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					Interest Rate Risk


    Finance 129




                     Drake Fin 129
                     DRAKE UNIVERSITY
    Review of Key Factors             Drake
          Impacting                   Drake University

                                       Fin 129
    Interest Rate Volatility
Federal Reserve and Monetary Policy
  Discount Window

  Reserve Requirements

  Open Market Operations
Review of Key Factors Impacting         Drake
                                        Drake University

     Interest Rate Volatility            Fin 129



 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
    Saving and Investment                      Drake
                                                Drake University

          Decisions                              Fin 129


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
                                     Drake
     Borrowing Decisions             Drake University

                                      Fin 129



Borrowing Decision
  Marginal Productivity of Capital
  Expected Inflation
  Other
                                              Drake
     Equilibrium in the Market                Drake University

                                               Fin 129
Original Equilibrium             Decrease in Income
                S
                                                        S




             D                                     D


Increase in Marg. Prod Cap   Increase in Inflation Exp.
                                                        S
             S




                                               D
            D
                                         Drake
   Loanable Funds Theory                 Drake University

                                          Fin 129



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,
                                         Drake
        Loanable Funds                   Drake University

                                          Fin 129



Now equilibrium extends through all markets
– money markets, bonds markets and
investment market.
Inflation expectations can also influence the
supply of funds.
                                            Drake
 Liquidity Preference Theory                Drake University

                                             Fin 129




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
                                         Drake
         The Yield Curve                 Drake University

                                          Fin 129



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
Three Explanations of the Yield   Drake
                                  Drake University

            Curve                  Fin 129



The Expectations Theories
Segmented Markets Theory
Preferred Habitat Theory
                                                 Drake
  Pure Expectations Theory                       Drake University

                                                  Fin 129


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)
    Pure Expectations Theory:                Drake
                                             Drake University

     A Simplified Illustration                Fin 129



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.
   Investing in successive one              Drake
                                            Drake University

          period bonds                       Fin 129



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
                                            Drake
       The 2 Period Return                  Drake University

                                             Fin 129


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
                                           Drake
Set the two equal to each other            Drake University

                                            Fin 129



               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
                                                Drake
          Expectations Hypothesis               Drake University


             R2t = (Rt+Ret+1)/2                  Fin 129


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.
                                          Drake
Problems with Pure Expectations            Drake University

                                            Fin 129


 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.
                                        Drake
           Price Risk                    Drake University

                                          Fin 129



The return on the bond with a 5 year maturity
is known with certainty the other two are not.
                                        Drake
    Reinvestment rate risk               Drake University

                                          Fin 129



Now assume the investor is considering a
short term investment then reinvesting for the
remainder of the five years or investing for
five years.
                                                       Drake
      Local Expectations                               Drake University

                                                        Fin 129



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
Return to maturity expectations           Drake
                                          Drake University

          hypothesis                       Fin 129



 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.
    Expectations Theory and               Drake
                                          Drake University

        Forward Rates                      Fin 129


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.
                                       Drake
 Segmented Markets Theory              Drake University

                                        Fin 129


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.
                                           Drake
Biased Expectations Theories               Drake University

                                            Fin 129


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)
                                             Drake
 Liquidity Preference Theory                 Drake University

                                              Fin 129


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.
                                             Drake
   Preferred Habitat Theory                   Drake University

                                               Fin 129


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.
    Preferred Habitat Theory               Drake
                                           Drake University

and The 3 Empirical Observations            Fin 129


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.
                                            Drake
   Preferred Habitat Theory                 Drake University

                                             Fin 129


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.
                                           Drake
  Preferred Habitat Theory                 Drake University

                                            Fin 129



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.
                                                               Drake
            Yield Curves Previous Month                        Drake University

                                                                Fin 129


    0.053




    0.048




    0.043
Yield




    0.038


                    8/8/2007          8/15/2007           8/22/2007
    0.033           8/29/2007         9/5/2007            9/12/2007


                                    Maturity (Years)
    0.028
         0.00    5.00      10.00   15.00     20.00     25.00          30.00
                                                              Drake
        Yield Curves Previous 6 Months                        Drake University

