11-Interest Rate Risk
Review
 Interest Rates are determined by supply and
demand, are moving all the time, and can be
difficult to forecast.
 The yield curve is generally upward sloping
 Interest Rate Risk: The uncertainty
surrounding future interest rates.
Our Focus
 Unforeseen parallel shifts in the yield curve
 Unforeseen changes in the slope of the yield curve
Where We are Going
 Dollar Gap
 Method to understand the impact of interest rate
risk on bank profits
 Simple, and requires some ad-hoc assumptions
 Not discussed in Bodie-Kane-Marcus
 Duration
 Method to understand the impact of interest rate
risk on the value of bank shareholder equity
 More elegant and mathematically intense
 The focus of the reading in Bodie-Kane-Marcus
 Used extensively as well by bond traders
Interest Rate Risk
 Banks assets
 Generally long-term, fixed rate
 Bank liabilities
 Generally short term, variable rate
 Impact on profits:
 Rates increase
 Interest received stays fixed
 Interest paid increases
 Profits decrease
Interest-Rate Sensitive
 An asset or liability whose rate is reset within
some “short period of time”
e.g. 0-30 days, 1-year, etc.
 Interest rate sensitive assets:
 Short-term bond rolled over into other short-term
bonds
 Variable rate loans
 Interest rate sensitive liabilities:
 Short-term deposits
Interest-Rate Risk Example
 Assets: 50 billion
 5B IRS; rate = 8% per year
 Liabilities: 40 billion
 24B IRS; rate = 5% per year
 Profits ($billion)
 50(.08)-40*(.05) = 2
Parallel Shift in Yield Curve
 Suppose all rates increase by 1%
 Assets
 IRS (5B): rate = 9%
 Not IRS (45B): rate = 8%
 Liabilities
 IRS (24B): rate = 6%
 Not IRS (16B): rate = 5%
Profits After Rate Increase
 Interest Expenses ($billion)
 16(.05)+24(.06)=2.24
 Interest revenues ($billion)
 5*(.09)+45*(.08)=4.05
 Profits
 Before: 2 billion
 After:4.05-2.24=1.81 billion
 Decrease: 0.19 billion or 9.5% drop in profits
(1.81/2)-1=.095
Gap Analysis
 Gap = IRSA – IRSL
 IRSA = dollar value of interest rate
sensitive assets
 IRSL = dollar value of interest rate
sensitive liabilities
Gap
 From Previous Example
 IRSA (millions) = 5
 IRSL (millions) = 24
 Dollar Gap (millions): 5 – 24 = -19
 Change in profits= Dollar Gap Di
 From previous example:
 Change in profits (millions)
-19 .01 = -0.19
Gap Analysis
 If the horizon is long enough,
virtually all assets are IRS
 If the horizon is short enough,
virtually all assets become non-IRS
 No standard horizon
Dollar Gap Summary
Dollar GAP
Negative
Negative
Positive
Positve
Zero
Di
Increase
Decrease
Increase
Decrease
Either
DProfits
Decrease
Increase
Increase
Decrease
Zero
What is the right GAP?
 One of the most difficult questions bank
managers face
 Defensive Management
 Reduce volatility of net interest income
 Make Gap as close to zero as possible
 Aggressive Management
 Forecast future interest rate movements
 If forecast is positive, make Gap positive
 If forecast is negative, make Gap negative
Problems with Gap
 Time horizon to determine IRS is
ambiguous
 Ignores differences in rate sensitivity
due to time horizon
 Focus on profits rather than
shareholder wealth
Building a Bank

Suppose you are in the process of creating a
bank portfolio.

Shareholder equity: $25 million
You’ve raised $75M in deposits (liabilities)
You’ve purchased $100M in 30-yr annual coupon
bonds (assets)


Bank Equity and Interest Rates
Assets
 30-yr bonds
–
–
–
FV: $100M
Coupon rate: 1.8%
YTM=1.8%
Liabilities
 Deposits
–
–
$75M
Paying 1% per year
Annual profits: $1.8M - $0.75M = $1.05M
Rate on bonds is fixed – no matter how rates change.
Rate on deposits resets every year.
Gap Analysis

IRSA – IRSL = 0 – 75M = -75M
Assume rates increase by 10 basis points
We must now pay 1.1% on deposits

Change in profits: -75M(.001) = -75,000


–
Profits down 7% = 75K/1.05M
Gap Analysis

One Solution: To protect profits from interest rate
increases, sell your holdings in the long term bonds
and buy shorter term bonds

But since yield curve is usually upward sloping
(liquidity risk-premiums), shorter term bonds will
usually earn lower yields.

Result: Lower profits
Manager’s Objective

Managers should probably not be concerned
about protecting profits.

