2. Measuring Interest Rate Risk with Duration
GAP
Economic Value of Equity Analysis
Focuses on changes in stockholders’
equity given potential changes in
interest rates
Duration GAP Analysis
Compares the price sensitivity of a
bank’s total assets with the price
sensitivity of its total liabilities to
assess the impact of potential changes
in interest rates on stockholders’
equity.
3. Recall from Chapter 4
Duration is a measure of the effective
maturity of a security.
Duration incorporates the timing and
size of a security’s cash flows.
Duration measures how price sensitive
a security is to changes in interest
rates.
The greater (shorter) the duration, the
greater (lesser) the price sensitivity.
4. Duration and Price Volatility
Duration as an Elasticity Measure
Duration versus Maturity
Consider the cash flows for these two
securities over the following time line
0 5 10 15 20
$1,000
01 5 10 15 20
900 $100
5. Duration versus Maturity
The maturity of both is 20 years
Maturity does not account for the differences in
timing of the cash flows
What is the effective maturity of both?
The effective maturity of the first security is:
(1,000/1,000) x 1 = 20 years
The effective maturity of the second security is:
[(900/1,000) x 1]+[(100/1,000) x 20] = 2.9 years
Duration is similar, however, it uses a
weighted average of the present values of the
cash flows
6. Duration versus Maturity
Duration is an approximate measure of
the price elasticity of demand
% Change in Quantity Demanded
Price Elasticity of Demand -
% Change in Price
7. Duration versus Maturity
The longer the duration, the larger the
change in price for a given change in
interest rates.
P
Duration - P
i
(1 i)
i
P - Duration P
(1 i)
8. Measuring Duration
Duration is a weighted average of the
time until the expected cash flows
from a security will be received,
relative to the security’s price
Macaulay’s Duration
k n
CFt (t) CFt (t)
(1+ r)t (1+ r)t
D = t=1
k
t =1
CFt Price of the Security
(1+ r)t
t =1
9. Measuring Duration
Example
What is the duration of a bond with a
$1,000 face value, 10% annual coupon
payments, 3 years to maturity and a
12% YTM? The bond’s price is $951.96.
100 1 100 2 100 3 1,000 3
1
+ 2
+ 3
+
(1.12) (1.12) (1.12) (1.12) 3 2,597.6
D 3
= 2.73 years
100 1000 951.96
(1.12) t (1.12) 3
t =1
+
10. Measuring Duration
Example
What is the duration of a bond with a
$1,000 face value, 10% coupon, 3 years
to maturity but the YTM is 5%?The
bond’s price is $1,136.16.
100 * 1 100 * 2 100 * 3 1,000 * 3
1
+ 2
+ 3
+
(1.05) (1.05) (1.05) (1.05) 3 3,127.31
D = 2.75 years
1136.16 1,136.16
11. Measuring Duration
Example
What is the duration of a bond with a
$1,000 face value, 10% coupon, 3 years
to maturity but the YTM is 20%?The
bond’s price is $789.35.
100 * 1 100 * 2 100 * 3 1,000 * 3
1
+ 2
+ 3
+
(1.20) (1.20) (1.20) (1.20) 3 2,131.95
D = 2.68 years
789.35 789.35
12. Measuring Duration
Example
Whatis the duration of a zero coupon
bond with a $1,000 face value, 3 years
to maturity but the YTM is 12%?
1,000 * 3
(1.12) 3 2,135.34
D = 3 years
1,000 711.78
(1.12) 3
By definition, the duration of a zero
coupon bond is equal to its maturity
13. Duration and Modified Duration
The greater the duration, the greater
the price sensitivity
Modified Duration gives an estimate of
price volatility:
Macaulay' s Duration
Modified Duration
(1 i)
P
- Modified Duration i
P
14. Effective Duration
Effective Duration
Used to estimate a security’s price
sensitivity when the security contains
embedded options.
Compares a security’s estimated price in
a falling and rising rate environment.
15. Effective Duration
Pi- - Pi
Effective Duration -
P0 (i - i )
Where:
Pi- = Price if rates fall
Pi+ = Price if rates rise
P0 = Initial (current) price
i+ = Initial market rate plus the increase in rate
i- = Initial market rate minus the decrease in rate
16. Effective Duration
Example
Consider a 3-year, 9.4 percent semi-
annual coupon bond selling for $10,000
par to yield 9.4 percent to maturity.
Macaulay’s Duration for the option-free
version of this bond is 5.36 semiannual
periods, or 2.68 years.
The Modified Duration of this bond is
5.12 semiannual periods or 2.56 years.