                                                               Fin 129


    0.052




    0.047
Yield




    0.042




    0.037          6/15/2007         5/15/2007           6/15/2007
                   7/16/2007         8/15/2007           9/12/2007
                                   Maturity (Years)
    0.032
         0.00   5.00      10.00   15.00     20.00     25.00       30.00
                                                                  Drake
    Yield Curves Previous 6 quarters                              Drake University

                                                                   Fin 129

   0.055




        0.05




   0.045
Yield




        0.04




   0.035
                          6/15/2006           9/15/2006       12/15/2006
                          3/15/2007           6/15/2007       9/12/2007
                                  Maturity (Years)
        0.03
            0.00   5.00               10.00           15.00         20.00
                 US Treasury Yields                                             Drake
                                                                                Drake University

                Jan1989 -June 2006                                               Fin 129
 0.1
                                    1-mo           3-mo           6-mo          1-yr
0.09                                2-yr           3-yr           5-yr          7-yr
                                    10-yr          20-yr          30-yr
0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

   0
 12/8/1989   9/3/1992   5/31/1995      2/24/1998     11/20/2000     8/17/2003    5/13/2006
                      US Treas Rates                                              Drake
                                                                                  Drake University

                   May 1990 – Sept 2007                                            Fin 129
 0.1

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01        3-mo          6-mo          10-yr       20-yr
  0
 5/7/1990     1/31/1993    10/28/1995   7/24/1998   4/19/2001   1/14/2004   10/10/2006
Impact of Interest Rate Volatility       Drake
                                          Drake University

   on Financial Institutions               Fin 129



 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
     Static GAP Analysis                       Drake
                                               Drake University

    (The repricing model)                       Fin 129



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.
                                          Drake
  Commercial Banks & GAP                  Drake University

                                              Fin 129



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
                                         Drake
          GAP Analysis                   Drake University

                                          Fin 129



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.
                                          Drake
    Interest Sensitive GAP                Drake University

                                              Fin 129




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

                                             Fin 129


        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
                                          Drake
          GAP Analysis                     Drake University

                                            Fin 129


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
  If interest rates h NII will
  If interest rates i NII will
Would you expect a commercial bank to be
asset or liability sensitive for 6 mos? 5 years?
                                           Drake
  Important things to note:                Drake University

                                            Fin 129



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.
                                                Drake
      Steps in Calculating GAP                  Drake University

                                                 Fin 129



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.
                                             Drake
Alternative measures of GAP                  Drake University

                                              Fin 129



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
Other useful measures using               Drake
                                           Drake University

            GAP                             Fin 129


Relative Interest sensitivity GAP (GAP ratio)
  GAP / Bank Size
  The higher the number the higher the risk that
  is present
Interest Sensitivity Ratio
                                          Drake
   What is “Rate Sensitive”               Drake University

                                           Fin 129



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
                                            Drake
   What about Core Deposits?                Drake University

                                             Fin 129


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)
                                        Drake
Expectations of Rate changes            Drake University

                                            Fin 129



If you expect rates to increase would you
want GAP to be positive or negative?
 Unequal changes in interest              Drake
                                           Drake University

           rates                            Fin 129



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
                                            Drake
       Strengths of GAP                     Drake University

                                             Fin 129



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.
                                            Drake
  Weaknesses of Static GAP                   Drake University

                                              Fin 129


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.
                                            Drake
 Weaknesses of Static GAP                   Drake University

                                             Fin 129



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)
                                             Drake
      Weaknesses of GAP                      Drake University

                                              Fin 129



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
                                             Drake
Other Factors Impacting NII                  Drake University

                                              Fin 129



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:
                                          Drake
Other Factors Impacting NII                Drake University

                                            Fin 129


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)
                                         Drake
     Extending Basic GAP                 Drake University

                                          Fin 129



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
                                                 Drake
       The Maturity Model                        Drake University

                                                  Fin 129


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
                                          Drake
 Weighted Average Maturity                Drake University

                                           Fin 129



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


M i  Wi1M i1  Wi 2 M i 2      Win M in
                                         Drake
          Maturity GAP                   Drake University

                                          Fin 129



Given the weighted average maturity of the
assets and liabilities you can calculate the
maturity GAP
                                           Drake
     Maturity Gap Analysis                 Drake University

