Managing Interest Rate Risk:GAP and Earnings SensitivityChapter 5Bank Management, 6th edition. Timothy W. Koch and S. Scott MacDonaldCopyright 2006 by South-Western, a division of Thomson Learning

Interest Rate RiskInterest Rate RiskThe potential loss from unexpected changes in interest rates which can significantly alter a banks profitability and market value of equity.

Interest Rate Risk: GAP & Earnings SensitivityWhen a banks 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 stockholders equity

Interest Rate RiskBanks 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) targetingEarnings Sensitivity AnalysisEarnings 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 ALCOs primary responsibility is interest rate risk management. The ALCO coordinates the banks strategies to achieve the optimal risk/reward trade-off.

Two Types of Interest Rate RiskSpread Risk (reinvestment rate risk)Changes in interest rates will change the banks cost of funds as well as the return on their invested assets. They may change by different amounts.Price RiskChanges in interest rates may change the market values of the banks 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 banks interest income but also increases the banks interest expense.

Static GAP Analysis considers the impact of changing rates on the banks 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 GAPExample: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 banks initial spread is 4%.

What is the banks risk?

Sheet1

$10,000 Car loan

4 year Car loan at8.50%

1 year CD at4.50%

Spread4.00%

4 year Car Loan8.50%

1 Year CD4.50%

4.00%

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Measuring Interest Rate Risk with GAPTraditional Static GAP Analysis GAPt = RSAt -RSLtRSAtRate Sensitive AssetsThose assets that will mature or reprice in a given time period (t)RSLtRate Sensitive LiabilitiesThose liabilities that will mature or reprice in a given time period (t)

Measuring Interest Rate Risk with GAPTraditional Static GAP AnalysisWhat is the banks 1-year GAP with the auto loan?RSA1yr = $0RSL1yr = $10,000GAP1yr = $0 - $10,000 = -$10,000The banks one year funding GAP is -10,000If interest rates rise in 1 year, the banks margin will fall. The opposite is also true that if rates fall, the margin will rise.

Measuring Interest Rate Risk with GAPTraditional Static GAP AnalysisFunding GAPFocuses on managing net interest income in the short-runAssumes 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 AnalysisDevelop an interest rate forecastSelect a series of time buckets or intervals for determining when assets and liabilities will repriceGroup 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 maturesIt represents and interim, or partial, principal paymentIt can be repricedThe interest rate applied to the outstanding principal changes contractually during the intervalThe 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 repricedAny full or partial principal payments within 90 days will be repricedFloating and variable rate instrumentsIf 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 IncomeChanges in the level of interest ratesChanges in the composition of assets and liabilitiesChanges in the volume of earning assets and interest-bearing liabilities outstandingChanges in the relationship between the yields on earning assets and rates paid on interest-bearing liabilities

Factors Affecting Net Interest Income: An ExampleConsider the following balance sheet:

Sheet1

Expected Balance Sheet for Hypothetical Bank

AssetsYieldLiabilitiesCost

Rate sensitive$5008.0%$6004.0%

Fixed rate$35011.0%$2206.0%

Non earning$150$100

$920

Equity

$80

Total$1,000$1,000

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

NII = 78.5 - 37.2 = 41.3

NIM = 41.3 / 850 = 4.86%

GAP = 500 - 600 = -100

78.5

37.2

41.3

4.86%

-100

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Examine the impact of the following changesA 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 costsA proportionate doubling in size of the bank.

1% increase in short-term ratesWith a negative GAP, more liabilities than assets reprice higher; hence NII and NIM fall

Sheet1

Expected Balance Sheet for Hypothetical Bank

AssetsYieldLiabilitiesCost

Rate sensitive$5009.0%$6005.0%

Fixed rate$35011.0%$2206.0%

Non earning$150$100

$920

Equity

$80

Total$1,000$1,000

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

NII = 83.5 - 43.2 = 40.3

NIM = 40.3 / 850 = 4.74%

GAP = 500 - 600 = -100

83.5

43.2

40.3

4.74%

-100

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1% decrease in the spreadNII and NIM fall (rise) with a decrease (increase) in the spread. Why the larger change?

