64 171,93
Cha nge in
pri ce
0,96% 9,15% 17,46%
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
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,most of which come years before the
bond?s 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,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
bond?s 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 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。
0
1
P
tCPV
D
T
t
t?
tCPV t
0P
T
8%Bond Timeyears Payment PV of CF(10%) Weight C1 XC4
.5 40 38.095,0395,0197
1 40 36.281,0376,0376
1.5
2.0
40
1040
sum
34.553
855.611
964.540
.0358
.8871
1.000
.0537
1.7742
1.8852
Duration Calculation,Example using
Table 16.3
Duration
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
T
t
tCPVP
1
0
T
t
t t
P
CPVD
1 0
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 risk
Duration 和股票价格变化之间的关系
这里 表示债券价格的变化
是债券的初始价格
是到期收益的变化
是初始的到期收益
y
yD
P
P
1
P?
P
y?
y
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
Maturity
Zero coupon bond
15% 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,duration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
yy?1
2.3 Convexity
As a measure of interest rate sensitivity,
duration clearly is a key tool in fixed
income portfolio management.
The duration rule for the impact of
interest rates on bond prices is only an
approximation.
Yield
Price
Duration
Pricing Error
from convexity
Duration and Convexity
The duration rule is a good approximation for
small changes in bond yield,but it is less
accurate for larger changes.
The duration approximation always
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 convexity as exhibiting
higher curvature in the price yield
relationship.
Convexity allows us to improve the
duration approximation for bond price
changes.
Correction for Convexity
n
t
t
t tt
y
CF
yP
C o n v e x i t y
1
2
2 )()1()1(
1
Correction for Convexity:
])([21 2yC o n v e ix it yyDPP
Ex,A 30-year maturity,an 8% 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 bond?s yield
increase from 8% to 10%,the bond price will
fall to $811.46,a decline of 18.85%.
The duration rule would predict
The duration-with-convexity rule
y
%52.2202.026.11* yDPP
%27.1802.04.2125.002.026.11
])([21
2
2
yC o n v e ix it yyD
P
P
Why do investors like convexity?
y_y?y
Bond A
Bond B
Bond A has greater price increases and
smaller price decreases when interest
rates fluctuate by larger amounts.
2.4 Passive bond management
Passive 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 funds
Create a portfolio that mirrors the composition of
an index that measures the broad market.
债券市场指标
Lehman Brothers,Merrill Lynch,Salomon Brothers
Number of issues
maturity of included bonds
excluded issues
weighting
reinvestment
Daily 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.
Sector
T-t-M
Treasury Agency Mortgage
-Backed
Industrial Finance Utility Yankee
<1 year 12.1%
1-3 years 5.4%
3-5 years 4.1%
5-7 years
7-10
years
0.1%
10-15
years
15-30
years
9.2% 3.4%
30+
years
Immunization
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 worth 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 1000000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%
121 ww
278.21 21 ww
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
28.1469308.110000 5
Terminal value of a bond portfolio after 5 years (all
proceeds reinvested)
A,rates remain at 8%
1 4 80 0
4
08.1? = 10 8 8,39
2 3 80 0
3
08.1? 10 0 7,77
3 2 80 0
2
08.1? 93 3,1 2
4 1 80 0
1
08.1? 86 4
5 0 80 0 80 0
Sa l e of b o nd 0 10 8 00/ 1,08 10 0 00
14 6 93,2 8
Terminal value of a bond portfolio after 5 years (all
proceeds reinvested)
B,rates fall to 7%
1 4 80 0
4
07.1?
2 3 80 0
3
07.1?
3 2 80 0
2
07.1?
4 1 80 0
1
07.1?
5 0 80 0
Sa l e of b o nd 0 10 8 00/ 1,07
14 6 94,0 5
Terminal value of a bond portfolio after 5 years (all
proceeds reinvested)
C,rates increase to 9%
1 4 80 0
4
09.1?
2 3 80 0
3
09.1?
3 2 80 0
2
09.1?
4 1 80 0
1
09.1?
5 0 80 0
Sa l e of b o nd 0 10 8 00/ 1,09
14 6 96,0 2
For a horizon equal to the portfolio?s
duration,price risk and reinvestment
risk exactly cancel out.
Accumulated value of invested funds
funds
0 t* D t
value
10000
6000
8% interest rate
Coupon
bond
Single
payment
obligation
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。
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。
即使利率不变，当时间变化时，久期也会发生变化
Rebalancing immunized portfolio
Even 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 will become
unmatched and the goals of immunization will
not be realized.
Immunization 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,
例子：假设证券组合经理在 7年后有一笔
19487元的债务，以现在 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。
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(3
5.0
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
第 2年，即使利率不变，经理也需要调整策略
611)1(2
95
An appropriate compromise must be
established between the desire for perfect
immunization and the need to control
trading costs.
Immunization 在实际中 的问题
The notion of duration is strictly valid only for a
flat yield curve,
multiple nonparallel shifts in a nonhorizontal
yield curve
通货膨胀
On 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.
2.5 Active bond management
Interest rate forecasting
identification of relative mispricing
