Managing Interest Rate Risk: GAP and Earnings Sensitivity Chapter 5 Bank Management 6th edition....

Managing Interest Rate Risk:GAP and Earnings Sensitivity

Chapter 5

Bank ManagementBank Management, 6th edition.6th edition.Timothy W. Koch and S. Scott MacDonaldTimothy W. Koch and S. Scott MacDonaldCopyright © 2006 by South-Western, a division of Thomson Learning

Interest Rate Risk

Interest Rate Risk The potential loss from unexpected

changes in interest rates which can significantly alter a bank’s profitability and market value of equity.

Interest Rate Risk: GAP & Earnings Sensitivity

When a bank’s assets and liabilities do not reprice at the same time, the result is a change in net interest income. The change in the value of assets and

the change in the value of liabilities will also differ, causing a change in the value of stockholder’s equity

Interest Rate Risk

Banks typically focus on either: Net interest income or The market value of stockholders' equity

GAP Analysis A static measure of risk that is commonly

associated with net interest income (margin) targeting

Earnings Sensitivity Analysis Earnings sensitivity analysis extends GAP

analysis by focusing on changes in bank earnings due to changes in interest rates and balance sheet composition

Asset and Liability Management Committee (ALCO)

The ALCO’s primary responsibility is interest rate risk management.

The ALCO coordinates the bank’s strategies to achieve the optimal risk/reward trade-off.

Two Types of Interest Rate Risk

Spread Risk (reinvestment rate risk) Changes in interest rates will change

the bank’s cost of funds as well as the return on their invested assets. They may change by different amounts.

Price Risk Changes in interest rates may change

the market values of the bank’s assets and liabilities by different amounts.

Interest Rate Risk: Spread (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, ceteris paribus,

increases a bank’s interest income but also increases the bank’s interest expense.

Static GAP Analysis considers the impact of changing rates on the bank’s net interest income.

Interest Rate Risk: Price Risk

If interest rates change, the market values of assets and liabilities also change. The longer is duration, the larger is 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.

Measuring Interest Rate Risk with GAP

Example: A bank makes a $10,000 four-year car

loan to a customer at fixed rate of 8.5%. The bank initially funds the car loan with a one-year $10,000 CD at a cost of 4.5%. The bank’s initial spread is 4%.

What is the bank’s risk?

4 year Car Loan 8.50%1 Year CD 4.50%

4.00%


Traditional Static GAP AnalysisGAPt = RSAt -RSLt

RSAt

Rate Sensitive Assets Those assets that will mature or reprice in

a given time period (t)

RSLt

Rate Sensitive Liabilities Those liabilities that will mature or reprice

in a given time period (t)


Traditional Static GAP Analysis What is the bank’s 1-year GAP with the

auto loan? RSA1yr = $0

RSL1yr = $10,000

GAP1yr = $0 - $10,000 = -$10,000 The bank’s one year funding GAP is -10,000 If interest rates rise in 1 year, the bank’s

margin will fall. The opposite is also true that if rates fall, the margin will rise.


Traditional Static GAP Analysis Funding GAP

Focuses on managing net interest income in the short-run

Assumes a ‘parallel shift in the yield curve,’ or that all rates change at the same time, in the same direction and by the same amount.Does this ever happen?

Traditional Static GAP Analysis Steps in GAP Analysis

Develop an interest rate forecast Select a series of “time buckets” or

intervals for determining when assets and liabilities will reprice

Group assets and liabilities into these “buckets ”

Calculate the GAP for each “bucket ” Forecast the change in net interest

income given an assumed change in interest rates

What Determines Rate Sensitivity (Ignoring Embedded Options)?

An asset or liability is considered rate sensitivity if during the time interval: It matures It represents and interim, or partial, principal

payment It can be repriced

The interest rate applied to the outstanding principal changes contractually during the interval

The outstanding principal can be repriced when some base rate of index changes and management expects the base rate / index to change during the interval

What are RSAs and RSLs?

Considering a 0-90 day “time bucket,” RSAs and RSLs include: Maturing instruments or principal payments

If an asset or liability matures within 90 days, the principal amount will be repriced

Any full or partial principal payments within 90 days will be repriced

Floating and variable rate instruments If the index will contractually change within 90

days, the asset or liability is rate sensitive The rate may change daily if their base rate

changes. Issue: do you expect the base rate to change?

Factors Affecting Net Interest Income

Changes in the level of interest rates Changes in the composition of assets

and liabilities Changes in the volume of earning

assets and interest-bearing liabilities outstanding

Changes in the relationship between the yields on earning assets and rates paid on interest-bearing liabilities

Factors Affecting Net Interest Income: An Example

Consider the following balance sheet:

Assets Yield Liabilities CostRate sensitive 500$ 8.0% 600$ 4.0%Fixed rate 350$ 11.0% 220$ 6.0%Non earning 150$ 100$

920$ Equity

80$ Total 1,000$ 1,000$

GAP = 500 - 600 = -100

NII = (0.08 x 500 + 0.11 x 350) - (0.04 x 600 + 0.06 x 220)

NIM = 41.3 / 850 = 4.86%NII = 78.5 - 37.2 = 41.3

Expected Balance Sheet for Hypothetical Bank

Examine the impact of the following changes

A 1% increase in the level of all short-term rates?

A 1% decrease in the spread between assets yields and interest costs such that the rate on RSAs increases to 8.5% and the rate on RSLs increase to 5.5%?

Changes in the relationship between short-term asset yields and liability costs

A proportionate doubling in size of the bank.

1% increase in short-term rates


920$ Equity

80$ Total 1,000$ 1,000$

GAP = 500 - 600 = -100

NII = (0.09 x 500 + 0.11 x 350) - (0.05 x 600 + 0.06 x 220)

NIM = 40.3 / 850 = 4.74%NII = 83.5 - 43.2 = 40.3


With a negative GAP, more liabilities than assets reprice higher; hence NII and NIM fall

1% decrease in the spread


920$ Equity

80$ Total 1,000$ 1,000$

GAP = 500 - 600 = -100

NII = (0.085 x 500 + 0.11 x 350) - (0.055 x 600 + 0.06 x 220)

NIM = 34.8 / 850 = 4.09%NII = 81 - 46.2 = 34.8


NII and NIM fall (rise) with a decrease (increase) in the spread. Why the larger change?

Changes in the Slope of the Yield Curve

If liabilities are short-term and assets are long-term, the spread will widen as the yield curve increases in

slope narrow when the yield curve

decreases in slope and/or inverts

Proportionate doubling in size

Assets Yield Liabilities CostRate sensitive 1,000$ 8.0% 1,200$ 4.0%Fixed rate 700$ 11.0% 440$ 6.0%Non earning 300$ 200$

1,840$ Equity

160$ Total 2,000$ 2,000$

GAP = 1000 - 1200 = -200

NII = (0.08 x 1000 + 0.11 x 700) - (0.04 x 1200 + 0.06 x 440)

NIM = 82.6 / 1700 = 4.86%NII = 157 - 74.4 = 82.6


NII and GAP double, but NIM stays the same. What has happened to risk?

Changes in the Volume of Earning Assets and Interest-Bearing Liabilities

Net interest income varies directly with changes in the volume of earning assets and interest-bearing liabilities, regardless of the level of interest rates

RSAs increase to $540 while fixed-rate assets decrease to $310 and RSLs decrease to $560 while fixed-rate liabilities increase to $260


920$ Equity

80$ Total 1,000$ 1,000$

GAP = 540 - 560 = -20

NII = (0.08 x 540 + 0.11 x 310) - (0.04 x 560 + 0.06 x 260)

NIM = 39.3 / 850 = 4.62%NII = 77.3 - 38 = 39.3


Although the bank’s GAP (and hence risk) is lower, NII is also lower.