                                            Fin 129



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.
                                                      Drake
 The Balance Sheet and MGap                           Drake University

                                                       Fin 129


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
                                        Drake
      Matching Maturity                  Drake University

                                          Fin 129



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?).
                                         Drake
                                          Drake University

            Duration                       Fin 129



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.
                                         Drake
    Duration Mathematics                  Drake University

                                           Fin 129



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.
                                                                 Drake
        Duration Mathematics                                     Drake University

                                                                  Fin 129



    CP       CP          CP                CP          MV
P                                          
   (1  r) (1  r) 2
                       (1  r) 3
                                         (1  r) n
                                                     (1  r) n

P (-1)CP (-2)CP (-3)CP                       (-n)CP        (-n)MV
                                             n 1
                                                          
r (1  r) 2
               (1  r) 3
                           (1  r) 4
                                             (1  r)        (1  r) n 1

P      1  1CP        2CP       3CP               nCP       nMV 
                                                  
r    1  r  (1  r) (1  r) 2 (1  r) 3
                                                 (1  r) n (1  r) n 
                                                                      
  The approximate price change for a small change in r
                                                                         Drake
            Duration Mathematics                                         Drake University

                                                                          Fin 129



 P     1  1CP          2CP         3CP                 nCP         nMV 
           (1  r)  (1  r) 2  (1  r) 3      (1  r) n  (1  r) n 
 r    1 r                                                                  
 To find the % price change divide both sides by the original
 Price

P 1     1  1CP          2CP         3CP                 nCP         nMV  1
            (1  r)  (1  r) 2  (1  r) 3      (1  r) n  (1  r) n  P
r P    1 r                                                                  

        The RHS is referred to as the Modified Duration
    Which is the % change in price for a small change in yield
               Duration Mathematics                                Drake
                                                                    Drake University

                Macaulay Duration                                    Fin 129



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

          change in price
           original price   Change in Price    Original yield  P (1  r)
D MAC                                        Original price   r P
                                               
          change in yield  Change in Yield
                                              
                                                                 
                                                                 
           original yield
                Duration Mathematics                                       Drake
                                                                           Drake University

                 Macaulay Duration                                          Fin 129



           change in price
            original price   Change in Price      Original yield  P (1  r)
 D MAC                                         
                                                   Original price   r P
           change in yield  Change in Yield
                                                 
                                                                    
                                                                    
            original yield
                                 substitute


   P      1  1CP        2CP       3CP               nCP       nMV 
                                                     
   r    1  r  (1  r) (1  r) 2 (1  r) 3
                                                    (1  r) n (1  r) n 
                                                                         

           1  1CP          2CP         3CP                 nCP         nMV  (1  r)
DMAC          (1  r)  (1  r) 2  (1  r) 3      (1  r) n  (1  r) n  P
          1 r                                                                  
                                                               Drake
                                                                Drake University

   Macaulay Duration of a bond                                   Fin 129



           1CP      2CP       3CP             nCP         nMV  1
                                              
                                                          (1  r) n  P
DMAC
           (1  r) (1  r) (1  r)
                           2        3
                                              (1  r) n
                                                                    


                      N
                        t(CP) N(MV)
                    (1  r) t  (1  r) N
   DMAC          t 1
                     N
                          CP       MV
                    (1  r) t  (1  r) N
                   t 1
                                              Drake
              Duration Example                Drake University

                                               Fin 129



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

         60
               t ($50) 60($1000)
         (1  .06)t  (1  .06)60
DMAC    60
         t 1
                                     17.3895 periods
                   50     $1000
          (1  .06)t  (1  .06)60
          t 1
                                          Drake
      Example continued                   Drake University

                                           Fin 129



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.
                Using Duration                                Drake
                                                              Drake University

          to estimate price changes                            Fin 129



           P (1  r)      Rearrange          P            r
D MAC                                           D MAC
           r P                                P          (1  r)
                                     % Change in Price

Estimate the % price change for a 1 basis point increase in yield

           P            r               .0001
               D MAC          8.69925        0.000776
            P          (1  r)             1.12
            The estimated price change is then
              -0.000776(838.8357)=-0.6515
                                         Drake
  Using Duration Continued               Drake University