17. Effective Duration
Example
Assume, instead, that the bond is
callable at par in the near-term .
If rates fall, the price will not rise much
above the par value since it will likely
be called
If rates rise, the bond is unlikely to be
called and the price will fall
18. Effective Duration
Example
If rates rise 30 basis points to 5%
semiannually, the price will fall to
$9,847.72.
If rates fall 30 basis points to 4.4%
semiannually, the price will remain at
par
$10,000 - $9,847.72
Effective Duration 2.54
$10,000( 0.05 - 0.044)
19. Duration GAP
Duration GAP Model
Focuseson either managing the
market value of stockholders’ equity
The bank can protect EITHER the
market value of equity or net interest
income, but not both
Duration GAP analysis emphasizes the
impact on equity
20. Duration GAP
Duration GAP Analysis
Compares the duration of a bank’s
assets with the duration of the bank’s
liabilities and examines how the
economic value stockholders’ equity
will change when interest rates
change.
21. Two Types of Interest Rate Risk
Reinvestment Rate Risk
Changes in interest rates will change
the bank’s cost of funds as well as the
return on invested assets
Price Risk
Changes in interest rates will change
the market values of the bank’s assets
and liabilities
22. Reinvestment Rate Risk
If interest rates change, the bank will
have to reinvest the cash flows from
assets or refinance rolled-over
liabilities at a different interest rate in
the future
An increase in rates increases a bank’s
return on assets but also increases the
bank’s cost of funds
23. Price Risk
If interest rates change, the value of
assets and liabilities also change.
The longer the duration, the larger the
change in value for a given change in
interest rates
Duration GAP considers the impact of
changing rates on the market value of
equity
24. Reinvestment Rate Risk and Price Risk
Reinvestment Rate Risk
Ifinterest rates rise (fall), the yield from
the reinvestment of the cash flows
rises (falls) and the holding period
return (HPR) increases (decreases).
Price risk
If interest rates rise (fall), the price falls
(rises). Thus, if you sell the security
prior to maturity, the HPR falls (rises).
25. Reinvestment Rate Risk and Price Risk
Increases in interest rates will increase
the HPR from a higher reinvestment
rate but reduce the HPR from capital
losses if the security is sold prior to
maturity.
Decreases in interest rates will
decrease the HPR from a lower
reinvestment rate but increase the
HPR from capital gains if the security
is sold prior to maturity.
26. Reinvestment Rate Risk and Price Risk
An immunized security or portfolio is
one in which the gain from the higher
reinvestment rate is just offset by the
capital loss.
For an individual security,
immunization occurs when an
investor’s holding period equals the
duration of the security.
27. Steps in Duration GAP Analysis
Forecast interest rates.
Estimate the market values of bank assets,
liabilities and stockholders’ equity.
Estimate the weighted average duration of
assets and the weighted average duration of
liabilities.
Incorporate the effects of both on- and off-
balance sheet items. These estimates are
used to calculate duration gap.
Forecasts changes in the market value of
stockholders’ equity across different
interest rate environments.
28. Weighted Average Duration of Bank Assets
Weighted Average Duration of Bank
Assets (DA)
n
DA w iDai
i
Where
wi = Market value of asset i divided by
the market value of all bank assets
Dai = Macaulay’s duration of asset i
n = number of different bank assets
29. Weighted Average Duration of Bank Liabilities
Weighted Average Duration of Bank
Liabilities (DL)
m
DL z jDlj
j
Where
zj = Market value of liability j divided by
the market value of all bank liabilities
Dlj= Macaulay’s duration of liability j
m = number of different bank liabilities
30. Duration GAP and Economic Value of Equity
Let MVA and MVL equal the market values
of assets and liabilities, respectively.
If: ΔEVE ΔMVA ΔMVL
and
Duration GAP
DGAP DA - (MVL/MVA)DL
Then:
y
ΔEVE - DGAP MVA
(1 y)
where y = the general level of interest
rates
31. Duration GAP and Economic Value of Equity
To protect the economic value of
equity against any change when rates
change , the bank could set the
duration gap to zero:
y
ΔEVE - DGAP MVA
(1 y)
33. Calculating DGAP
DA
($700/$1000)*2.69 + ($200/$1000)*4.99 = 2.88
DL
($620/$920)*1.00 + ($300/$920)*2.81 = 1.59
DGAP
2.88 - (920/1000)*1.59 = 1.42 years
What does this tell us?
The average duration of assets is greater than the
average duration of liabilities; thus asset values
change by more than liability values.
34. 1 percent increase in all rates.
1 Par Years Market
$1,000 % Coup Mat. YTM Value Dur.
