Interest Rate Risk and ALM
第二組組員
財研一 張涵媁
財研一 陳彥旭
財研一 梅原一哲
小叮嚀:同學列印
時,請記得選取“純
粹黑白”功能,即可
出現“白底黑字”,
避免背景過暗的情
況
影響利率波動的因素
(1)財政政策
(2)貨幣政策
(3)通貨膨脹
(4)企業需求和家庭需求
利率風險的概念
公司價值對利率隨機變動之敏感度
利率風險的來源
1.資產與負債到期日的不平衡
2.利率的不確定性
3.利率變動造成金融資產及負債未來現金
流量的不確定
銀行利率風險
(1) 直接利率風險
因資產與負債到期日不平衡所產生的利率風險
(2) 間接利率風險
因利率變化而引起存款人提前解約、貸款人提前還款的
風險。
銀行利率風險管理步驟
銀行利率風險管理步驟
(一)定義銀行風險管理的目標
狹義目標 :
利息淨邊際收益率(利差)變異數最小的前提下，追求最大
的利息淨邊際收益率
廣義目標 :
淨值報酬率變異數最小的前提下，追求最大淨值報酬率
The maturity model (到期模型)
到期模型
市場價值來表示資產負債科目
F+C 100+10
P1 = (1+R) = 1.1 =100
C1
F+C2
P2 =
+
2 =100
(1+R) (1+R)
利率至11%
100+10
P1 =
= 99.10
Δ P1 = -0.90%
1.11
10
100+10
P2 = 1.11 + (1.11) 2 = 98.29 Δ P2 = -1.71%
Δ P1
Δ P2
<
<
‧‧‧<
ΔR
ΔR
Δ Pn
ΔR
資產或負債與利率間之關係
FI’s fixed-income assets and liabilities
1) 市場利率提高(降低)通常導致金融機構資產
及負債市值的減少(增加)
2) 具有固定收益(成本)的資產(負債)到期日愈
長，則利率上升(下降)所導致的資產或負債
市值之減少(增加)量愈大
3) 利率下降時，較長期資產或負債科目市值下
降的比率遞減
Maturity Gap
MA :the weighted-average maturity of an FI’s assets
 ML :the weighted-average maturity of an FI’s liabilities
Mi=Wi1Mi1+Wi2Mi2+…+WinMin
Mi = The weighted-average maturity of an FI’s assets(liabilities)，
i=A or L
 Wij =The importance of each asset(liability) in the asset(liability)
portfolio as measured by the market value of that asset(liability)
position relative to the market value of all the asset(liability)
Mij=The maturity of the jth asset (or liability) j=1,2…n
Assets
A=$100 (MA=3 years)
$100
Liabilities
L=$90 (ML=1 year)
E= 10
$100
利率上升1%
Assets
A=$97.56 (MA=3 years)
$97.56
Liabilities
L=$89.19 (ML=1 year)
E= 8.37
$97.56
ΔE(change in FI net worth)= ΔA - ΔL
- $1.63 = (-$2.44) - (-$0.81)
如果要免除或規避利率風險的暴露
MA-ML=0
Maturity matching does not always
protect an FI against interest rate risk
1) The degree of leverage in the FI’s balance sheet
2) The duration or average life of asset or liability
cash flows rather than the maturity of assets and
liabilities
The maturity Model的缺點
用到期模型完全測量金融機構的利
率 風 險是十分困難的。因 為 到 期 模 型 忽
略 了資 產 及 負 債 現 金 流 量。
The repricing (or funding gap) model
The repricing model is essentially a book
value accounting cash flow analysis of the
repricing gap
the repricing gap is the difference between
assets whose interest rates will be repriced or
changed over some future period and liabilities
whose interest rates will be repriced or changed
over some future period
資金缺口的例子
TA B L E 1.R epricing G ap (in m illio ns o f do llars)
1
2
3
4
A ssets
L iabilities
G aps
C um ulative
G ap
1.O ne day
$ 20
$30
$－ 10
$－ 10
－ 10
－ 20
－ 15
－ 35
2.M o re than o ne day-three m o nths
30
40
3.M o re than three m o nths-six m o nths
70
85
4.M o re than 6m o nths-12m o nths
90
70
+ 20
5.M o re than o ne year-five years
40
30
+ 10
6.O ver five years
10
5
+5
$260
$260
－ 15
－5
0
重新定價模型的優點
銀行試圖了解各個不同到期日的資金組
