Interest Rate Risk

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Interest Rate Risk

• Finance 129

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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

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Total Assets of Federal Reserve
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Federal Reserve Assets - Detailed
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Current Balance Sheet Sept 2011
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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)

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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

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Borrowing Decisions

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

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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
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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.

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Loanable Funds

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

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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)

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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

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Three Explanations of the Yield Curve

• The Expectations Theories
• Segmented Markets Theory
• Preferred Habitat Theory

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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.

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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.

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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

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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
•  

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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

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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)

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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.

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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.

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Preferred Habitat Theoryand The 3 Empirical
Observations

• The biased expectation theories can explain all
  three empirical facts.

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Yield Curves Feb 2012 Aug 2012
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US Treasury Rates May 1990 -Sept 2011
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Maturity Yield Spreads1990 - 2011
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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

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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.

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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

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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.

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Interest Sensitive GAP

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

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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

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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?

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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.

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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.

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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

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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

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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

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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)

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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.

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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

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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.

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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.

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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)

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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

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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

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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)

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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)

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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

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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

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Maturity GAP

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

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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.

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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.

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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?).

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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.

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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.

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

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Duration MathematicsMacaulay Duration
substitute
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Macaulay Duration of a bond
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Duration Example

• 10 30 year coupon bond, current rates 12, semi
  annual payments

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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.

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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
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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

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

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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.

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Duration and Convexity
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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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Basic Duration Gap

• Duration Gap

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Basic DGAP Conintued
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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

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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

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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.

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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

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DGAP and equity

• Let DMVE DMVA DMVL
• We can find DMVA DMVL using duration
• From our definition of duration

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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.

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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.

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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.

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Duration in Practice

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

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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

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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.

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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.

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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.

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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

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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.

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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.