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Expected Balance Sheet for Hypothetical Bank

AssetsYieldLiabilitiesCost

Rate sensitive$5008.5%$6005.5%

Fixed rate$35011.0%$2206.0%

Non earning$150$100

$920

Equity

$80

Total$1,000$1,000

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

NII = 81 - 46.2 = 34.8

NIM = 34.8 / 850 = 4.09%

GAP = 500 - 600 = -100

81

46.2

34.8

4.09%

-100

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Changes in the Slope of the Yield CurveIf liabilities are short-term and assets are long-term, the spread will widen as the yield curve increases in slopenarrow when the yield curve decreases in slope and/or inverts

Proportionate doubling in sizeNII and GAP double, but NIM stays the same. What has happened to risk?

Sheet1

Expected Balance Sheet for Hypothetical Bank

AssetsYieldLiabilitiesCost

Rate sensitive$1,0008.0%$1,2004.0%

Fixed rate$70011.0%$4406.0%

Non earning$300$200

$1,840

Equity

$160

Total$2,000$2,000

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

NII = 157 - 74.4 = 82.6

NIM = 82.6 / 1700 = 4.86%

GAP = 1000 - 1200 = -200

157

74.4

82.6

4.86%

-200

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Changes in the Volume of Earning Assets and Interest-Bearing LiabilitiesNet 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 $260Although the banks GAP (and hence risk) is lower, NII is also lower.

Sheet1

Expected Balance Sheet for Hypothetical Bank

AssetsYieldLiabilitiesCost

Rate sensitive$5408.0%$5604.0%

Fixed rate$31011.0%$2606.0%

Non earning$150$100

$920

Equity

$80

Total$1,000$1,000

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

NII = 77.3 - 38 = 39.3

NIM = 39.3 / 850 = 4.62%

GAP = 540 - 560 = -20

77.3

38

39.3

4.62%

-20

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Changes in Portfolio Composition and RiskTo 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 GAPIf there is a parallel shift in the yield curve:

It is rare, however, when the yield curve shifts parallelIf 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?

Summary of GAP and the Change in NII

Sheet1

GAP Summary

GAPChange in Interest IncomeChange in Interest IncomeChange in Interest ExpenseChange in Net Interest Income

PositiveIncreaseIncrease>IncreaseIncrease

PositiveDecreaseDecrease>DecreaseDecrease

NegativeIncreaseIncrease 100,0001.94.012.97.91.227.9

FF purchased0.0

NOW9.69.6

Savings1.91.9

DD13.513.5

Other liabilities1.01.0

Equity7.07.0

Total Liab & Eq.5.011.030.324.43.04.821.5100.0

Periodic GAP1.34.0-20.3-14.46.030.2

Cumulative GAP1.35.3-15.0-29.4-23.46.8

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Rate sensit

Balance Sheet Comp. VariancesMayJuneDifference

Earning Assets

U.S. Trea. & agency$10.00$9.50$(0.50)

Money mkt inv$3.50$3.00$(0.50)

Municipals$11.50$11.500.0

FF sold & repo's$4.00$5.00$1.00

Comm loans$42.00$42.50$0.50

Install loans$13.80$13.800.0

$84.80$85.30

Liabilities and Equity

MMDAs$16.90$17.30$0.40

Super NOWs$2.20$2.200.0

CDs < $100,000$20.60$19.60$(1.00)

CDs > $100,000$29.10$30.10$1.00

NOWs & savings$9.30$9.300.0

DD & other$14.50$14.500.0

Equity$6.90$7.00$0.10

Total volume effect$99.50$100.00

Average Interest Rate Variances

Assets:

U.S. Trea. & agency9.15%9.25%0.10%

Money mkt inv9.65%9.70%0.05%

FF sold & repo's9.90%10.05%0.15%

Comm loans15.05%15.07%0.02%

Weighted Average13.39%

Liabilities and Equity:

