1. Interest Rate Risk
Finance 129

2. Review of Key Factors Impacting Interest Rate Volatility
Federal Reserve and Monetary Policy
Discount Window
Reserve Requirements
Open Market Operations

3. Review of Key Factors Impacting Interest Rate Volatility
Fisher model of the Savings Market
Two main participants: Households and Business
Households supply excess funds to Businesses who are short of funds
The Saving or supply of funds is upward sloping
The investment or demand for funds is downward sloping

4. Saving and Investment Decisions
Saving Decision
Marginal Rate of Time Preference
Trading current consumption for future consumption
Expected Inflation
Income and wealth effects
Generally higher income – save more
Federal Government
Money supply decisions
Business
Short term temporary excess cash.
Foreign Investment

5. Borrowing Decisions Borrowing Decision
Marginal Productivity of Capital
Expected Inflation
Other

6. Equilibrium in the Market
Original Equilibrium
Decrease in Income S S D D
Increase in Marg. Prod Cap
Increase in Inflation Exp. S S D D

7. Loanable Funds Theory
Expands suppliers and borrowers of funds to include business, government, foreign participants and households.
Interest rates are determined by the demand for funds (borrowing) and the supply of funds (savings).
Very similar to Fisher in the determination of interest rates,

8. Loanable Funds
Now equilibrium extends through all markets – money markets, bonds markets and investment market.
Inflation expectations can also influence the supply of funds.

9. Liquidity Preference Theory
Two assets, money and financial assets
Equilibrium in one implies equilibrium in other
Supply of Money is controlled by Central Bank and is not related to level of interest rates

10. The Yield Curve
Three things are observed empirically concerning the yield curve:
Rates across different maturities move together
More likely to slope upwards when short term rates are historically low, sometimes slope downward when short term rates are historically high
The yield curve usually slope upward

11. Three Explanations of the Yield Curve
The Expectations Theories
Segmented Markets Theory
Preferred Habitat Theory

12. Pure Expectations Theory
Long term rates are a representation of the short term interest rates investors expect to receive in the future. In other words the forward rates reflect the future expected rate.
Assumes that bonds of different maturities are
In other words, the expected return from holding a one year bond today and a one year bond next year is the same as buying a two year bond today. (the same process that is used to calculate forward rates)

13. Pure Expectations Theory: A Simplified Illustration
Let
Rt = today’s time t interest rate on a one period bond
Ret+1 = expected interest rate on a one period bond in the next period
R2t = today’s (time t) yearly interest rate on a two period bond.

14. Investing in successive one period bonds
If the strategy of buying the one period bond in two consecutive years is followed the return is:
(1+Rt)(1+Ret+1) – 1 which equals
Rt+Ret+1+ (Rt)(Ret+1)
Since (Rt)(Ret+1) will be very small we will ignore it

15. The 2 Period Return
If the strategy of investing in the two period bond is followed the return is:
(1+R2t)(1+R2t) - 1 = 1+2R2t+(R2t)2 - 1
(R2t)2 is small enough it can be dropped
which leaves

16. Set the two equal to each other
2R2t = Rt+Ret+1
R2t = (Rt+Ret+1)/2
In other words, the two period interest rate is the average of the two one period rates

17. Expectations Hypothesis R2t = (Rt+Ret+1)/2
When the yield curve is upward sloping (R2t>R1t) it is expected that short term rates will be increasing (the average future short term rate is above the current short term rate).
Likewise when the yield curve is downward sloping the average of the future short term rates is below the current rate. (Fact 2)
As short term rates increase the long term rate will also increase and a decrease in short term rates will decrease long term rates. (Fact 1)
This however does not explain Fact 3 that the yield curve usually slopes up.

18. Problems with Pure Expectations
The pure expectations theory ignores the fact that there is reinvestment rate risk and different price risk for the two maturities.
Consider an investor considering a 5 year horizon with three alternatives:
buying a bond with a 5 year maturity
buying a bond with a 10 year maturity and holding it 5 years
buying a bond with a 20 year maturity and holding it 5 years.

19. Price Risk
The return on the bond with a 5 year maturity is known with certainty the other two are not.

20. Reinvestment rate risk
Now assume the investor is considering a short term investment then reinvesting for the remainder of the five years or investing for five years.

21. Cox, Ingersoll, and Ross 1981 Journal of Finance
Local Expectations
Similarly owning the bond with each of the longer maturities should also produce the same 6 month return of 2%.
The key to this is the assumption that the forward rates hold. It has been shown that this interpretation is the only one that can be sustained in equilibrium.*
Cox, Ingersoll, and Ross 1981 Journal of Finance

22. Return to maturity expectations hypothesis
This theory claims that the return achieved by buying short term and rolling over to a longer horizon will match the zero coupon return on the longer horizon bond. This eliminates the reinvestment risk.

