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固定收益课题 4 ：
债券定价和风险 / Valuation and Risk
陈国辉博士 ／ 南洋理工大学商学院
债券定价模式
C = coupon rate p.a.
m = number of coupon payment per annum
y = yield to maturity p.a.
n = number of period
P = price of bond
价格和收益率的关系 (Price-Yield Relationship)
Price
Yield
•价格和收益率成反向关系 (Inverse relation)
•高收益率使到价格下滑 (High yield causes price to decrease)
•凸性 (convexity)
到期收益率 (Yield-to-Maturity,YTM) 的局限性
• YTM 不是真正能实现的收益率• YTM is not the realized rate of return• 实现 YTM 的 2 个条件 (2 conditions for realizing YTM
as return) ：• 持有债券到到期日 (Hold the security up to maturity)• 利息的再投资率必须等于 YTM (Reinvestment rate has to equal
to YTM)
• 债券投资者更关心总收益 (total return) ：资本收益 (capital gain or price return) + 票息收益 coupon return) + 再投资收益 (reinvestment return)
总收益 Total Return
• 假设一投资者的投资期限是 2 年。再假设他买了 9 ％票息的 2 年期债券，价格为 98.26 (ytm = 10%)
• 请问此投资的总收益是多少 , 假设其再投资收益率是 10 ％？• Future value of his investment
• First coupon: $9(1.1)• Second coupon $9• Maturity value = 100• Total future value = 9(1.1) + 9 + 100 = 118.9
• Today’s investment = 98.26• Total return:
• 118.9/(1+R)2=98.26• R = 10%
• 如果再投资收益率是 9 ％，那总收益是多少？• 如果再投资收益率是 11 ％，那总收益是多少？
Reinvestment Risk
总收益 Total Return
• 假设一投资者的投资期限是 1 年。再假设他买了 9 ％票息的 2 年期债券，价格为 98.26 (ytm = 10%)
• 请问他的总收益率是多少？• 假设他卖出债券时的价格的 YTM 是 10%, ie., 109/(1+0.1) or
99.0909• 或假设以低于 99.0909 的价格卖出？• 或假设以高过 99.0909 的价格卖出？
Price Risk
平均到期期限 Average Maturity• 投资者在乎到期日吗？• 假设 3 个债券
• Debt A: Pay $10,000 in 5 years• Debt B: Pay $1 in 0.25 year and $9,999 in 5 years.• Debt C: Pay $9,999 in 0.25 year and $1 in 5 years.
• 注意付息的时间和额度
久期 Duration
• 为了更准确的计算平均到期期限，我们可以考虑用时间来权重每个现金流。得到的结果叫久期 (duration) 或麦考利久期 Macaulay Duration 。• 假设债券每年付 $C ， YTM=y 。 先假设每年复利一次，那么久期：
• 因为其分母是债券的价值，所以我们也可以用以下公式表示：nn
nn
y
C
yC
yC
yC
y
Cn
y
Cn
yC
yC
D)1(
)100(
)1()1()1(
)1(
)100(
)1(
)1(
)1(2
)1(1
)1(2
)1(2
...
...
PD
nn y
Cn
y
Cn
yC
yC
)1(
)100(
)1(
)1(
)1(2
)1(1
)1(2 ...
久期：债券价格对利率变动的敏感度
假设年票息
nn y
C
yC
yC
yCP
)1(
)100(
)1()1()1( )1(2 ...
dp / dy ：
)1(
1
...1
...
)1(
)100(
)1(
)1(
)1(2
)1(
)1(
)100(
)1(
)1(
)1(2
)1(
)1(21
132
y
PD
dy
dP
PDdy
dPy
dy
dPy
dy
dP
nn
nn
y
Cn
y
Cn
yC
yC
y
Cn
y
Cn
yC
yC
久期：债券价格对利率变动的敏感度
• dp/dy ＝实值久期 (Dollar Duration) (Bloomberg calls it RISK.)
