# Managing Interest Rate Risk: Duration GAP and Economic Value ...

Managing Interest Rate Risk(II): Duration GAP and Economic Value of Equity 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 banks total assets with the price sensitivity of its total liabilities to assess the impact of potential changes in interest rates on stockholders equity. Duration GAP Duration GAP Model Focuses on 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 Compares 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. 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 offbalance 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) n

DA w iDai Where i wi = Market value of asset i divided by the market value of all bank assets Da = Macaulays duration of asset i i n = number of different bank assets Weighted Average Duration of Bank Liabilities Weighted Average Duration of Bank Liabilities (DL) m DL z jDl j Where j zj = Market value of liability j divided by the market value of all bank liabilities Dl = Macaulays duration of liability j j m = number of different bank liabilities

Duration GAP and Economic Value of Equity Let MVA and MVL equal the market values of assets and liabilities, respectively. EVE E V E EVE M V A EVE M V L If: and Duration GAP D G A P D A - (M V L /M V A )D L Then: y EVE E V E - DGA P MVA (1 y ) where y = the general level of interest rates To 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 1 Assets Cash Earning assets 3-yr Commercial loan 6-yr Treasury bond 84 1

Total Earning Assets Non-cash earning assets 1 (1.12) Total assets D Liabilities Interest bearing liabs. 1-yr Time deposit 3-yr Certificate of deposit Tot. Int Bearing Liabs. Tot. non-int. bearing Total liabilities Total equity Total liabs & equity Par \$1,000 % Coup Years Mat. \$100 \$ \$ 700 12.00% 3 \$ 200 8.00% 6 84 3 \$ 84 9002 \$ 2 3

( 1 . 12 ) ( 1 . 12 ) \$ 1,000 700 \$ 620 \$ 300 \$ 920 \$ \$ 920 \$ 80 \$ 1,000 YTM Market Value 5.00% 7.00% 1 3

Dur. 100 12.00% \$ 700 8.00% \$ 200 700 3 \$ 900 11.11% (10.00% 1.12) 3 \$\$ 1,000 2.69 4.99 5.00% \$ 620 7.00% \$ 300 5.65% \$ 920 \$ 5.65% \$ 920 \$ 80 \$ 1,000 1.00 2.81 2.88 1.59 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. 1 percent increase in all rates. 1 Par \$1,000 % Coup Years Mat. Assets Cash \$ 100 Earning assets 3-yr Commercial loan \$ 700 12.00% 3 6-yr Treasury bond \$ 200

8.00% 6 Total Earning Assets \$ 900 3 84 Non-cash earning assets \$ PV t 1 t Total assets \$ 1,000 1.13 Liabilities Interest bearing liabs. 1-yr Time deposit 3-yr Certificate of deposit Tot. Int Bearing Liabs. Tot. non-int. bearing Total liabilities Total equity Total liabs & equity \$ 620 \$ 300 \$ 920 \$ \$ 920 \$ 80 \$ 1,000

5.00% 7.00% 1 3 YTM Market Value Dur. \$ 100 13.00% \$ 9.00% \$ 12.13% \$ 700 \$ 10.88%3 \$ 1.13 683 191 875 975 2.69 4.97 6.00% \$ 8.00% \$ 6.64% \$

\$ 6.64% \$ \$ \$ 614 292 906 906 68 975 1.00 2.81 2.86 1.58 Change in the Market Value of Equity y EVEEVE - DGAP[ ]MVA (1 y) In this case: .01 EVEEVE - 1.42[ ]\$ 1,000 \$ 12 .91 1 .10 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 DGAP Summary Positive Positive Change in Interest Rates Increase Decrease Decrease > Decrease Decrease Increase > Increase Increase

Negative Negative Increase Decrease Decrease < Decrease Increase Increase < Increase Decrease Zero Zero Increase Decrease Decrease = Decrease Increase = Increase DGAP Assets Liabilities Equity None None 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 / ( MVL/MVA) = 1.42 / (\$920 / \$1,000) = 1.54 years Immunized Portfolio 1 Par Years \$1,000 % Coup Mat. Assets Cash \$ 100 Earning assets 3-yr Commercial loan \$ 700 6-yr Treasury bond \$ 200 Total Earning Assets \$ 900 Non-cash earning assets\$ Total assets \$ 1,000 Liabilities Interest bearing liabs. 1-yr Time deposit \$ 340 3-yr Certificate of deposit\$ 300 6-yr Zero-coupon CD* \$ 444 Tot. Int Bearing Liabs. \$ 1,084

Tot. non-int. bearing \$ Total liabilities \$ 1,084 Total equity \$ 80 YTM Market Value \$ 12.00% 8.00% 5.00% 7.00% 0.00% 3 6 1 3 6 100 12.00% \$ 700 8.00% \$ 200 11.11% \$ 900 \$ 10.00% \$ 1,000 5.00% 7.00%

8.00% 6.57% \$ \$ \$ \$ \$ 6.57% \$ \$ DGAP = 2.88 0.92 (3.11) 0 Dur. 340 300 280 920 920 80 2.69 4.99 2.88 1.00 2.81 6.00 3.11 Immunized Portfolio with a 1% increase in rates 1

Par \$1,000 Assets Cash \$ 100.0 Earning assets 3-yr Commercial loan \$ 700.0 6-yr Treasury bond \$ 200.0 Total Earning Assets \$ 900.0 Non-cash earning assets\$ Total assets \$ 1,000.0 Liabilities Interest bearing liabs. 1-yr Time deposit \$ 340.0 3-yr Certificate of deposit\$ 300.0 6-yr Zero-coupon CD* \$ 444.3 Tot. Int Bearing Liabs. \$ 1,084.3 Tot. non-int. bearing \$ Total liabilities \$ 1,084.3 Total equity \$ 80.0 Years % Coup Mat. YTM

Market Value Dur. \$ 100.0 12.00% 8.00% 5.00% 7.00% 0.00% 3 6 1 3 6 13.00% \$ 683.5 9.00% \$ 191.0 12.13% \$ 874.5 \$ 10.88% \$ 974.5 6.00% 8.00% 9.00% 7.54% \$ 336.8 \$ 292.3 \$ 264.9 \$ 894.0 \$ 7.54% \$ 894.0

\$ 80.5 2.69 4.97 2.86 1.00 2.81 6.00 3.07 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. 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 Strengths and Weaknesses: DGAP and EVESensitivity 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 EVESensitivity 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

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