Interest Rate RiskInterest rate changes have significant effects on many financial firms’ net income, asset value, liability value and equity value (net difference between assets and liabilities).

Three Traditional Ways to Measure Interest Rate Risk

1. Repricing Gap - focuses on net interest income changes.

2. Maturity Gap - focuses on equity value changes - ignores cash flow timing.

3. Duration Gap - focuses on equity value including cash flow timing.

•Duration Gap is the most complete and precise measure.

Repricing GapThe repricing gap is the dollar value of the difference between the book values of assets and liabilities with a certain range of maturity (called a bucket).

Steps to Calculate the Repricing Gap and Cumulative Gap

1. List the firm’s assets and liabilities by bucket.

2. Repricing Gap = (assets - liabilities) by bucket.

3. Cumulative Gap = sum of Repricing Gaps.

The effect of interest rate changes on a firm’s net income is

NII = (Gap) R

where NII is the annualized change in net interest income and R is the annual interest rate change.

Repricing Gap ExampleTime Period Assets Liabilities Gap Cm. Gap1 day 20 30 -10 -101 day - 3 months 30 40 -10 -203 - 6 months 70 85 -15 -356 - 12 months 90 70 20 -151 - 5 years 40 30 10 -5Over 5 years 10 5 5 0

Note: Demand deposits are excluded from liabilities because the interest rates paid (zero) do not change.

Question: If interest rates rise by 1 percentage point today, over the next three months, what is the approximate

annualized change in net interest income?

NII = (-20 million) (.01) = -200,000.

Weaknesses of Repricing Gap

1. It ignores market value changes of assets and liabilities.

2. Aggregation of assets and liabilities can be misleading when their distributions within a bucket differ.

3. Runoff problems - some assets or liabilities may mature partially or completely before the stated maturity date - e.g., 30 year mortgages seldom last 30 years.

4. Runoffs may be sensitive to interest rate changes.

5. Ignores the effect of off-balance-sheet items.

See SLM Holdings 10Q (3/2000) Edgar filing for example.

Example: Chap 8 - Prob. 9Consider the following balance sheet.Cash 10 Overnight Repos 1701 mon, 7.05% Tbill 75 7-yr 8.55% Sub. Deb. 1503 mon, 7.25% Tbill 752-yr, 7.5% Tnote 508-yr, 8.96% Tnote 1005-yr, 8.2%, muni 25(reset - 6 months) Equity 15

Total Assets 335 Total Liab. + Equity 335

a. 30 day repricing gap = 75 - 170 = -95

91 day repricing gap = (75 + 75) - 170 = -20

2-yr repricing gap = (75 + 75 + 50 + 25) - 170 = 55

b. 30 day impact of a .5% rise or a .75% drop in all rates.

NII = (-95 million) (.005) = -475,000.

NII = (-95 million) (-.0075) = 712,500

c. Assume one-year runoffs of $10 million for 2-yr Tnote and $20 million for 8-year Tnote.

1-yr repricing gap = (75 + 75 + 10 + 20 + 25) - 170 = 35

d. Redo part b.

NII = (35 million) (.005) = 175,000.

NII = (35 million) (-.0075) = -262,500

Maturity Gap ModelThe Maturity Gap measures the difference between a firm’s weighted average asset maturity (MA) and weighted average liability maturity (ML).

Maturity Gap = (MA - ML)

MA = WA1MA1 + WA2MA2 + WA3MA3 + … + WAnMAn

ML = WL1ML1 + WL2ML2 + WL3ML3 + … + WLnMLn

WAi = (market value of asset i)/(market value of total assets).WLi = (market value of liability j)/(market value of total liab.)MAi is the maturity of asset i.MLi is the maturity of liability j.

Maturity Gap and the Effect of Interest Rates on Equity Value• When (MA - ML) > 0 then an increase (decrease) in interest

rates is expected to decrease (increase) a financial firm’s equity.

• When (MA - ML) < 0 then an increase (decrease) in interest rates is expected to increase (decrease) a financial firm’s equity.

Equity = Assets - Liabilities

or in change form,

Equity = Assets - Liabilities

Equity, Assets and Liabilities are measured in market value.

