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 (2) 間接利率風險 因資產與負債到期日不平衡所產生的利率風險 (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 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) 市場價值來表示資產負債科目

F+C (1+R) 100+10 1.1 P1 = = =100 C1 (1+R) F+C2 (1+R) P2 = + =100 利率至11% P1 = = 99.10 Δ P1 = -0.90% 100+10 1.11 P2 = + = 98.29 Δ P2 = -1.71% 10 (1.11) 2 Δ P1 Δ R Δ P2 Δ Pn < < ‧‧‧<

FI’s fixed-income assets and liabilities 資產或負債與利率間之關係 FI’s fixed-income assets and liabilities 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

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 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 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

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 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 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 rate-sensitive assets RSA 利率敏感性資產 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

累積資金缺口 CGAP 兩點涵義 CGAP=RSA - RSL 累積資金缺口佔銀行總資產額百分比 =(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 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

CGAP與R與NII的影響

Rate changes on RSAs generally differ from those on RSLs 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

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 -50 3 4 5 6 解決方法： 1.縮小分隔時點 2.

Runoff問題 (the problem of Runoffs) Assets Liability Item 原始 $Amount Runoff in less than one year $Amount Runoff in more than one year 1.Short-term consumer loans $50 　0 1.Equity $20 2.Long-term consumer loans 25 5 20 2.Demand deposits 40 $30 10 3.3-month T-bills 30 3.passbook savings 15 4.6-month T-bills 35 0　 4.3-month CDs 5.3-year notes 70 60 5.3-month BA 6.10-year mortgages 2 18 6.6-month CP 7.30-year floating-rate mortgages 7.1-year time deposits 8.2-year time deposits $270 $172 $98 $205 $65

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。

例如MA=ML 皆是一年，但Assets與Liability的現金流量不同。 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 50+7.5 53.75 L CD 100 -115

Maturity gap=0 仍有利率風險 利率 15％ 12％ Cash Flow at ½ year Principal 50 50 利率 15％ 12％ 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（duration）的意義 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（duration）的意義 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*

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。

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。 8％ 7％ 9％ Interest rate 8％ 7％ 9％ Coupon,5*80 400 Reinvestment income 69 60 78 Proceeds from sale of bond at end of the fifth year 1000 1009 991 Total cash flow 1469

存續期間的公式 Macaulay duration 　　　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：

修 正 後 存 續 期 間 （Modified Duration） 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） 其中：D dol  價格存續期間

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 .假設債券不具凸性

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 非固定 1 80 0.9259 74.07 2 0.8573 68.59 137.18 0.8448 67.58 135.16 3 0.7938 63.51 190.53 0.7637 61.10 183.3 4 0.7350 58.80 235.20 0.6880 55.04 220.16 5 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

存續期間假設產生的問題 2.凸性（convexity）存在

不同商品的Duration 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

Demand deposits and passbook savings 非RSL的理由 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) 是RSL的理由 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) 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 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） 到期期間 票 面 利 率 6% 8% 票　面　利　率 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

Duration的影響因子 由上表可以看到 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 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

Difficulties in Applying the Duration Model to Real-World FI balance sheet b. Convexity and duration (回憶barbell strategy)

Difficulties in Applying the Duration Model to Real-World FI balance sheet c. All fixed-income securities are convex

convexity increase with bond maturity CX=28 CX=130 CX=312

Convexity varies with coupon B N=6 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 CX=28 CX=25.72

Assets are more convex than liabilities

Hedging Interest Rate Risk Figure 24-2 The Effects of Hedging on Risk and Expected Return (1)Microhedging Using a futures (forward) contract to hedge a specific asset or liability (2)Macrohedge Hedging the entire duration gap of an FI

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

Macrohedging with futures Fully hedge

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

Hedging with options (3) To hedge net worth exposure (2) (1)FI’s net worth exposure to an interest rate shock (2) (3) To hedge net worth exposure Example 25-1

Example 25-1 Cost= Np * Put premium per contract 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 Money Center 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)

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。