Instead, should be concerned about
protecting value of shareholder equity: the
value shareholders would get if they sold
their shares.
Market Value of Bank Assets


Before Rates Increase: Assets =$100M
After rate increase?
–
–
-
Bank is earning 1.8% on 30-yr bonds
Other similar 30-year bonds are paying a YTM of 1.9%
Market value of bank assets:
-
-
N=30, YTM=1.9%, PMT=1.8M, FV=100M
Value = $97.73M
Market Value of Bank Liabilities

Liability of $75.75M due in one year
–
–



Principal and interest
Depositors will “redeposit” principal with you at new rate in
1-year
Market Value of liabilities = amount I would have to
put away now at current rates to pay off liability in
one year = present value
Before rates increase: 75.75/1.01=$75M
After rates increase: 75.75/1.011=$74.93M
Market Value of Equity

PV(assets) – PV(liabilities)

One way to think of it:
–
–
–
–
Assume 1 individual were to purchase the bank
After purchasing the bank she plans to liquidate
When she sells assets, she will get PV(assets)
But of these assets, she will have to set aside some
cash to pay off liabilities due in 1 year, PV(liabilities)
Market Value of Equity

One way to think of it (continued)
–
–
–
When considering a purchase price, she shouldn’t
pay more than PV(assets)-PV(liabilities)
But current shareholders also have the option to
liquidate rather than sell the bank
Current shareholders shouldn’t take anything less
than PV(assets) - PV(liabilities)
Interest Rates and Bank Equity

Before rates increase: equity=$25M
After rates increase: 97.73-74.93=$22.8M

Change in equity: 22.8M-25M = -2.2M

A 10 basis point increase in rates leads to a
drop in equity of 8.8% (22.8/25-1=-.088)

Solutions

Sell long term bonds and buy short-term bonds
–
–
–

Problem: Many assets of banks are non-tradable loans
(fixed term) – more on this later.
Some bank loans are tradeable: securitized mortgages
How much do we want to hold in long vs. short bonds?
Refuse to grant long-term fixed rate loans
–
–
Problem: No clients – no “loan generation fees”
Bank wants to act as loan broker
Solutions




What should be our position in long versus
short-term bonds?
How much interest rate risk do we want?
Longer term bonds earn higher yields, but the
PVs of such bonds are very sensitive to interest
rate changes.
We need a simple way to measure the
sensitivity of PV to interest rate changes.
PV and Yields
$30,000,000.00
$25,000,000.00
$20,000,000.00
PV
$15,000,000.00
$10,000,000.00
$5,000,000.00
$0.00
0
0.02
0.04
0.06
y
0.08
0.1
0.12
Modified Duration
DPV
  D*  PV
Dy
or rather
DPV   D*  PV  Dy 


Duration: a measure of the sensitivity of PV to changes
in interest rates: larger the duration, the more sensitive
Bank managers choose bank portfolio to target the
duration of bank equity.
Duration and Change in PV

Let DPV = “change in present value”

The change in PV for any asset or liability is
approximately
DPV   D*  PV  Dy 
D* " Modified Duration"
Dy  change in yields (parallel shift of yield curve)
Example

From before:
–
–


Original PV of 30-year bonds: $100M
When YTM increased 10 bp, PV dropped to 97.73
DPV=97.73M -100M = -2.27M
Duration Approximation
PV  100 M
Dy  0.001
D*  23.02 (you don' t know how to find D* yet)
ΔPV  -23.02  .001  100 M  2.30 M
Example

From before:
–
–


Original PV of liabilities: $75M
When rate increased 10 bp, PV dropped to
74.93M
DPV=74.93M -75M = -0.074M
Duration Approximation
PV  75 M
Dy  0.001
D*  0.99 (you don' t know how to find D* yet)
ΔPV  -0.99  .001  75 M  0.074 M
Example

Change in bank equity using duration
approximation:
DE  DA  DL
 2.30 M  ( 0.07 M )  2.23M

Before, the change in equity was -2.20.
Modified Duration

Modified Duration is defined as
D
D 
1 y
*
where “D” is called “Macaulay’s Duration”
Macaulay’s Duration

Let t be the time each cash flow is received (paid)

Then duration is simply a weighted sum of t
T
D   t  wt
t 1

The weights are defined as
CFt /(1  y )t
wt 
Bond Price
Example

Annual coupon paying bond
–
–


Price=$1000
Time when cash is received:
–

matures in 2 years, par=1000,
coupon rate =10%, YTM=10%
t1=1 ($100 is received), t2=2 ($1100 is received)
weights:
100 /(1.10)
w1 
 0.0909,
1000
1100 /(1.10)2
w2 
 0.9091
1000
Example

Macaulay’s Duration:
D  0.0909 1  0.9091 2  1.91

Modified Duration:
1.91
D 
 1.74
1.10
*