Assets
Cash $ 100 $ 100
Earning assets
3-yr Commercial loan $ 700 12.00% 3 13.00% $ 683 2.69
6-yr Treasury bond $ 200 8.00% 6 9.00% $ 191 4.97
Total Earning Assets $ 900 12.13% $ 875
84 700 $
3
Non-cash earning assets $ -
PV -
t 1 t
Total assets $ 1,000 1.13 1.133 $
10.88% 975 2.86
Liabilities
Interest bearing liabs.
1-yr Time deposit $ 620 5.00% 1 6.00% $ 614 1.00
3-yr Certificate of deposit $ 300 7.00% 3 8.00% $ 292 2.81
Tot. Int Bearing Liabs. $ 920 6.64% $ 906
Tot. non-int. bearing $ - $ -
Total liabilities $ 920 6.64% $ 906 1.58
Total equity $ 80 $ 68
Total liabs & equity $ 1,000 $ 975
35. Calculating DGAP
DA
($683/$974)*2.68 + ($191/$974)*4.97 = 2.86
DA
($614/$906)*1.00 + ($292/$906)*2.80 = 1.58
DGAP
2.86 - ($906/$974) * 1.58 = 1.36 years
What does 1.36 mean?
The average duration of assets is greater than the
average duration of liabilities, thus asset values
change by more than liability values.
36. Change in the Market Value of Equity
y
ΔEVE - DGAP[ ]MVA
(1 y)
In this case:
.01
ΔEVE - 1.42[ ]$1,000 $12.91
1.10
37. Positive and Negative Duration GAPs
Positive DGAP
Indicates that assets are more price sensitive
than liabilities, on average.
Thus, when interest rates rise (fall), assets will
fall proportionately more (less) in value than
liabilities and EVE will fall (rise) accordingly.
Negative DGAP
Indicates that weighted liabilities are more
price sensitive than weighted assets.
Thus, when interest rates rise (fall), assets will
fall proportionately less (more) in value that
liabilities and the EVE will rise (fall).
39. An Immunized Portfolio
To immunize the EVE from rate
changes in the example, the bank
would need to:
decrease the asset duration by 1.42
years or
increase the duration of liabilities by
1.54 years
DA / ( MVA/MVL)
= 1.42 / ($920 / $1,000)
= 1.54 years
41. Immunized Portfolio with a 1% increase in rates
1 Par Years Market
$1,000 % Coup Mat. YTM Value Dur.
Assets
Cash $ 100.0 $ 100.0
Earning assets
3-yr Commercial loan $ 700.0 12.00% 3 13.00% $ 683.5 2.69
6-yr Treasury bond $ 200.0 8.00% 6 9.00% $ 191.0 4.97
Total Earning Assets $ 900.0 12.13% $ 874.5
Non-cash earning assets$ - $ -
Total assets $ 1,000.0 10.88% $ 974.5 2.86
Liabilities
Interest bearing liabs.
1-yr Time deposit $ 340.0 5.00% 1 6.00% $ 336.8 1.00
3-yr Certificate of deposit 300.0
$ 7.00% 3 8.00% $ 292.3 2.81
6-yr Zero-coupon CD* $ 444.3 0.00% 6 9.00% $ 264.9 6.00
Tot. Int Bearing Liabs. $ 1,084.3 7.54% $ 894.0
Tot. non-int. bearing $ - $ -
Total liabilities $ 1,084.3 7.54% $ 894.0 3.07
Total equity $ 80.0 $ 80.5
42. Immunized Portfolio with a 1% increase in rates
EVE changed by only $0.5 with the
immunized portfolio versus $25.0
when the portfolio was not immunized.
43. Stabilizing the Book Value of Net Interest Income
This can be done for a 1-year time horizon,
with the appropriate duration gap measure
DGAP* MVRSA(1- DRSA) - MVRSL(1- DRSL)
where:
MVRSA = cumulative market value of RSAs
MVRSL = cumulative market value of RSLs
DRSA = composite duration of RSAs for the
given time horizon
Equal to the sum of the products of each asset’s
duration with the relative share of its total asset
market value
DRSL = composite duration of RSLs for the
given time horizon
Equal to the sum of the products of each liability’s
duration with the relative share of its total liability
market value.
44. Stabilizing the Book Value of Net Interest Income
If DGAP* is positive, the bank’s net interest
income will decrease when interest rates
decrease, and increase when rates increase.
If DGAP* is negative, the relationship is
reversed.
Only when DGAP* equals zero is interest
rate risk eliminated.