23. Expectations Theory and Forward Rates
The forward rate represents a “break even” rate since it the rate that would make you indifferent between two different maturities
The pure expectations theory and its variations are based on the idea that the forward rate represents the market expectations of the future level of interest rates.
However the forward rate does a poor job of predicting the actual future level of interest rates.

24. Segmented Markets Theory
Interest Rates for each maturity are determined by the supply and demand for bonds at each maturity.
Different maturity bonds are not perfect substitutes for each other.
Implies that investors are not willing to accept a premium to switch from their market to a different maturity.

25. Biased Expectations Theories
Both Liquidity Preference Theory and Preferred Habitat Theory include the belief that there is an expectations component to the yield curve.
Both theories also state that there is a risk premium which causes there to be a difference in the short term and long term rates. (in other words a bias that changes the expectations result)

26. Liquidity Preference Theory
This explanation claims that the since there is a price risk and liquidity risk associated with the long term bonds, investor must be offered a premium to invest in long term bonds
Therefore, the long term rate reflects both an expectations component and a risk premium.
This tends to imply that the yield curve will be upward sloping as long as the premium is large enough to outweigh a possible expected decrease.

27. Preferred Habitat Theory
Like the liquidity theory this idea assumes that there is an expectations component and a risk premium.
In other words the bonds are substitutes, but savers might have a preference for one maturity over another (they are not perfect substitutes).
However the premium associated with long term rates does not need to be positive.
If there are demand and supply imbalances then investors might be willing to switch to a different maturity if the premium produces enough benefit.

28. Preferred Habitat Theory and The 3 Empirical Observations
Thus according to Preferred Habitat theory a rise in short term rates still causes a rise in the average of the future short term rates. This occurs because of the expectations component of the theory.

29. Preferred Habitat Theory
The explanation of Fact 2 from the expectations hypothesis still works. In the case of a downward sloping yield curve, the term premium (interest rate risk) must not be large enough to compensate for the currently high short term rates (Current high inflation with an expectation of a decrease in inflation). Since the demand for the short term bonds will increase, the yield on them should fall in the future.

30. Preferred Habitat Theory
Fact three is explained since it will be unusual for the term premium to be so small or negative, therefore the the yield curve usually slopes up.

31. Yield Curves Previous Month

32. Yield Curves Previous 6 Months

33. Yield Curves Previous 6 quarters

34. US Treasury Yields Jan1989 -June 2006

35. US Treas Rates May 1990 – Sept 2007

36. Impact of Interest Rate Volatility on Financial Institutions
The market value of assets and liabilities is tied to the level of interest rates
Interest income and expense are both tied to the level of interest rates

37. Static GAP Analysis (The repricing model)
Repricing GAP
The difference between the value of interest sensitive assets and interest sensitive liabilities of a given maturity.
Measures the amount of rate sensitive (asset or liability will be repriced to reflect changes in interest rates) assets and liabilities for a given time frame.

38. Commercial Banks & GAP
Commercial banks are required to report quarterly the repricing Gaps for the following time frames
One day
More than one day less than 3 months
More than 3 months, less than 6 months
More than 6 months, less than 12 months
More than 12 months, less than 5 years
More than five years

39. GAP Analysis
Static GAP-- Goal is to manage interest rate income in the short run (over a given period of time)
Measuring Interest rate risk – calculating GAP over a broad range of time intervals provides a better measure of long term interest rate risk.

40. Interest Sensitive GAP
Given the Gap it is easy to investigate the change in the net interest income of the financial institution.

41. Example Over next 6 Months: Rate Sensitive Liabilities = $120 million
Rate Sensitive Assets = $100 Million
If rate are expected to decline by 1%
Change in net interest income

42. GAP Analysis Asset sensitive GAP (Positive GAP)
RSA – RSL > 0
If interest rates h NII will
If interest rates i NII will
Liability sensitive GAP (Negative GAP)
RSA – RSL < 0
Would you expect a commercial bank to be asset or liability sensitive for 6 mos? 5 years?

43. Important things to note:
Assuming book value accounting is used only the income statement is impacted, the book value on the balance sheet remains the same.
The GAP varies based on the bucket or time frame calculated.
It assumes that all rates move together.

44. Steps in Calculating GAP
Select time Interval
Develop Interest Rate Forecast
Group Assets and Liabilities by the time interval (according to first repricing)
Forecast the change in net interest income.