Changes in Portfolio Composition and Risk

To reduce risk, a bank with a negative GAP would try to increase RSAs (variable rate loans or shorter maturities on loans and investments) and decrease RSLs (issue relatively more longer-term CDs and fewer fed funds purchased)

Changes in portfolio composition also raise or lower interest income and expense based on the type of change

Changes in Net Interest Income are directly proportional to the size of the GAP

If there is a parallel shift in the yield curve:

It is rare, however, when the yield curve shifts parallel If rates do not change by the same

amount and at the same time, then net interest income may change by more or less.

We can figure out how much. How?

expexp i GAP ΔNII

Summary of GAP and the Change in NII

GAPChange in

Interest Income

Change in Interest Income

Change in Interest Expense

Change in Net Interest

IncomePositive Increase Increase > Increase IncreasePositive Decrease Decrease > Decrease Decrease

Negative Increase Increase < Increase DecreaseNegative Decrease Decrease < Decrease Increase

Zero Increase Increase = Increase NoneZero Decrease Decrease = Decrease None

GAP Summary

Rate, Volume, and Mix Analysis

Banks often publish a summary of how net interest income has changed over time. They separate changes over time to:

shifts in assets and liability composition and volume

changes associated with movements in interest rates.

The purpose is to assess what factors influence shifts in net interest income over time.

Measuring Interest Rate Risk: Synovus

Interest earned on: Volume Yield/Rate Net Change Volume Yield/Rate Net ChangeTaxable loans, net 149,423$ (117,147) 32,276 161,222 36,390 197,612 Tax-exempt loans, net† 1,373 (586) 787 1,108 (450) 658 Taxable investment securities (5,313) (916) (6,229) 4,507 2,570 7,077 Tax-exempt investment securities† 2,548 74 2,622 2,026 (206) 1,820 Interest earning deposits with banks 223 (176) 47 28 48 76 Federal funds sold and securities purchased under resale agreements

406 (1,745) (1,339) 1,447 1,410 2,857

Mortgage loans held for sale 7,801 (1,680) 6,121 (113) 549 436 Total interest income 156,461 (122,176) 34,285 170,225 40,311 210,536

Interest paid on: Interest bearing demand deposits 6,074 (12,517) (6,443) 1,537 5,433 6,970 Money market accounts 21,380 (36,244) (14,864) 4,654 13,888 18,542 Savings deposits (369) (3,307) (3,676) (660) (67) (727) Time deposits 32,015 (22,545) 9,470 38,824 32,812 71,636 Federal funds purchased and securities sold under repurchase agreements

(6,165) (29,744) (35,909) 23,148 15,870 39,018

Other borrowed funds 21,318 (4,272) 17,046 21,960 3,361 25,321 Total interest expense 74,253 (108,629) (34,376) 89,463 71,297 160,760

Net interest income 82,208 (13,547) 68,661 80,762 (30,986) 49,776

2004 Compared to 2003 2003 Compared to 2002 Change Due to * Change Due to *

Interest Rate-Sensitivity Reports Classifies a bank’s assets and liabilities into time intervals according to the minimum number of days until each instrument is expected to be repriced.

GAP values are reported a periodic and cumulative basis for each time interval. Periodic GAP

Is the Gap for each time bucket and measures the timing of potential income effects from interest rate changes

Cumulative GAP It is the sum of periodic GAP's and measures aggregate

interest rate risk over the entire period Cumulative GAP is important since it directly measures

a bank’s net interest sensitivity throughout the time interval.

Measuring Interest Rate Risk with GAP1-7

Days8-30Days

31-90Days

91-180Days

181-365Days

Over1 year

Not RateSensitive Total

AssetsU.S. Treas & ag 0.7 3.6 1.2 0.3 3.7 9.5 MM Inv 1.2 1.8 3.0 Municipals 0.7 1.0 2.2 7.6 11.5 FF & Repo's 5.0 5.0 Comm loans 1.0 13.8 2.9 4.7 4.6 15.5 42.5 Install loans 0.3 0.5 1.6 1.3 1.9 8.2 13.8 Cash 9.0 9.0 Other assets 5.7 5.7 Total Assets 6.3 15.0 10.0 10.0 9.0 35.0 14.7 100.0

Liabilities and EquityMMDA 5.0 12.3 17.3 Super NOW 2.2 2.2 CD's < 100,000 0.9 2.0 5.1 6.9 1.8 2.9 19.6 CD's > 100,000 1.9 4.0 12.9 7.9 1.2 27.9 FF purchased - NOW 9.6 9.6 Savings 1.9 1.9 DD 13.5 13.5 Other liabilities 1.0 1.0 Equity 7.0 7.0 Total Liab & Eq. 5.0 11.0 30.3 24.4 3.0 4.8 21.5 100.0

Periodic GAP 1.3 4.0 -20.3 -14.4 6.0 30.2Cumulative GAP 1.3 5.3 -15.0 -29.4 -23.4 6.8

Advantages and Disadvantages of Static GAP Analysis

Advantages Easy to understand Works well with small changes in interest rates

Disadvantages Ex-post measurement errors Ignores the time value of money Ignores the cumulative impact of interest rate

changes Typically considers demand deposits to be

non-rate sensitive Ignores embedded options in the bank’s assets

and liabilities

Measuring Interest Rate Risk with the GAP Ratio

GAP Ratio = RSAs/RSLs A GAP ratio greater than 1 indicates a

positive GAP A GAP ratio less than 1 indicates a

negative GAP

What is the ‘Optimal GAP’

There is no general optimal value for a bank's GAP in all environments.

Generally, the farther a bank's GAP is from zero, the greater is the bank's risk.

A bank must evaluate its overall risk and return profile and objectives to determine its optimal GAP

GAP and Variability in Earnings

Neither the GAP nor GAP ratio provide direct information on the potential variability in earnings when rates change. Consider two banks, both with $500 million in total

assets. Bank A: $3 mil in RSAs and $2 mil in RSLs.

GAP = $1 mil and GAP ratio = 1.5 mil Bank B: $300 mil in RSAs and $200 mil RSLs.

GAP equals $100 mill and 1.5 GAP ratio. Clearly, the second bank assumes greater interest

rate risk because its net interest income will change more when interest rates change.

Link Between GAP and Net Interest Margin

Many banks will specify a target GAP to earning asset ratio in the ALCO policy statements

rates interest in change % Expected

NIM) tedNIM)(Expec in Change % (Allowable

assets Earning

Gap Target

Establishing a Target GAP: An Example

Consider a bank with $50 million in earning assets that expects to generate a 5% NIM.

The bank will risk changes in NIM equal to plus or minus 20% during the year Hence, NIM should fall between 4% and

6%.

Establishing a Target GAP: An Example (continued)

If management expects interest rates to vary up to 4 percent during the upcoming year, the bank’s ratio of its 1-year cumulative GAP (absolute value) to earning assets should not exceed 25 percent.

Target GAP/Earning assets = (.20)(0.05) / 0.04 = 0.25

Management’s willingness to allow only a 20 percent variation in NIM sets limits on the GAP, which would be allowed to vary from $12.5 million to $12.5 million, based on $50 million in earning assets.

Speculating on the GAP

Many bank managers attempt to adjust the interest rate risk exposure of a bank in anticipation of changes in interest rates.

This is speculative because it assumes that management can forecast rates better than the market.

Can a Bank Effectively Speculate on the GAP?

Difficult to vary the GAP and win as this requires consistently accurate interest rate forecasts

A bank has limited flexibility in adjusting its GAP; e.g., loan and deposit terms

There is no adjustment for the timing of cash flows or dynamics of the changing GAP position

Earnings Sensitivity Analysis

Allows management to incorporate the impact of different spreads between asset yields and liability interest costs when rates change by different amounts.

Steps to Earnings Sensitivity Analysis

Forecast future interest rates Identify changes in the composition of

assets and liabilities in different rate environments

Forecast when embedded options will be exercised

Identify when specific assets and liabilities will reprice given the rate environment

Estimate net interest income and net income Repeat the process to compare forecasts of

net interest income and net income across different interest rate environments.

