Rate Interest Risk
Interest 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 Gap
The 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 firms 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 firms 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 Example
Time Period
1 day
1 day - 3 months
3 - 6 months
6 - 12 months
1 - 5 years
Over 5 years
Assets
20
30
70
90
40
10
Liabilities
30
40
85
70
30
5
Gap
-10
-10
-15
20
10
5
Cm. Gap
-10
-20
-35
-15
-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. 9
Consider the following balance sheet.
Cash
10
Overnight Repos
170
1 mon, 7.05% Tbill 75
7-yr 8.55% Sub. Deb. 150
3 mon, 7.25% Tbill 75
2-yr, 7.5% Tnote
50
8-yr, 8.96% Tnote
100
5-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 Model
The Maturity Gap measures the difference between a firms
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.)
M is the maturity of asset i.
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
firms equity.
When (MA - ML) < 0 then an increase (decrease) in interest
rates is expected to increase (decrease) a financial
firms 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 Mortgage
County Bank has the following Balance sheet:
Cash
$20 Demand Deposits
$100
15-yr, 10% Loan
160
5-yr, 6% CD Balloon 210
30-yr, 8% Bond
300
20-yr, 7% Debenture 120
Equity
50
Total 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).
Duration Gap Model
Duration is a better measure of asset or liability interest rate
risk than maturity. The duration formula is
CFt (t )
t
t =1 (1 + Y )
D= T
CFt
t
t =1 (1 + Y )
T
= time weight x (discount cash flows)/(Bond Price)
D
CFt
Y
T
= duration
= cash flow in time period t
= yield to maturity (interest rate) per period
= maturity in periods - usually semi-annual
A Shorter Way to Calculate a
Coupon Bond's Duration
(1 + Y ) (1 + Y ) + T (c Y )
D=
Y
c [(1 + Y ) T 1] + Y
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
EXAMPLE: 30 year treasury bond - 12% coupon (paid
semi-annually) 9% - yield
(1+.045) [(1+.045) + 60(.06.045)]
D=
.045
[.06[(1+.045) 60 1]+.045]
= 20.87 semi-annual periods or 10.44 annual periods
Note: Yield and interest rate are used interchangeably here
because a bonds interest rate is called its yield.
Using Duration to Estimate
Bond Price Change
Interest 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 Yo )
(1 + Y )
% P = D x
= D x
(1 + Y )
(1 + Yo )
Yn
Yo
D
= the new semi-annual yield
= the old semi-annual yield
= duration in semi-annual periods
EXAMPLE: 30 yr Treasury
12% coupon (paid semiannually)
Duration = 20.87 semi-annual periods
Old yield = 9% annual - New Yield = 8.5% annual
(.0425.045)
% P = 20.87 x
(1+.045)
= .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
Duration Gap
Similar to the Maturity Gap, Duration Gap measures the
difference between a firms 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.)
D is the duration of asset i.
Duration and the Effect of
Interest Rates on Equity Value
A more precise measure of the effect of an interest rate
change on a financial firms 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 firms equity (called immunization), some have
suggested setting
Maturity Gap = (MA - ML) = 0
Duration Gap = (DA - DL) = 0.
or
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, 20
The balance sheet of Gotbucks Bank is
Cash
30
8%, 2-yr Deposits
8.5% Fed. Funds
20
8.5% Fed. Funds
11% Float Loan
105
9% Euro CD
12%, 5-yr Loan
65
Equity
Total Assets
220
Total Liabilities
a. Fixed Loan Duration
20
50
130
20
220
(1 + .12) [(1 + .12) + 5(.12 .12)]
D=
= 4.03
.12
[.12[(1 + .12) 1] + .12]
5
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
D=
(1 + .08) [(1 + .08) + 2(.08 .08)]
= 1.925
2
.08
[.08[(1 + .08) 1] + .08]
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:
(1 + Y )
(Y Y )
% P = D x
= D x
+ .5CX (Y Y )
(1 + Y )
(1 + Y )
n
o
n
o
The change in equity value becomes:
Equity = -[DA - kDL]A(Yn - Yo)/(1 + Yo)
+ .5[CXA - kCXL]A(Yn - Yo)2
o
2
Example of Using Convexity
Husky 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 Huskys
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.