群因利率變動所造成的淨利風險暴露時
•簡易可行
•具有資訊價值
資金淨利的變動量
ΔNIIi=Change in net interest income in the ith bucket
GAPi=Dollar size of the gap between the book value of
rate-sensitive assets and rate-sensitive liabilities in maturity
bucket i
ΔRi=The change in the level of interest rates impacting
assets and liabilities in the ith bucket
ΔNIIi=(GAPi)×(ΔRi)
=(RSAi-RSLi)×(ΔRi)
ΔNIIi=(GAPi)×(ΔRi)
=(RSAi-RSLi)×(ΔRi)
Ex.一天期的資金缺口GAP:負一千萬美元
利率
資金淨利
一天期的資金缺口GAP:正
利率
資金淨利
•累積資金缺口CGAP
CGAP = (-10)+(-10)+(-15)+20= -15million
利率上升1%
ΔNIIi = (CGAP) × (ΔRi)
= (-15million) × (0.01) = - $150000
RSA and RSL
Rate sensitivity
An asset or liability is repriced at or near current
market interest rates within a maturity bucket
rate-sensitive assets RSA
利率敏感性資產
rate-sensitive liabilities RSL
利率敏感性負債
A s s e ts
TA B L E 2 . S im p le F I B a la n c e S h e e t (in m illio n s o f d o lla rs )
L ia b ilitie s
1 .S h o rt-te rm c o n s u m e r lo a n s
(o n e -ye a r m a tu rity)
$50
1 .E q u ity c a p ita l (fix e d )
$20
2 .L o n g -te rm c o n s u m e r
lo a n s (tw o -ye a r m a tu rity)
25
2 .D e m a n d d e p o s its
40
3 .T h re e -m o n th Tre a s u ry b ills
30
3 .P a s s b o o k s a vin g s
30
4 .S ix -m o n th Tre a s u ry n o te s
35
4 .T h re e -m o n th C D s
40
5 .T h re e -ye a r Tre a s u ry b o n d s
70
20
6 .1 0 -ye a r,fix e d -ra te m o rtg a g e s
20
5 .T h re e -m o n th b a n k e rs
a c c e p ta n c e s
6 .S ix -m o n th c o m m e rc ia l
pa p e r
7 .O n e -ye a r tim e d e p o s its
20
$ 2 7 0 8 .Tw o -ye a r tim e d e p o s its
40
7 .3 0 -ye a r,flo a tin g -ra te m o rtg a g e s 4 0
(ra te a d ju s te d e ve ry n in e m o n th s )
60
$270
累積資金缺口 CGAP
 CGAP=RSA - RSL
=(50+30+35+40) - (40+20+60+20)=15百萬美元
 累積資金缺口佔銀行總資產額百分比
CGAP/A=15百萬美元/270百萬美元=5.6%
 兩點涵義
1) The direction of the interest rate exposure
2) The scale of that exposure as indicated by dividing
the gap by the asset size of the institution
CGAP Effect
 CGAP Effect
ΔNIIi=〈GAPi〉×〈ΔRi〉
利率變動為正向時
讓CGAP為正
利率變動為負向時
讓CGAP為負
CGAP與R與NII的影響
Row CGAP △R
△interest
revenue
△interest
expense
△NII
1
>0
↑
↑
>
↑
↑
2
>0
↓
↓
>
↓
↓
3
<0
↑
↑
<
↑
↑
4
<0
↓
↓
<
↓
↓
Spread Effect
Rate changes on RSAs generally
differ from those on RSLs
CGAP Effect + Spread Effect
ΔNII =(RSA × Δ RRSA)-(RSL × ΔRRSL)
=($155million ×1.2%)-($155million ×1.0%)
=$310000
spread增加
資金淨利增加
spread減少
資金淨利減少
CGAP與R與Spread與NII的影響
Row
1
2
3
4
5
6
7
8
CGAP
>0
>0
>0
>0
<0
<0
<0
<0
△R
↑
↑
↓
↓
↑
↑
↓
↓
△Spread
↑
↓
↑
↓
↑
↓
↑
↓
△NII
↑
↑↓
↑↓
↓
↑↓
↓
↑
↑↓
重新定價模型的缺點
1.忽略市價的改變(ignores market value effects)
2.過度加總問題 (overaggregation)
3.Runoff問題 (the problem of Runoffs)
4.資產負債表外的現金流動
(cash flows from off-balance-sheet activities)
overaggregation
+50
0
3
4
5
6
-50
解決方法：
1.縮小分隔時點
2.
NII  RSA 1  R A 