Earnings Sensitivity Analysis and the Exercise of Embedded Options

Many bank assets and liabilities contain different types of options, both explicit and implicit: Option to refinance a loan Call option on a federal agency bond

the bank owns Depositors have the option to withdraw

funds prior to maturity Cap (maximum) rate on a floating-rate

loan

Earnings Sensitivity Analysis Recognizes that Different Interest Rates Change by Different Amounts at Different Times It is well recognized that banks are

quick to increase base loan rates but are slow to lower base loan rates when rates fall.

Recall the our example from before:

GAP1Yr = $0 - $10,000 = -$10,000 What if rates increased?

1 year GAP Position

4 year Car Loan 8.50%1 Year CD 4.50%

4.00%

Change in Rates Base Change in Rates

-3 -2 -1 GAP1yr +1 +2 +3

-1,000 -2,000 -8,000 -10,000 -10,000 -10,000 -10,000

Re-finance the auto loans All CD’s will mature

What about the 3 Month GAP Position?

Base GAP3m = $10,000 - $10,000 = 0

3 Month GAP Position

Change in Rates Base Change in Rates

-3 -2 -1 GAP3m +1 +2 +3

+8,000 +6,000 +2,000 0 -1,000 -3,000 -6,000Re-finance auto loans, and

less likely to “pull” CD’sPeople will “pull” the

CD’s for higher returns

The implications of embedded options

Does the bank or the customer determine when the option is exercised? How and by what amount is the bank being

compensated for selling the option, or how much must it pay to buy the option?

When will the option be exercised? This is often determined by the economic

and interest rate environment

Static GAP analysis ignores these embedded options

Earnings Sensitivity Analysis (Base Case)Example

Assets3 Months >3-6 >6-12 >1-3 >3-5 >5-10 >10-20 >20

Total or Less Months Months Years Years Years Years Years

LoansPrime Based 100,000 100,000Equity Credit Lines 25,000 25,000Fixed Rate >1 yr 170,000 18,000 18,000 36,000 96,000 2,000Var Rate Mtg I Yr 55,000 13,750 13,750 27,50030-Yr Fix Mortgage 250,000 5,127 5,129 9,329 32,792 28,916 116,789 51,918Consumer 100,000 6,000 6,000 12,000 48,000 28,000Credit Card 25,000 3,000 3,000 6,000 13,000

InvestmentsEurodollars 80,000 80,000CMOs FixRate 35,000 2,871 2,872 5,224 13,790 5,284 4,959US Treasury 75,000 5,000 5,000 25,000 40,000Fed Funds Sold 25,000 25,000

Cash & Due From Banks 15,000 15,000Loan Loss Reserve -15,000 -15,000Non-earning Assets 60,000 60,000 Total Assets 1,000,000 278,748 53,751 101,053 228,582 104,200 121,748 51,918 60,000

Earnings Sensitivity Analysis (Base Case)Example

Liabilities and GAP Measures3 Months >3-6 >6-12 >1-3 >3-5 >5-10 >10-20 >20

Total or Less Months Months Years Years Years Years Years

DepositsMMDAs 240,000 240,000Retail CDs 400,000 60,000 60,000 90,000 160,000 30,000Savings 35,000 35,000NOW 40,000 40,000DDA Personal 55,000 55,000Comm'l DDA 60,000 24,000 36,000

BorrowingsTT&L 25,000 25,000L-T notes FR 50,000 50,000Fed Funds Purch 0

NIR Liabilities 30,000 30,000Capital 65,000 65,000 Tot Liab & Equity 1,000,000 349,000 60,000 90,000 160,000 30,000 50,000 0 261,000

Swaps- Pay Fixed 50,000 -25,000 -25,000

GAP -20,252 -6,249 11,053 43,582 49,200 71,748 51,918 -201,000CUMULATIVE GAP -20,252 -26,501 -15,448 28,134 77,334 149,082 201,000 0

Interest Rate Forecasts

Most LikelyForecast andRate RampsDec. 20056

5

4

3

2

011 1

20063 5 7 9 11 1

20073 5 7 9 12

Fed Funds Forecast vs. Implied Forward Rates

Time (month)

4.50

4.25

4.00

3.75

3.50

3.25

3.001 3 5 7 9 11 13

Market Implied Rates

Most LikelyForecast

15 17 19 21 23

2

(.5)

1.0

.5

ALCO Guideline

Board Limit(1.0)

(1.5)

Cha

nge

inN

II($

MM

)

(2.0)

(2.5)

(3.0)- 300 -200 -100 +100 +200 +300ML

Ramped Change in Rates from Most Likely (Basis Points)

Sensitivity of Earnings: Year Two

1.0

.5

2

ALCO Guideline

Board Limit(1.0)

(.5)

(1.5)C

hang

ein

NII

($M

M)

(2.0)

(2.5)

(3.0)

(3.5)-300 -200 -100 +100 +200 +300ML

Ramped Change in Rates from Most Likely (Basis Point)

Sensitivity of Earnings: Year One

Earnings Sensitivity Analysis Results

For the bank: The embedded options can potentially

alter the bank’s cash flows Interest rates change by different

amounts at different times Summary results are known as

Earnings-at-Risk or Net Interest Income Simulation

Earnings Sensitivity Analysis

Earnings-at-Risk The potential variation in net interest income

across different interest rate environments, given different assumptions about balance sheet composition, when embedded options will be exercised, and the timing of repricings.

Demonstrates the potential volatility in earnings across these environments

The greater is the potential variation in earnings (earnings at risk), the greater is the amount of risk assumed by a bank , or

The greater is the maximum loss, the greater is risk

Income Statement GAP

Income Statement GAP Forecasts the change in net interest

income given a 1% rise or fall in the bank’s benchmark rate over the next year.

It converts contractual GAP data to figures evidencing the impact of a 1% rate movement.

Income statement GAP is also know in the industry as Beta GAP analysis

Income Statement GAP Adjusts the Balance Sheet GAP to Incorporate the Earnings Change Ratio

The Earnings Change Ratio This ratio indicates how the yield on

each asset and rate paid on each liability is assumed to change relative to a 1 percent move in the benchmark rate.

Balance Income Balance IncomeSheet Statement Sheet StatementGAP* GAP GAP* GAP

A B A X B C D C x D

Fixed Rate $5,661 100% $5,661 $5,661 100% $5,661 Floating Rate 3,678 100% 3,678 3,678 100% 3,678

Principal Cash FlowsAgencies 200 71% 142 200 71% 142 Agy Callables 2,940 71% 2,087 300 60% 180 CMO Fixed 315 58% 183 41 51% 21 Fed Funds Sold 2,700 96% 2,592 2,700 96% 2,592 Floating Rate

$15,494 $14,343 $12,580 $12,274

Savings $1,925 75% $1,444 $1,925 5% $96 Money Mkt Accts 11,001 60% 6,601 11,001 40% 4,400 NOW 2,196 80% 1,757 2,196 20% 439 Fed Funds Purch/Repo 0 96% 0 0 96% 0 CDs - IOOM 3,468 85% 2,948 3,468 85% 2,948 CDs < 100M 4,370 84% 3,671 4,370 84% 3,671

$22,960 $16,420 $22,960 $11,554

($7,466) ($2,077) ($10,380) $719

$29,909 $29,909 $29,909 $29,909 -24.96% -6.94% -34.71% 2.40%

($20.8) $7.2 0.07% 0.02%5.20% 5.20%1.34% 0.46%

Amounts In Thousands Prime Down 100bp Prime Up 100bp

ECRt ECRt

Rate-Sensitive AssetsLoans

Securities

Total Rate-Sensitive Assets

Rate-Sensitive Liabilities

Total Rate-Sensitive LiabilitiesRate Sensitivity Gap (Assets-Liab)Total Assets

Percentage Change in Net

GAP as a Percent of Total AssetsChange in Net Interest Change in Net Interest Net Interest Margin

Inco

me

Sta

tem

ent

GA

P

Managing the GAP and Earnings Sensitivity Risk

Steps to reduce risk Calculate periodic GAPs over short

time intervals. Fund repriceable assets with matching

repriceable liabilities so that periodic GAPs approach zero.

