1. Prof. Dr. Rainer Stachuletz Banking Academy of Vietnam
Based upon: Bank Management, 6th edition. Timothy W. Koch and S. Scott MacDonald
Managing Interest Rate Risk: Duration GAP and Economic Value of Equity
Chapter 6
Prof. Dr. Rainer Stachuletz – Banking Academy of Vietnam - Hanoi

2. Measuring Interest Rate Risk with Duration GAP
Economic Value of Equity Analysis
Focuses on changes in stockholders’ equity given potential changes in interest rates
Duration GAP Analysis
Compares the price sensitivity of a bank’s total assets with the price sensitivity of its total liabilities to assess the impact of potential changes in interest rates on stockholders’ equity.

3. Recall from Chapter 4
Duration is a measure of the effective maturity of a security.
Duration incorporates the timing and size of a security’s cash flows.
Duration measures how price sensitive a security is to changes in interest rates.
The greater (shorter) the duration, the greater (lesser) the price sensitivity.

4. Duration and Price Volatility
Duration as an Elasticity Measure
Duration versus Maturity
Consider the cash flows for these two securities over the following time line

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

6. Duration versus Maturity
Duration is an approximate measure of the price elasticity of demand

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

8. Measuring Duration
Duration is a weighted average of the time until the expected cash flows from a security will be received, relative to the security’s price
Macaulay’s Duration

9. Measuring Duration Example
What is the duration of a bond with a $1,000 face value, 10% annual coupon payments, 3 years to maturity and a 12% YTM? The bond’s price is $

10. Measuring Duration Example
What is the duration of a bond with a $1,000 face value, 10% coupon, 3 years to maturity but the YTM is 5%?The bond’s price is $1,

11. Measuring Duration Example
What is the duration of a bond with a $1,000 face value, 10% coupon, 3 years to maturity but the YTM is 20%?The bond’s price is $

12. 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

13. Duration and Modified Duration
The greater the duration, the greater the price sensitivity
Modified Duration gives an estimate of price volatility:

14. Effective Duration Effective Duration
Used to estimate a security’s price sensitivity when the security contains embedded options.
Compares a security’s estimated price in a falling and rising rate environment.

15. Effective Duration Where: Pi- = Price if rates fall
Pi+ = Price if rates rise
P0 = Initial (current) price
i+ = Initial market rate plus the increase in rate
i- = Initial market rate minus the decrease in rate

16. Effective Duration Example
Consider a 3-year, 9.4 percent semi-annual coupon bond selling for $10,000 par to yield 9.4 percent to maturity.
Macaulay’s Duration for the option-free version of this bond is 5.36 semiannual periods, or 2.68 years.
The Modified Duration of this bond is 5.12 semiannual periods or 2.56 years.

17. Effective Duration Example
Assume, instead, that the bond is callable at par in the near-term .
If rates fall, the price will not rise much above the par value since it will likely be called
If rates rise, the bond is unlikely to be called and the price will fall

18. Effective Duration Example
If rates rise 30 basis points to 5% semiannually, the price will fall to $9,
If rates fall 30 basis points to 4.4% semiannually, the price will remain at par

19. 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

20. Duration GAP Duration GAP Analysis
Compares the duration of a bank’s assets with the duration of the bank’s liabilities and examines how the economic value stockholders’ equity will change when interest rates change.

21. Two Types of Interest Rate Risk
Reinvestment Rate Risk
Changes in interest rates will change the bank’s cost of funds as well as the return on invested assets
Price Risk
Changes in interest rates will change the market values of the bank’s assets and liabilities

22. Reinvestment Rate Risk
If interest rates change, the bank will have to reinvest the cash flows from assets or refinance rolled-over liabilities at a different interest rate in the future
An increase in rates increases a bank’s return on assets but also increases the bank’s cost of funds

23. Price Risk
If interest rates change, the value of assets and liabilities also change.
The longer the duration, the larger the change in value for a given change in interest rates
Duration GAP considers the impact of changing rates on the market value of equity

24. Reinvestment Rate Risk and Price Risk
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).

25. Reinvestment Rate Risk and Price Risk
Increases in interest rates will increase the HPR from a higher reinvestment rate but reduce the HPR from capital losses if the security is sold prior to maturity.
Decreases in interest rates will decrease the HPR from a lower reinvestment rate but increase the HPR from capital gains if the security is sold prior to maturity.