MMDAs9.25%9.35%0.10%

Super NOWs8.00%8.10%0.10%

CDs < $100,00010.00%10.05%0.05%

CDs > $100,00010.35%10.60%0.25%

Weighted Average9.90%

Spread3.49%

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Variance

Interest Income and Expense Variances (30-day Difference)

Difference in Actual Dollars

Interest Income

U.S. Trea. & agency$(2,979.45)=9.5x0.0925x(30/360)

Money mkt inv$(3,842.47)-10x0.0915x(30/360)

FF sold & repo's$8,753.42

Comm loans$6,883.56

Total change$8,815.07

Interest Expense

MMDAs$4,463.01

Super NOWs$180.82

CDs < $100,000$(7,413.70)

CDs > $100,000$14,691.78

Total change$11,921.92

Net Interest Income Change$(3,106.85)

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Break even

Calculate Break Even Asset YieldAnnualized Average Rate

Rollover of RSA and RSL's$ amount

Rates Unchanged

Repriceable assets21,300,00014.10%

Repriceable liabilities16,000,0009.50%

GAP5,300,000

Interest income (next 30 days)246,847=21.3mx0.141x(30/360)

Interest expense (next 30days)124,932=16mx0.095x(30/360)

Net interest return121,915

Forecasted Break-even Yield on Assets

"New" Int exp. on existing RSL-2.00%122,3019.30%

Int exp on new money1.00 mill8,54810.40%

Target net spread on repriceables121,915

Required interest income252,764

Break even asset yield (annualied)252,764x(30/365) =13.81%

21300000+1000000(1-0.03)

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Advantages and Disadvantages of Static GAP AnalysisAdvantagesEasy to understandWorks well with small changes in interest ratesDisadvantagesEx-post measurement errorsIgnores the time value of moneyIgnores the cumulative impact of interest rate changesTypically considers demand deposits to be non-rate sensitiveIgnores embedded options in the banks assets and liabilities

Measuring Interest Rate Risk with the GAP RatioGAP Ratio = RSAs/RSLsA GAP ratio greater than 1 indicates a positive GAPA GAP ratio less than 1 indicates a negative GAP

What is the Optimal GAPThere 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 EarningsNeither 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 milBank 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

Establishing a Target GAP: An ExampleConsider 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 banks 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 = 0.25Managements 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 GAPMany 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 forecastsA bank has limited flexibility in adjusting its GAP; e.g., loan and deposit termsThere is no adjustment for the timing of cash flows or dynamics of the changing GAP position

Earnings Sensitivity AnalysisAllows 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 AnalysisForecast future interest rates Identify changes in the composition of assets and liabilities in different rate environmentsForecast when embedded options will be exercisedIdentify when specific assets and liabilities will reprice given the rate environment Estimate net interest income and net incomeRepeat 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 OptionsMany bank assets and liabilities contain different types of options, both explicit and implicit:Option to refinance a loanCall option on a federal agency bond the bank ownsDepositors have the option to withdraw funds prior to maturityCap (maximum) rate on a floating-rate loan

Earnings Sensitivity Analysis Recognizes that Different Interest Rates Change by Different Amounts at Different TimesIt 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 = -$10,000What if rates increased?1 year GAP Position

Change in RatesBaseChange in Rates-3-2-1GAP1yr+1+2+3-1,000-2,000-8,000-10,000-10,000-10,000-10,000Re-finance the auto loansAll CDs will mature

Sheet1

$10,000 Car loan

4 year Car loan at8.50%

1 year CD at4.50%

Spread4.00%

4 year Car Loan8.50%

1 Year CD4.50%

4.00%

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What about the 3 Month GAP Position?Base GAP3m = $10,000 - $10,000 = 0 3 Month GAP Position

Change in RatesBaseChange in Rates-3-2-1GAP3m+1+2+3+8,000+6,000+2,0000-1,000-3,000-6,000Re-finance auto loans, and less likely to pull CDsPeople will pull the CDs for higher returns