tA
1 tA
1  K A 
 1  R A   RSL 1  R L 


tL
1 tL
1  K L 
 1  R L 

Runoff問題
(the problem of Runoffs)
Assets
Liability
原始
$Amount
Run
off
in
less
than
one
year
$Amount
Run
off
in
mor
e
than
one
year
$20
0
$20
原始
$Amount
Runo
ff in
less
than
one
year
$Amount
Runoff
in
more
than
one
year
1.Short-term consumer loans
$50
$50
0
2.Long-term consumer loans
25
5
20
2.Demand deposits
40
$30
10
3.3-month T-bills
30
30
0
3.passbook savings
30
15
15
4.6-month T-bills
35
35
0
4.3-month CDs
40
40
0
5.3-year notes
70
10
60
5.3-month BA
20
20
0
6.10-year mortgages
20
2
18
6.6-month CP
60
60
0
7.30-year floating-rate
mortgages
40
40
0
7.1-year time deposits
20
20
0
8.2-year time deposits
40
20
20
Item
Item
1.Equity
Runoff問題
(the problem of Runoffs)
1.原本的CGAP
RSA=50+30+35+40(FRN)=115
RSL=40+20+60+20=140
CGAP=115-140=-25
2.Runoff調整後的CGAP
RSA=50+5+30+35+10+2+40=172
RSL=30+15+40+20+60+20+20=205
CGAP=172-205=-33
其中RSA中mortgages在利率下降時，數值會變大。
Maturity model 的缺點
例如MA=ML 皆是一年，但Assets與Liability的現
金流量不同。
A
t=0
t=1/2
1 year
Loan -100
L
CD
100
50+7.5
53.75
-115
Maturity gap=0 仍有利率風險
利率
15％
• Cash Flow at ½ year
Principal
50
Interest
7.5
• Cash Flow at 1 year
Principal
50
Interest
3.75
Reinvestment income
4.3125
• Total cash flow
115.5625
12％
50
7.5
50
3.75
3.45
114.7
存續期間（duration）的意義
1. 存續期間是債券持有人收到現金流量的加
權平均發生時間，即債券的加權平均到期
期限。
2. 存續期間為利率變動對債券價格之彈性觀
念，故為一債券利率風險的衡量指標。
3. 存續期間是債券現金流量之平衡點，故也
是進行投資組合免疫策略時不可缺少的工
具。
存續期間模型
Duration Model
存續期間（duration）的意義
1. 存續期間是債券持有人收到現金流量的加權平均
發生時間，即債券的加權平均到期期限。