26. Reinvestment Rate Risk and Price Risk
An immunized security or portfolio is one in which the gain from the higher reinvestment rate is just offset by the capital loss.
For an individual security, immunization occurs when an investor’s holding period equals the duration of the security.

27. Steps in Duration GAP Analysis
Forecast interest rates.
Estimate the market values of bank assets, liabilities and stockholders’ equity.
Estimate the weighted average duration of assets and the weighted average duration of liabilities.
Incorporate the effects of both on- and off-balance sheet items. These estimates are used to calculate duration gap.
Forecasts changes in the market value of stockholders’ equity across different interest rate environments.

28. Weighted Average Duration of Bank Assets
Weighted Average Duration of Bank Assets (DA)
Where
wi = Market value of asset i divided by the market value of all bank assets
Dai = Macaulay’s duration of asset i
n = number of different bank assets

29. Weighted Average Duration of Bank Liabilities
Weighted Average Duration of Bank Liabilities (DL)
Where
zj = Market value of liability j divided by the market value of all bank liabilities
Dlj= Macaulay’s duration of liability j
m = number of different bank liabilities

30. Duration GAP and Economic Value of Equity
Let MVA and MVL equal the market values of assets and liabilities, respectively.
If:
and
Duration GAP
Then:
where y = the general level of interest rates

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

32. Hypothetical Bank Balance Sheet

33. Calculating DGAP DA DL DGAP
($700/$1000)* ($200/$1000)*4.99 = 2.88 DL
($620/$920)* ($300/$920)*2.81 = 1.59
DGAP
(920/1000)*1.59 = 1.42 years
What does this tell us?
The average duration of assets is greater than the average duration of liabilities; thus asset values change by more than liability values.

34. 1 percent increase in all rates.

35. Calculating DGAP DA DGAP ($683/$974)*2.68 + ($191/$974)*4.97 = 2.86
($614/$906)* ($292/$906)*2.80 = 1.58
DGAP
($906/$974) * 1.58 = 1.36 years
What does 1.36 mean?
The average duration of assets is greater than the average duration of liabilities, thus asset values change by more than liability values.

36. Change in the Market Value of Equity
In this case:

37. Positive and Negative Duration GAPs
Positive DGAP
Indicates that assets are more price sensitive than liabilities, on average.
Thus, when interest rates rise (fall), assets will fall proportionately more (less) in value than liabilities and EVE will fall (rise) accordingly.
Negative DGAP
Indicates that weighted liabilities are more price sensitive than weighted assets.
Thus, when interest rates rise (fall), assets will fall proportionately less (more) in value that liabilities and the EVE will rise (fall).

38. DGAP Summary

39. An Immunized Portfolio
To immunize the EVE from rate changes in the example, the bank would need to:
decrease the asset duration by 1.42 years or
increase the duration of liabilities by 1.54 years
DA / ( MVA/MVL) = 1.42 / ($920 / $1,000) = 1.54 years

40. Immunized Portfolio
DGAP = 2.88 – 0.92 (3.11) ≈ 0

41. Immunized Portfolio with a 1% increase in rates

42. Immunized Portfolio with a 1% increase in rates
EVE changed by only $0.5 with the immunized portfolio versus $25.0 when the portfolio was not immunized.

43. Stabilizing the Book Value of Net Interest Income
This can be done for a 1-year time horizon, with the appropriate duration gap measure
DGAP* MVRSA(1- DRSA) - MVRSL(1- DRSL) where:
MVRSA = cumulative market value of RSAs
MVRSL = cumulative market value of RSLs
DRSA = composite duration of RSAs for the given time horizon
Equal to the sum of the products of each asset’s duration with the relative share of its total asset market value
DRSL = composite duration of RSLs for the given time horizon
Equal to the sum of the products of each liability’s duration with the relative share of its total liability market value.

44. Stabilizing the Book Value of Net Interest Income
If DGAP* is positive, the bank’s net interest income will decrease when interest rates decrease, and increase when rates increase.
If DGAP* is negative, the relationship is reversed.
Only when DGAP* equals zero is interest rate risk eliminated.
Banks can use duration analysis to stabilize a number of different variables reflecting bank performance.

