Solution manual bank management and financial services 9th edition by rose, peter chap007

Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

CHAPTER 7
RISK MANAGEMENT FOR CHANGING INTEREST RATES: ASSET-LIABILITY
MANAGEMENT AND DURATION TECHNIQUES
Goals of This Chapter: The purpose of this chapter is to explore the options bankers have today
for dealing with risk–especially the risk of loss due to changing interest rates–and to see how a
bank’s management can coordinate the management of its assets with the management of its
liabilities in order to achieve the institution’s goals.
Key Topics In This Chapter
•
•
•
•
•
•

Asset, Liability, and Funds Management
Market Rates and Interest Rate Risk
The Goals of Interest Rate Hedging
Interest-Sensitive Gap Management
Duration Gap Management
Limitations of Interest Rate Risk Management Techniques
Chapter Outline

I.
II.

III.

IV.

Introduction: The Necessity for Coordinating Bank Asset and Liability Management
Decisions
Asset-Liability Management Strategies
A.
Asset Management Strategy
B.
Liability Management Strategy
C.
Funds Management Strategy
Interest Rate Risk: One of the Greatest Management Challenges
A.
Forces Determining Interest Rates
B.
The Measurement of Interest Rates
1.
Yield to Maturity
2.
Bank Discount Rate
C.
The Components of Interest Rates
1.
Risk Premiums
2.
Yield Curves
3.
The Maturity Gap and the Yield Curve
D.
Responses to Interest Rate Risk
1.

Asset-Liability Committee (ALCO)
One of the Goals of Interest Rate Hedging: Protect the Net Interest Margin
A.
The Net Interest Margin
B.
Interest-Sensitive Gap Management as a Risk-Management Tool
1.
Asset-Sensitive Gap
2.
Liability-Sensitive Gap
3.
Dollar Interest-Sensitive Gap
4.
Relative Interest Sensitive Gap

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

V.

VI.

VII
VIII.

5.
Interest Sensitivity Ratio
6.

Computer-Based Techniques
7.
Cumulative Gap
8.
Strategies in Gap Management
C.
Problems with Interest-Sensitive GAP Management
The Concept of Duration as a Risk-Management Tool
A.
Definition of Duration
B.
Calculation of Duration
C.
Net Worth and Duration
D.
Price Sensitivity to Changes in Interest Rates and Duration
E.
Convexity and Duration
Using Duration to Hedge Against Interest Rate Risk
A.
Duration Gap
1.
Dollar Weighted Duration of Assets
2.
Dollar Weighted Duration of Liabilities
3.
Positive Duration Gap
4.
Negative Duration Gap
B.

Change in the Bank’s Net Worth
The Limitations of Duration Gap Management
Summary of the Chapter
Concept Checks

7-1. What do the following terms mean: Asset management? Liability management? Funds
management?
Asset management refers to a banking strategy where management has control over the
allocation of bank assets but believes the bank's sources of funds (principally deposits) are
outside its control. The key decision area for management is not deposits and other borrowings
but assets. The financial manager exercises control over the allocation of incoming funds by
deciding who is granted loans and what the terms on those loans will be.
Liability management is a strategy wherein greater control towards bank liabilities is exercised.
This is done mainly by opening up new sources of funding and monitoring the volume, mix and
cost of their deposits and non-deposit items.
Funds management combines both asset and liability management approaches into a balanced
liquidity management strategy. Effective coordination in managing assets and liabilities will help
to maximize the spread between revenues and costs and control risk exposure.
7-2. What factors have motivated financial institutions to develop funds management
techniques in recent years?
The necessity to find new sources of funds in the 1970s and the risk management problems
encountered with troubled loans and volatile interest rates in the 1970s and 1980s led to the
concept of planning and control over both sides of a bank's balance sheet—the essence of funds
management.

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

The maturing of liability management techniques, coupled with more volatile interest rates and
greater risk, eventually gave birth to the funds management approach,
7-3. What forces cause interest rates to change? What kinds of risk do financial firms face
when interest rates change?
Interest rates are determined, not by individual banks, but by the collective borrowing and
lending decisions of thousands of participants in the money and capital markets. They are also
impacted by changing perceptions of risk by participants in the money and capital markets,
especially the risk of borrower default, liquidity risk, price risk, reinvestment risk, inflation risk,
term or maturity risk, marketability risk, and call risk.
Financial institutions can lose income or value no matter which way interest rates go. As market
interest rates move, financial firms typically face at least two major kinds of interest rate risk—
price risk and reinvestment risk. Price risk arises when market interest rates rise. Rising interest
rates can lead to losses on security instruments and on fixed-rate loans as the market values of
these instruments fall. Rising interest rates will also cause a loss to income if an institution has
more rate-sensitive liabilities than rate-sensitive assets. Reinvestment risk rears its head when
market interest rates fall. Falling interest rates will usually result in capital gains on fixed-rate
securities and loans but an institution will lose income if it has more rate-sensitive assets than
liabilities. Also, financial firms will be forced to invest incoming funds in lower-yielding earning
assets, lowering their expected future income. A big part of managing assets and liabilities
consists of finding ways to deal effectively with these two forms of risk.
7-4.