Banks can use duration analysis to stabilize
a number of different variables reflecting
bank performance.
45. Economic Value of Equity Sensitivity Analysis
Effectively involves the same steps as
earnings sensitivity analysis.
In EVE analysis, however, the bank
focuses on:
The relative durations of assets and
liabilities
How much the durations change in
different interest rate environments
What happens to the economic value of
equity across different rate environments
46. Embedded Options
Embedded options sharply influence the
estimated volatility in EVE
Prepayments that exceed (fall short of)
that expected will shorten (lengthen)
duration.
A bond being called will shorten duration.
A deposit that is withdrawn early will
shorten duration.
A deposit that is not withdrawn as
expected will lengthen duration.
47. First Savings Bank Economic Value of Equity
Market Value/Duration Report as of 12/31/04
Most Likely Rate Scenario-Base Strategy
Assets
Book Value Market Value Book Yield Duration*
Loans
Prime Based Ln $ 100,000 $ 102,000 9.00%
Equity Credit Lines $ 25,000 $ 25,500 8.75% -
Fixed Rate > I yr $ 170,000 $ 170,850 7.50% 1.1
Var Rate Mtg 1 Yr $ 55,000 $ 54,725 6.90% 0.5
30-Year Mortgage $ 250,000 $ 245,000 7.60% 6.0
Consumer Ln $ 100,000 $ 100,500 8.00% 1.9
Credit Card $ 25,000 $ 25,000 14.00% 1.0
Total Loans $ 725,000 $ 723,575 8.03% 2.6
Loan Loss Reserve $ (15,000) $ 11,250 0.00% 8.0
Net Loans $ 710,000 $ 712,325 8.03% 2.5
Investments
Eurodollars $ 80,000 $ 80,000 5.50% 0.1
CMO Fix Rate $ 35,000 $ 34,825 6.25% 2.0
US Treasury $ 75,000 $ 74,813 5.80% 1.8
Total Investments $ 190,000 $ 189,638 5.76% 1.1
Fed Funds Sold $ 25,000 $ 25,000 5.25% -
Cash & Due From $ 15,000 $ 15,000 0.00% 6.5
Non-int Rel Assets $ 60,000 $ 60,000 0.00% 8.0
Total Assets $ 100,000 $ 100,000 6.93% 2.6
48. First Savings Bank Economic Value of Equity
Market Value/Duration Report as of 12/31/04
Most Likely Rate Scenario-Base Strategy
Liabilities
Book Value Market Value Book Yield Duration*
Deposits
MMDA $ 240,000 $ 232,800 2.25% -
Retail CDs $ 400,000 $ 400,000 5.40% 1.1
Savings $ 35,000 $ 33,600 4.00% 1.9
NOW $ 40,000 $ 38,800 2.00% 1.9
DDA Personal $ 55,000 $ 52,250 8.0
Comm'l DDA $ 60,000 $ 58,200 4.8
Total Deposits $ 830,000 $ 815,650 1.6
TT&L $ 25,000 $ 25,000 5.00% -
L-T Notes Fixed $ 50,000 $ 50,250 8.00% 5.9
Fed Funds Purch - - 5.25% -
NIR Liabilities $ 30,000 $ 28,500 8.0
Total Liabilities $ 935,000 $ 919,400 2.0
Equity $ 65,000 $ 82,563 9.9
Total Liab & Equity $ 1,000,000 $ 1,001,963 2.6
Off Balance Sheet Notional
lnt Rate Swaps - $ 1,250 6.00% 2.8 50,000
Adjusted Equity $ 65,000 $ 83,813 7.9
49. Duration Gap for First Savings Bank EVE
Market Value of Assets
$1,001,963
Duration of Assets
2.6 years
Market Value of Liabilities
$919,400
Duration of Liabilities
2.0 years
50. Duration Gap for First Savings Bank EVE
Duration Gap
= 2.6 – ($919,400/$1,001,963)*2.0
= 0.765 years
Example:
A 1% increase in rates would reduce
EVE by $7.2 million
= 0.765 (0.01 / 1.0693) * $1,001,963
Recall that the average rate on assets
is 6.93%
51. Sensitivity of EVE versus Most Likely (Zero Shock)
Interest Rate Scenario
20.0
Change in EVE (millions of dollars)
13.6
8.8 8.2
10.0
2
(10.0)
ALCO Guideline (8.2)
Board Limit
(20.0)
(20.4)
(30.0)
(36.6)
(40.0)
-300 -200 -100 0 +100 +200 +300
Shocks to Current Rates
Sensitivity of Economic Value of Equity measures the change in the economic value of
the corporation’s equity under various changes in interest rates. Rate changes are
instantaneous changes from current rates. The change in economic value of equity is
derived from the difference between changes in the market value of assets and changes
in the market value of liabilities.