45. Economic Value of Equity Sensitivity Analysis
Effectively involves the same steps as earnings sensitivity analysis.
In EVE analysis, however, the bank focuses on:
The relative durations of assets and liabilities
How much the durations change in different interest rate environments
What happens to the economic value of equity across different rate environments

46. Embedded Options
Embedded options sharply influence the estimated volatility in EVE
Prepayments that exceed (fall short of) that expected will shorten (lengthen) duration.
A bond being called will shorten duration.
A deposit that is withdrawn early will shorten duration.
A deposit that is not withdrawn as expected will lengthen duration.

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

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

49. Duration Gap for First Savings Bank EVE
Market Value of Assets
$1,001,963
Duration of Assets
2.6 years
Market Value of Liabilities
$919,400
Duration of Liabilities
2.0 years

50. Duration Gap for First Savings Bank EVE
= 2.6 – ($919,400/$1,001,963)*2.0 = years
Example:
A 1% increase in rates would reduce EVE by $7.2 million = (0.01 / ) * $1,001,963
Recall that the average rate on assets is 6.93%

51. Sensitivity of EVE versus Most Likely (Zero Shock) Interest Rate Scenario
Sensitivity of Economic Value of Equity measures the change in the economic value of the corporation’s equity under various changes in interest rates. Rate changes are instantaneous changes from current rates. The change in economic value of equity is derived from the difference between changes in the market value of assets and changes in the market value of liabilities.

52. Effective “Duration” of Equity
By definition, duration measures the percentage change in market value for a given change in interest rates
Thus, a bank’s duration of equity measures the percentage change in EVE that will occur with a 1 percent change in rates:
Effective duration of equity yrs. = $8,200 / $82,563

53. Asset/Liability Sensitivity and DGAP
Funding GAP and Duration GAP are NOT directly comparable
Funding GAP examines various “time buckets” while Duration GAP represents the entire balance sheet.
Generally, if a bank is liability (asset) sensitive in the sense that net interest income falls (rises) when rates rise and vice versa, it will likely have a positive (negative) DGAP suggesting that assets are more price sensitive than liabilities, on average.

54. Strengths and Weaknesses: DGAP and EVE-Sensitivity Analysis
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

55. Strengths and Weaknesses: DGAP and EVE-Sensitivity Analysis
It is difficult to compute duration accurately
“Correct” duration analysis requires that each future cash flow be discounted by a distinct discount rate
A bank must continuously monitor and adjust the duration of its portfolio
It is difficult to estimate the duration on assets and liabilities that do not earn or pay interest
Duration measures are highly subjective

56. Speculating on Duration GAP
It is difficult to actively vary GAP or DGAP and consistently win
Interest rates forecasts are frequently wrong
Even if rates change as predicted, banks have limited flexibility in vary GAP and DGAP and must often sacrifice yield to do so

57. Gap and DGAP Management Strategies Example
Cash flows from investing $1,000 either in a 2-year security yielding 6 percent or two consecutive 1-year securities, with the current 1-year yield equal to 5.5 percent.

58. Gap and DGAP Management Strategies Example
It is not known today what a 1-year security will yield in one year.
For the two consecutive 1-year securities to generate the same $120 in interest, ignoring compounding, the 1-year security must yield 6.5% one year from the present.
This break-even rate is a 1-year forward rate, one year from the present:
6% + 6% = 5.5% + x so x must = 6.5%

59. Gap and DGAP Management Strategies Example
By investing in the 1-year security, a depositor is betting that the 1-year interest rate in one year will be greater than 6.5%
By issuing the 2-year security, the bank is betting that the 1-year interest rate in one year will be greater than 6.5%

60. Yield Curve Strategy
When the U.S. economy hits its peak, the yield curve typically inverts, with short-term rates exceeding long-term rates.
Only twice since WWII has a recession not followed an inverted yield curve
As the economy contracts, the Federal Reserve typically increases the money supply, which causes the rates to fall and the yield curve to return to its “normal” shape.

61. Yield Curve Strategy
To take advantage of this trend, when the yield curve inverts, banks could:
Buy long-term non-callable securities
Prices will rise as rates fall
Make fixed-rate non-callable loans
Borrowers are locked into higher rates
Price deposits on a floating-rate basis
Lengthen the duration of assets relative to the duration of liabilities

62. Interest Rates and the Business Cycle The general level of interest rates and the shape of the yield curve appear to follow the U.S. business cycle. T i m e I n t r s R a ( P c ) E x p o C L g - S h k u
In expansionary stages rates rise until they reach a peak as the Federal Reserve tightens credit availability.
In contractionary stages rates fall until they reach a trough when the U.S. economy falls into recession.

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