What makes it so difficult to correctly forecast interest rate changes?

Interest rates cannot be set by an individual bank or even by a group of banks. They are
determined by thousands of investors trading in the credit markets. Moreover, each market rate
of interest has multiple components—the risk-free real interest rate plus various risk premiums.
A change in any of these rate components can cause interest rates to change. This makes it
virtually impossible to accurately forecast interest rate changes.
To consistently forecast market interest rates correctly would require bankers to correctly

anticipate changes in the risk-free real interest rate and in all rate components. Another important
factor is the timing of the changes. To be able to take full advantage of their predictions, they
also need to know when the changes will take place.
7-5.

What is the yield curve, and why is it important to know about its shape or slope?

The yield curve is the graphic picture of how interest rates vary with different maturities of loans
viewed at a single point in time (and assuming that all other factors, such as credit risk, are held
constant).
The slope of the yield curve determines the spread between long-term and short-term interest
rates. In banking most of the long-term rates apply to loans and securities (i.e., bank assets) and

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

most of the short-term interest rates are attached to bank deposits and money market borrowings
(i.e., bank liabilities).
If the yield curve is upward sloping, then revenues from longer-term assets will outstrip expenses
from shorter term liabilities. The result will normally be a positive net interest margin (interest
revenues greater than interest expenses), which tends to generate higher earnings. In contrast, a
relatively flat (horizontal) or negatively sloped yield curve often generates a small or even
negative net interest margin, putting downward pressure on the earnings of financial firms that
borrow short and lend long.
Thus, the shape or slope of the yield curve has a profound influence on a bank's net interest
margin or spread between asset revenues and liability costs.
7-6. What is it that a lending institution wishes to protect from adverse movements in interest
rates?

Changes in market interest rates can damage a financial firm’s profitability by increasing its cost
of funds, by lowering its returns from earning assets and by reducing the value of the owners’
investment. Therefore, a financial institution wishes to protect both the value of assets and
liabilities, and the revenues and costs generated by both assets and liabilities from adverse
movements in interest rates.
7-7.

What is the goal of hedging?

The goal of hedging in banking is to freeze the spread between asset returns and liability costs
and to offset declining values on certain assets by profitable transactions so that a target rate of
return is assured.
7-8. First National Bank of Bannerville has posted interest revenues of $63 million and
interest costs from all of its borrowings of $42 million. If this bank possesses $700 million in
total earning assets, what is First National’s net interest margin? Suppose the bank’s interest
revenues and interest costs double, while its earning assets increase by 50 percent. What will
happen to its net interest margin?
The bank’s net interest margin is 3 percent computed as follows:
Net Interest Margin =

$63million-$42 million
= 0.03 or 3 percent
$700 million

If interest revenues and interest costs double while earning assets grow by 50 percent, the net
interest margin will change as follows:
Net Interest Margin =

( $63million - $42 million ) × 2
= 0.04 or 4 percent

$700 million × ( 1.50 )

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

Clearly the net interest margin increases—in this case by one third.
7-9.

Can you explain the concept of gap management?

Gap management requires the management to perform analysis of the maturities and repricing
opportunities associated with interest-bearing assets and with interest-bearing liabilities. When
more assets are subject to repricing or will reach maturity in a given period than liabilities or vice
versa, the bank has a gap between assets and liabilities and is exposed to loss from adverse
interest-rate movements based on the gap's size and direction. If an organization is over exposed
to interest rate fluctuation, the management will try and match the volume of assets that can be
repriced, with the volume of liabilities.
7-10

When is a financial firm asset sensitive? Liability sensitive?

