Saunders & Cornett, Financial Institutions Management, 4th ed. 1 “If Max gets to Heaven, he...

Saunders & Cornett, Financial Institutions Management, 4th ed.

1

“If Max gets to Heaven, he won’t last long. He will be chucked out for trying to pull off a

merger between Heaven and Hell…after having secured a controlling interest in key

subsidiary companies in both places, of course.”

H.G. Wells


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The Impact of Unanticipated Changes in Interest Rates:

• On Profitability– Net Interest Income (NII) = Interest Income minus

Interest Expense– Interest rate risk of NII is measured by the repricing

model. (chap. 8)

• On Market Value of Equity– Market Value of Equity = Market Value of Assets

minus Market Value of Debt– Interest rate risk of equity MV is measured by the

duration model. (chap. 9)


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The Repricing Model

• Rate Sensitive Assets (Liabilities) RSA/RSL: are repriced within a period of time called a maturity bucket.– Repricing occurs whenever either maturity or a roll

date is reached.– The roll date is the reset date specified in floating rate

instruments that determines the new market benchmark rate used to set cash flows (eg., coupon payments).

• The Federal Reserve set the following 6 maturity buckets: 1 day; 1day-3 months; 3-6 months; 6-12 months; 1-5 yrs; > 5 yrs.


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The Repricing Model

• Repricing Gap (GAP) = RSA – RSLR = interest rate shockNII = GAP x R for each maturity

bucket i• Cumulative Gap (CGAP) = i GAPi

NII = CGAP x Ri where Ri is the average interest rate change on RSA & RSL

• Gap Ratio = CGAP/Assets


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Example of Repricing Model

Assets $m Liabilities & Net Worth $m Short term (1 yr fixed rate)

consumer loans 50 Equity Capital (fixed) 20

Long term (2 yrs fixed rate) consumer loans

25 Demand Deposits 40

3 mo. T-bills 30 Passbook savings 30 6 mo. T-bills 35 3 mo. CDs 40 3 yr. T-bonds 70 3 mo. bankers acceptances 20

10 yr, fixed rate mortgages 20 6 mo. commercial paper 60 30 yr. floating rate mortgages (9

mo. adjustment period) 40 1 yr. time deposits

2 yr. time deposits 20 40

TOTAL 270 TOTAL 270


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Repricing Ex. (cont.)• 1 day GAP = 0 – 0 = 0 (DD & passbook excluded)• (1day-3mo] GAP = 30 – (40+20) = -$30m• (3mo-6mo] GAP = 35 – 60 = -$25m• (6mo-12mo] GAP = (50+40) - 20 = $70m • (1yr-5yr] GAP = (25+70) – 40 = $55m• >5 yr GAP = 20 – (20+40+30) = -$70m• 1 yr CGAP = 0-30-25+70 = $15m• 1 yr Gap Ratio = 15/270 = 5.6%• 5 yr CGAP = 0-30-25+70+55 = $70m• 5 yr Gap Ratio = 70/270 = 25.9%


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Assume an across the board 1% increase in interest rates

• 1 day NII = 0(.01) = 0• (1day-3mo] NII = -$30m(.01) = -$300,000• (3mo-6mo] NII = -$25m(.01) = -$250,000• (6mo-12mo] NII = $70m(.01) = $700,000 • (1yr-5yr] NII = $55m(.01) = $550,000• >5 yr GAP = -$70m(.01) = -$700,000• 1 yr CNII = $15m(.01) = $150,000• 5yr CNII = $70m(.01) = $700,000


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Unequal Shifts in Interest Rates

NII = (RSA x RRSA) – (RSL x RRSL)• Even if GAP=0 (RSA=RSL) unequal shifts

in interest rates can cause NII.• Must compare relative size of RSA and

RSL (GAPs) to relative size of interest rate shocks (RRSA- RRSL = spread).

• The spread can be positive or negative = Basis Risk.


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Strengths of Repricing Model

• Simplicity

• Low data input requirements

• Used by smaller banks to get an estimate of cash flow effects of interest rate shocks.


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Weaknesses of the Repricing Model

• Ignores market value effects.• Overaggregation within maturity buckets• Runoffs – even fixed rate instruments pay off principal

and interest cash flows which must be reinvested at market rates. Must estimate cash flows received or paid out during the maturity bucket period. But assumes that runoffs are independent of the level of interest rates. Not true for mortgage prepayments.

• Ignores cash flows from off-balance sheet items. Usually are marked to market.


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Measuring the Impact of Unanticipated Interest Rate Shocks on Market Values

E = A - L• What determines price sensitivity to changes in interest

rates?• The longer the time to maturity, the greater the price impact

of any given interest rate shock.• This can be viewed in the positively sloped yield curve. See

Appendix 8A.• But, yield curves must be drawn using pure discount yields. • The correct statement is: The longer the DURATION, the

greater the price impact of any given interest rate shock.