The implications of embedded optionsDoes 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 environmentStatic GAP analysis ignores these embedded options

Earnings Sensitivity Analysis (Base Case)ExampleAssets

Sheet1

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

Totalor LessMonthsMonthsYearsYearsYearsYearsYears

Loans

Prime Based100,000100,000

Equity Credit Lines25,00025,000

Fixed Rate >1 yr170,00018,00018,00036,00096,0002,000

Var Rate Mtg I Yr55,00013,75013,75027,500

30-Yr Fix Mortgage250,0005,1275,1299,32932,79228,916116,78951,918

Consumer100,0006,0006,00012,00048,00028,000

Credit Card25,0003,0003,0006,00013,000

Investments

Eurodollars80,00080,000

CMOs FixRate35,0002,8712,8725,22413,7905,2844,959

US Treasury75,0005,0005,00025,00040,000

Fed Funds Sold25,00025,000

Cash & Due From Banks15,00015,000

Loan Loss Reserve-15,000-15,000

Non-earning Assets60,00060,000

Total Assets1,000,000278,74853,751101,053228,582104,200121,74851,91860,000

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

Totalor LessMonthsMonthsYearsYearsYearsYearsYears

Deposits

MMDAs240,000240,000

Retail CDs400,00060,00060,00090,000160,00030,000

Savings35,00035,000

NOW40,00040,000

DDA Personal55,00055,000

Comm'l DDA60,00024,00036,000

Borrowings

TT&L25,00025,000

L-T notes FR50,00050,000

Fed Funds Purch0

NIR Liabilities30,00030,000

Capital65,00065,000

Tot Liab & Equity1,000,000349,00060,00090,000160,00030,00050,0000261,000

Swaps- Pay Fixed50,000-25,000-25,000

GAP-20,252-6,24911,05343,58249,20071,74851,918-201,000

CUMULATIVE GAP-20,252-26,501-15,44828,13477,334149,082201,0000

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Earnings Sensitivity Analysis (Base Case)ExampleLiabilities and GAP Measures

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

Totalor LessMonthsMonthsYearsYearsYearsYearsYears

Loans

Prime Based100,000100,000

Equity Credit Lines25,00025,000

Fixed Rate >1 yr170,00018,00018,00036,00096,0002,000

Var Rate Mtg I Yr55,00013,75013,75027,500

30-Yr Fix Mortgage250,0005,1275,1299,32932,79228,916116,78951,918

Consumer100,0006,0006,00012,00048,00028,000

Credit Card25,0003,0003,0006,00013,000

Investments

Eurodollars80,00080,000

CMOs FixRate35,0002,8712,8725,22413,7905,2844,959

US Treasury75,0005,0005,00025,00040,000

Fed Funds Sold25,00025,000

Cash & Due From Banks15,00015,000

Loan Loss Reserve-15,000-15,000

Non-earning Assets60,00060,000

Total Assets1,000,000278,74853,751101,053228,582104,200121,74851,91860,000

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

Totalor LessMonthsMonthsYearsYearsYearsYearsYears

Deposits

MMDAs240,000240,000

Retail CDs400,00060,00060,00090,000160,00030,000

Savings35,00035,000

NOW40,00040,000

DDA Personal55,00055,000

Comm'l DDA60,00024,00036,000

Borrowings

TT&L25,00025,000

L-T notes FR50,00050,000

Fed Funds Purch0

NIR Liabilities30,00030,000

Capital65,00065,000

Tot Liab & Equity1,000,000349,00060,00090,000160,00030,00050,0000261,000

Swaps- Pay Fixed50,000-25,000-25,000

GAP-20,252-6,24911,05343,58249,20071,74851,918-201,000

CUMULATIVE GAP-20,252-26,501-15,44828,13477,334149,082201,0000

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Interest Rate Forecasts

Earnings Sensitivity Analysis ResultsFor the bank:The embedded options can potentially alter the banks cash flowsInterest rates change by different amounts at different timesSummary results are known as Earnings-at-Risk or Net Interest Income Simulation

Earnings Sensitivity AnalysisEarnings-at-RiskThe 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 environmentsThe greater is the potential variation in earnings (earnings at risk), the greater is the amount of risk assumed by a bank , orThe greater is the maximum loss, the greater is risk

Income Statement GAPIncome Statement GAPForecasts the change in net interest income given a 1% rise or fall in the banks 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 RatioThe Earnings Change RatioThis 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.