Example: Ch 8. 17 - Bond Instead of MortgageCounty Bank has the following Balance sheet:Cash $20 Demand Deposits $10015-yr, 10% Loan 160 5-yr, 6% CD Balloon 21030-yr, 8% Bond 300 20-yr, 7% Debenture 120

Equity 50Total Assets 480 Total Liab. And Eq. 480

a. What is the Maturity Gap?

MA = [0(20) + 15(160) + 30(300)]/480 = 23.75

ML = [0(100) + 5(210) + 20(120)]/480 = 8.02

MGAP = 23.75 - 8.02 = 15.73 years

b. What is the gap if all interest rates rise by 1%?

Loan Value = 16[PVA 15,.11] + 160[PV 15,.11] = 148.49

Bond Value = 24 [PVA 30,.09] + 300[PV 30,.09] = 269.08

MA = [0(20) + 15(148.49) + 30(269.08)]/437.6 = 23.53

CD Value = 12.6[PVA 5,.07] + 210[PV 5,.07] = 201.39

Debenture Value = 8.4[PVA 20,.08] + 120[PV 20,.08] = 108.22

ML = [0(100) + 5(201.39) + 20(108.22)]/409.61 = 7.99

MGAP = 23.53 - 7.99 = 15.54

c. Market Value of Equity falls by 22 to 28 (437.6 - 409.61).

d. If rates rose 2%, equity would be about 6 - barely solvent.

Duration Gap ModelDuration is a better measure of asset or liability interest rate risk than maturity. The duration formula is

= time weight x (discount cash flows)/(Bond Price)

D = durationCFt = cash flow in time period tY = yield to maturity (interest rate) per periodT = maturity in periods - usually semi-annual

D

CF t

YCF

Y

t

Tt

t

t

Tt

t

1

1

1

1

( )

( )

( )

A Shorter Way to Calculate a Coupon Bond's Duration

where T is the number of payments - for a thirty year bond with semi-annual coupons T =

60

c is the coupon rate per period - for a 12% coupon paid semi-annually, c = .06.

Y is the yield to maturity per period - for a 9% yield with semi-annual coupons Y = .045

DY

Y

Y T c Y

c Y YT

( ) ( ) ( )

[( ) ]

1 1

1 1

EXAMPLE: 30 year treasury bond - 12% coupon (paid semi-annually) - 9% yield

= 20.87 semi-annual periods or 10.44 annual periods

Note: Yield and interest rate are used interchangeably here because a bond’s “interest rate” is called its “yield.”

D

( . )

.

[( . ) (. . )]

[. [( . ) ] . ]

1 045

045

1 045 60 06 045

06 1 045 1 04560

Using Duration to Estimate Bond Price ChangeInterest rate changes affect the value of promised payments and the value of additional income from reinvested payments. Duration measures both effects.

Duration is the elasticity (from economics) of the asset or liability price with respect to a yield change.

For a bond paying semi-annual coupons:

Yn = the new semi-annual yieldYo = the old semi-annual yieldD = duration in semi-annual periods

)1(

)(

)1(

)1( %

o

on

Y

YYxD

Y

YxDP

EXAMPLE: 30 yr Treasury12% coupon (paid semiannually) Duration = 20.87 semi-annual periodsOld yield = 9% annual - New Yield = 8.5% annual

= .05 = 5%

QUESTION: Suppose two bonds are identical except that one pays annual coupons and the other pays semi-annual coupons. Do they have the same duration? If not, which is larger? - Annual

%P x

20 87

0425 045

1 045.

(. . )

( . )

Duration GapSimilar to the Maturity Gap, Duration Gap measures the difference between a firm’s weighted average asset Duration (DA) and weighted average liability Duration (DL).

Duration Gap = (DA - DL)

DA = WA1DA1 + WA2DA2 + WA3DA3 + … + WAnDAn

DL = WL1DL1 + WL2DL2 + WL3DL3 + … + WLnDLn

WAi = (market value of asset i)/(market value of total assets).WLi = (market value of liability j)/(market value of total liab.)DAi is the duration of asset i.DLi is the duration of liability j.