A financial firm is asset sensitive when it has more interest-rate sensitive assets maturing or
subject to repricing during a specific time period than rate-sensitive liabilities. A liability
sensitive position, in contrast, would find the financial institution having more interest-rate
sensitive deposits and other liabilities than rate-sensitive assets for a particular planning period.
7-11. Commerce National Bank reports interest-sensitive assets of $870 million and
interest-sensitive liabilities of $625 million during the coming month. Is the bank asset sensitive
or liability sensitive? What is likely to happen to the bank’s net interest margin if interest rates

rise? If they fall?
Because interest-sensitive assets are larger than liabilities by $245 million, the bank is asset
sensitive.
If interest rates rise, the bank's net interest margin should rise as asset revenues increase more
than the resulting increase in liability costs. On the other hand, if interest rates fall, the bank's net
interest margin will fall as asset revenues decline faster than liability costs.
7-12. Peoples’ Savings Bank has a cumulative gap for the coming year of + $135 million, and
interest rates are expected to fall by two and a half percentage points. Can you calculate the
expected change in net interest income that this thrift institution might experience? What change
will occur in net interest income if interest rates rise by one and a quarter percentage points?
For the decrease in interest rates:
Expected Change in Net Interest Income = $135 million × ( − 0.025) = − $3.38 million
The net interest income will decrease by $3.38 million.
For the increase in interest rates:
Expected Change in Net Interest Income = $135 million × ( + 0.0125) = + $1.69 million

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

The net interest income will increase by $1.69 million.
7-13 How do you measure the dollar interest-sensitive gap? The relative interest-sensitive gap?
What is the interest sensitivity ratio?
The dollar interest-sensitive gap is measured by taking the repriceable (interest-sensitive) assets
minus the repriceable (interest-sensitive) liabilities over some set planning period. Common
planning periods include 3 months, 6 months and 1 year.
The relative interest-sensitive gap is the dollar interest-sensitive gap divided by the size of a
financial institution (often total assets).
The interest-sensitivity ratio is just the ratio of interest-sensitive assets to interest sensitive

liabilities.
Regardless of which measure you use, the results should be consistent. If you find a positive
(negative) gap for dollar interest-sensitive gap, you should also find a positive (negative) relative
interest-sensitive gap and an interest sensitivity ratio greater (less) than one.
7-14 Suppose Carroll Bank and Trust reports interest-sensitive assets of $570 million and
interest-sensitive liabilities of $685 million. What is the bank’s dollar interest-sensitive gap? Its
relative interest-sensitive gap and interest-sensitivity ratio?
Dollar Interest-Sensitive Gap = Interest-Sensitive Assets – Interest Sensitive Liabilities
= $570 mill. − $685 mill. = − $115 mill.
Relative Gap

=

Interest-Sensitivity
Ratio

IS Gap
Bank Size
(i.e. total
assets)
=

= − $115
$570

= − 0.2018

Interest-Sensitive Assets
Interest-Sensitive Liabilities

=

$570
$685

= 0.8321

7-15 Explain the concept of weighted interest-sensitive gap. How can this concept aid
management in measuring a financial institution’s real interest-sensitive gap risk exposure?
Weighted interest-sensitive gap is based on the idea that not all interest rates change at the same
speed and magnitude. Some are more sensitive than others. Interest rates on bank assets may
change more slowly than interest rates on liabilities and both of these may change at a different
speed and amount than those interest rates determined in the open market.
In the weighted interest-sensitive gap methodology, all interest-sensitive assets and liabilities are
given a weight based on their speed and magnitude (sensitivity) relative to some market interest
rate.

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

Fed Fund’s loans, for example, have an interest rate which is determined in the market and which
would have a weight of 1. All other loans, investments and deposits would have a weight based
on their sensitivity relative to the Fed Fund’s rate. To determine the interest-sensitive gap, the
dollar amount of each type of asset or liability would be multiplied by its weight and added to the
rest of the interest-sensitive assets or liabilities. Once the weighted total of the assets and
liabilities is determined, a weighted interest-sensitive gap can be determined by subtracting the
interest-sensitive liabilities from the interest-sensitive assets.
This weighted interest-sensitive gap should be more accurate than the unweighted interestsensitive gap. The interest-sensitive gap may change from negative to positive or vice versa and

may change significantly the interest rate strategy pursued by the bank.
7-16. What is duration?
Duration is a value- and time-weighted measure of maturity that considers the timing of all cash
inflows from earning assets and all cash outflows associated with liabilities. It measures the
average maturity of a promised stream of future cash payments. It is a direct measure of price
risk.
7-17. How is a financial institution’s duration gap determined?
A bank's duration gap is determined by taking the difference between the dollar-weighted
duration of a bank's assets portfolio and the dollar-weighted duration of its liabilities. The
duration of the bank’s assets can be determined by taking a weighted average of the duration of
all of the assets in the bank’s portfolio. The weight is the dollar amount of a particular type of
asset out of the total dollar amount of the assets of the bank. The duration of the liabilities can be
determined in a similar manner.
7-18. What are the advantages of using duration as an asset-liability management tool as
opposed to interest-sensitive gap analysis?
Interest-sensitive gap only looks at the impact of changes in interest rates on the bank’s net
income. It does not take into account the effect of interest rate changes on the market value of the
bank’s equity capital position. Whereas, duration provides a single number which tells the bank
their overall exposure to interest rate risk. Duration can be used for hedging against the interest
rate risk, and it can also measure the sensitivity of the market value of financial instruments to
changes in interest rates.
7-19. How can you tell if you are fully hedged using duration gap analysis?
You are fully hedged when the dollar weighted duration of the assets portfolio of the bank equals
the dollar weighted duration of the liability portfolio. This means that the bank has a zero
duration gap position when it is fully hedged. Of course, because the bank usually has more
assets than liabilities the duration of the liabilities needs to be adjusted by the ratio of total
liabilities to total assets to be entirely correct.

