Bond Price Volatility
1
Price-Yield Relationship
2
Price-Yield Relationship(Continued)
• Bond price volatility是衡量interest rate risk
• 就option-free bonds而言，債券價格與收益率是呈convex關係 (非
線型關係)
• 一既定幅度的收益率變動，對不同債券的影響不相同 ；(非均一)
• 同一絕對值的收益率變動，收益率上升與收益率下降所影響的價
格變動絕對值不同，後著所引起的價格變動幅度價大；(不對稱)
• 收益率變動幅度愈小時，債券價格變動不對稱的價差愈小，反之
收益率變動幅度愈大，債券價格變動不對稱的價差就愈大
3
債券價格波動度的衡量
• Full Valuation Approaches and Partial Valuation Approaches
• Full Valuation Approaches: 如債券評價模型以及effective duration
• Partial Valuation Approaches: 如DV01 (PVBP), Macaulay duration,
modified duration, and key rate duration etc.
4
債券評價模型

• 設收益率由y變動為 y，以債券評價模型就變動前與變動後的收益
率分別評價，其相對應之價格為 P 與 P '
T
c
M

t
(1  y )T
t 1 (1  y )
p0  
T
c
M

t
(1  y ) T
t 1 (1  y )
p0  
• 債券波動度可計算如下
p0  p0
y  y
5
Effective Duration
• 債券評價模型或是partial valuation models只適用於純債券
• 對於cash flows不確定的債券，callable bonds或是putable bonds等，
應使用effective duration
• Effective duration同時考慮收益率上升與下降的不對稱性
• Effective duration可納入利率波動度的影響因素
• Effective duration亦可同時處理expected cash flow問題。
6
Effective Duration (Continued)
• 假設收益率下降
比為
 y 時，每一basis
point所引起債券價格變動百分
P  P0
P0 ( y )100
• 反之，若收益率上升 ，每一basis point所引起債券價格變動百
分比為
P0  P
P0 ( y )100
• 平均每一basis point的債券價格變動百分比為
P0  P 
1  P  P0
P  P




2  P0 ( y)100 P0 ( y)100  2 P0 ( y)100
7
Price Value of a Basis Point (PVBP)
• 又稱為dollar value of an 01或簡稱為DV01
• 這是衡量收益率變動一個bp(由於變動幅度很小，所以是收益率上
升或下降並無影響)，所引起的價格變動，亦即
T
c
M

t
(1  y)T
t 1 (1  y)
p0  
y  y  0.01%
8
Macaulay Duration
• 債券價格波動度是衡量收益率變動所引起債券價格變動的程度，
此一觀念可以表示為 dp
dy
• 因此可以就上述的債券評價模型微分如下
dp
 c1
 2c2 (1  y )  3c3 (1  y ) 2
 TcT (1  y )T 1




dy (1  y ) 2
(1  y ) 4
(1  y ) 6
(1  y ) 2T

3c3
 1  c1
2c2
TcT 







(1  y )  (1  y ) (1  y ) 2 (1  y )3
(1  y )T 
 1  T t  ct 

 

(1  y )  t 1 (1  y )t 
9
Macaulay Duration (Continued)
• 等號左右兩邊各除上p
 1
dp 1
 1  T t  ct  1
1  T
 
 



t

pv




t
dy p (1  y )  t 1 (1  y )t  p (1  y )  t 1
 p

pvt 
1  T
1  T

  t 


t

w


t
(1  y )  t 1
p  (1  y )  t 1

• 或表示呈
dp
1
 D 
p
dy
(1  y )
• Macaulay duration:
T
 T tCt 
D  
/ P   t  wt
t
t 1
 t 1 (1  y ) 
10
Modified Duration
•
Modified duration:
dp
  D*  p
dy
D*  D /(1  y ) 
•
dp 1

dy p
Dollar duration: (以金額表示的修正存續期間)
dp
  D*  p
dy
•
Duration in years = duration in m periods per year / m
11
Modified Duration (Continued)
• 以價格變動金額表示:
dp   D*  p  dy
• 以價格變動百分比表示:
dp
  D*  dy
p
12
存續期間的解讀
• Duration is the “first derivative.”
• 期初投資成本回收所需時間
• 以現金流量現值占債券價格的比例為權數的加權平均到期期限
• 收益率變動一個百分點(上升或下降同)，債券價格變動的情形，
可以價格變動的金額或是價格變動的百分比表示，此一解釋是最
主要的。
13
存續期間的特性
• 其他條件不變，coupon rate愈高，duration愈小
• 其他條件不變，yield愈高，duration愈小
• 其他條件不變，maturity愈大，duration愈大
14
Zero-coupon bonds的存續期間
• Zero-coupon bonds的duration = maturity
• 零息債券的利率風險大，但無再投資風險
15
永續債券的存續期間
• Consols的maturity無限，但其duration = (1+y)/y ，說明如下
P
C
y
dP
D