Fund long-term assets with matching noninterest-bearing liabilities.

Use off-balance sheet transactions to hedge.

Adjust the Effective Rate Sensitivity of a Bank’s Assets and Liabilities

Objective Approaches

Reduce asset sensitivity

Buy longer-term securities.Lengthen the maturities of loans.Move from floating-rate loans to term loans.

Increase asset sensitivity

Buy short-term securities.Shorten loan maturities.Make more loans on a floating-rate basis.

Reduce liability sensitivity

Pay premiums to attract longer-term deposit instruments.

Issue long-term subordinated debt.

Increase liability sensitivity

Pay premiums to attract short-term deposit instruments.

Borrow more via non-core purchased liabilities.

Managing Interest Rate Risk:Duration GAP and Economic Value of Equity

Chapter 6


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.

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.

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

0 5

900

10 15 201

$100

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 20 = 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


Duration is an approximate measure of the price elasticity of demand

Price in Change %

Demanded Quantity in Change % - Demand of Elasticity Price


The longer the duration, the larger the change in price for a given change in interest rates.

i)(1i

PP

- Duration

Pi)(1

iDuration - P

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

Security the of Pricer)+(1(t)CF

r)+(1CF

r)+(1(t)CF

=D

n

1=tt

t

k

1=tt

t

k

1=tt

t

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.

years 2.73 = 951.96

2,597.6

(1.12)1000

+ (1.12)

100(1.12)

31,000 +

(1.12)3100

+ (1.12)

2100+

(1.12)1100

D3

1=t3t

332

1

Measuring Duration


$1,000 face value, 10% coupon, 3 years to maturity but the YTM is 5%?The bond’s price is $1,136.16.

years2.75 = 1,136.16

3,127.31

1136.16(1.05)

3*1,000 +

(1.05)3*100

+ (1.05)

2*100+

(1.05)1*100

D332

1

Measuring Duration


$1,000 face value, 10% coupon, 3 years to maturity but the YTM is 20%?The bond’s price is $789.35.

years2.68 = 789.35

2,131.95

789.35(1.20)

3*1,000 +

(1.20)3*100

+ (1.20)

2*100+

(1.20)1*100

D332

1

Measuring Duration

Example What is the duration of a zero coupon

bond with a $1,000 face value, 3 years to maturity but the YTM is 12%?

By definition, the duration of a zero coupon bond is equal to its maturity

years3 = 711.78

2,135.34

(1.12) 1,000

(1.12)3*1,000

D

3

3

Duration and Modified Duration

The greater the duration, the greater the price sensitivity

Modified Duration gives an estimate of price volatility:

i Duration Modified - P

P

i)(1

Duration sMacaulay' Duration Modified

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.

Effective Duration

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

)i (iP

P P Duration Effective

-0

i-i

-

-

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.

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

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

5420

.0.044) .05$10,000(

9,847.72$ $10,000 Duration Effective

-

-

Duration GAP

Duration GAP Model Focuses on 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

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.

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

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

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

Reinvestment Rate Risk and Price Risk

Reinvestment Rate Risk If interest 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).


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.


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.

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.

Weighted Average Duration of Bank Assets

Weighted Average Duration of Bank Assets (DA)

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

n

iiiDawDA

Weighted Average Duration of Bank Liabilities

Weighted Average Duration of Bank Liabilities (DL)

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

m

jjjDlzDL

Duration GAP and Economic Value of Equity

Let MVA and MVL equal the market values of assets and liabilities, respectively.

If:

and

Duration GAP

Then:

where y = the general level of interest rates

L(MVL/MVA)D -DA DGAP

MVAy)(1

yDGAP- ΔEVE

ΔMVLΔMVAΔEVE

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:

MVAy)(1

yDGAP- ΔEVE

1 Par Years Market$1,000 % Coup Mat. YTM Value Dur.

AssetsCash $100 100$ Earning assets

3-yr Commercial loan 700$ 12.00% 3 12.00% 700$ 2.696-yr Treasury bond 200$ 8.00% 6 8.00% 200$ 4.99 Total Earning Assets 900$ 11.11% 900$ Non-cash earning assets -$ -$

Total assets 1,000$ 10.00% 1,000$ 2.88

LiabilitiesInterest bearing liabs.

1-yr Time deposit 620$ 5.00% 1 5.00% 620$ 1.003-yr Certificate of deposit 300$ 7.00% 3 7.00% 300$ 2.81 Tot. Int Bearing Liabs. 920$ 5.65% 920$ Tot. non-int. bearing -$ -$ Total liabilities 920$ 5.65% 920$ 1.59

Total equity 80$ 80$ Total liabs & equity 1,000$ 1,000$

Hypothetical Bank Balance Sheet

700)12.1(3700

)12.1(384

)12.1(284

)12.1(184

3321

D

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.


AssetsCash 100$ 100$ Earning assets


Total assets 1,000$ 10.88% 975$ 2.86


1-yr Time deposit 620$ 5.00% 1 6.00% 614$ 1.003-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$

1 percent increase in all rates.

3

3

1t t 1.13

700

1.13

84PV

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.

Change in the Market Value of Equity

In this case:

MVA]y)(1

yDGAP[- ΔEVE

91120001101

01.$,$]

.

.1.42[- ΔEVE

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).

DGAP Summary

Assets Liabilities Equity

Positive Increase Decrease > Decrease → DecreasePositive Decrease Increase > Increase → Increase

Negative Increase Decrease < Decrease → IncreaseNegative Decrease Increase < Increase → Decrease

Zero Increase Decrease = Decrease → NoneZero Decrease Increase = Increase → None

DGAP Summary

DGAPChange in

Interest Rates

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


AssetsCash 100$ 100$ Earning assets


Total assets 1,000$ 10.00% 1,000$ 2.88


1-yr Time deposit 340$ 5.00% 1 5.00% 340$ 1.003-yr Certificate of deposit 300$ 7.00% 3 7.00% 300$ 2.816-yr Zero-coupon CD* 444$ 0.00% 6 8.00% 280$ 6.00 Tot. Int Bearing Liabs. 1,084$ 6.57% 920$ Tot. non-int. bearing -$ -$ Total liabilities 1,084$ 6.57% 920$ 3.11

Total equity 80$ 80$

Immunized Portfolio

DGAP = 2.88 – 0.92 (3.11) ≈ 0


AssetsCash 100.0$ 100.0$ Earning assets

3-yr Commercial loan 700.0$ 12.00% 3 13.00% 683.5$ 2.696-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


1-yr Time deposit 340.0$ 5.00% 1 6.00% 336.8$ 1.003-yr Certificate of deposit 300.0$ 7.00% 3 8.00% 292.3$ 2.816-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$

Immunized Portfolio with a 1% increase in rates

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.

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.

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.

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

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.