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What is Duration?• Duration is the weighted-average time to maturity on an

investment.• Duration is the investment’s interest elasticity - measures the

change in price for any given change in interest rates.• Duration (D) equals time to maturity (M) for pure discount

instruments only. • Duration of Floating Rate Instrument = time to first roll date.• For all other instruments, D < M• Duration decreases as:

– Coupon payments increase– Time to maturity decreases– Yields increase.


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The Spreadsheet Method of Calculating Duration

Ex. 1:5 yr. 10% p.a. coupon par values Cs y PV(Cs) tPV(Cs)

1 100 0.1 90.90909 90.909092 100 0.1 82.64463 165.28933 100 0.1 75.13148 225.39444 100 0.1 68.30135 273.20545 1100 0.1 683.0135 3415.067

Price= 1000 4169.865Duration 4.169865


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Ex. 2: Interest Rates Decrease to 9% p.a.

s Cs y PV(Cs) tPV(Cs)1 100 0.09 91.74312 91.743122 100 0.09 84.168 168.3363 100 0.09 77.21835 231.6554 100 0.09 70.84252 283.37015 1100 0.09 714.9245 3574.623

Price= 1038.897 4349.727Duration 4.186872


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Ex. 3: Interest Rates Increase to 11% p.a.

s Cs y PV(Cs) tPV(Cs)1 100 0.11 90.09009 90.090092 100 0.11 81.16224 162.32453 100 0.11 73.11914 219.35744 100 0.11 65.8731 263.49245 1100 0.11 652.7965 3263.982

Price= 963.041 3999.247Duration 4.152727


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The Duration Model

• Modified Duration = MD = D/(1+R)• Price sensitivity (interest elasticity): P -D(P)R/(1+R)• Consider a 1% increase in interest rates:• Ex. 1: P -(4.17)(1000)(.01)/1.10 = -$37.91

- New Price = 1000 - 37.91 = $962.09 Exact $963.04• Ex. 2: P -(4.19)(1038.897)(.01)/1.09 = -$39.94

– New Price = 1038.897 – 39.94 = $998.96 Exact $1000• Ex. 3: P -(4.15)(963.041)(.01)/1.11 = -$36.01

– New Price = 963.041 – 36.01 = $927.03 Exact $927.90


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The Duration Model: Using Duration to Measure the FI’s

Interest Rate Risk ExposureE = A - LA = -(DAA)RA/(1+RA)

L = -(DLL)RL/(1+RL)

• Assume that RA/(1+RA) = RL/(1+RL)

E/A -DG(R)/(1+R) where

• DG = DA – (L/A)DL

• DA = i=A wiDi DL = j=L wjDj


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Consider a 2% increase in all interest rates (ie, R/(1+R) = .02)

• FI with DG = +5 yrs. E/A -10%• FI with DG = +2 yrs. E/A -4%• FI with DG = +0.5 yrs E/A -1%• FI with DG = 0 E/A 0% Immunization• FI with DG = -0.5 yrs E/A +1%• FI with DG = - 2 yrs E/A +4%• FI with DG = - 5 yrs E/A +10%


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Convexity• Second order approximation• Measures curvature in the price/yield relationship.• More precise than duration’s linear approximation.• Duration is a pessimistic approximator

– Overstates price declines and understates price increases.– Convexity adjustment is always positive.– Long term bonds have more convexity than short term

bonds. Zero coupon less convex than coupon bonds of same duration.

P -D(P)(R)/(1+R) + .5(P)(CX)(R)2


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The Spreadsheet Method to Calculate Convexity Ex. 1

s Cs y PV(Cs) tPV(Cs) t(t+1)PV(Cs)1 100 0.1 90.90909 90.90909 181.81822 100 0.1 82.64463 165.2893 495.86783 100 0.1 75.13148 225.3944 901.57784 100 0.1 68.30135 273.2054 1366.0275 1100 0.1 683.0135 3415.067 20490.4

Price= 1000 4169.865 23435.69Duration 4.169865 19.36834 CX


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Ex. 2


Price= 1038.897 4349.727 24479.7Duration 4.186872 19.83265 CX


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Ex. 3




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How Do We Forecast Interest Rate Shocks?

• Expectation Hypothesis– Upward (downward) sloping yield curve

forecasts increasing (decreasing) interest rates.– (1+0R2)2 = (1+0R1)(1+1R1)– Spot rates: 0R2= 5.5% p.a. 0R1=4%

Implied forward rate: 1R1 = 7.02% p.a. Forecasts 3.02% increase in 1 yr rates in 1 yr.

• Liquidity Premium Hypothesis • Market Segmentation Hypothesis


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Appendix 8A Calculating the Forward Zero Yield Curve for Valuation

• Three steps:– Decompose current spot yield curve on risk-

free (US Treasury) coupon bearing instruments into zero coupon spot risk-free yield curve.