45. Alternative measures of GAP
Cumulative GAP
Totals the GAP over a range of of possible maturities (all maturities less than one year for example).
Total GAP including all maturities

46. Other useful measures using GAP
Relative Interest sensitivity GAP (GAP ratio)
GAP / Bank Size
The higher the number the higher the risk that is present
Interest Sensitivity Ratio

47. What is “Rate Sensitive”
Any Asset or Liability that matures during the time frame
Any principal payment on a loan is rate sensitive if it is to be recorded during the time period
Assets or liabilities linked to an index
Interest rates applied to outstanding principal changes during the interval

48. What about Core Deposits?
Against Inclusion
Demand deposits pay zero interest
NOW accounts etc do pay interest, but the rates paid are sticky
For Inclusion
Implicit costs
If rates increase, demand deposits decrease as individuals move funds to higher paying accounts (high opportunity cost of holding funds)

49. Expectations of Rate changes
If you expect rates to increase would you want GAP to be positive or negative?

50. Unequal changes in interest rates
So far we have assumed that the change the level of interest rates will be the same for both assets and liabilities.
If it isn’t you need to calculate GAP using the respective change.
Spread effect – The spread between assets and liabilities may change as rates rise or decrease

51. Strengths of GAP Easy to understand and calculate
Allows you to identify specific balance sheet items that are responsible for risk
Provides analysis based on different time frames.

52. Weaknesses of Static GAP
Market Value Effects
Basic repricing model the changes in market value. The PV of the future cash flows should change as the level of interest rates change. (ignores TVM)
Over aggregation
Repricing may occur at different times within the bucket (assets may be early and liabilities late within the time frame)
Many large banks look at daily buckets.

53. Weaknesses of Static GAP
Runoffs
Periodic payment of principal and interest that can be reinvested and is itself rate sensitive.
You can include runoff in your measure of rate sensitive assets and rate sensitive liabilities.
Note: the amount of runoffs may be sensitive to rate changes also (prepayments on mortgages for example)

54. Weaknesses of GAP Off Balance Sheet Activities
Basic GAP ignores changes in off balance sheet activities that may also be sensitive to changes in the level of interest rates.
Ignores changes in the level of demand deposits

55. Other Factors Impacting NII
Changes in Portfolio Composition
An aggressive position is to change the portfolio in an attempt to take advantage of expected changes in the level of interest rates. (if rates are h have positive GAP, if rates are i have negative GAP)
Problem:

56. Other Factors Impacting NII
Changes in Volume
Bank may change in size so can GAP along with it.
Changes in the relationship between ST and LT
We have assumes parallel shifts in the yield curve. The relationship between ST and LT may change (especially important for cumulative GAP)

57. Extending Basic GAP
You can repeat the basic GAP analysis and account for some of the problems
Include
Forecasts of when embedded options will be exercised and include them

58. The Maturity Model
In this model the impact of a change in interest rates on the market value of the asset or liability is taken into account.
The securities are marked to market
Keep in Mind the following:
The longer the maturity of a security the larger the impact of a change in interest rates
An increase in rates generally leads to a fall in the value of the security
The decrease in value of long term securities increases at a diminishing rate for a given increase in rates

59. Weighted Average Maturity
You can calculate the weighted average maturity of a portfolio. The same three principles of the change in the value of the portfolio (from last slide) will apply

60. Maturity GAP
Given the weighted average maturity of the assets and liabilities you can calculate the maturity GAP

61. Maturity Gap Analysis
If Mgap is + the maturity of the FI assets is longer than the maturity of its liabilities. (generally the case with depository institutions due to their long term fixed assets such as mortgages).
This also implies that its assets are more rate sensitive than its liabilities since the longer maturity indicates a larger price change.

62. The Balance Sheet and MGap
The basic balance sheet identity state that:
Asset = Liabilities + Owners Equity or
Owners Equity = Assets - Liabilities
Technically if Liab >Assets the institution is insolvent
If MGAP is positive and interest rate decrease then the market value of assets increases more than liabilities.
Likewise, if MGAP is negative an increase in interest rates would cause

63. Matching Maturity
By matching maturity of assets and liabilities owners can be immunized form the impact of interest rate changes.
However this does not always completely eliminate interest rate risk. Think about duration and funding sources (does the timing of the cash flows match?).

64. Duration
Duration: Weighted maturity of the cash flows (either liability or asset)
Weight is a combination of timing and magnitude of the cash flows
The higher the duration the more sensitive a cash flow stream is to a change in the interest rate.

65. Duration Mathematics
Taking the first derivative of the bond value equation with respect to the yield will produce the approximate price change for a small change in yield.