P
dy
(1  y )
dP
1 C
y 1 y
D
 (1  y )   2  (1  y )  
dy
P y
C
y
16
投資組合的存續期間
•
D p  w1D1  w2 D2    wk Dk
• 上述投資組合存續期間是衡量yield curve parallel shift的影響
• 若收益率變動為nonparallel時 ，則需以key rate duration 衡量之
17
Convexity Adjustment
• Duration model 是linear，但yield-price relationship是
nonlinear(positive convex)，所以會產生誤差
• Duration model所產生的誤差可以convexity彌補，量化之convexity
稱為convexity measure
• 純債券的convexity measure為一正數，但是如callable bonds or
mortgage-backed securities (MBSs)等之convexity measure 則不一定
為正數
18
Convexity Adjustment (Continued)
d 2 P n t (t  1)CFt

2
t 2
dy
t 1 (1  y )
1  n t (t  1)CFt 



(1  y ) 2  t 1 (1  y )t 
• Dollar convexity measure:
d 2P
 convexity measure  P0
2
dy
• Convexity measure:
d 2P 1
convexity measure  2 
P
dy
19
Convexity Adjustment (Continued)
By Taylor expansion, the change in the price of a bond is approximated
as
dP
1 d 2P
2
dP 
dy 
(
dy
)