D
n

dP
P
dr

t  CFt
t 1 (1  r)
P
(1  r)
2. 加入收帳機率，Pi*C Ft=CF t*
t
n

 t  Wt
t 1
存續期間（duration）的意義
3. 存續期間為利率變動對債券價格之彈性觀念，故
為一債券利率風險的衡量指標。

D 
dP
P
dr
(1  r )
存續期間（duration）的意義
4. 存續期間是債券現金流量之平衡點，故也是進行投
資組合免疫策略時不可缺少的工具。
Interest rate
8％
7％
9％
Coupon,5*80
400
400
400
69
60
78
Proceeds from sale of bond at end
of the fifth year
1000
1009
991
Total cash flow
1469
1469
1469
Reinvestment income
存續期間的公式 Macaulay duration

n

dP
P
dr
D 

t 1
t  CFt
(1  r)
P
t
n

 t  Wt
t 1
(1  r)
其中：D  存續期間
CFt  債券在第t期的現金流量
n  債券的到期時間
r  債券的殖利率
P  債券目前的價格
Wt  第t期債券現金流量現值占債券價格（各期現金流量現值加總）
之比例，即各期現金流量現值之權重，可表示為：
CFt
Wt 
(1  r)
P
n
t
 Wt
t 1
1
修正後存續期間
（Modified Duration）
dP
D mod
•
D
P


d(1  r)
1 r
其中：D mod  修正後存續期間
D  Macaulay存續期間
價格存續期間
（Dollar Duration）
Ddol 
dP
d(1  r)
 Dmod  P  D 
dP  Ddol  dr
• 其中：D dol  價格存續期間
P
1 r
存續期間的假設
• 假設殖利率曲線為水平線，或是利率不同
變動比率相同
R 1
1  R1

R 2
1 R2
 ....... 
• .假設債券不具凸性
R n
1 Rn
存續期間假設產生的問題
1.殖利率曲線並非水平或同比率變動
(比較真實與假設狀況計算的存續期間差異)
T
CF
DF
固定8％
CF*DF
CF*DF*T
DF
非固定
CF*DF
CF*DF*T
1
80
0.9259
74.07
74.07
0.9259
74.07
74.07
2
80
0.8573
68.59
137.18
0.8448
67.58
135.16
3
80
0.7938
63.51
190.53
0.7637
61.10
183.3
4
80
0.7350
58.80
235.20
0.6880
55.04
220.16
5
80
0.6806
54.45
272.25
0.6153
49.22
246.1
6
1080
0.6302
680.58
4083.48
0.5553
599.75
3598.50
1000
4992.71
906.76
4457.29
D
4992.71
1000
 4.993
D 
*
4457.29
906.76
 4.91562
存續期間假設產生的問題
2.凸性（convexity）存在
dP
P
債
券
價
格
 D 
dr
(1  r)