• 如果价格变化是以百分比来表示，那么叫修正久期 (modified duration) ：
)1(
/
y
D
dy
PdP
国债的久期
CwPD
wnwnww y
Cwn
y
Cwn
y
Cw
y
Cw
21
)1(
)100()1(
)1(
)2(
)1(
)1(
)1(
)1(
... 121
21
21
221
21
21
121
21
21
21
21
21
在付息期间 （ w ＝从今天到下个付息日的部分时间）：
PD
nn y
Cn
y
Cn
y
C
y
C
)1(
)100(
)1(
)1(
)1(
1
)1(21
21
21
121
21
21
22121
2121
21
...
在付息日（ w ＝ 1 ）：
久期 Duration
假设以下 3 个债券：
Bond Yr 1 Yr 2 Yr 3 Yr 4 Yr 5 Duration
A 5 5 5 5 105 4.5 Yr
B 20 20 20 20 120 3.8 Yr
C 0 0 0 0 100 5.0 Yr
久期的特性 （ 1 ）• 给予既定的到期期限，票息越高，久期越短 (Holding maturity
unchanged, increasing the coupon reduces duration)• 这是因为更高的现金流在短期内实现• This follows because more cash flows are paid sooner.
• 给予既定的到期期限和票息， 到期收益率越高，久期更短 (Holding maturity and coupon unchanged, increasing yield leads to lower duration)• 这是因为高 YTM ，意味着高折现率，更远的现金流的现在值就越低了• This follows because higher yield discount distant cash payment more
heavily.• 给予既定的票息和 YTM ，更长的到期期限，更长的久期 ( Holding
coupon and yield unchanged, increasing maturity leads to higher duration)• 看似很容易理解，但不是所有的债券都有此特性• Seems clear but not true all the time. It only applies to premium bonds
(see next page).
Zero coupon bond
Discount bond
Par bond
Premium bond
Perpetual bond
Maturity
Duration
久期的特性 （ 2 ）• 随着到期日的贴近，久期下滑 (Duration declines as
bond approaches maturity)• 开始时慢速下滑• 期限低过 5 年后，下滑的速度加快。• 最后以 1 对 1 的比例下滑
• 成锯齿状变化 (Sawtooth fluctuation)• 形状来自于定期付票息：
• 在付息日，久期上升 (On coupon payment date, duration increases due to payment of the near-term coupon)
• 期限更长的债券，其上升的幅度更大 (Long term bond has bigger jump (see table on next slide))
Duration (in year) of a 12%, 12% yield 20 years bond
Duration
Coupon
Remaining Term to Maturity
Immediately before coupon
Immediately after coupon
Chang in duration
1 19.5 7.48 7.92 0.442 19.0 7.43 7.87 0.443 18.5 7.37 7.81 0.444 18.0 7.31 7.75 0.445 17.5 7.25 7.68 0.4310 15.0 6.89 7.30 0.4115 12.5 6.39 6.78 0.3920 10.0 5.74 6.08 0.3425 7.5 4.86 5.15 0.2930 5.0 3.68 3.90 0.2235 2.5 2.11 2.23 0.1236 2.0 1.74 1.84 0.1037 1.5 1.34 1.42 0.0838 1.0 0.92 0.97 0.0539 0.5 0.47 0.50 0.03
久期实用案例 1• 债券免疫 Bond Immunization
• 平衡再投资风险和价格风险 (Balancing reinvestment risk and price risk is called immunization)
• 可通过匹配久期和投资期限来达到免疫 (This can be done using duration matching: matching duration with the investor’s horizon (See proof in class.))
久期实用案例 2
• 久期衡量债券价格的波动率（即对利率的敏感度）(Duration as a measure of volatility to estimate price change due to change in interest rate):
yy
DPP
y
D
dy
PdP
21
21
1
)1(
/
案例 : 19.5 year 12% coupon bond priced at 12% yield. A 10 bp change in yield.