2
dy
2 dy
 [ D*  P ]( dy ) 
(mod ified duration 
1
convexity measure  P (dy ) 2  
2
dP 1
dP
 , or dollar duration 
)
dy P
dy
20
Duration vs. Convexity
• 當收益率變動幅度大時，duration approach所衡量的價格波動度誤
差擴大，應加上convexity以調整之
• 由於純債券的convexity是positive，所以同時考量duration與
convexity的債券價格波動度會較只考量duration者為大
• 因此就純債券而言，convexity具有價值，亦即相同duration但不同
convexity的二種債券，convexity較高者應有較高的債券價格
21
Measuring Yield Curve Risk
• Yield curve risk: the exposure of a portfolio or position to a change in
the term structure.
• Yield curve risk can be measured by changing the spot rate for a
particular key maturity and determining the sensitivity of a security or
portfolio to this change holding the spot rate for the other key
maturities constant.
• The sensitivity of the change in value to a particular change in spot rate
is called rate duration.
• There is a rate duration for every point on the spot rate curve.
• The most popular version of this approach is that developed by
Thomas Ho in 1992.
22
Key Rate Durations
• Thomas Ho, 1992. “Key Rate Durations: Measures of Interest Rate
Risks,” The Journal of Fixed Income.
• The spot curve rarely moves in a parallel fashion , which makes
effective duration not precise enough in many bond portfolio
strategies, such as hedging or immunization.
• Key rate durations define the risk of the changing shape of the spot
curve.
• Key rate durations are a vector representing the price sensitivity of
a security to each key rate change.
• Key rate durations are not a single measure as is effective duration.
• The sum of the key rate durations is identical to the effective
duration.
23
Key Rate Durations
Let R(t) be the initial spot yield curve, YTM of the default-free zerocoupon bond maturing at time t.
Let R*(t) be the spot yield curve shifted from the initial yield curve
instantaneously.
Define S(t) to be the difference between the shifted spot curve and the
initial spot curve,
S(t) = R*(t) – R(t)
S(t) is not necessarily a constant function.
24
Key Rate Durations
• We want to find a simple representation of the shift by assuming
that the shift can be approximated by a piecewise linear function.
• Ho’s approach focuses on 11 key maturities of the spot rate curve.
• These key rates include 3 months, 1 year, 2 years, 3 years, 5 years,
7 years, 10 years, 15 years, 20 years, 25 years, and 30 years.
• Denote these key rates by t(i) where i = 1, 2, …, 11 and the shift at
each key rate is S[t(i)]
• Then the yield curve shift along the maturity range S(t) can be
approximated by linear interpolation of the shifts of each key rate.
25
Basic Key Rate Shifts
• In general, the ith key rate shift is defined to be zero shift for
maturities shorter than the (i-1)th key rate and longer than the
(i+1)th key rate.
• Let b[t;i,d(i)] be the ith basic key rate shift of term t, with the level
of shift being d(i). Then
• b[t;1,d(1)] = d(1)
0 < t < 0.25
= d(1) *(1-t) / (1.0-0.25)
0.25 < t < 1.0
=0
1.0 < t < 30
• b[t;2,d(2)] = 0
0 < t < 0.25
= d(2) * (t-0.25) / (1.0-0.25) 0.25 < t < 1.0
= d(2) * (2-t) / (2.0-1.0)
1.0 < t < 2.0
=0
2.0 < t < 30
Other basic key rate shifts are defined analogously.
26
Basic Key Rate Shifts
• The eleven independent basic key rate shifts together can
approximate all the yield curve shifts, S(.).
S[t;d(1), ….,d(11)] = b[t;1,d(1)] + …+b[t;11,d(11)]
• Let the security price initially be P.
• Suppose the price is changed to P* with the shift of the key rate.
• Then the ith key rate duration D(i) is defined as:
P* - P = - PD(i)d(i)
where d(i) is the shift defining the ith basic key rate shift.
27
Basic Key Rate Shifts
• Given that each infinitesimal change in a key rate contributes to the
proportional change in price, and the shift of the yield curve can be
approximated by the sum of all the basic key rate shifts, we have:
P* - P = -P[D(1)d(1)+D(2)d(2)+…+D(11)d(11)]
D(i) be the ith key rate duration.
• That is, the total proportional change in price is the sum of the
effect of each key rate change on the security.
28
Proposition 1:
•
Key rate durations are in fact a linear decomposition of effective duration.
Let the effective duration of a bond be D, and the ith key rate duration of
the bond be D(i). Then
D = D(1) + D(2) + … + D(11)
•
•
By definition, the sum of any set of key rate durations is identical to
effective duration.
After all, a parallel shift of the spot yield curve is identical to all the key
rates shifting by the same amount. Therefore,
d(i) = d
for all i = 1,…..,11
and
P* - P = - P[D(1) + D(2) + … + D(11)]d
29
Numerical Example
• Assume tat change in the spot curve can be defined by a set of key
rates with terms of two years, sixteen years and thirty years.
• Consider two portfolios. At the beginning of the period, Portfolio I
is made up of $50 market value of two-year zero-coupon bonds,
and $50 market value of thirty-year zero-coupon bonds.
• Portfolio II consists of $100 market value of sixteen-year zerocoupon bonds.
30
Numerical Example
Portfolio
2-year issue
16-year issue
30-year issue
I
$50
$0
$50
II
$0
$100
$0
Issue
D(1)
D(2)
D(3)
Effective
duration
2-year
2
0
0
2
16-year
0
16
0
16
30-year
0
0
30
30
31
Numerical Example
•
•
•
D(1): key rate duration for the 2-year part of the curve
D(2): key rate duration for the 16-year part of the curve
D(3): key rate duration for the 30-year part of the curve
•
Portfolio I:
D(1)  (50 / 100 )  2  (0 / 100 )  0  (50 / 100 )  0  1
D(2)  (50 / 100 )  0  (0 / 100 )  16  (50 / 100 )  0  0
D(3)  (50 / 100 )  0  (0 / 100 )  0  (50 / 100 )  30  15
•
Effective duration = (0 / 100)  2  (100 / 100)  16  (0 / 100)  30  16
•
Portfolio II:
D(1)  (0 / 100 )  2  (100 / 100 )  0  (0 / 100 )  0  0
D(2)  (0 / 100 )  0  (100 / 100 )  16  (0 / 100 )  0  16
D(3)  (0 / 100 )  0  (100 / 100 )  0  (0 / 100 )  30  0
•
Effective duration = (50 / 100 )  2  (0 / 100 )  16  (50 / 100 )  30  16
32
Scenario 1: All spot rates shift down 10 basis points.
•
Portfolio I:
(1)  (0.1%)  (0)  (0.1%)  (15)  (0.1%)  1.6%
•
Portfolio II:
(0)  (0.1%)  (16)  (0.1%)  (0)  (0.1%)  1.6%
33
Scenario 2:The 2-Year Rate Shifts Up 10 bps and 30Year Shifts Down 10 bps
•
Portfolio I:
(1)  (0.1%)  (15)  (0.1%)  1.4%
•
Portfolio II:
(0)  (0.1%)  (0)  (0.1%)  0%
34
Scenario 3:The 2-Year Rate Shifts Down 10 bps and
30-Year Shifts UP 10 bps
•
Portfolio I:
(1)  (0.1%)  (15)  (0.1%)  1.4%
•
Portfolio II:
(0)  (0.1%)  (0)  (0.1%)  0%
35
Numerical Example
• A one-factor model, using only the effective duration D, will have
the same factor attribute for both portfolios, which implies that both
portfolios have the same price sensitivity.
• This numerical example shows how the key rate durations can be
used to show that the two portfolios are different in interest rate risk
exposure, although they each have the same effective duration.
36
Key Rate Durations in a Portfolio Context
• The concept of key rate durations for a single security can be
applied in a portfolio context.
• Suppose the bond portfolio consists of m bond positions.
• Let the value of the jth bond position be V(j).
• The sum of all the bond position values must equal the portfolio
value V.
• Some of these positions can have negative values because the
portfolio may have short positions in certain bonds.
• The proportion of the bon position to the portfolio value is called
the weight w(j).
w(j) = V(j) / V,
for j = 1, 2, …, m
37
Proposition 2:
• The Key rate durations of a portfolio are the weighted sum of the
key rate durations of each bond position, where the weights are
w(j).
D(i) = w(1)D(1,i) + w(2)D(2,i) + … + w(m)D(m,i)
for each i = 1….11 where
D(i) is the portfolio ith key rate duration, and
D(j,i) is the jth bond position ith key rate duration
38
Key Rate Durations
• A ladder portfolio is one with approximately equal dollar amounts in each
maturity sector.
• A barbell portfolio has considerably greater weights given to the shorter and
longer maturity bonds than to the intermediate maturity bonds.
• A bullet portfolio has greater weights concentrated in the intermediate
maturity relative to the short and longer maturities.
• Although these portfolios have the same effective duration, their key rate
durations are different.
39