66. Duration Mathematics
The approximate price change for a small change in r

67. Duration Mathematics
To find the % price change divide both sides by the original
Price
The RHS is referred to as the Modified Duration
Which is the % change in price for a small change in yield

68. Duration Mathematics Macaulay Duration
Macaulay Duration is the price elasticity of the bond (the % change in price for a percentage change in yield).
Formally this would be:

69. Duration Mathematics Macaulay Duration
substitute

70. Macaulay Duration of a bond

71. Duration Example
10% 30 year coupon bond, current rates =12%, semi annual payments

72. Example continued
Since the bond makes semi annual coupon payments, the duration of periods must be divided by 2 to find the number of years.
/ 2 = years
This interpretation of duration indicates the average time taken by the bond, on a discounted basis, to pay back the original investment.

73. Using Duration to estimate price changes
Rearrange
% Change in Price
Estimate the % price change for a 1 basis point increase in yield
The estimated price change is then
( )=

74. Using Duration Continued
Using our 10% semiannual coupon bond, with 30 years to maturity and YTM = 12%
Original Price of the bond =
If YTM = 12.01% the price is
This implies a price change of
Our duration estimate was

75. Modified Duration From before, modified duration was defined as
Macaulay Duration

76. Modified Duration
Using Macaulay Duration

77. Duration Keeping other factors constant the duration of a bond will:
Increase with the maturity of the bond
Decrease with the coupon rate of the bond
Will decrease if the interest rate is floating making the bond less sensitive to interest rate changes
Decrease if the bond is callable, as interest rates decrease (increasing the likelihood of call) duration increases

78. Duration and Convexity
Using duration to estimate the price change implies that the change in price is the same size regardless of whether the price increased or decreased.
The price yield relationship shows that this is not true.

79. Duration and Convexity

80. Basic Duration Gap
Duration Gap

81. Basic DGAP Conintued

82. Basic DGAP If the Basic DGAP is + If Rates h
in the value of assets > in value of liab
If Rate i

83. Basic DGAP If the Basic DGAP is (-) If Rates h
in the value of assets < in value of liab
If Rate i

84. Basic DGAP
Does that imply that if DA = DL the financial institution has hedged its interest rte risk?
No, because the $ amount of assets > $ amount of liabilities otherwise the institution would be insolvent.

85. DGAP
Let MVL = market value of liabilities and MVA = market value of assets
Then to immunize the balance sheet we can use the following identity:

86. DGAP and equity Let DMVE = DMVA – DMVL
We can find DMVA & DMVL using duration
From our definition of duration:

88. DGAP Analysis If DGAP is (+) If DGAP is (-)
An in rates will cause MVE to
If DGAP is (-)
The closer DGAP is to zero the smaller the potential change in the market value of equity.

89. Weaknesses of DGAP
It is difficult to calculate duration accurately (especially accounting for options)
Each CF needs to be discounted at a distinct rate can use the forward rates from treasury spot curve
Must continually monitor and adjust duration
It is difficult to measure duration for non interest earning assets.

90. More General Problems Interest rate forecasts are often wrong
To be effective management must beat the ability of the market to forecast rates
Varying GAP and DGAP can come at the expense of yield
Offer a range of products, customers may not prefer the ones that help GAP or DGAP –

91. Duration in Practice Impact of convexity Shape of the yield curve
Default Risk
Floating Rate Instruments
Demand Deposits
Mortgages
Off Balance Sheet items

92. Convexity Revisited
The more convexity the asset or portfolio has, the more protection against rate increases and the greater the possible gain for interest rate falls.
The greater the convexity the greater the error possible if simple duration is calculated.
All fixed income securities have convexity
The larger the change in rates, the larger the impact of convexity

93. Flat Term Structure
Our definition of duration assumes a flat term structure and that the all shirts in the yield curve are parallel.
Discounting using the spot yield curve will provide a slightly different measure of inflation.

94. Default Risk
Our measures assume that the risk of default is zero. Duration can be recalculated by replacing each cash flow by the expected cash flow which includes the probability that the cash flow will be received.

95. Floating Rates
If an asset or liability carries a floating interest rate it readjusts its payments so the future cash flows are not known.
Duration is generally viewed as being the time until the next resetting of the interest rate.

96. Demand Deposits Simulation
Deposits have an open ended maturity. You need to define the maturity to define duration.
Method 1
Look at turnover of deposits (or run). If deposits turn over 5 times a year then they have an average maturity of 73 days (365/5).
Method 2
Think of them as a puttable bond with a duration of 0
Method 3
Look at the % change in demand deposits for a given level of interest rate changes.
Simulation

97. Mortgages
Mortgages and mortgage backed securities have prepayment risk associated with them. Therefore we need to model the prepayment behavior of the mortgage to understand the cash flow.

98. Off Balance Sheet Items
The value of derivative products also are impacted by duration changes. They should be included in any portfolio duration estimate or GAP analysis.