interest rate risk 3

8/13/2019 Interest Rate Risk 3

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Saunders & Cornett, FinancialInstitutions Management, 4th ed.

1

If Max gets to Heaven, he wont last long.He will be chucked out for trying to pull off a

merger between Heaven and Hellafter

having secured a controlling interest in keysubsidiary companies in both places, of

course.

H.G. Wells


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2

The Impact of Unanticipated

Changes in Interest Rates: On Profitability

Net Interest Income (NII) = Interest Income minusInterest Expense

Interest rate risk of NII is measured by the repricingmodel. (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 theduration model. (chap. 9)


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4

The Repricing Model

Repricing Gap (GAP) = RSARSL

R = interest rate shock

NII = GAP x R for each maturitybucket i

Cumulative Gap (CGAP) = iGAPi

NII = CGAP x Ri where Riis theaverage interest rate change on RSA & RSL

Gap Ratio = CGAP/Assets


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5

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 (9mo. adjustment period) 40 1 yr. time deposits2 yr. time deposits 2040

TOTAL 270 TOTAL 270


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6

Repricing Ex. (cont.) 1 day GAP = 00 = 0 (DD & passbook excluded)

(1day-3mo] GAP = 30(40+20) = -$30m (3mo-6mo] GAP = 3560 = -$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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7

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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8

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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9

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.

k f h


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10

Weaknesses of the

Repricing Model Ignores market value effects.

Overaggregation within maturity buckets

Runoffseven fixed rate instruments pay offprincipal and interest cash flows which must be

reinvested at market rates. Must estimate cashflows received or paid out during the maturitybucket period. But assumes that runoffs areindependent of the level of interest rates. Not truefor mortgage prepayments.

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


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

an investment.

Duration is the investments interest elasticity -measures the change in price for any given change ininterest 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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13

The Spreadsheet Method of

Calculating Duration

Ex. 1:5 yr. 10% p.a. coupon par value

s Cs y PV(Cs) tPV(Cs)

1 100 0.1 90.90909 90.90909

2 100 0.1 82.64463 165.2893

3 100 0.1 75.13148 225.3944

4 100 0.1 68.30135 273.2054

5 1100 0.1 683.0135 3415.067Price= 1000 4169.865

Duration 4.169865


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14

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


1 100 0.09 91.74312 91.74312

2 100 0.09 84.168 168.336

3 100 0.09 77.21835 231.655

4 100 0.09 70.84252 283.3701

5 1100 0.09 714.9245 3574.623Price= 1038.897 4349.727

Duration 4.186872


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15

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


1 100 0.11 90.09009 90.09009

2 100 0.11 81.16224 162.3245

3 100 0.11 73.11914 219.3574

4 100 0.11 65.8731 263.4924

5 1100 0.11 652.7965 3263.982Price= 963.041 3999.247

Duration 4.152727


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16

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 = 1038.89739.94 = $998.96 Exact $1000

Ex. 3: P -(4.15)(963.041)(.01)/1.11 = -$36.01

New Price = 963.04136.01 = $927.03 Exact $927.90


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17

The Duration Model:

Using Duration to Measure the FIs

Interest Rate Risk Exposure E = A - L

A = -(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=AwiDi DL= j=LwjDj


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18

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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19

Convexity

Second order approximation Measures curvature in the price/yield relationship.

More precise than durations 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 ofsame duration.

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


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20

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 181.8182

2 100 0.1 82.64463 165.2893 495.8678

3 100 0.1 75.13148 225.3944 901.5778

4 100 0.1 68.30135 273.2054 1366.027

5 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


1 100 0.09 91.74312 91.74312 183.4862

2 100 0.09 84.168 168.336 505.008

3 100 0.09 77.21835 231.655 926.6202

4 100 0.09 70.84252 283.3701 1416.85

5 1100 0.09 714.9245 3574.623 21447.74

Price= 1038.897 4349.727 24479.7Duration 4.186872 19.83265 CX


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Ex. 3


1 100 0.11 90.09009 90.09009 180.1802

2 100 0.11 81.16224 162.3245 486.9735

3 100 0.11 73.11914 219.3574 877.4297

4 100 0.11 65.8731 263.4924 1317.462

5 1100 0.11 652.7965 3263.982 19583.89



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23

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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24

Appendix 8A Calculating the Forward Zero

Yield Curve for Valuation

Three steps:

Decompose current spot yield curve on risk-

free (US Treasury) coupon bearing instrumentsinto 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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Saunders & Cornett, FinancialInstitutions Management, 4th ed. 26

6.47%

Yield toMaturity

p.a.

6Mos.

1Yr.

2Yr.

3Yr.

Maturity

CY CRF

2.5Yr.

1.5Yr.

6.25%

6.09%5.98%

5.511%

5.322%

Figure 6.6

Maturity

Yield toMaturity p.a.

gure .

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

CY CRF

ZY CRF

5.511%

5.98% 6.09%

6.25%

0.0647%

5.322%

5.5136%

5.9353%

6.1079%

6.2755%

7.6006%

ZY CRF

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 ForwaFY CRF

FY CR


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Step 2: Calculating the Forward Yields

Use the expectations hypothesis to calculate6 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) = (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 FYCRin Figure 6.8.

Table 6.8 - Credit Spreads For Aaa Bonds

Maturity (in years, compounded annually) Credit Spread, si2 0.007071

3 0.008660

5 0.011180

10 0.01581115 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


1 100 0.1 90.90909 90.90909 181.81822 100 0.102 82.34492 164.6898 494.0695

3 100 0.107 73.71522 221.1456 884.5826

4 100 0.115 64.69944 258.7978 1293.989

5 1100 0.12 624.1695 3120.848 18725.09



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Saunders & Cornett, FinancialI tit ti M t 4th d 31

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% payield. 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.

interest rate risk 3

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