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Interest Rate Risk

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Title: Interest Rate Risk


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
  • New Liquidity Facilities
  • Quantitative Easing
  • Operation Twist

3
Total Assets of Federal Reserve
4
Federal Reserve Assets - Detailed
5
Current Balance Sheet Sept 2011
6
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
    (saving increases as interest rates increase)
  • The investment or demand for funds is downward
    sloping (demand for funds decease as interest
    rates increase)

7
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

8
Borrowing Decisions
  • Borrowing Decision
  • Marginal Productivity of Capital
  • Expected Inflation
  • Other

9
Equilibrium in the Market
Decrease in Income
Original Equilibrium
S
S
S
D
D
Increase in Marg. Prod Cap
Increase in Inflation Exp.
S
S
S
D
D
D
D
10
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, except the markets for the supply
    and demand for funds is expanded.

11
Loanable Funds
  • Now equilibrium extends through all markets
    money markets, bonds markets and investment
    market.
  • Inflation expectations can also influence the
    supply of funds.

12
Liquidity Preference Theory
  • Liquidity Preference
  • 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 (A
    vertical line)

13
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

14
Three Explanations of the Yield Curve
  • The Expectations Theories
  • Segmented Markets Theory
  • Preferred Habitat Theory

15
Pure Expectations Theory
  • Long term rates are a representation of the short
    term interest rates investors expect to receive
    in the future. (forward rates reflect the future
    expected rate).
  • Assumes that bonds of different maturities are
    perfect substitutes
  • 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.

16
Pure Expectations Theory A Simplified
Illustration
  • Let
  • Rt todays time t interest rate on a one
    period bond
  • Ret1 expected interest rate on a one period
    bond in the next period
  • R2t todays (time t) yearly interest rate on a
    two period bond.

17
Investing in successive one period bonds
  • If the strategy of buying the one period bond in
    two consecutive years is followed the return is
  • (1Rt)(1Ret1) 1 which equals
  • RtRet1 (Rt)(Ret1)
  • Since (Rt)(Ret1) will be very small we will
    ignore it
  • that leaves
  • RtRet1

18
The 2 Period Return
  • If the strategy of investing in the two period
    bond is followed the return is
  • (1R2t)(1R2t) - 1 12R2t(R2t)2 - 1
  • (R2t)2 is small enough it can be dropped
  • which leaves
  • 2R2t
  •  

19
Set the two equal to each other
  • 2R2t RtRet1
  • R2t (RtRet1)/2
  • In other words, the two period interest rate is
    the average of the two one period rates

20
Expectations Hypothesis R2t (RtRet1)/2
  • Fact 1 and Fact 2 are explained well by the
    expectations hypothesis
  • However it does not explain Fact 3, that the
    yield curve usually slopes up.

21
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.

22
Price Risk
  • The return on the bond with a 5 year maturity is
    known with certainty the other two are not.
  • The longer the maturity the greater the price risk

23
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.
  • Again the 5 year return is known with certainty,
    but the others are not.

24
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.

25
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.

26
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.

27
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.
  • Therefore the shape of the yield curve depends
    upon the asset liability constraints and goals of
    the market participants.

28
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)

29
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.
  • The yield curve will be upward sloping as long as
    the premium is large.

30
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.

31
Preferred Habitat Theoryand The 3 Empirical
Observations
  • The biased expectation theories can explain all
    three empirical facts.

32
Yield Curves Feb 2012 Aug 2012
33
US Treasury Rates May 1990 -Sept 2011
34
Maturity Yield Spreads1990 - 2011
35
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

36
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 assets and
    liabilities (asset or liability will be repriced
    to reflect changes in interest rates) for a given
    time frame.

37
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

38
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.

39
Interest Sensitive GAP
  • Given the Gap it is easy to investigate the
    change in the net interest income of the
    financial institution.

40
Example
  • Over next 6 Months
  • Rate Sensitive Liabilities 120 million
  • Rate Sensitive Assets 100 Million
  • GAP 100M 120M - 20 Million
  • If rate are expected to decline by 1
  • Change in net interest income
  • (-20M)(-.01) 200,000

41
GAP Analysis
  • Asset sensitive GAP (Positive GAP)
  • RSA RSL gt 0
  • If interest rates h NII will h
  • If interest rates i NII will i
  • Liability sensitive GAP (Negative GAP)
  • RSA RSL lt 0
  • If interest rates h NII will i
  • If interest rates i NII will h
  • Would you expect a commercial bank to be asset or
    liability sensitive for 6 mos? 5 years?

42
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.

43
Steps in Calculating GAP
  1. Select time Interval
  2. Develop Interest Rate Forecast
  3. Group Assets and Liabilities by the time interval
    (according to first repricing)
  4. Forecast the change in net interest income.