Book Value Market Value Book Yield Duration*

LoansPrime 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% 1.1Var Rate Mtg 1 Yr $ 55,000 $ 54,725 6.90% 0.530-Year Mortgage $ 250,000 $ 245,000 7.60% 6.0Consumer Ln $ 100,000 $ 100,500 8.00% 1.9Credit Card $ 25,000 $ 25,000 14.00% 1.0Total Loans $ 725,000 $ 723,575 8.03% 2.6Loan Loss Reserve $ (15,000) $ 11,250 0.00% 8.0 Net Loans $ 710,000 $ 712,325 8.03% 2.5InvestmentsEurodollars $ 80,000 $ 80,000 5.50% 0.1CMO Fix Rate $ 35,000 $ 34,825 6.25% 2.0US 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% 6.5Non-int Rel Assets $ 60,000 $ 60,000 0.00% 8.0 Total Assets $ 100,000 $ 100,000 6.93% 2.6

First Savings Bank Economic Value of Equity Market Value/Duration Report as of 12/31/04 Most Likely Rate Scenario-Base Strategy

Ass

ets

Book Value Market Value Book Yield Duration*

DepositsMMDA $ 240,000 $ 232,800 2.25% -Retail CDs $ 400,000 $ 400,000 5.40% 1.1Savings $ 35,000 $ 33,600 4.00% 1.9NOW $ 40,000 $ 38,800 2.00% 1.9DDA Personal $ 55,000 $ 52,250 8.0Comm'l DDA $ 60,000 $ 58,200 4.8 Total Deposits $ 830,000 $ 815,650 1.6TT&L $ 25,000 $ 25,000 5.00% -L-T Notes Fixed $ 50,000 $ 50,250 8.00% 5.9Fed 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 Notionallnt Rate Swaps - $ 1,250 6.00% 2.8 50,000

Adjusted Equity $ 65,000 $ 83,813 7.9

First Savings Bank Economic Value of Equity Market Value/Duration Report as of 12/31/04 Most Likely Rate Scenario-Base Strategy

Liab

ilitie

s

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

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%

Sensitivity of EVE versus Most Likely (Zero Shock) Interest Rate Scenario

2

(10.0)

20.0

10.0 8.8 8.2

(8.2)

(20.4)

(36.6)

13.6

ALCO G uide lineBoard Limit(20.0)

(30.0)

Ch

ang

e in

EV

E (

mill

ion

s o

f d

olla

rs)

(40.0)-300 -200 -100 +100 +200 +3000

Shocks to Curre nt 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.

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

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.

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

Duration analysis takes a longer term view than static gap analysis

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

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

Gap and DGAP Management StrategiesExample

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

$60 $60

Two-Year Security

0 1 2

$55 ?

One-Year Security & then another One-Year Security


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%


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%

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.

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

Interest Rates and the Business CycleThe general level of interest rates and the shape of the yield curve appear to follow the U.S. business cycle.

In expansionary stages rates rise until they reach a peak as the Federal Reserve tightens credit availability.

Time

ExpansionContraction

Expansion

Long-TermRates

Short-TermRatesPeak

Trough

DATE WHEN 1-YEAR RATE FIRST EXCEEDS 10-YEAR RATE

LENGTH OF TIME UNTIL START OF NEXT RECESSION

Apr. ’68 20 months (Dec. ’69)Mar. ’73 8 months (Nov. ’73)Sept. ’78 16 months (Jan. ’80)Sept. ’80 10 months (July ’81)Feb. ’89 17 months (July ’90)Dec. ’00 15 months (March ’01)

The inverted yield curve has predicted the last five recessions

In contractionary stages rates fall until they reach a trough when the U.S. economy falls into recession.

Using Derivatives to Manage Interest Rate Risk

Chapter 7


Derivatives

A derivative is any instrument or contract that derives its value from another underlying asset, instrument, or contract.

Managing Interest Rate Risk

Derivatives Used to Manage Interest Rate Risk Financial Futures Contracts Forward Rate Agreements Interest Rate Swaps Options on Interest Rates

Interest Rate Caps Interest Rate Floors

Characteristics of Financial Futures

Financial Futures Contracts A commitment, between a buyer and a

seller, on the quantity of a standardized financial asset or index

Futures Markets The organized exchanges where

futures contracts are traded Interest Rate Futures

When the underlying asset is an interest-bearing security


Buyers A buyer of a futures contract is said to

be long futures Agrees to pay the underlying futures

price or take delivery of the underlying asset

Buyers gain when futures prices rise and lose when futures prices fall

Note that prices and interest rates move inversely, so buyers gain when rates fall.


Sellers A seller of a futures contract is said to

be short futures Agrees to receive the underlying

futures price or to deliver the underlying asset

Sellers gain when futures prices fall and lose when futures prices rise

The same for sellers, so they gain when rates rise.


Cash or Spot Market Market for any asset where the buyer

tenders payment and takes possession of the asset when the price is set

Forward Contract Contract for any asset where the buyer

and seller agree on the asset’s price but defer the actual exchange until a specified future date


Forward versus Futures Contracts Futures Contracts

Traded on formal exchanges Examples: Chicago Board of Trade and the

Chicago Mercantile Exchange Involve standardized instruments Positions require a daily marking to

market Positions require a deposit equivalent

to a performance bond


Forward versus Futures Contracts Forward contracts

Terms are negotiated between parties Do not necessarily involve

standardized assets Require no cash exchange until

expiration No marking to market

Types of Futures Traders

Speculator Takes a position with the objective of

making a profit Tries to guess the direction that prices

will move and time trades to sell (buy) at higher (lower) prices than the purchase price.


Hedger Has an existing or anticipated position in the

cash market and trades futures contracts to reduce the risk associated with uncertain changes in the value of the cash position

Takes a position in the futures market whose value varies in the opposite direction as the value of the cash position when rates change

Risk is reduced because gains or losses on the futures position at least partially offset gains or losses on the cash position.


Hedger versus Speculator The essential difference between a

speculator and hedger is the objective of the trader.

A speculator wants to profit on trades A hedger wants to reduce risk

associated with a known or anticipated cash position


Commission Brokers Execute trades for other parties

Locals Trade for their own account

Locals are speculators Scalper

A speculator who tries to time price movements over very short time intervals and takes positions that remain outstanding for only minutes


Day Trader Similar to a scalper but tries to profit

from short-term price movements during the trading day; normally offsets the initial position before the market closes such that no position remains outstanding overnight

Position Trader A speculator who holds a position for a

longer period in anticipation of a more significant, longer-term market move.


Spreader versus Arbitrageur Both are speculators that take

relatively low-risk positions Futures Spreader

May simultaneously buy a futures contract and sell a related futures contract trying to profit on anticipated movements in the price difference

The position is generally low risk because the prices of both contracts typically move in the same direction


Arbitrageur Tries to profit by identifying the same asset

that is being traded at two different prices in different markets at the same time

Buys the asset at the lower price and simultaneously sells it at the higher price

Arbitrage transactions are thus low risk and serve to bring prices back in line in the sense that the same asset should trade at the same price in all markets

Margin Requirements

Initial Margin A cash deposit (or U.S. government

securities) with the exchange simply for initiating a transaction

Initial margins are relatively low, often involving less than 5% of the underlying asset’s value

Maintenance Margin The minimum deposit required at the

end of each day

Margin Requirements

Unlike margin accounts for stocks, futures margin deposits represent a guarantee that a trader will be able to make any mandatory payment obligations

Same effect as a performance bond

Margin Requirements

Marking-to-Market The daily settlement process where at

the end of every trading day, a trader’s margin account is:

Credited with any gains Debited with any losses

Variation Margin The daily change in the value of margin

account due to marking-to-market

Expiration and Delivery

Expiration Date Every futures contract has a formal

expiration date On the expiration date, trading stops

and participants settle their final positions

Less than 1% of financial futures contracts experience physical delivery at expiration because most traders offset their futures positions in advance

Example

90-Day Eurodollar Time Deposit Futures The underlying asset is a Eurodollar

time deposit with a 3-month maturity. Eurodollar rates are quoted on an

interest-bearing basis, assuming a 360-day year.

Each Eurodollar futures contract represents $1 million of initial face value of Eurodollar deposits maturing three months after contract expiration.

Example

90-Day Eurodollar Time Deposit Futures Forty separate contracts are traded at

any point in time, as contracts expire in March, June, September and December each year

Buyers make a profit when futures rates fall (prices rise)

Sellers make a profit when futures rates rise (prices fall)

Example

90-Day Eurodollar Time Deposit Futures Contracts trade according to an index

that equals 100% - the futures interest rate

An index of 94.50 indicates a futures rate of 5.5 percent

Each basis point change in the futures rate equals a $25 change in value of the contract (0.001 x $1 million x 90/360)

The first column indicates the settlement month and year

Each row lists price and yield data for a distinct futures contract that expires sequentially every three months

The next four columns report the opening price, high and low price, and closing settlement price.