52. Effective “Duration” of Equity
By definition, duration measures the
percentage change in market value for
a given change in interest rates
Thus,a bank’s duration of equity
measures the percentage change in
EVE that will occur with a 1 percent
change in rates:
Effective duration of equity
9.9 yrs. = $8,200 / $82,563
53. Asset/Liability Sensitivity and DGAP
Funding GAP and Duration GAP are NOT
directly comparable
Funding GAP examines various “time
buckets” while Duration GAP represents
the entire balance sheet.
Generally, if a bank is liability (asset)
sensitive in the sense that net interest
income falls (rises) when rates rise and
vice versa, it will likely have a positive
(negative) DGAP suggesting that assets
are more price sensitive than liabilities, on
average.
54. Strengths and Weaknesses: DGAP and EVE-
Sensitivity Analysis
Strengths
Duration analysis provides a
comprehensive measure of interest rate
risk
Duration measures are additive
This allows for the matching of total
assets with total liabilities rather than the
matching of individual accounts
Durationanalysis takes a longer term
view than static gap analysis
55. Strengths and Weaknesses: DGAP and EVE-
Sensitivity Analysis
Weaknesses
It is difficult to compute duration
accurately
“Correct” duration analysis requires that
each future cash flow be discounted by a
distinct discount rate
A bank must continuously monitor and
adjust the duration of its portfolio
It is difficult to estimate the duration on
assets and liabilities that do not earn or
pay interest
Duration measures are highly subjective
56. Speculating on Duration GAP
It is difficult to actively vary GAP or
DGAP and consistently win
Interest rates forecasts are frequently
wrong
Even if rates change as predicted,
banks have limited flexibility in vary
GAP and DGAP and must often
sacrifice yield to do so
57. Gap and DGAP Management Strategies
Example
Cash flows from investing $1,000 either
in a 2-year security yielding 6 percent or
two consecutive 1-year securities, with
the current 1-year yield equal to 5.5
percent. 0 1 2
Two-Year Security
$60 $60
0 1 2
One-Year Security & then
another One-Year Security
$55 ?
58. Gap and DGAP Management Strategies
Example
It is not known today what a 1-year
security will yield in one year.
For the two consecutive 1-year
securities to generate the same $120
in interest, ignoring compounding, the
1-year security must yield 6.5% one
year from the present.
This break-even rate is a 1-year
forward rate, one year from the
present:
6% + 6% = 5.5% + x
so x must = 6.5%
59. Gap and DGAP Management Strategies
Example
By investing in the 1-year security, a
depositor is betting that the 1-year
interest rate in one year will be greater
than 6.5%
By issuing the 2-year security, the
bank is betting that the 1-year interest
rate in one year will be greater than
6.5%
60. Yield Curve Strategy
When the U.S. economy hits its peak,
the yield curve typically inverts, with
short-term rates exceeding long-term
rates.
Only twice since WWII has a recession
not followed an inverted yield curve
As the economy contracts, the Federal
Reserve typically increases the money
supply, which causes the rates to fall
and the yield curve to return to its
“normal” shape.
61. Yield Curve Strategy
To take advantage of this trend, when
the yield curve inverts, banks could:
Buy long-term non-callable securities
Prices will rise as rates fall
Make fixed-rate non-callable loans
Borrowers are locked into higher rates
Price deposits on a floating-rate basis
Lengthen the duration of assets
relative to the duration of liabilities
62. Interest Rates and the Business Cycle
The general level of interest rates and the shape of the yield curve
appear to follow the U.S. business cycle.
Peak In expansionary Short-TermRates
stages rates rise until
they reach a peak as
)
t the Federal Reserve Long-TermRates
n tightens credit
e
c
r
availability.
e
P
( Expansion The inverted yield curve has predicted the last
s five recessions
e
t
a Contraction
DATE WHEN 1-YEAR RATE LENGTH OF TIME UNTIL
R
FIRST EXCEEDS 10-YEAR RATE START OF NEXT RECESSION
t Expansion Apr. ’68 20 months (Dec. ’69)
s Mar. ’73 8 months (Nov. ’73)
e contractionary
In
rstages rates fall until Sept. ’78 16 months (Jan. ’80)
e
tthey reach a trough Sept. ’80 10 months (July ’81)
n Trough Feb. ’89 17 months (July ’90)
Iwhen the U.S. Dec. ’00 15 months (March ’01)
economy falls into
recession.
Time