Income Statement GAP

Sheet1

Amounts In ThousandsPrime Down 100bpPrime Up 100bp

BalanceECRtIncomeBalanceECRtIncome

SheetStatementSheetStatement

GAP*GAPGAP*GAP

Rate-Sensitive AssetsABA X BCDC x D

Loans

Fixed Rate$5,661100%$5,661$5,661100%$5,661

Floating Rate3,678100%3,6783,678100%3,678

Securities

Principal Cash Flows

Agencies20071%14220071%142

Agy Callables2,94071%2,08730060%180

CMO Fixed31558%1834151%21

Fed Funds Sold2,70096%2,5922,70096%2,592

Floating Rate

Total Rate-Sensitive Assets$15,494$14,343$12,580$12,274

Rate-Sensitive Liabilities

Savings$1,92575%$1,444$1,9255%$96

Money Mkt Accts11,00160%6,60111,00140%4,400

NOW2,19680%1,7572,19620%439

Fed Funds Purch/Repo096%0096%0

CDs - IOOM3,46885%2,9483,46885%2,948

CDs < 100M4,37084%3,6714,37084%3,671

Total Rate-Sensitive Liabilities$22,960$16,420$22,960$11,554

Rate Sensitivity Gap (Assets-Liab)($7,466)($2,077)($10,380)$719

Total Assets$29,909$29,909$29,909$29,909

GAP as a Percent of Total Assets-24.96%-6.94%-34.71%2.40%

Change in Net Interest Income($20.8)$7.2

Change in Net Interest Margin0.07%0.02%

Net Interest Margin5.20%5.20%

Percentage Change in Net Interest Margin1.34%0.46%

*One year balance sheet GAP includes all balances that may change in rate in the next 12 months.

tthe Earnings Change Ratio (ECR) is an estimate of the change in rate of a rate-sensitive instrument per 100bp move in prime.

Sheet2

Sheet3

Managing the GAP and Earnings Sensitivity RiskSteps to reduce riskCalculate 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 Banks Assets and Liabilities

ObjectiveApproachesReduce asset sensitivityBuy longer-term securities.Lengthen the maturities of loans.Move from floating-rate loans to term loans.Increase asset sensitivityBuy short-term securities.Shorten loan maturities.Make more loans on a floating-rate basis.Reduce liability sensitivityPay premiums to attract longer-term deposit instruments.Issue long-term subordinated debt.Increase liability sensitivityPay premiums to attract short-term deposit instruments.Borrow more via non-core purchased liabilities.

Managing Interest Rate Risk:Duration GAP and Economic Value of EquityChapter 6Bank Management, 6th edition. Timothy W. Koch and S. Scott MacDonaldCopyright 2006 by South-Western, a division of Thomson Learning

Measuring Interest Rate Risk with Duration GAPEconomic Value of Equity AnalysisFocuses on changes in stockholders equity given potential changes in interest ratesDuration GAP AnalysisCompares the price sensitivity of a banks 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 4Duration is a measure of the effective maturity of a security.Duration incorporates the timing and size of a securitys 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 VolatilityDuration as an Elasticity MeasureDuration versus MaturityConsider the cash flows for these two securities over the following time line

0

5

10

15

20

$1,000

0

5

900

10

15

20

1

$100

Duration versus MaturityThe maturity of both is 20 yearsMaturity does not account for the differences in timing of the cash flowsWhat is the effective maturity of both?The effective maturity of the first security is:(1,000/1,000) x 20 = 20 yearsThe effective maturity of the second security is: [(900/1,000) x 1]+[(100/1,000) x 20] = 2.9 yearsDuration is similar, however, it uses a weighted average of the present values of the cash flows

Duration versus MaturityDuration is an approximate measure of the price elasticity of demand

Duration versus MaturityThe longer the duration, the larger the change in price for a given change in interest rates.