                                          Fin 129



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
                                                                            Drake
             Modified Duration                                               Drake University

                                                                              Fin 129



     From before, modified duration was defined as


P 1     1  1CP          2CP         3CP                 nCP         nMV  1
            (1  r)  (1  r) 2  (1  r) 3      (1  r) n  (1  r) n  P
r P    1 r                                                                  

                                    Macaulay Duration


                  Modified    Macaulay Duration
                           
                  Duration         (1  r)
                                                    Drake
  Modified Duration                                 Drake University

                                                     Fin 129


         Using Macaulay Duration

 P            r               .0001
     D MAC          8.69925        0.000776
  P          (1  r)             1.12

     Modified    Macaulay Duration
              
     Duration         (1  r)
P            r     D MAC
    D MAC                r  D MODIFIEDr
 P          (1  r) (1  r)
      8.69925
             (.0001)  0.000776
        1.12
                                           Drake
                                            Drake University


              Duration                       Fin 129


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
                                          Drake
   Duration and Convexity                 Drake University

                                           Fin 129



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.
                                                            Drake
             Duration and Convexity                         Drake University

                                                             Fin 129

             3000


             2500


             2000
Bond Value




             1500


             1000


             500


               0
                    0   0.05      0.1          0.15   0.2
                               Interest Rate
                          Drake
     Basic Duration Gap   Drake University

                           Fin 129



Duration Gap
                                                        Drake
           Basic DGAP Conintued                          Drake University

                                                          Fin 129

                          $ Weighted Duration             N
                                                  DA   w i Da i
                            of Asset Portfolio           i 1

                                               Asset i
                         where w i 
                                     Market Value of All Assets
                         Da i  Macaulay Duration of asset i
$ Weighted Duration            N
                        DL   w jDl j
of Liability Portfolio        j1

                        Asset j
where w j 
            Market Value of All Liabilitie s
Dl j  Macaulay Duration of Liability j
                                               Drake
           Basic DGAP                           Drake University

                                                 Fin 129



If the Basic DGAP is +
  If Rates h
   in the value of assets > in value of liab

  If Rate i
    in the value of assets > in value of liab
                                               Drake
           Basic DGAP                          Drake University

                                                Fin 129



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

  If Rate i
   in the value of assets < in value of liab
                                           Drake
           Basic DGAP                      Drake University

                                            Fin 129



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.
                                        Drake
              DGAP                      Drake University

                                         Fin 129



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

                                                Fin 129



      Let DMVE = DMVA – DMVL
We can find DMVA & DMVL using duration
    From our definition of duration:
                Δi
      ΔP  D         P Applying the formula
              (1  i)
                                        Drake
                                        Drake University

                                         Fin 129



ΔMVE  ΔMVA - ΔMVL
             Δy              Δy      
       -DA      MVA - - DL      MVL 
            1 y            1 y     
                               Δy
       -(DA)MVA - (DL)MVL 
                              1 y
                     MVL  Δy
      - (DA) - (DL)      1  y MVA
                     MVA 
                 Δy
ΔMVE  -DGAP        MVA
               1 y
                                          Drake
          DGAP Analysis                   Drake University

                                           Fin 129



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

                                             Fin 129



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.
                                          Drake
   More General Problems                  Drake University

                                           Fin 129



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 –
                            Drake
     Duration in Practice   Drake University

                             Fin 129



Impact of convexity
Shape of the yield curve
Default Risk
Floating Rate Instruments
Demand Deposits
Mortgages
Off Balance Sheet items
                                          Drake
      Convexity Revisited                  Drake University

                                            Fin 129



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
                                         Drake
      Flat Term Structure                Drake University

                                          Fin 129



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.
                                        Drake
          Default Risk                   Drake University

                                          Fin 129



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.
                                        Drake
         Floating Rates                  Drake University

                                          Fin 129



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.
                                                  Drake
        Demand Deposits                           Drake University

                                                   Fin 129


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
                                      Drake
          Mortgages                   Drake University

                                       Fin 129



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.
                                         Drake
   Off Balance Sheet Items               Drake University

                                          Fin 129



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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