• Before coupon payment when duration = 7.48
75.0)001.0()12.0(1
48.797.105
1 21
21
yy
DPP
久期实用案例 3
• 当久期在债券付息日升高时，是否意味债券的风险增加了？ When duration jumps on coupon payment day, does it mean that the bond suddenly begins to impose higher market risk on the investment?• 付息后，久期＝ 7.92 (duration = 7.92)• 价格变化：
• 注意有现金流，而现金的久期等于零。投资经理必须决定如何投资收到的现金。这将影响其“组合”久期。 However, duration-matched manager has to decide how to reinvest the cash. His/her decision will change the “portfolio” duration.
75.0)001.0()12.0(1
92.7100
1 21
21
yy
DPP
久期实用案例 4• 随着期限的变短，久期也在下降 (Duration of the bond
change as time passes)• 举例：
• 假设初始投资时： 20-year bond ， 12% coupon, 12% yield ， duration = 8 years = 投资期限 (investment horizon)
• 4 年之后，投资期限＝ 4 years ， 但是久期 ＝ 7.5 years• 所以必须重新平衡组合 rebalance the “portfolio”.• 在什么时候最适合做重新平衡 When is the best time to do
rebalancing?• 在付息日。因此国债在 Feb 15, May 15, Aug 15, 及 Nov 15 的交易
量特高。
久期实用案例 5• 债券市场用 DV01 来衡量久期 :
• That is, how much the price of bond will change for 1 bp change in yield.
• DV01 = (Modified Duration x Full Price) x 0.0001for $100 face value bond.
久期实用案例 6 – 组合• 一个债券组合的久期等于其加权平均久期。权重以债券市值计算。
(A portfolio’s duration is simply the weighted average duration of the bonds in a portfolio. The weight is calculated by market value.)
Bond Market Value (mil $)
Weight Mod Duration
Weight x Mod Dur
A 10 0.1 4 0.4
B 40 0.4 7 2.8
C 30 0.3 6 1.8
D 20 0.2 2 0.4
Total 100 1 5.4
If yield curve shift up by 100 bp, what is the percentage value change of portfolio?
凸性 Convexity• 凸性测量在利率和价格关系线上的弧线部分
Convexity measures the curvature (nonlinearity) in a bond-yield curve
• 所有没带期权的债券都存在正凸性 All non-callables exhibit some degree of positive convexity
• 正凸性意味在同量的利率下降和上升情况下， 债券价格的上升比下降来的大 Positive convexity indicates a bond’s price rises more for a given decline in yield but fall less for a given increase in yield
• 凸性来自于债券的定价模式 Convexity exists because of bond price-yield relationship
• 凸性就像是在期权里的 Gamma Convexity is like Gamma in option
• 凸性越高越好 Convexity is desirable
从 Dp/Dy 到 D2p/Dy2
• From duration, we know:
• The second derivative:
121
21
21
21
32121
221
21
)1(
)100(
)1(
)1(
)1(
2
)1(...
2
1
nn y
Cn
y
Cn
y
C
y
C
dy
dp
221
21
2)1(21
21
421
21
321
21
)1(
)100()1(
)1(
)1()1(
)1(
32
)1(
21
2
2
...2
1
2
1
nn y
Cnn
y
Cnn
y
C
y
C
dy
pd
2212
21
21
)1(
)100()1(
1)1(
)()1(
2
2
4
1nt y
nnn
ty
Ctt
dy
pd
测量凸性 Measure Convexity
yearsindy
Pd
PConvexity
2
21
yearsindy
Pd
PConvexity
2
21
2
1
有些人用以下定义：
我们对凸性的定义 :
• 注意：我们测量凸性时收益是以小数表示。如果收益率是以％表示，那么当凸性＝ 100 时，凸性应写成1
凸性的特性• 久期越长，凸性更大
Higher the duration, higher the convexity (see next slide).
