Interest Rate Risk and ALM

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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. 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

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Interest Rate Risk and ALM

• 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

(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

• 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

(1)直接利率風險

(2) 間接利率風險

(一)定義銀行風險管理的目標

• 狹義目標 :

• 廣義目標 :

### The maturity model (到期模型)

F+C

(1+R)

100+10

1.1

100+10

1.11

P1 = = 99.10 Δ P1 = -0.90%

Δ P1

Δ R

Δ P2

Δ R

Δ Pn

Δ R

< < ‧‧‧<

10

1.11

100+10

(1.11)

P2 = + = 98.29 Δ P2 = -1.71%

2

C1

(1+R)

F+C2

(1+R)

P2 = + =100

2

P1 = = =100

### 資產或負債與利率間之關係

• 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)
• 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
• 利率下降時，較長期資產或負債科目市值下降的比率遞減

FI’s fixed-income assets and liabilities

### 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

Liabilities

L=\$90 (ML=1 year)

E= 10

\$100

Assets

A=\$97.56 (MA=3 years)

\$97.56

Liabilities

L=\$89.19 (ML=1 year)

E= 8.37

\$97.56

Assets

A=\$100 (MA=3 years)

\$100

ΔE(change in FI net worth)= ΔA - ΔL

- \$1.63 = (-\$2.44) - (-\$0.81)

Maturity matching does not always protect an FI against interest rate risk

MA-ML=0

• The degree of leverage in the FI’s balance sheet
• The duration or average life of asset or liability cash flows rather than the maturity of assets and liabilities

### 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)

Ex.一天期的資金缺口GAP:負一千萬美元

ΔNIIi=(GAPi)×(ΔRi)

=(RSAi-RSLi)×(ΔRi)

• 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

rate-sensitive liabilities RSL

### 累積資金缺口 CGAP

CGAP=RSA - RSL

=(50+30+35+40) - (40+20+60+20)=15百萬美元

CGAP/A=15百萬美元/270百萬美元=5.6%

The direction of the interest rate exposure

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

ΔNII =(RSA ×Δ RRSA)-(RSL ×ΔRRSL)

=(\$155million ×1.2%)-(\$155million ×1.0%)

=\$310000

Rate changes on RSAs generally differ from those on RSLs

### 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)

+50

0

-50

3 4 5 6

overaggregation

1.縮小分隔時點

2.

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

### Maturity model 的缺點

A t=0 t=1/2 1 year

Loan -100 50+7.5 53.75

L

CD 100 -115

Maturity gap=0 仍有利率風險

• Cash Flow at ½ year

Principal 50 50

Interest 7.5 7.5

• Cash Flow at 1 year

Principal 50 50

Interest 3.75 3.75

Reinvestment income 4.3125 3.45

• Total cash flow 115.5625 114.7

• 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。
• 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。
• 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。
• 加入收帳機率，Pi*C Ft=CF t*

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。

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。

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：

• 其中：D mod 修正後存續期間

D  Macaulay存續期間

• 其中：D dol 價格存續期間

• 假設殖利率曲線為水平線，或是利率不同變動比率相同
• .假設債券不具凸性

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)

2.凸性（convexity）存在

• zero-coupon bond D=M
• consol bond (perpetuities) D=1+
• FRN (Floating-Rate Note) D=付息期間
• Demand deposits and passbook savings
• Mortgages and mortgage-backed securities

1.按規定不須付息

2.雖然NOW有付息，但是相對穩定

3.數量眾多，且相對的穩定，類似FI的核心存款(長期資金來源)

1.有間接的費用，但是銀行並沒有其他來源填補。

2.利率上升時，存戶會提款運用於其他工具。

(MMMF)

Demand deposits and passbook savings

2.D=0

3.算出利率對上述兩項目的影響

Prepayment and Liquidity Risk

Liquidity risk

Prepayment risk

Liquidity risk

Prepayment risk

CGAP>0

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

（YTM = 8%，半年付息一次）

Duration的影響因子

• Duration and Maturity
• Duration and Coupon Interest
• Duration and yield(直接對Duration 微分可得)
Duration and Immunization
• (1)Duration Gap

a. the leverage adjusted duration gap=

b. the size of the FI：A

c. the size of the interest rate shock =

(2) Immunization
• the leverage adjusted duration gap=

=0

a. Reduce DA

b. Reduce DA and increase DL

c. Change k and DL

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

(1)Duration Matching Can Be Costly restructuring the B/S is time-consuming and costly take hedging positions in the markets for derivative securities

(2)Immunization is a dynamic problem trade-off between being perfectly immunized the transaction costs of maintaining an immunized B/S

(3)Large Interest Rate Changes and Convexity

characteristics of convexity

a. Convexity is desirable

b. Convexity and duration

(回憶barbell strategy)

c. All fixed-income securities are convex

Hedging Interest Rate Risk

(1)Microhedging

Using a futures (forward) contract to hedge a specific asset or liability

(2)Macrohedge

Hedging the entire duration gap of an FI

Figure 24-2

The Effects of Hedging on Risk and Expected Return

Macrohedging with futures
• FI’s net worth exposure to interest rate shocks
• The sensitivity of the price of a futures contract depends on the duration of the deliverable bond underlying the contract
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%

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：

DF=9.5年, PF=\$97000

On Balance Sheet

Off Balance Sheet

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)

Cost= Np * Put premium per contract

Cost= 537* \$2500 =\$1342500

Interest Rate Swaps

• The Savings Bank
• Assets : \$100m

Fixed-rate mortgages

• Liabilities :\$100m

Short-term CDs(one year)

• Money Center Bank
• Assets : \$100m

C&I loans(rate indexed to LIBOR)

• Liabilities :\$100m

Medium-term notes(coupons fixed)

Securitization

• Asset-Liability Management：

• 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。