Measuring DurationDuration is a weighted average of the time until the expected cash flows from a security will be received, relative to the securitys priceMacaulays Duration

Measuring DurationExampleWhat 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 bonds price is $951.96.

Measuring DurationExampleWhat is the duration of a bond with a $1,000 face value, 10% coupon, 3 years to maturity but the YTM is 5%?The bonds price is $1,136.16.

Measuring DurationExampleWhat is the duration of a bond with a $1,000 face value, 10% coupon, 3 years to maturity but the YTM is 20%?The bonds price is $789.35.

Measuring DurationExampleWhat 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

Duration and Modified DurationThe greater the duration, the greater the price sensitivityModified Duration gives an estimate of price volatility:

Effective DurationEffective DurationUsed to estimate a securitys price sensitivity when the security contains embedded options.Compares a securitys estimated price in a falling and rising rate environment.

Effective Duration

Where: Pi- = Price if rates fallPi+ = Price if rates riseP0 = Initial (current) pricei+ = Initial market rate plus the increase in ratei- = Initial market rate minus the decrease in rate

Effective DurationExampleConsider a 3-year, 9.4 percent semi-annual coupon bond selling for $10,000 par to yield 9.4 percent to maturity. Macaulays 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 DurationExampleAssume, 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 calledIf rates rise, the bond is unlikely to be called and the price will fall

Effective DurationExampleIf 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

Duration GAPDuration GAP ModelFocuses on either managing the market value of stockholders equityThe bank can protect EITHER the market value of equity or net interest income, but not bothDuration GAP analysis emphasizes the impact on equity

Duration GAPDuration GAP AnalysisCompares the duration of a banks assets with the duration of the banks liabilities and examines how the economic value stockholders equity will change when interest rates change.

Two Types of Interest Rate RiskReinvestment Rate RiskChanges in interest rates will change the banks cost of funds as well as the return on invested assetsPrice RiskChanges in interest rates will change the market values of the banks 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 futureAn increase in rates increases a banks return on assets but also increases the banks 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 ratesDuration GAP considers the impact of changing rates on the market value of equity

Reinvestment Rate Risk and Price RiskReinvestment Rate RiskIf interest rates rise (fall), the yield from the reinvestment of the cash flows rises (falls) and the holding period return (HPR) increases (decreases).Price riskIf interest rates rise (fall), the price falls (rises). Thus, if you sell the security prior to maturity, the HPR falls (rises).

Reinvestment Rate Risk and Price RiskIncreases 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.

Reinvestment Rate Risk and Price RiskAn 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 investors holding period equals the duration of the security.

Steps in Duration GAP AnalysisForecast 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 AssetsWeighted Average Duration of Bank Assets (DA)

Wherewi = Market value of asset i divided by the market value of all bank assetsDai = Macaulays duration of asset in = number of different bank assets

Weighted Average Duration of Bank LiabilitiesWeighted Average Duration of Bank Liabilities (DL)

Wherezj = Market value of liability j divided by the market value of all bank liabilitiesDlj= Macaulays duration of liability jm = number of different bank liabilities

Duration GAP and Economic Value of EquityLet MVA and MVL equal the market values of assets and liabilities, respectively. If:andDuration GAP

Then:

where y = the general level of interest rates

Duration GAP and Economic Value of EquityTo protect the economic value of equity against any change when rates change , the bank could set the duration gap to zero:

Hypothetical Bank Balance Sheet

DGAP

1ParYearsMarket

$1,000% CoupMat.YTMValueDur.