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

7-20. What are the principal limitations of duration gap analysis? Can you think of some way
of reducing the impact of these limitations?
There are several limitations with duration gap analysis. It is often difficult to find assets and
liabilities of the same duration to fit into the financial-service institution’s portfolio. In addition,
some accounts such as deposits and others don’t have well defined patterns of cash flows which
make it difficult to calculate duration for these accounts. Duration is also affected by
prepayments by customers as well as defaults. Duration gap models assume that a linear
relationship exists between the market values (prices) of assets and liabilities and interest rates,
which is not strictly true. Finally, duration analysis works best when interest rate changes are
small and short and long term interest rates change by the same amount. If this is not true,
duration analysis is not as accurate.
Recent research suggests that duration balancing can still be effective, even with moderate
violations of the technique’s underlying assumptions. In this age of mergers and continuing
financial-services industry consolidation, the duration gap concept remains a valuable
managerial tool despite its limitations.
7-21. Suppose that a savings institution has an average asset duration of 2.5 years and an
average liability duration of 3.0 years. If the savings institution holds total assets of $560 million
and total liabilities of $467 million, does it have a significant leverage-adjusted duration gap? If
interest rates rise, what will happen to the value of its net worth?
$467 million 
Liabilities

= 2.5 years 3.0 years ì
ữ
$560 million
Assets

= 2.5 years 2.5018 years
= − 0.0018 years

Duration Gap = DA – DL ×

This bank has a very slight negative duration gap; so small in fact that we could consider it
insignificant. If interest rates rise, the bank's liabilities will fall slightly more in value than its
assets, resulting in a small increase in net worth.
7-22. Stilwater Bank and Trust Company has an average asset duration of 3.25 years and an
average liability duration of 1.75 years. Its liabilities amount to $485 million, while its assets
total $512 million. Suppose that interest rates were 7 percent and then rise to 8 percent. What
will happen to the value of the Stilwater bank's net worth as a result of a decline in interest rates?
First, we need an estimate of Stilwater's duration gap. This is:
Duration Gap = 3.25 years – 1.75 years ×

$485 mill.
= + 1.5923 years
$512 mill.

Then, the change in net worth if interest rates rise from 7 percent to 8 percent will be:

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques


 

+.01

+.01
× $512 mill  -  -1.75 years ×
× $485 mill.
Change in NW = -3.25 years ×
(1 + .07)
(1 + .07)

 

= − $7.62 million.
The value of Stilwater bank's net worth would rise by about $7.62 million if the interest rate rises
by 1 percentage points.
Problems and Projects
7-1. A government bond is currently selling for $1,195 and pays $75 per year in interest for 14
years when it matures. If the redemption value of this bond is $1,000, what is its yield to
maturity if purchased today for $1,195?
The yield to maturity ( x ) equation for this bond would be:
$1,195 =

$75
$75
$75
$75
$75
$1,075
+
+
+
+ ... +
+

1
2
3
4
13
(1 + YTM) (1 + YTM)
(1 + YTM) (1 + YTM)
(1 + YTM)
(1 + YTM)14

Using a financial calculator the YTM = 5.4703 percent.
7-2. Suppose the government bond described in problem 1 above is held for five years and
then the savings institution acquiring the bond decides to sell it at a price of $940. Can you figure
out the average annual yield the savings institution will have earned for its five-year investment
in the bond?
$1,195 =

$75
$75
$75
$75
$75
+
+
+
+
1
2
3
4

(1 + HPY) (1 + HPY)
(1 + HPY)
(1 + HPY)
(1 + HPY ) 5

Using a financial calculator, the HPY is 2.19 percent.
7-3. U.S. Treasury bills are available for purchase this week at the following prices (based
upon $100 par value) and with the indicated maturities:
a.
b.
c.

$97.25, 182 days.
$95.75, 270 days.
$98.75, 91 days.

