Interest Rate Risk
Finance 129
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
Total Assets of Federal Reserve
www.federalreserve.gov/monetarypolicy/bst_recenttrends.htm
Federal Reserve Assets - Detailed
www.federalreserve.gov/releases/h41
Current Balance Sheet Sept 2011
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)
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
Borrowing Decisions
Borrowing Decision
Marginal Productivity of Capital
Expected Inflation
Other
Equilibrium in the Market
Original Equilibrium
S
Decrease in Income
S’
S
D
D
Increase in Marg. Prod Cap
Increase in Inflation Exp.
S’
S
D
D’
D
S
D’
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.
Loanable Funds
Now equilibrium extends through all markets –
money markets, bonds markets and investment
market.
Inflation expectations can also influence the
supply of funds.
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)
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
Three Explanations of the Yield Curve
The Expectations Theories
Segmented Markets Theory
Preferred Habitat Theory
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.
Pure Expectations Theory:
A Simplified Illustration
Let
Rt = today’s time t interest rate on a one
period bond
Ret+1 = expected interest rate on a one period
bond in the next period
R2t = today’s (time t) yearly interest rate on a two
period bond.
Investing in successive one period bonds
If the strategy of buying the one period bond in
two consecutive years is followed the return is:
(1+Rt)(1+Ret+1) – 1 which equals
Rt+Ret+1+ (Rt)(Ret+1)
Since (Rt)(Ret+1) will be very small we will ignore
it
that leaves
Rt+Ret+1
The 2 Period Return
If the strategy of investing in the two period bond
is followed the return is:
(1+R2t)(1+R2t) - 1 = 1+2R2t+(R2t)2 - 1
(R2t)2 is small enough it can be dropped
which leaves
2R2t
Set the two equal to each other
2R2t = Rt+Ret+1
R2t = (Rt+Ret+1)/2
In other words, the two period interest rate is the
average of the two one period rates
Expectations Hypothesis
R2t = (Rt+Ret+1)/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.
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.
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
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.
Local Expectations
Similarly owning the bond with each of the
longer maturities should also produce the same
6 month return of 2%.
The key to this is the assumption that the
forward rates hold. It has been shown that this
interpretation is the only one that can be
sustained in equilibrium.*
Cox, Ingersoll, and Ross 1981 Journal of Finance
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.
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.
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.
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)
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.
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.
Preferred Habitat Theory
and The 3 Empirical Observations
The biased expectation theories can explain all
three empirical facts.
Yield Curves Feb 2012 – Aug 2012
data from www.ustreas.gov
US Treasury Rates
May 1990 -Sept 2011
data from www.ustreas.gov
Maturity Yield Spreads
1990 - 2011
data from www.ustreas.gov
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
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.
Commercial Banks & GAP
Commercial banks are required to report
quarterly the repricing Gaps for the following
time frames
One day
More than
More than
More than
More than
More than
one day less than 3 months
3 months, less than 6 months
6 months, less than 12 months
12 months, less than 5 years
five years
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.
Interest Sensitive GAP
GAP  Rate Sensistive Assets - Rate Sensistive Liabilitie s
Given the Gap it is easy to investigate the
change in the net interest income of the
financial institution.
Change in NII  (GAP)(Chan ge in Rates)
NII  (GAP)( R)
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
GAP Analysis
Asset sensitive GAP (Positive GAP)
RSA – RSL > 0
If interest rates h NII will h
If interest rates i NII will i
Liability sensitive GAP (Negative GAP)
RSA – RSL < 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?
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.
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.
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
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
Rate Sensitive Assets
Rate Sensitive Liabilitie s
 1  Liability Sensitive
 1  Asset Sensitive
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
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)
Expectations of Rate changes
If you expect rates to increase would you want
GAP to be positive or negative?
Positive – the increase in assets > increase in
liabilities so net interest income will increase.
Unequal changes in interest rates
So far we have assumed that the change the
level of interest rates will be the same for both
assets and liabilities.
If it isn’t you need to calculate GAP using the
respective change.
Spread effect – The spread between assets and
liabilities may change as rates rise or decrease
NII  (RSA)( R assets ) - (RSL)( R liabilties)
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.
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.
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)
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
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
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)
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)
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
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
M i  Wi1 M i1  Wi 2 M i 2      Win M in
Maturity GAP
Given the weighted average maturity of the
assets and liabilities you can calculate the
maturity GAP
MGap  M assets  M liabilities
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.
The Balance Sheet and MGap
The basic balance sheet identity state that:
Asset = Liabilities + Owners Equity or
Owners Equity = Assets - Liabilities
Technically if Liab >Assets the institution is insolvent
If MGAP is positive and interest rate decrease then
the market value of assets increases more than
liabilities and owners equity increases.
Likewise, if MGAP is negative an increase in
interest rates would cause a decrease in owners
equity.
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?).
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.
Duration Mathematics
Bond 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.
Duration Mathematics
CP
CP
CP
CP
MV
P


  

2
3
n
(1  r) (1  r) (1  r)
(1  r)
(1  r) n
P (-1)CP (-2)CP (-3)CP
(-n)CP
(-n)MV



  

2
3
4
n 1
r (1  r)
(1  r)
(1  r)
(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
Duration Mathematics
P
1  1CP
2CP
3CP
nCP
nMV 



  


2
3
n
r
1  r  (1  r) (1  r)
(1  r)
(1  r)
(1  r) n 
To find the % price change divide both sides by the original
Price
P 1
1  1CP
2CP
3CP
nCP
nMV  1