Assets00.00

Cash$100$100

Earning assets0.00.0000

3-yr Commercial loan$70012.00%312.00%$7002.691.8830357142857150.09333333333333334

6-yr Treasury bond$2008.00%68.00%$2004.990.99854200741561780.017777777777777778

Total Earning Assets$90011.11%$900

Non-cash earning assets0.00.0

Total assets$1,00010.00%$1,0002.88

Liabilities0.00.00

Interest bearing liabs.0.00.0000

1-yr Time deposit$6205.00%15.00%$6201.000.67391304347826080.03369565217391304

3-yr Certificate of deposit$3007.00%37.00%$3002.810.91565809810615670.022826086956521743

Tot. Int Bearing Liabs.$9205.65%$920

Tot. non-int. bearing0.00.0

Total liabilities$9205.65%$9201.59

Total equity$80$80

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

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Cha DGAP

Duration Gap Calculations

Duration of Assets2.882

Duration of Liabilities1.590

Liabilities / Assets ratio0.920

DGAP = Dur assets - (assets/liab)*Dur liab1.419

DGAP* = Dur assets - Dur liab1.462

Approximate Impact from a Change in Rates

Expected change in rates1.00%

Average rate on Earning Assets11.11%

Average rate on Int Bearing Liab5.65%

Approx Change in Mkt Value of Assets-25.93

Approx Change in Mkt Value of Liabilities-13.84

Approx Change in Mkt Value of Equity (Difference)-12.09

Approx Change in Mkt Value of Equity-12.77

using Duration GAP method

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Calculating DGAPDA($700/$1000)*2.69 + ($200/$1000)*4.99 = 2.88DL($620/$920)*1.00 + ($300/$920)*2.81 = 1.59DGAP 2.88 - (920/1000)*1.59 = 1.42 yearsWhat 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.

1 percent increase in all rates.

DGAP

1ParYearsMarket

$1,000% CoupMat.YTMValueDur.

Assets00.00

Cash$100$100

Earning assets0.00.0000

3-yr Commercial loan$70012.00%313.00%$6832.691.88400074846416080.10160244885174138

6-yr Treasury bond$2008.00%69.00%$1914.970.97360036307989670.019659843102640576

Total Earning Assets$90012.13%$875

Non-cash earning assets0.00.0

Total assets$1,00010.88%$9752.86

Liabilities0.00.00

Interest bearing liabs.0.00.0000

1-yr Time deposit$6205.00%16.00%$6141.000.67755695912674270.04065341754760451

3-yr Certificate of deposit$3007.00%38.00%$2922.810.90456238542203020.02579544326986065

Tot. Int Bearing Liabs.$9206.64%$906

Tot. non-int. bearing0.00.0

Total liabilities$9206.64%$9061.58

Total equity$80$68

Total liabs & equity$1,000$975

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Cha DGAP

Duration Gap Calculations

Duration of Assets2.858

Duration of Liabilities1.582

Liabilities / Assets ratio0.930

DGAP = Dur assets - (assets/liab)*Dur liab1.386

DGAP* = Dur assets - Dur liab1.472

Approximate Impact from a Change in Rates

Expected change in rates1.00%

Average rate on Earning Assets12.13%

Average rate on Int Bearing Liab6.64%

Approx Change in Mkt Value of Assets-24.84

Approx Change in Mkt Value of Liabilities-13.45

Approx Change in Mkt Value of Equity (Difference)-11.39

Approx Change in Mkt Value of Equity-12.05

using Duration GAP method

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Calculating DGAPDA($683/$974)*2.68 + ($191/$974)*4.97 = 2.86 DA($614/$906)*1.00 + ($292/$906)*2.80 = 1.58DGAP2.86 - ($906/$974) * 1.58= 1.36 yearsWhat 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:

Positive and Negative Duration GAPsPositive 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 DGAPIndicates 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

Sheet1

GAP Summary

GAPChange in Interest IncomeChange in Interest IncomeChange in Interest ExpenseChange in Net Interest Income

PositiveIncreaseIncrease>IncreaseIncrease

PositiveDecreaseDecrease>DecreaseDecrease

NegativeIncreaseIncreaseIncreaseIncrease

NegativeIncreaseDecrease