The next column, the change in settlement price from the previous day.

The two columns under Yield convert the settlement price to a Eurodollar futures rate as:100 - Settlement Price

= Futures Rate

Eurodollar Futures

Eurodollar (CME)-$1,000,000; pts of 100%

OPEN HIGH LOW SETTLE CHA YIELD CHA OPEN

INT Mar 96.98 96.99 96.98 96.99 — 3.91 — 823,734 Apr 96.81 96.81 96.81 96.81 _.01 3.19 .01 19,460 June 96.53 96.55 96.52 96.54 — 3.46 — 1,409,983 Sept 96.14 96.17 96.13 96.15 _.01 3.05 .01 1,413,496 Dec 95.92 95.94 95.88 95.91 _.01 4.09 .01 1,146,461 Mr06 95.78 95.80 95.74 95.77 _.01 4.23 .01 873,403 June 95.64 95.60 95.62 95.64 _.01 4.34 .01 567,637 Sept 95.37 95.58 95.53 95.54 _.01 4.44 .01 434,034 Dec 95.47 95.50 95.44 95.47 — 4.53 — 300,746 Mr07 95.42 95.44 95.37 95.42 — 4.58 — 250,271 June 95.31 95.38 95.31 95.37 .01 4.63 _.01 211,664 Sept 95.27 95.32 95.23 95.31 .02 4.69 _.02 164,295 Dec 95.21 95.27 95.18 95.26 .03 4.74 _.03 154,123 Mr08 95.16 95.23 95.11 95.21 .04 4.79 _.04 122,800 June 95.08 95.17 95.07 95.14 .05 4.84 _.05 113,790 Sept 95.03 95.13 95.01 95.11 .06 4.89 _.06 107,792 Dec 94.95 95.06 94.94 95.05 .07 4.95 _.07 96,046 Mr09 94.91 95.02 94.89 95.01 .08 4.99 _.07 81,015 June 94.05 94.97 94.84 94.97 .08 5.03 _.08 76,224 Sept 94.81 94.93 94.79 94.92 .08 5.08 _.08 41,524 Dec 94.77 94.38 94.74 94.87 .08 5.15 _.08 40,594 Mr10 94.77 94.64 94.70 94.83 .09 5.27 _.09 17,481 Sept 94.66 94.76 94.62 94.75 .09 5.25 _.09 9,309 Sp11 94.58 94.60 94.47 94.60 .09 5.40 _.09 2,583 Dec 94.49 94.56 94.43 94.56 .09 5.44 _.09 2,358 Mr12 94.48 94.54 94.41 94.53 .09 5.47 _.09 1,392 Est vol 2,082,746; vol Wed 1,519,709; open int 8,631,643, _160,422.

The Basis

The basis is the cash price of an asset minus the corresponding futures price for the same asset at a point in time For financial futures, the basis can be

calculated as the futures rate minus the spot rate

It may be positive or negative, depending on whether futures rates are above or below spot rates

May swing widely in value far in advance of contract expiration

4.50

4.09

3.00

1.76

1.090March 10, 2005 August 23, 2005 Expiration

December 20, 2005

Basis Futures Rate- Cash Rate

Cash Rate

December 2005 Futures Rate

Rate

(Per

cent

)

The Relationship Between Futures Rates and Cash Rates - One Possible Pattern on March 10

Speculation versus Hedging

A speculator takes on additional risk to earn speculative profits Speculation is extremely risky

A hedger already has a position in the cash market and uses futures to adjust the risk of being in the cash market The focus is on reducing or avoiding

risk


Example Speculating

You believe interest rates will fall, so you buy Eurodollar futures

If rates fall, the price of the underlying Eurodollar rises, and thus the futures contract value rises earning you a profit

If rates rise, the price of the Eurodollar futures contract falls in value, resulting in a loss


Example Hedging

A bank anticipates needing to borrow $1,000,000 in 60 days. The bank is concerned that rates will rise in the next 60 days

A possible strategy would be to short Eurodollar futures.

If interest rates rise (fall), the short futures position will increase (decrease) in value. This will (partially) offset the increase (decrease) in borrowing costs


With financial futures, risk often cannot be eliminated, only reduced. Traders normally assume basis risk in

that the basis might change adversely between the time the hedge is initiated and closed

Perfect Hedge The gains (losses) from the futures

position perfectly offset the losses (gains) on the spot position at each price

Profit Diagrams for the December 2005 Eurodollar Futures Contract: Mar 10, 2005

Profit

FuturesPrice95.91 95.91

A. Speculation

Loss

1. Buy Dec. 2005 Eurodollar Futures at $95.91

1

0

2

Profit

FuturesPrice

Loss

2. Sell Dec. 2005 Eurodollar Futures at $95.91

1

0

2

Profit

Price Price

Cash Futures

Futures

Hedge Result Hedge Result

Cash

95.91 95.91

B. Hedging

Loss

Hedge: Long Futures--Cash Loss WhenRates Fall

1

0

2

Profit

Loss

Hedge: Short Futures--Cash Loss WhenRates Rise

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Steps in Hedging

Identify the cash market risk exposure to reduce Given the cash market risk, determine whether a

long or short futures position is needed Select the best futures contract Determine the appropriate number of futures

contracts to trade. Buy or sell the appropriate futures contracts Determine when to get out of the hedge position,

either by reversing the trades, letting contracts expire, or making or taking delivery

Verify that futures trading meets regulatory requirements and the banks internal risk policies

A Long Hedge

A long hedge (buy futures) is appropriate for a participant who wants to reduce spot market risk associated with a decline in interest rates

If spot rates decline, futures rates will typically also decline so that the value of the futures position will likely increase.

Any loss in the cash market is at least partially offset by a gain in futures

Long Hedge Example

On March 10, 2005, your bank expects to receive a $1 million payment on November 8, 2005, and anticipates investing the funds in 3-month Eurodollar time deposits The cash market risk exposure is that the

bank will not have access to the funds for eight months.

In March 2005, the market expected Eurodollar rates to increase sharply as evidenced by rising futures rates.

Long Hedge Example

In order to hedge, the bank should buy futures contracts The best futures contract will generally

be the December 2005, 3-month Eurodollar futures contract, which is the first to expire after November 2005.

The contract that expires immediately after the known cash transactions date is generally best because its futures price will show the highest correlation with the cash price.

Long Hedge Example

The time line of the bank’s hedging activities would look something like this:

March 10, 2005 November 8, 2005 December 20, 2005

Cash: Anticipated investmentFutures: Buy a futures contract

Invest $1 millionSell the futures contract

Expiration of Dec. 2005 futures contract

Long Hedge Example

3.99%90

360

$1,000,000

$9,975return Effective

Date Cash Market Futures Market Basis

3/10/05 Bank anticipates investing $1 million Bank buys one December 2005 4.09% - 3.00% = 1.09%

(Initial futures in Eurodollars in 8 months; current Eurodollar futures contract at position) cash rate = 3.00% 4.09%; price = 95.91 11/8/05 Bank invests $1 million in 3-month Bank sells one December 2005

4.03% - 3.93% = 0.10% (Close futures Eurodollars at 3.93% Eurodollar futures contract at position) 4.03%; price = 95.97% Net effect Opportunity gain: Futures profit: Basis change: 0.10% - 1.09% 3.93% - 3.00% = 0.93%; 4.09% - 4.03% = 0.06%; = -0.99% 93 basis points worth 6 basis points worth $25 each = $2,325 $25 each = $150 Cumulativee investment income: Interest at 3.93% = $1,000,000(.0393)(90/360) = $9,825 Profit from futures trades = $ 150

Total = $9,975

A Short Hedge

A short hedge (sell futures) is appropriate for a participant who wants to reduce spot market risk associated with an increase in interest rates

If spot rates increase, futures rates will typically also increase so that the value of the futures position will likely decrease.