Calculate the bank discount rate (DR) on each bill if it is held to maturity. What is the equivalent
yield to maturity (sometimes called the bond-equivalent or coupon-equivalent yield) on each of
these Treasury Bills?
The discount rates and equivalent yields to maturity (bond-equivalent or coupon-equivalent
yields) on each of these Treasury bills are:

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

Discount Rates

Equivalent Yields to Maturity

a.
100 - 97.25
100

×

360
182

= 5.44%

100 - 97.25
97.25

× 365
182

= 5.67%

100 - 95.75
100

×

360
270

= 5.67%

100 - 95.75
95.75

× 365
270

= 6.00%

100 - 98.75
100

×

360
91

100 - 98.75
98.75

× 365
91

= 5.08%

b

c.
= 4.95%

7-4. Farmville Financial reports a net interest margin of 2.75 percent in its most recent

financial report, with total interest revenue of $95 million and total interest costs of $82 million.
What volume of earning assets must the bank hold? Suppose the bank’s interest revenues rise by
5 percent and its interest costs and earnings assets increase by 9 percent. What will happen to
Farmville’s net interest margin?
The relevant formula is:
Net interest margin = 0.0275 =

$95 mill. − $82 mill.
Total earning assets

Then, total earning assets must be $473 million.
If revenues rise by 5 percent, and interest costs and earnings assets rise by 9 percent, net interest
margin is:
Net interest margin =

$95(1.05) − $82(1.09)
473(1.09)
=

99.75 − 89.38
515.57

= 0.0201 or 2.01 percent
7-5. If a credit union’s net interest margin, which was 2.50 percent, increases 10 percent and
its total assets, which stood originally at $575 million, rise by 20 percent, what change will occur
in the bank's net interest income?
The correct formula is:

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

Net interest margin =

Net interest income
Total earning assets

Original net interest income = Net interest margin × Total earning assets
= 2.5% × $575 million = $14.375 million
New net interest income:
0.025 ×1.10 =

Net interest income
$575 million×1.20

Net Interest Income = 0.0275 × 690
= $18.975 million
Change in net interest income = New net interest income – Original net interest income
= $18.975 million - $14.375 million = $4.6 million.
7-6. The cumulative interest rate gap of Poquoson Savings Bank increases 60 percent from an
initial figure of $25 million. If market interest rates rise by 25 percent from an initial level of 3
percent, what changes will occur in this thrift’s net interest income?
New net interest income = New market interest rate × Increase in assets
= 3.75 percent × $40 million = $1.5 million
Initial net interest income = Initial market interest rate × Initial assets
= 3 percent × $25 million = $0.75 million
Percent change in net interest income = ($1.5 million – $0.75 million)/ $0.75 million
= 100 percent
Thus, the bank's net interest income will increase by 100 percent.

7-7. New Comers State Bank has recorded the following financial data for the past three years
(dollars in millions):
Interest revenues
Interest expenses
Loans (excluding nonperforming)
Investments
Total deposits
Money market borrowings

Current Year
$82
64
450
200
450
150

7-11

Previous Year
$80
66
425
195
425
125

Two Years Ago
$78
68

400
200
400
100


Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

What has been happening to the bank’s net interest margin? What do you think caused the
changes you have observed? Do you have any recommendations for New Comers’ management
team?
Net interest margin (NIM) = Net interest income/Total earning assets
Where,
Net interest income = Net interest revenues - Net interest expenses
Total earning assets = Loans + Investments
NIM (Current) = ($82-64)/ (450 + 200) = 18/650 = 0.028 or 2.77%
NIM (Previous) = ($80-66)/ (425 + 195) = 14/620 = 0.0226 or 2.26%
NIM (Two years ago) = ($78-68)/ (400 + 200) = 10/600 = 0.0167 or 1.67%
The net interest margin has been increasing over the years. As interest revenues and expenses as
well as the bank’s assets have increased consistently over the years, there has been a constant
increase in the net interest margin. If the bank can further cut down on its interest expenses and
increase its assets in the next years, the net interest margin will increase at a higher rate.
7-8
The First National Bank of Dogsville finds that its asset and liability portfolio contains
the following distribution of maturities and repricing opportunities:

Loans
Securities
Interest-sensitive assets
Transaction deposits

Time accts.
Money market borrowings
Interest-sensitive liabilities

Coming
Week
$200.00
21.00

Next 30
Days
$300.00
26.00

Next 31-90
Days
$475.00
40.00

More Than
90 Days
$525.00
70.00

$320.00
100.00
136.00

$ 0.00
290.00

140.00

$ 0.00
196.00
100.00

$ 0.00
100.00
65.00

When and by how much is the bank exposed to interest rate risk? For each maturity or repricing
interval, what changes in interest rates will be beneficial and which will be damaging, given the
current portfolio position?