Duration and the Effect of Interest Rates on Equity ValueA more precise measure of the effect of an interest rate change on a financial firm’s equity value is:

Equity = -[DA - kDL]A(Yn - Yo)/(1 + Yo)

where k=L/A and [DA - kDL] is the leverage-adjusted Duration Gap, hereafter referred to as just the Duration Gap.

To eliminate the effect of interest rate changes on the value of a firm’s equity (called immunization), some have suggested setting

Maturity Gap = (MA - ML) = 0 or

Duration Gap = (DA - DL) = 0.

A more precise way to “immunize” equity value is by setting

[DA - kDL] = 0.

• A typical situation is that the dollar amount of assets (A) and liabilities (L) are given, then we select particular assets and liabilities with durations DA and DL so [DA - kDL] = 0.

• For solvent firms, we know that (A - L) = E > 0 and k < 1 so that equity immunization requires DA < DL.

•Many financial firms have DA > DL ,which implies that they are not immunized.

• To immunize equity as a percent of assets (E/A), setting DA = DL is the proper method.

Example: Ch. 9, 20The balance sheet of Gotbucks Bank isCash 30 8%, 2-yr Deposits 208.5% Fed. Funds 20 8.5% Fed. Funds 5011% Float Loan 105 9% Euro CD 13012%, 5-yr Loan 65 Equity 20Total Assets 220 Total Liabilities 220

a. Fixed Loan Duration

b. Assuming Floating Rate and Fed Funds have .36 duration

Asset Duration = [30(0) + 65(4.03) + 125(.36)]/220 = 1.4

c. Deposits Duration

03.4]12.]1)12.1[(12[.

)]12.12(.5)12.1[(

12.

)12.1(5

D

925.1]08.]1)08.1[(08[.

)]08.08(.2)08.1[(

08.

)08.1(2

D

d. Assuming the Euro CD has .401 duration,

Liab. Duration = [20(1.925) + 180(.401)]/200 = .5535

e. Duration Gap = 1.4 - (200/220)(.5535) = .8938 years.

f. An 1% increase in interest rates decreases equity by

E = -.8938(.01)*220 = -1,966,360

g. A decrease of .5% in interest rates increases equity by

E = -.8938(-.005)*220 = 983,180

h. To eliminate the effects on equity, the bank can increase liability its duration to 1.54 [x – (200/220)(.5535) = 0], decrease its asset duration to .5032 [1.4 – (200/220)(x) = 0], or some combination of the two.

Criticisms of Duration and Equity Immunization

1. As interest rates change, durations change, so one must constantly rebalance assets and liabilities to keep immunized. Transactions costs may be large.

2. We have assumed all interest rates change by the same amount but this is seldom true.

3. We have ignored default risk. Default or payment rescheduling can increase or decrease duration.

4. Durations of floating rate instruments and demand deposits are unclear. For floating rate instruments we usually assume duration equals the time to repricing. Demand deposits’ duration is assumed to be zero or small.

5. The most significant criticism is that duration is an approximation and works best for small changes in yields. Convexity (CX) is a measure of the duration error when yield changes are large. To get a better approximation to price changes due to interest rate changes, one can adjust an earlier price change equation to:

2)(5.)1(

)(

)1(

)1( %

on

o

on YYCXY

YYxD

Y

YxDP

The change in equity value becomes:

Equity = -[DA - kDL]A(Yn - Yo)/(1 + Yo)

+ .5[CXA - kCXL]A(Yn - Yo)2

Example of Using ConvexityHusky Financial has $100 million of assets with a weighted average duration of 8.5, a weighted average convexity of 200 and a yield of 10%. It also has $80 million of liabilities with a weighted average duration of 6, a weighted average convexity of 40 and a yield of 10%. If market yields rise by 2 percentage points, what is the expected change in Husky’s equity value if convexity is ignored? How about if one considers convexity?

Equity = -[8.5 - .8(6)]100(.02)/(1 + .10) = -$6.7 MM

with convexity

Equity = -$6.7 + .5[200 - .8(40)]100(.02)2 = -$3.26 M

Here, ignoring convexity overestimates the negative change.