  


2
3
n
r P
1  r  (1  r) (1  r) (1  r)
(1  r)
(1  r) n  P
The RHS is referred to as the Modified Duration
Which is the % change in price for a small change in yield
Duration Mathematics
Macaulay Duration
Macaulay Duration is the price elasticity of the
bond (the % change in price for a percentage
change in yield).
Formally this would be:
D MAC
change in price
 Change in Price
original price

 
change in yield  Change in Yield
original yield
 Original yield  P (1  r)

 
 Original price  r P
Duration Mathematics
Macaulay Duration
D MAC
change in price
 Change in Price
original price

 
change in yield  Change in Yield
original yield
 Original yield  P (1  r)

 
 Original price  r P
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 
DMAC
1  1CP
2CP
3CP
nCP
nMV  (1  r)



  


2
3
n
1  r  (1  r) (1  r) (1  r)
(1  r)
(1  r) n  P
Macaulay Duration of a bond
DMAC
 1CP
2CP
3CP
nCP
nMV  1
 


  

2
3
n
n 
(1

r)
(1

r)
(1

r)
(1

r)
(1

r)

P
N
DMAC
t(CP) N(MV)


t
N
(1  r)
t 1 (1  r)
 N
CP
MV


t
N
(1  r)
t 1 (1  r)
Duration Example
10% 30 year coupon bond, current rates
=12%, semi annual payments
60
DMAC
t ($50) 60($1000)


t
60
(1  .06)
t 1 (1  .06)
 60
 17.3895 periods
50
$1000


t
60
(1  .06)
t 1 (1  .06)
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.
Using Duration
to estimate price changes
D MAC
P (1  r)

r P
Rearrange
P
r
  D MAC
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
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
Modified Duration
From before, modified duration was defined as
P 1
1  1CP
2CP
3CP
nCP
nMV  1



  


2
3
n
r P
1  r  (1  r) (1  r)
(1  r)
(1  r)
(1  r) n  P
Macaulay Duration
Modified
Macaulay Duration

Duration
(1  r)
Modified Duration
Using Macaulay Duration
P
r
.0001
  D MAC
 8.69925
 0.000776
P
(1  r)
1.12
Modified
Duration

Macaulay Duration
(1  r)
D MAC
P
r
  D MAC

r  D MODIFIEDr
P
(1  r) (1  r)
8.69925

(.0001)  0.000776
1.12
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
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.
Duration and Convexity
3000
2500
Bond Value
2000
1500
1000
500
0
0
0.05
0.1
Interest Rate
0.15
0.2
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.
3500
Impact of yield on Duration
Estimate of Price change
3000
2500
2000
1500
1000
500
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
3500
Change in duration outlines
the price yield relationship
3000
2500
2000
1500
1000
500
0
0
0.05
0.1
0.15
0.2
0.25
0.3
3500
Duration and the Convexity of
the Price - Yield Relationship
3000
2500
2000
1500
1000
500
0
0
0.05
0.1
0.15
0.2
0.25
Duration and the Convexity of
the Price - Yield Relationship
3500
3000
2500
2000
1500
1000
500
0
0
0.05
0.1
0.15
0.2
0.25
Basic Duration Gap
Duration Gap
$ Weighted Duration $ Weighted Duration
Basic DGAP 

of Asset Portfolio
of Libaility Portfolio
Basic DGAP  DA  DL
Basic DGAP Conintued
$ Weighted Duration
of Asset Portfolio
N
 DA   w i Da i
i 1
Asset i
where w i 
Market Value of All Assets
Da i  Macaulay Duration of asset i
N
$ Weighted Duration
 DL   w jDl j
of Liability Portfolio
j1
where w j 
Asset j
Market Value of All Liabilitie s
Dl j  Macaulay Duration of Liability j
Basic DGAP
If the Basic DGAP is +
If Rates h
i in the value of assets > i in value of liab
Owners equity will decrease
If Rate i
h in the value of assets > h in value of liab
Owners equity will increase
Basic DGAP
If the Basic DGAP is (-)
If Rates h
i in the value of assets < i in value of liab
Owners equity will increase
If Rate i
h in the value of assets < h in value of liab
Owners equity will decrease
Basic DGAP
Does that imply that if DA = DL the financial
institution has hedged its interest rte risk?
No, because the $ amount of assets > $
amount of liabilities otherwise the institution
would be insolvent.
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:
MVL
DA  DL
MVA
MVL
DGAP  DA  DL
MVA
DGAP and equity
Let MVE = MVA – MVL
We can find MVA & MVL using duration
From our definition of duration:
Δi
ΔP   D
P Applying the formula
(1  i)
Δy
MVA  -DA
MVA
1 y
Δy
MVL  -DL
MVL
1 y
ΔMVE  ΔMVA - ΔMVL


Δy
Δy
 -DA
MVA - - DL
MVL 
1 y
1 y


Δy
 -(DA)MVA - (DL)MVL 
1 y
MVL  Δy

 - (DA) - (DL)
MVA

MVA  1  y

Δy
ΔMVE  -DGAP
MVA
1 y
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.
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.
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.
Duration in Practice
Impact of convexity
Shape of the yield curve
Default Risk
Floating Rate Instruments
Demand Deposits
Mortgages
Off Balance Sheet items
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
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.
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.
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.
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
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.
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.