– Calculate one year forward risk-free yield curve.

– Add on fixed credit spreads for each maturity and for each credit rating.


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Step 1: Calculation of the Spot Zero Coupon Risk-free Yield Curve Using a No Arbitrage Method

• Figure 6.6 shows spot yield curve for coupon bearing US Treasury securities.

• Assuming par value coupon securities:

• Figure 6.7 shows the zero coupon spot yield curve.

Six Month Zero: 100 = C+F = C+F = 100+(5.322/2) 1+0r1 1+0z1 1 + (.05322/2) Therefore, the six month zero riskfree rate is: 0z1 = 5.322 percent per annum One Year Zero: 100 = C + C+F = C + C+F 1+0r2 (1+0r2)

2 1+0z1 (1+0z2)2

100 = (5.511/2) + 100+(5.511/2) = (5.511/2) + 100+(5.511/2) 1+(.05511/2) (1+.05511/2)2 1+(.05322/2) (1+.055136/2)2


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6.47%

Yield toMaturity

p.a.

6Mos.

1Yr.

2Yr.

3Yr.

Maturity

CYC RF

2.5Yr.

1.5Yr.

6.25%

6.09%

5.98%

5.511%

5.322%

Figure 6.6

Maturity

Yield toMaturity p.a.

Figure 6.7

6 Mos 1 Yr 1.5 Yrs 2 Yrs 2.5 Yrs 3 Yrs

CYC RF

ZYCRF

5.511%

5.98%6.09%

6.25%

0.0647%

5.322%

5.5136%

5.9353%

6.1079%

6.2755%

7.6006%

ZYCRF

Maturity

Yield toMaturity p.a.

Figure 6.8

6 Mos 1 Yr 1.5 Yrs 2 Yrs 2.5 Yrs 3 Yrs

5.322% 5.5136%

5.9353%6.1079%6.7813%

6.6264%

6.9475%

14.3551%

7.2813%

7.1264%

7.4475%

14.8551%

6.2755%

7.6006%

1 Year Forward

1 Year ForwardFYC RF

FYC R


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Step 2: Calculating the Forward Yields• Use the expectations hypothesis to calculate

6 month maturity forward yields:(1 + 0z2)

2 = (1 + 0z1)(1 + 1z1) (1+(.055136/2)2 = (1+.05322/2)(1+1z1) Therefore, the rate for six months forward delivery of 6-month maturity US Treasury securities is expected to be: 1z1 = 5.7054 percent p.a. (1 + 0z3)

3 = (1 + 0z2)2(1 + 2z1)

(1+(.059961/2)3 = (1+.055136/2)2(1+2z1) Therefore, the rate for one year forward delivery of 6-month maturity US Treasury securities is expected to be: 2z1 = 6.9645 percent p.a.


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Use the 6 month maturity forward yields to calculate the 1 year forward risk-free yield curve

Figure 6.8

(1 + 2z2)2 = (1 + 2z1)(1 + 3z1)

Therefore, the rate for 1 year maturity US Treasury securities to be delivered in 1 year is: 2z2 = 6.703 percent p.a. (1 + 2z3)

3 = (1 + 2z1)(1 + 3z1)(1 + 4z1) Therefore, the rate for 18-month maturity US Treasury securities to be delivered in 1 year is: 2z3 = 6.7148 percent p.a. (1 + 2z4)

4 = (1 + 2z1)(1 + 3z1)(1 + 4z1)(1 + 5z1) Therefore, the rate for 2 year maturity US Treasury securities to be delivered in 1 year is: 2z4 = 6.7135 percent p.a.


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Step 3: Add on Credit Spreads to Obtain the Risky 1 Year Forward Zero Yield Curve

• Add on credit spreads (eg., from Bridge Information Systems) to obtain FYCR in Figure 6.8.

Table 6.8 - Credit Spreads For Aaa Bonds Maturity (in years, compounded annually) Credit Spread, si

2 0.007071 3 0.008660 5 0.011180

10 0.015811 15 0.019365 20 0.022361


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Calculating Duration if the Yield Curve is not Flat

Ex. 1 with upward sloping yield curve




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The Barbell Strategy

• Convexity of Zero Coupon Securities: CX = T(T+1)/(1+R)2

• Strategy 1: Invest in 15 yr zero coupon with 8% pa yield. D=15, CX = 15(16)/1.082=206

• Strategy 2: Invest 50% in overnite FF D=0, CX =0 and 50% in 30 yr zero coupon with 8% yield D=30, CX = 30(31)/1.082 = 797 Portfolio CX = .5(0) + .5(797) = 398.5 > 206 Invest in Strategy 2. But the cost of Strategy 2>Stategy 1 if CX priced.

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