Loans
Securities
Total IS Assets
Transaction deposits
Time Accts.
Money Mkt. Borr.
Total IS Liab.

Coming Week Next 30 Days Next 31-90 Days More Than 90 Days
$200
$300
$475
$525
21
26
40

70
$221
$326
$515
$595
$320
100
136
$556

$−
290
140
$430

7-12

$−
196
100
$296

$−
100
65
$165


Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

GAP

−335

−104

+219

+430

Cumulative GAP

−335

−439

−220

+210

First National has a negative gap in the nearest period and therefore would benefit if interest
rates fall. In the next period it has a slightly negative gap and would therefore benefit of interest
rate rise. However, its cumulative gap is still negative. The third period is positive gap and hence
the bank would benefit if interest rates rise. In the final period the gap is positive and the bank
would benefit if interest rates rise. Its cumulative gap is slightly positive and also shows that
rising interest rates would be beneficial to the bank overall.
7-9
Sunset Savings Bank currently has the following interest-sensitive assets and liabilities
on its balance sheet with the interest-rate sensitivity weights noted.
Interest-Sensitive Assets

Federal fund loans
Security holdings
Loans and leases
Interest-Sensitive Liabilities
Interest-bearing deposits
Money-market borrowings

$ Amount
$ 50.00
50.00
350.00
$ Amount
$ 250.00
90.00

Rate Sensitivity Index
1.00
1.20
1.45
Rate Sensitivity Index
0.75
0.95

What is the bank’s current interest-sensitive gap? Adjusting for these various interest rate
sensitivity weights what is the bank’s weighted interest-sensitive gap? Suppose the federal funds
interest rate increases or decreases 50 basis points. How will the bank’s net interest income be
affected (a) given its current balance sheet makeup and (b) reflecting its weighted balance sheet
adjusted for the foregoing rate-sensitivity indexes?
Dollar IS Gap

= ISA - ISL = ($50 + $50 + $350) − ($250 + $90)

= $110

Weighted IS Gap = ( 1) × ( $50 ) + ( 1.20 ) × ( 50 ) + ( 1.45 ) × ( 350 )  − ( 0.75 ) × ( $250 ) + ( 0.95 ) × ( $90 ) 
= $50 + $60 + $507.5 − $187.5 + $85.5
= $617.5 − $273
= $344.5

a.)

Change in Bank’s Income = IS Gap × Change in interest rates
= ($110) (0.005) = $0.55 million

Using the regular IS Gap; net income will change by plus or minus $550,000
b.)

Change in Bank’s Income = Weighted IS Gap × Change in interest rates
= ($344.50) (0.005) = $1.72250

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

Using the weighted IS Gap; net income will change by plus or minus $1,722,500.
7-10 Sparkle Savings Association has interest-sensitive assets of $400 million, interestsensitive liabilities of $325 million, and total assets of $500 million. What is the bank’s dollar
interest-sensitive gap? What is Sparkle’s relative interest-sensitive gap? What is the value of its
interest-sensitivity ratio? Is it asset sensitive or liability sensitive? Under what scenario for
market interest rates will Sparkle experience a gain in net interest income? A loss in net interest

income?
Dollar Interest-Sensitive Gap = ISA – ISL = $400 million − $325 million = $75 million
Relative IS Gap

=

Interest-Sensitivity Ratio

=

ISA
ISL

ISA – ISL
Bank Size
=

=

$400
$325

$75
$500

= 0.15

= 1.23

Here, the interest sensitivity gap is positive and asset sensitive as the interest sensitive assets are

greater than interest sensitive liabilities. Sparkle Savings Association, being an asset sensitive
financial firm, will have a positive relative IS gap and an interest-sensitivity ratio greater than 1.
In case of a positive IS gap, there will be a gain in net interest income if the market interest rates
are rising. For a positive IS gap, there will be a loss in net interest income, if the market interest
rates are falling.
7-11 Snowman Bank, N.A., has a portfolio of loans and securities expected to generate cash
inflows for the bank as follows:
Expected Cash Inflows of
Principal and Interest
Payments
$1,275,600
746,872
341,555
62,482
9,871

Annual Period in Which Cash Receipts
Are Expected
Current year
Two years from today
Three years from today
Four years from today
Five years from today

Deposits and money market borrowings are expected to require the following cash outflows:
Expected Cash Outflows of
Principal and Interest
Payments
$1,295,500
831,454

123,897

Annual Period during Which Cash
Payments Must Be Made
Current year
Two years from today
Three years from today

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

1,005
-----

Four years from today
Five years from today

If the discount rate applicable to the previous cash flows is 4.25 percent, what is the duration of
the Snowman’s portfolio of earning assets and of its deposits and money market borrowings?
What will happen to the bank's total returns, assuming all other factors are held constant, if
interest rates rise? If interest rates fall? Given the size of the duration gap you have calculated, in
what type of hedging should Snowman engage? Please be specific about the hedging transactions
needed and their expected effects.
Snowman has asset duration of:

( $1,275,600 × 1 ) + ( $746,872 × 2 ) + ( $341,555 × 3 )
DA =

=

+

( $62,482 × 4 )

+

( $9,871 × 5 )

(1 + 0.0425)
(1 + 0.0425)
(1 + 0.0425)
(1 + 0.0425)
(1 + 0.0425) 5
( $1,275,600 ) + ( $746,872 ) + ( $341,555 ) + ( $62,482 ) + ( $9,871 )
(1 + 0.0425)1 (1 + 0.0425) 2 (1 + 0.0425) 3 (1 + 0.0425) 4 (1 + 0.0425) 5
1

2

3

4

$3, 754,097 / $2,273,192= 1.6515 years

Snowman has a liability duration of:

( $1,295,500 × 1 ) + ( $831,454 × 2 ) + ( $123,897 × 3 )

DL =

+

( $1,005 × 4 )

(1 + 0.0425)
(1 + 0.0425)
(1 + 0.0425)
(1 + 0.0425) 4
( $1,295,500 ) + ( $831,454 ) + ( $123,897 ) + ( $1,005 )
(1 + 0.0425)1 (1 + 0.0425) 2 (1 + 0.0425) 3 (1 + 0.0425) 4
1

2

3

= $3,104,236 / $2,117,934 = 1.4657 years
Snowman's duration gap = Dollar-weighted duration of asset portfolio − Dollar-weighted
duration of liability portfolio = 1.6515 − 1.4657 = 0.1858 years.
Because Snowman's Asset Duration is greater than its Liability Duration, the bank has a positive
duration gap, which means that the bank's net worth will decrease if interest rates rise, because
the value of the liabilities will decline by less than the value of the assets. On the other hand, if
interest rates were to fall, this positive duration gap will increase the net worth. In this case, the
value of the assets will rise by a greater amount than the value of the liabilities.
Given the magnitude of the duration gap, the management of Snowman Bank, needs to do a
combination of things to close its duration gap between assets and liabilities. If the interest rates
are rising, it probably needs to try to shorten asset duration and lengthen liability duration to
move towards a negative duration gap. The opposite is true if interest rates are expected to fall.

The bank can use financial futures or options to deal with whatever asset-liability gap exists at
the moment. The bank may want to consider securitization or selling some of its assets,
reinvesting the cash flows in maturities that will more closely match its liabilities' maturities. The

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

bank may also consider negotiating some interest-rate swaps to change the cash flow patterns of
its liabilities to more closely match its asset maturities.
7-12. Given the cash inflow and outflow figures in Problem 11 for Snowman Bank, N.A.,
suppose that interest rates began at a level of 4.25 percent and then suddenly rise to 4.75 percent.
If the bank has total assets of $20 billion and total liabilities of $18 billion, by how much would
the value of Snowman’s net worth change as a result of this movement in interest rates? Suppose,
on the other hand, that interest rates decline from 4.25 percent to 3.5 percent. What happens to
the value of Snowman’s net worth in this case and by how much in dollars does it change? What
is the size of its duration gap?
From Problem #11 we find that Snowman's average asset duration is 1.6515 years and average
liability duration is 1.5223 years. If total assets are $20 billion and total liabilities are $18 billion,
then Snowman’ has a leverage-adjusted duration gap of:
1.6515 – 1.4657 ×

$18 bill.
= 0.1163
$20 bill.

The change in Snowman's net worth would be calculated as:

 


Δr
Δr
× A  −  −D L ×
× L
Change in Value of Net Worth =  − DA ×
(1+r)
(1+r)

 

If interest rates increase from 4.25 to 4.75 percent, change in net worth will be:

 

0.005
0.005
=  −1.6515 ×
× 20  −  −1.4657 ×
× 18
(1+0.0425)
(1+0.0425)

 

= −0.1584 − (−0.1265)
= − 0.0319 billion
There is a decrease in the net worth of Snowman with the increase in the interest rate.
If interest rates fall from 4.25 percent to 3.5 percent, change in net worth will be:


 

−0.0075
−0.0075
=  −1.6515 ×
× 20  −  −1.4657 ×
× 18
(1+0.0425)
(1+0.0425)

 

= 0.2376 − 0.1898
= + 0.0478 billion.
When the interest rates fall, Snowman’s net worth will increase.
7-13. Conway Thrift Association reports an average asset duration of 7 years and an average
liability duration of 4 years. In its latest financial report, the association recorded total assets of
$1.8 billion and total liabilities of $1.5 billion. If interest rates began at 5 percent and then
suddenly climbed to 6 percent, what change will occur in the value of Conway’s net worth? By