Any loss in the cash market is at least partially offset by a gain in the futures market

Short Hedge Example

On March 10, 2005, your bank expects to sell a six-month $1 million Eurodollar deposit on August 15, 2005 The cash market risk exposure is that

interest rates may rise and the value of the Eurodollar deposit will fall by August 2005

In order to hedge, the bank should sell futures contracts

Short Hedge Example

The time line of the bank’s hedging activities would look something like this:

March 10, 2005 August 17, 2005 September 20, 2005

Cash: Anticipated sale of investment

Futures: Sell a futures contract

Sell $1 million Eurodollar Deposit

Buy the futures contract

Expiration of Sept. 2005 futures contract

Short Hedge Example

Date Cash Market Futures Market Basis 3/10/05 Bank anticipates selling Bank sells one Sept. 3.85% - 3.00% = 0.85% $1 million Eurodollar 2005 Eurodollar futures deposit in 127 days; contract at 3.85%; current cash rate price = 96.15 = 3.00% 8/17/05 Bank sells $1 million Bank buys one Sept. 4.14% - 4.00% = 0.14% Eurodollar deposit at 2005 Eurodollar futures 4.00% contract at 4.14%; price = 95.86 Net result: Opportunity loss. Futures profit: Basis change: 0.14% - 0.85% 4.00% - 3.00% = 1.00%; 4.14% - 3.85% 3 0.29%; =-0.71% 100 basis points worth 29 basis points worth $25 each = $2,500 $25 each = $725 Effective loss = $2,500 - $725 = $1,775 Effective rate at sale of deposit = 4.00% - 0.29% = 3.71% or 3.00% - (0.71%) = 3.71%

Change in the Basis

Long and short hedges work well if the futures rate moves in line with the spot rate

The actual risk assumed by a trader in both hedges is that the basis might change between the time the hedge is initiated and closed In the long hedge position above, the

spot rate increased by 0.93% while the futures rate fell by 0.06%. This caused the basis to fall by 0.99% (The basis fell from 1.09% to 0.10%, or by 0.99%)

Change in the Basis

Effective Return from a Hedge Total income from the combined cash

and futures positions relative to the investment amount

Effective return Initial Cash Rate - Change in Basis

In the long hedge example: 3.00% - (-0.99%) = 3.99%

Basis Risk and Cross Hedging

Cross Hedge Where a trader uses a futures contract

based on one security that differs from the security being hedged in the cash market

Example Using Eurodollar futures to hedge changes

in the commercial paper rate Basis risk increases with a cross

hedge because the futures and spot interest rates may not move closely together

Microhedging Applications

Microhedge The hedging of a transaction

associated with a specific asset, liability or commitment

Macrohedge Taking futures positions to reduce

aggregate portfolio interest rate risk

Microhedging Applications

Banks are generally restricted in their use of financial futures for hedging purposes Banks must recognize futures on a

micro basis by linking each futures transaction with a specific cash instrument or commitment

Many analysts feel that such micro linkages force microhedges that may potentially increase a firm’s total risk because these hedges ignore all other portfolio components

Creating a Synthetic Liability with a Short Hedge

3/10/05 7/3/05 9/30/05

Six-Month Deposit

Time Line

Three-Month Cash Eurodollar

3.25%

SyntheticSix-Month Deposit

3.00% 3.88%-0.48% 3.40%

Three-Month Synthetic Eurodollar

Profit =

All In Six-Month Cost = 3.20%

Creating a Synthetic Liability with a Short Hedge

Summary of Relevant Eurodollar Rates and Transactions March 10, 2005 3-month cash rate = 3.00%; bank issues a $1 million, 91-day Eurodollar deposit 6-month cash rate = 3.25% Bank sells one September 2005 Eurodollar futures; futures rate = 3.85% July 3, 2005 3-month cash rate = 3.88%; bank issues a $1 million, 91-day Eurodollar deposit Buy: One September 2005 Eurodollar futures; futures rate = 4.33%

Date Cash Market Futures Market Basis 3/10/05 Bank issues $1 million, 91-day Eurodollar time deposit Bank sells one September 2005 0.85%

at 3.00%; 3-mo. interest expense = $7,583. Eurodollar futures contract at 3.85% 7/3/05 Bank issues $1 million, 91-day Eurodollar time deposit Bank buys one September 2005 0.45%

at 3.88%; 3-mo. interest expense = $9,808 (increase in interest expense over previous period = $2,225).

Eurodollar futures contract at 4.33%; Net effect: 6-mo. interest expense = $17,391 Profit on futures = $1,200

3.20%182

360

$1,000,000

$1,200-$17,391cost borrowing Effective

Interest on 6-month Eurodollar deposit issued March 10 = $13,144 at 3.25%; vs. 3.20% from synthetic liability

The Mechanics of Applying a Microhedge

1. Determine the bank’s interest rate position

2. Forecast the dollar flows or value expected in cash market transactions

3. Choose the appropriate futures contract


4. Determine the correct number of futures contracts

Where NF = number of futures contracts A = Dollar value of cash flow to be hedged F = Face value of futures contract Mc = Maturity or duration of anticipated cash

asset or liability Mf = Maturity or duration of futures contract

bMfF

Mc ANF

contract futures onmovement rate Expected

instrument cash onmovement rate Expected b


5. Determine the Appropriate Time Frame for the Hedge

6. Monitor Hedge Performance

Macrohedging

Macrohedging Focuses on reducing interest rate risk

associated with a bank’s entire portfolio rather than with individual transactions

Macrohedging

Hedging: GAP or Earnings Sensitivity If GAP is positive, the bank is asset sensitive

and its net interest income rises when interest rates rise and falls when interest rates fall

If GAP is negative, the bank is liability sensitive and its net interest income falls when interest rates rise and rises when interest rates fall

Positive GAP Use a long hedge

Negative GAP Use a short hedge

Hedging: GAP or Earnings Sensitivity

Positive GAP Use a long hedge

If rates rise, the bank’s higher net interest income will be offset by losses on the futures position

If rates fall, the bank’s lower net interest income will be offset by gains on the futures position

Hedging: GAP or Earnings Sensitivity

Negative GAP Use a short hedge

If rates rise, the bank’s lower net interest income will be offset by gains on the futures position

If rates fall, the bank’s higher net interest income will be offset by losses on the futures position

Hedging: Duration GAP and EVE Sensitivity

To eliminate interest rate risk, a bank could structure its portfolio so that its duration gap equals zero

MVA]y)(1

yDGAP[- ΔEVE


Futures can be used to adjust the bank’s duration gap The appropriate size of a futures

position can be determined by solving the following equation for the market value of futures contracts (MVF), where DF is the duration of the futures contract

0i1

DF(MVF)

i1

DL(MVRSL)

i1

DA(MVRSA)

fla


Example: A bank has a positive duration gap of

1.4 years, therefore, the market value of equity will decline if interest rates rise. The bank needs to sell interest rate futures contracts in order to hedge its risk position

The short position indicates that the bank will make a profit if futures rates increase


Example: Assume the bank uses a Eurodollar

futures contract currently trading at 4.9% with a duration of 0.25 years, the target market value of futures contracts (MVF) is:

MVF = $4,024.36, so the bank should sell four Eurodollar futures contracts

0 (1.049)

0.25(MVF)

(1.06)

1.61($920)

(1.10)

2.88($900)


Example: If all interest rates increased by 1%, the

profit on the four futures contracts would total 4 x 100 x $25 = $10,000, which partially offset the $12,000 decrease in the economic value of equity associated with the increase in cash rates

Recall from Exhibit 6.2, the unhedged bank had a reduction in EVE of $12,000

Accounting Requirements and Tax Implications

Regulators generally limit a bank’s use of futures for hedging purposes If a bank has a dealer operation, it can use

futures as part of its trading activities In such accounts, gains and losses on these

futures must be marked-to-market, thereby affecting current income

Microhedging To qualify as a hedge, a bank must show that

a cash transaction exposes it to interest rate risk, a futures contract must lower the bank’s risk exposure, and the bank must designate the contract as a hedge

Using Forward Rate Agreements to Manage Interest Rate Risk

Forward Rate Agreements A forward contract based on interest rates based on a

notional principal amount at a specified future date Buyer

Agrees to pay a fixed-rate coupon payment (at the exercise rate) and receive a floating-rate payment

Seller Agrees to make a floating-rate payment and receive a

fixed-rate payment The buyer and seller will receive or pay cash when

the actual interest rate at settlement is different than the exercise rate

Forward Rate Agreements (FRA)

Similar to futures but differ in that they: Are negotiated between parties Do not necessarily involve

standardized assets Require no cash exchange until

expiration There is no marking-to-market

No exchange guarantees performance

Notional Principal

The two counterparties to a forward rate agreement agree to a notional principal amount that serves as a reference figure in determining cash flows. Notional

Refers to the condition that the principal does not change hands, but is only used to calculate the value of interest payments.