1
 凸性  dr
2
2
實際債券價格與殖利率關係
存續期間假設債券價格
與殖利率為直線關係
0
殖利率
不同商品的Duration
• zero-coupon bond
D=M
1
• consol bond (perpetuities)
D=1+
• FRN (Floating-Rate Note)
D=付息期間
R
• Demand deposits and passbook savings
• Mortgages and mortgage-backed securities
Demand deposits and passbook savings
非RSL的理由
是RSL的理由
1.按規定不須付息
1.有間接的費用，但是銀
行並沒有其他來源填
2.雖然NOW有付息，但
補。
是相對穩定
3.數量眾多，且相對的穩 2.利率上升時，存戶會提
款運用於其他工具。
定，類似FI的核心存
款(長期資金來源)
(MMMF)
解決方法：1.D=turnover per dollar
2.D=0
3.算出利率對上述兩項目的影響
Prepayment and Liquidity Risk
Prepayment
risk
Liquidity
risk
Prepayment
risk
CGAP>0
Liquidity
risk
CGAP<0
Prepayment risk：指利率下降時，長期貸款提前還款。
Liquidity risk：指利率上升時，活存減少。
Duration的影響因子
存續期間與票面利率、到期期間的關係
（YTM = 8%，半年付息一次）
票 面 利 率
到期期間
6%
8%
10%
1年
0.985
0.98
0.976
5年
4.361
4.218
4.095
10年
7.454
7.067
6.772
20年
10.922
10.292
9.870
永續債券
13.000
13.000
13.000
Duration的影響因子
由上表可以看到
• Duration and Maturity2
D
M
 0，
 D
 M
2
0
• Duration and Coupon Interest
D
C
0
• Duration and yield(直接對Duration 微分可
得) D
R
0
Duration and Immunization
• (1)Duration Gap
Di  Wi1Di1  Wi2Di2  .........  Win Din，i  A,L
E  (D A  D L
L
A
) A
R
1 R
L
a. the leverage adjusted duration gap= (D A  D L )
A
b. the size of the FI：A
R
c. the size of the interest rate shock =
(1  R )
(2) Immunization
 the leverage adjusted duration gap=
L
(D A  D L ) =0
A
a. Reduce DA
b. Reduce DA and increase DL
c. Change k and DL
其它的無法得到避險效果
Barbell Strategy and convexity
Strategy 1：D=15 CX=206
Strategy 2：D1=0 CX=0
D2=30 CX=797
D p=½(0)+½ (30)=15
CX p= ½(0)+½ (797)=398.5
(3) Immunization and Regulatory
Considerations
• Regulatory 可能會限制k，例如限制資本適
足率
此時避險的唯一選擇就是DA= DL
Difficulties in Applying the Duration
Model to Real-World FI balance
sheet
(1)Duration Matching Can Be Costly
restructuring the B/S is time-consuming
and costly take hedging positions in the
markets for derivative securities
解決之道：衍生性金融商品的運用
Difficulties in Applying the Duration
Model to Real-World FI balance
sheet
(2)Immunization is a dynamic problem trade-off
between being perfectly immunized the
transaction costs of maintaining an immunized
B/S
解決方法：訂出一個重新審核免疫策略的期間
Difficulties in Applying the Duration
Model to Real-World FI balance sheet
(3)Large Interest Rate Changes and Convexity
characteristics of convexity
a. Convexity is desirable
債
券
價
格
P3
P4
A
P0
P1
P2
凸性大債券
凸性小債券
0
y2
y0
y1
殖利率
Difficulties in Applying the Duration
Model to Real-World FI balance sheet
b. Convexity and duration
(回憶barbell strategy)
債
券
價
格
P3
P4
A
P0
P1
P2
凸性大債券
凸性小債券
0
y2
y0
y1
殖利率
Difficulties in Applying the Duration
Model to Real-World FI balance
sheet
c. All fixed-income securities are convex
P
P
 D
R
(1  R)