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

how much would Conway’s net worth change if, instead of rising, interest rates fell from 5
percent to 4.5 percent?
The key formula is:

 


Δr
Δr
× A  −  −D L ×
× L
Change in net worth =  − D A ×
(1+r)
(1+r)

 

For the change in interest rates from 5 to 6 percent, change in net worth will be:

 

0.01
0.01
=  −7 ×
× 1.8 −  −1.4657 ×
× 1.5
(1+0.05)
(1+0.05)

 

= – $0.12 billion – (–$0.05714 billion)
= – $0.06286 billion
On the other hand, if interest rates decline from 5 to 4.5 percent, change in net worth will be:

 


−0.005
−0.005
=  −7 ×
× 1.8 −  −1.4657 ×
× 1.5
(1+0.05)
(1+0.05)

 

= + $0.06 billion – $0.02857 billion
= + $0.03143 billion
7-14. A financial firm holds a bond in its investment portfolio whose duration is 15 years. Its
current market price is $975. While market interest rates are currently at 6 percent for
comparable quality securities, a decrease in interest rates to 5.75 percent is expected in the
coming weeks. What change (in percentage terms) will this bond’s price experience if market
interest rates change as anticipated?
Change in price is computed as follows:
∆P
∆r
≈ −D *
P
(1 + r)
−0.0025
= −15 ×
( 1 + 0.06 ) = 3.54 percent
This bond’s price will approximately increase by 3.54 percent or to $1,009.515.
7-15. A savings bank’s weighted average asset duration is 8 years. Its total liabilities amount to
$925 million, while its assets total 1.25 billion dollars. What is the dollar-weighted duration of

the bank’s liability portfolio if it has a zero leverage-adjusted duration gap?
Given the bank has a duration gap equal to zero:
Duration Gap = DA - DL ×
0 = 8 - DL ×

Total Liabilities
Total Assets

1.250
0.925

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

0 = 8 - DL × 1.351 or DL × 1.351 = 8
8
DL =
= 5.92 years
1.351
Hence, the dollar-weighted duration of the bank’s liability portfolio is 5.92 years.
7-16 Blue Moon National Bank holds assets and liabilities whose average durations and dollar
amounts are as shown in this table:
Asset and Liability Items
Investment Grade Bonds
Commercial Loans
Consumer Loans
Deposits
Nondeposit Borrowings

Avg. Duration
(years)
15.00
3.00
7.00
1.25
0.50

Dollar Amount
(millions)
$65.00
$400.00
$250.00
$600.00
$50.00

What is the weighted average duration of Blue Moon’s asset portfolio and liability portfolio?
What is the leverage-adjusted duration gap?
The weighted average duration of Blue Moon’s asset portfolio is calculated as follows:
65
400
250
D A = ∑ Wi × Di =
×15 +
×3+
× 7 = 1.3636 +1.6783 + 2.4476 = 5.4895 years
715
715
715

The weighted average duration of the liability portfolio is calculated as follows:
600
50
D L = ∑ Wi × Di =
×1.25 +
× 0.50 = 1.1538 + 0.0385 = 1.192 years
650
650
TL
650
= 5.4895 − 1.192 ×
= 4.4055 years
TA
715
Therefore, the leverage-adjusted duration gap is 4.4055 years.
Leverage-adjusted duration gap = D A − D L ×

7-17 A government bond currently carries a yield to maturity of 6 percent and a market price
of $1,168.49. If the bond promises to pay $100 in interest annually for five years, what is its
current duration?
The duration of the bond is computed as follows:
($100 × 1) ($100 × 2) ($100 × 3) ($100 × 4) ($1,100 × 5)
+
+
+
+
4,950.98
(1 + .06)1 (1 + .06) 2
(1 + .06)3
(1 + .06) 4

(1 + .06)5
D =
=
$100
$100
$100
$100
$1,100
1,168.5
+
+
+
+
1
2
3
4
5
(1 + .06) (1 + .06) (1 + .06) (1 + .06) (1 + .06)
Therefore, the current duration of the bond is 4.23 years.

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Chapter 07 - Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

7-18 Carter National Bank holds $15 million in government bonds having a duration of 12
years. If interest rates suddenly rise from 6 percent to 7 percent, what percentage change should
occur in the bonds’ market price?
The percentage change in market price is computed as follows:

∆P
∆r
≈ −D *
P
(1 + r)
.01
-12*
= -0.1132 or 11.32 percent
(1 + .06)
Therefore, the market price will change approximately by 11.32 percent.

7-19