Notional Principal

Buyer Agrees to pay a fixed-rate coupon

payment and receive a floating-rate payment against the notional principal at some specified future date.

Seller Agrees to pay a floating-rate payment

and receive the fixed-rate payment against the same notional principal.

Example: Forward Rate Agreements

Suppose that Metro Bank (as the seller) enters into a receive fixed-rate/pay floating-rating forward rate agreement with County Bank (as the buyer) with a six-month maturity based on a $1 million notional principal amount

The floating rate is the 3-month LIBOR and the fixed (exercise) rate is 7%


Metro Bank would refer to this as a “3 vs. 6” FRA at 7 percent on a $1 million notional amount from County Bank

The phrase “3 vs. 6” refers to a 3-month interest rate observed three months from the present, for a security with a maturity date six months from the present

The only cash flow will be determined in six months at contract maturity by comparing the prevailing 3-month LIBOR with 7%


Assume that in three months 3-month LIBOR equals 8% In this case, Metro Bank would receive from

County Bank $2,451. The interest settlement amount is $2,500:

Interest = (.08 - .07)(90/360) $1,000,000 = $2,500. Because this represents interest that would

be paid three months later at maturity of the instrument, the actual payment is discounted at the prevailing 3-month LIBOR:

Actual interest = $2,500/[1+(90/360).08]=$2,451


If instead, LIBOR equals 5% in three months, Metro Bank would pay County Bank: The interest settlement amount is $5,000

Interest = (.07 -.05)(90/360) $1,000,000 = $5,000 Actual interest = $5,000 /[1 + (90/360).05] = $4,938


The FRA position is similar to a futures position County Bank would pay

fixed-rate/receive floating-rate as a hedge if it was exposed to loss in a rising rate environment.

This is analogous to a short futures position


The FRA position is similar to a futures position Metro Bank would take its position as a

hedge if it was exposed to loss in a falling (relative to forward rate) rate environment.

This is analogous to a long futures position

Basic Interest Rate Swaps

Basic or Plain Vanilla Interest Rate Swap An agreement between two parties to

exchange a series of cash flows based on a specified notional principal amount

Two parties facing different types of interest rate risk can exchange interest payments


Basic or Plain Vanilla Interest Rate Swap One party makes payments based on a

fixed interest rate and receives floating rate payments

The other party exchanges floating rate payments for fixed-rate payments

When interest rates change, the party that benefits from a swap receives a net cash payment while the party that loses makes a net cash payment


Conceptually, a basic interest rate swap is a package of FRAs As with FRAs, swap payments are

netted and the notional principal never changes hands


Using data for a 2-year swap based on 3-month LIBOR as the floating rate This swap involves eight quarterly

payments. Party FIX agrees to pay a fixed rate Party FLT agrees to receive a fixed rate

with cash flows calculated against a $10 million notional principal amount


Firms with a negative GAP can reduce risk by making a fixed-rate interest payment in exchange for a floating-rate interest receipt

Firms with a positive GAP take the opposite position, by making floating-interest payments in exchange for a fixed-rate receipt


Basic interest rate swaps are used to: Adjust the rate sensitivity of an asset

or liability For example, effectively converting a

fixed-rate loan into a floating-rate loan Create a synthetic security

For example, enter into a swap instead of investing in a security

Macrohedge Use swaps to hedge the bank’s

aggregate interest rate risk


Swap Dealers Handle most swap transactions Make a market in swap contracts Offer terms for both fixed-rate and

floating rate payers and earn a spread for their services


Comparing Financial Futures, FRAs, and Basic Swaps

There is some credit risk with swaps in that the counterparty may default on the exchange of the interest payments Only the interest payment exchange is

at risk, not the principal

Objective Financial Futures FRAs & Basic SwapsProfit If Rates Rise Sell Futures Pay Fixed, Receive FloatingProfit If Rates Fall Buy Futures Pay Floating, Receive Fixed

Position

Interest Rate Caps and Floors

Interest Rate Cap An agreement between two

counterparties that limits the buyer’s interest rate exposure to a maximum limit

Buying a interest rate cap is the same as purchasing a call option on an interest rate

Bu

yin

g a

Cap

on

3-M

onth

LIB

OR

at

4 p

erce

nt

4 Percent

A. Cap5Long Call Option on Three-Month LIBOR

Dollar Payout(Three-month LIBOR

-4%)3 NotionalPrincipal Amount

1C

Three-MonthLIBOR

ValueDate

ValueDate

ValueDate

Time

B. Cap Payoff: Strike Rate5 4 Percent*

ValueDate

ValueDate

FloatingRate


Interest Rate Floor An agreement between two

counterparties that limits the buyer’s interest rate exposure to a minimum rate

Buying an interest rate floor is the same as purchasing a put option on an interest rate

Bu

yin

g a

Flo

or o

n 3

-Mon

th L

IBO

R a

t 4

per

cen

t

4 Percent

A. Floor= Long Put Option on Three-Month LIBOR

Dollar Payout(4%- Three-monthLIBOR)X NotionalPrincipal Amount

1P

Three-MonthLIBOR

ValueDate

ValueDate

ValueDate

Time

B. Floor Payoff: Strike Rate= 4 Percent*

ValueDate

ValueDate

FloatingRate


Interest Rate Collar The simultaneous purchase of an

interest rate cap and sale of an interest rate floor on the same index for the same maturity and notional principal amount

A collar creates a band within which the buyer’s effective interest rate fluctuates

It protects a bank from rising interest rates


Zero Cost Collar A collar where the buyer pays no net

premium The premium paid for the cap equals

the premium received for the floor Reverse Collar

Buying an interest rate floor and simultaneously selling an interest rate cap

It protects a bank from falling interest rates

Pricing Interest Rate Caps and Floors

The size of the premiums for caps and floors is determined by: The relationship between the strike

rate an the current index This indicates how much the index

must move before the cap or floor is in-the-money

The shape of yield curve and the volatility of interest rates

With an upward sloping yield curve, caps will be more expensive than floors

Pricing Interest Rate Caps and Floors

Term Bid Offer Bid Offer Bid OfferCaps1 year 24 30 3 7 1 22 years 51 57 36 43 10 153 years 105 115 74 84 22 295 years 222 240 135 150 76 57 years 413 433 201 324 101 11610 years 549 573 278 308 157 197

Floors1 year 1 2 15 19 57 552 years 1 6 31 37 84 913 years 7 16 40 49 128 1375 years 24 39 75 88 190 2057 years 38 60 92 106 228 25010 years 85 115 162 192 257 287

1.50% 2.00% 2.50%

A. Caps/Floors

4.00% 5.00% 6.00%

Managing Interest Rate Risk: GAP and Earnings Sensitivity Chapter 5 Bank Management 6th edition....

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Transcript of Managing Interest Rate Risk: GAP and Earnings Sensitivity Chapter 5 Bank Management 6th edition....