• 现金流越分散，凸性越大Higher the dispersion of cash flows, higher the convexity.• 比较两种债券：固定票息和零息，哪种债券的凸性较大？ (no
cash flow dispersion).• 杠铃 (Barbell) 组合的凸性比子弹的凸性来的高
Barbell has greater convexity than a bullet (draw a line between any two points on the duration-convexity curve to see the barbell has greater convexity.)
凸性的特性
Convexity of Zero vs Duration (Mod)
0
50
100
150
200
250
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Duration
Co
nv
ex
ity
Coupon bond’s convexity is sum of a duration-matched zero’s convexity.
凸性的特性• 利率波动率增高时，凸性会越大
Higher the yield volatility, higher the value of convexity.• 波动率增高会对正凸性更有利
Higher volatility enhances the expected performance of positively convex positions.
• 那么，有谁会愿意买子弹式债券呢？Why would anybody want to buy bullet then? No free lunch.
近似凸性 Approximate Convexity
20
0
)(P
P2PP Convexity eApproximat
y
应用久期和凸性测量价格变化
....)(2
1 22
2
dydy
Pddy
dy
dPdP
....)(1
2
11 22
2
dydy
Pd
Pdy
Pdy
dP
P
dP
2)(2
1)( dyConvexitydyDurationModfied
P
dP
应用泰勒系列关系式 Taylor series expansion ：
http://en.wikipedia.org/wiki/Taylor_series
Duration Approximation
Duration and Convexity Approximation
Yield
Price
举例• 假设修正久期＝ 4 年，凸性＝ 49 ，那么如果利率增加
100 基点，债券的价格会增加或降低多少？ Suppose Modified Duration is 4 years and convexity is 49, what is the % change of price for 100 bp increase in yield?
• -4x0.01+0.5x49x(.01)2=-0.03755 = -3.755%
近似久期 Approximate Duration
y
0
-
P2
PP Duration eApproximat
• 举例 : 25-yr, 6% coupon, 9% yield, P0 = 70.3570
• 步骤：– 提高 YTM 10 bp (ie., Δy=0.001), 用定价模式计算 P- = 69.6164
– 降低 YTM 10 bp (ie., Δy=0.001), 用定价模式计算 P+ = 71.1105
– 近似久期 Approximate duration = 10.62
久期实用案例 7 －相对久期Matching $Duration
• 假设你持有债券 X ，想置换债券 Y 。但是你期望保持其久期。Suppose you own Bond X and want to exchange for Bond Y. You want to ensure that after the swap, the $Duration is maintained.
Bond Price Par Amount
(PA)
Market Value
(MV)
Modified Duration
(MD)
Dollar Duration
($D)
X 80 10 mill 8 mill 5 8 mil x 5 / 100
= $400,000
(for 1% change in yield)
Y 90 ? ? 4 $400,000 (imposed)
久期实用案例 7－相对久期Matching $Duration
milPA
MV
MVMDD
Y
Y
YYY
11.11$9.0
000,000,10
000,000,10
100/400000$
债券免疫 Bond Immunization• 等配资产和负债的价值
Matching assets and liabilities value.• 要等配资产和负债的现金流是很难的
Cannot exactly match the cash flows.• 但是我们确保资产的价值比负债高。当需要现金时，我们售出的
资产大过负债的现金需求But can ensure the value of asset is greater than the value of liabilities. When fund is needed, can sell assets to cover cash needs.
资金不足和资金过分充足Under-funding and Over-funding
• 考虑一退休基金有以下的现金需求：Consider a pension fund has the following cash flow requirements:
Year Pension Payment Present value (5% yield)
1 10,000 9,523.81
2 10,000 9,070.29
3 10,000 8,638.38
4 10,000 8,227.02
5 10,000 7,835.26
Total 50,000 43,294.77
• 假设基金买了一只 5 ％的平价债券（ $43,294.77 ）Suppose the fund invested in a bond with coupon 5%. Par amount = $43,294.77.