P

P
8
CX  10 


P
P



1
2
CX(R)
2
convexity increase with bond
maturity
A
B
C
N=6
N=18
N=

R=8％
R=8％
R=8％
C=8％
C=8％
C=8％
D=5
D=10.12
D=13.5
CX=28
CX=130
CX=312
Convexity varies with coupon
A
B
N=6
N=6
R=8％
R=8％
C=8％
C=0％
D=5
D=6
CX=28
CX=36
For same duration, Zero-Coupon
Bonds less convex than Coupon
Bonds
A
B
N=6
N=5
R=8％
R=8％
C=8％
C=0％
D=5
D=5
CX=28
CX=25.72
Assets are more convex than
liabilities
價
值
Assets
Liabilities
0
y2
y0
y1
Interest rate
Hedging Interest Rate Risk
Figure 24-2
(1)Microhedging
Using a futures
(forward) contract to
hedge a specific
asset or liability
(2)Macrohedge
Hedging the entire
duration gap of an FI
The Effects of Hedging on Risk and
Expected Return
Macrohedging with futures
 FI’s net worth exposure to interest rate shocks
 E   [ D A  kD L ]  A 
R
1 R
 The sensitivity of the price of a futures contract depends
on the duration of the deliverable bond underlying the
contract
F
F
 DF
R
1 R
F   DF  F 
F  N
F
R
1 R
 PF
F   DF (N
F
 PF )
R
1 R
Macrohedging with futures
Fully hedge
F  E
 D F ( N F  PF )
R
1 R
NF 
  [ D A  kD L ]  A 
( D A  KD L ) A
D F  PF
R
1 R
Example24-1、24-2
Consider the following FI where :
DA=5年, DL=3年,
Assets=$100m, Liabilities=$90m,Equity=$10m
Expected Interest rates 10%11%
 E   [ D A  kD L ]  A 
 E   ( 5  0 . 9  3 )  100 
0 . 01
1 .1
R
1 R
  $ 2 . 091 m
Example24-1、24-2
Suppose the current futures price quote is $97 per $100 of face
value for the benchmark 20-year,8% coupon bond underlying the
nearby futures contract, the minimum contract size is $100000, and
the duration of the deliverable bond is 9.5 year.
That is：
On Balance Sheet
DF=9.5年, PF=$97000
NF 
NF 
( D A  KD L ) A
D F  PF
( 5  0 . 9  3 ) * $ 100 m
9 . 5  $ 97000
N F  249 . 59
 E   ( 5  0 . 9  3 )  100 
0 . 01
  $ 2 . 091 m
1 .1
Off Balance Sheet
 F   D F ( N F  PF ) 
R
1 R
  9 . 5 (  249  $ 97000 )(
0 . 01
)
1 .1
 $ 2 . 086 m
 E   F   $ 2 . 091 m  $ 2 . 086 m   $ 0 . 05 m
Hedging with options
(1)FI’s net worth exposure
to an interest rate shock
 E   [ D A  kD L ]  A 
R
(3) To hedge net worth
exposure
P  E
1 R
N
(2)
P  (N
p 
dp
dB
dB
p
 p)

dB
Example 25-1
  MD  B
dR
 p  [(   )  (  MD )  B   R ]
p
[  D  B ]
 R
  MD  dR
P  N

dR
B
dB
p
[ D A  kD L ]  A
 [  D  B 
R
1 R
]
Example 25-1
DA=5, DL=3, K=0.9 A=$100m
Rates are expected to rise :
10％11%
Suppose δ=0.5, B=$97000,
D=8.82（underlying bond of the put option)
Np 
$ 230000000
[ 0 . 5  8 . 82  $ 97000 ]
 537 . 672 contracts
 P  537  [ 0 . 5  8 . 82  $ 97000 
0 . 01
]  $ 2 . 09 m
1 .1
Cost= Np * Put premium per contract
Cost= 537* $2500 =$1342500
Interest Rate Swaps
 Money Center Bank
The Savings Bank
 Assets : $100m
Assets : $100m
C&I loans(rate indexed to LIBOR)
Fixed-rate mortgages
 Liabilities :$100m
Liabilities :$100m
Medium-term notes(coupons fixed)
Short-term CDs(one year)
Interest Rate Swaps
Securitization
證券化具有強化資金運用效率、提升銀行
自有資本適足率、降低資產負債管理成本
與利率風險，和促成銀行專業和分工等效
益。
傳統資產負債管理模式(ALM)與VaR系統
 Asset-Liability Management：
對資產與負債兩者間的利率風險、外匯風險、流動風險等
作針對性的管理措施。例如：購置資產時需考慮用什麼方
式融資，希望透過適當的管理方式來減低上述多方面的風
險。
 傳統的利率風險衡量方式，最多只考慮到當利率風險因子
變動對投資組合價值的影響，並未考慮到各風險因子本身
的波動程度及因子間的相關性。風險值(VaR)模型其主要
是利用各風險因子過去的變動，來衡量未來可能產生的風
險，不但考慮了傳統衡量方式的要件，並顧及風險因子的
波動性及相關性，因此較傳統方式具有優勢。