44
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

45
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

46
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

47
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)

48
Expectations of Rate changes
  • If you expect rates to increase would you want
    GAP to be positive or negative?
  • Positive the increase in assets gt increase in
    liabilities so net interest income will increase.

49
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 isnt you need to calculate GAP using the
    respective change.
  • Spread effect The spread between assets and
    liabilities may change as rates rise or decrease

50
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.

51
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.

52
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)

53
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

54
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 Forecasting is rarely accurate

55
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)

56
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
  • Include off balance sheet items
  • Recalculate across different interest rate
    assumptions (and repricing assumptions)

57
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

58
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

59
Maturity GAP
  • Given the weighted average maturity of the assets
    and liabilities you can calculate the maturity GAP

60
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.

61
The Balance Sheet and MGap
  • The basic balance sheet identity state that
  • Asset Liabilities Owners Equity or
  • Owners Equity Assets - Liabilities
  • Technically if Liab gtAssets the institution is
    insolvent
  • If MGAP is positive and interest rate decrease
    then the market value of assets increases more
    than liabilities and owners equity increases.
  • Likewise, if MGAP is negative an increase in
    interest rates would cause a decrease in owners
    equity.

62
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?).

63
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.

64
Duration MathematicsBond Example
  • 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.

65
Duration Mathematics
The approximate price change for a small change
in r
66
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
67
Duration MathematicsMacaulay Duration
  • Macaulay Duration is the price elasticity of the
    bond (the change in price for a percentage
    change in yield).
  • Formally this would be

68
Duration MathematicsMacaulay Duration
substitute
69
Macaulay Duration of a bond
70
Duration Example
  • 10 30 year coupon bond, current rates 12, semi
    annual payments

71
Example continued
  • Since the bond makes semi annual coupon payments,
    the duration of 17.3895 periods must be divided
    by 2 to find the number of years.
  • 17.3895 / 2 8.69475 years
  • This interpretation of duration indicates the
    average time taken by the bond, on a discounted
    basis, to pay back the original investment.

72
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
-0.000776(838.8357)-0.6515
73
Using Duration Continued
  • Using our 10 semiannual coupon bond, with 30
    years to maturity and YTM 12
  • Original Price of the bond 838.3857
  • If YTM 12.01 the price is 837.6985
  • This implies a price change of -0.6871
  • Our duration estimate was -0.6515

74
Modified Duration
From before, modified duration was defined as
Macaulay Duration
75
Modified Duration
Using Macaulay Duration
76
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

77
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.

78
Duration and Convexity
79
Duration and Yield Changes
  • Duration provides a linear approximation of the
    price change associated with a change in yield.
  • The duration of an asset will change depending
    upon the original yield used in its calculation.
  • As the yield decreases, the price change
    associated with a change in yield increases.
  • Likewise duration will increase as the yield of
    an option free bond decreases. This is
    illustrated as a steeper line approximately
    tangent to the price yield relationship.

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84
Basic Duration Gap
  • Duration Gap

85
Basic DGAP Conintued
86
Basic DGAP
  • If the Basic DGAP is
  • If Rates h
  • i in the value of assets gt i in value of liab
  • Owners equity will decrease
  • If Rate i
  • h in the value of assets gt h in value of liab
  • Owners equity will increase

87
Basic DGAP
  • If the Basic DGAP is (-)
  • If Rates h
  • i in the value of assets lt i in value of liab
  • Owners equity will increase
  • If Rate i
  • h in the value of assets lt h in value of liab
  • Owners equity will decrease

88
Basic DGAP
  • Does that imply that if DA DL the financial
    institution has hedged its interest rte risk?
  • No, because the amount of assets gt amount of
    liabilities otherwise the institution would be
    insolvent.

89
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

90
DGAP and equity
  • Let DMVE DMVA DMVL
  • We can find DMVA DMVL using duration
  • From our definition of duration

91

92
DGAP Analysis
  • If DGAP is ()
  • An h in rates will cause MVE to i
  • An i in rates will cause MVE to h
  • If DGAP is (-)
  • An h in rates will cause MVE to h
  • An i in rates will cause MVE to i
  • The closer DGAP is to zero the smaller the
    potential change in the market value of equity.

93
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.

94
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 Need to
    offer more attractive yields to entice this
    decreases profitability.

95
Duration in Practice
  • Impact of convexity
  • Shape of the yield curve
  • Default Risk
  • Floating Rate Instruments
  • Demand Deposits
  • Mortgages
  • Off Balance Sheet items

96
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

97
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.

98
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.

99
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.

100
Demand Deposits
  • 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

101
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.

102
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.
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