• 当利率上升后，基金的资产 <负债
Modified Duration
Value today If yield is 5% If yield is 6%
Assets 4.33 $43,294.77 $43,294.77 $41,471.04
Liabilities 2.76 $43,294.77 $43,294.77 $42,123.64
Difference 0 0 -$652.6 (under-funded)
债券免疫理论Bond Immunization Theorem
• Recall this:
20001 )(
2
1yPCyPMDPPP
Where
P1 = Value after yield change （利率变化后的价值）P0 = Value before yield change （利率变化前的价值）MD = Modified duration （修正久期）Δy = change in yield （利率变化）C = Convexity （凸性）
债券免疫理论Bond Immunization Theorem
• 对负债来说 :
20001 )(
2
1yACyAMDAAA AA
20001 )(
2
1yLCyLMDLLL LL
• 对资产来说 :
债券免疫理论Bond Immunization Theorem
• 我们的目的是：• A1 L1
• 假设 :• A0 = L0 和 • MDA = MDL
• 那么 :
• 假如我们能保证 CACL, 那么 A1 L1
20
2011 )(
2
1)(
2
1yLCyACLA LA
债券免疫理论Bond Immunization Theorem
• Back to the example. Suppose we invest in 2 bonds:• 2-year, value at par = 100, MD2 = 1.85941 number of bonds =
n2
• 5-year, value at par = 100, MD5= 4.329477, number of bonds = n5
• We can solve for the following equations:
77.43294100100 52 nn
764305.2329477.477.43297
10085941.1
77.43297
100 52 nn
债券免疫理论Bond Immunization Theorem• n2 = 274.3406
• n5 = 158.6071
• 你可以证明其总凸性大过负债的凸性You can verify that the total convexity is greater than the convexity of the liabilities (=12.08251)
• 你也应该发觉到，如果要得到更大的凸性，你应该选择现金流分散的债券或债券组合You should also notice that in general, to get higher convexity, you should choose a bond portfolio with higher cash flow dispersion.
债券免疫理论应用时的问题Bond Immunization in Practice• 实践中，可以供选择的债券很多
In practice, there are a large number of bonds to choose from, so there are many different combinations that may match the conditions: equal value, equal duration, and excess convexity.
• 要解决这个问题，我们需要多加其它的条件To handle this problem, we need additional criteria• 收益最大化 Maximize yield (aggressive approach)
Maximize Yield （收益率最大化）
• 要求收益最大化，但必须免疫（条件）• We can pick bond (n1,n2…nN) to maximize yA:
Subject to
i ii
i iiiA Dn
yDny
$
$max
i LiiLiiii
Li iiLiiii
i ii
i
CCnorCCA
Pn
DDnorMDMDA
Pn
LPnA
Nin
$$
$$
,...1,0
债券免疫理论应用时的问题－重新组合Bond Immunization in Practice － Rebalancing
• 在一段时间内，负债的久期和资产的久期不会产生一样的变化Change in duration of liabilities does not match change in duration of assets as time passes.• 举例
• 假设初开始投资期限为 8 年，我们购买了一只 20 年的债券（ 12 ％票息， 12 ％收益率），久期为 8 年Assume initial investment of a 20-year bond (12% coupon, 12% yield) with duration to match 8 years horizon.
• 4 年后， 20 年债券的久期掉到 7.5 年（假设）4 years later, horizon reduces to 4 years but 20-year bond’s duration only reduced to say 7.5 years duration.
• 因此我们必须重新组合Therefore he has to rebalance the “portfolio”.
为负债免疫的最佳方法是什么？What is the best way to immunize a liability?
• 假设其它条件不变，买零息债券最好Holding everything constant, buy zeros of same maturity period.
• 但是，零息债券的价格一般偏高（低收益），因此你需要更多的钱来达到目标收益However, zeros provide lower yield, hence higher price. That is, you need more money to satisfy the yield target.
