Finance 208
Seminar in Financial
Institutions
Professor A. Sinan Cebenoyan
Frank G. Zarb School of Business
Hofstra University
Overview – Risks
Set two
Copyright 2002, A.S. Cebenoyan
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Risks of Financial
Intermediation
• Interest Rate Risk: The risk incurred by an FI
when the maturity of its assets and liabilities are
mismatched.
0
0
Liabilities
1
1
2
Assets
Suppose the cost of Funds (liabilities) is 9 %, and interest return on
assets is 10%. Profit spread of 1%. But there is Refinancing
Risk -The Risk that the cost of rolling over or reborrowing funds will
rise above the returns being earned on asset investments.
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• Reinvestment Risk - The risk that the returns on
funds to be reinvested will fall below the cost of funds
1
0
2
Liabilities
0
Assets
1
FI borrows at 9%, and invests in an asset yielding 10%. But at what
rate will reinvestment take place?
Market Value Risk: As interest rates rise market value of assets or
liabilities will fall. Moreover, mismatching maturities by holding
longer term assets than liabilities implies when rates rise asset MVs fall
more than liabilities. This could lead to economic loss and insolvency.
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• Market Risk - The Risk incurred in the trading of
assets and liabilities due to changes in interest rates,
exchange rates, and other asset prices.
– Barings Bank lost $1.2 billion on its trading
position (buying Futures on the Nikkei index and
betting the index would rise)
• Credit Risk - The risk that the promised cash flows
from loans and securities held by FIs may not be paid
in full.
Virtually, all types of FIs face this risk. However,
those that make loans or buy bonds with longmaturities are more exposed (banks, thrifts, and life
insurance co.s). Default of a borrower puts both the
principal and the interest payments at risk.
– Diversification helps. Firm Specific Credit Risk is reduced,
while the FI is still exposed to Systematic Credit Risk
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• Off-Balance-Sheet Risk - The Risk incurred by an FI
due to activities related to contingent assets. While
all FIs, to some extent, engage in Off-Balance-Sheet
activities, mostly larger banks have drawn attention.
– For example: A letter of Credit which is a guaranty issued
by an FI for a fee (makes it attractive) on which payment is
contingent on the default of the agent that purchases the
letter of credit. Nothing appears on the B/S but the fee
appears on the income statement.
• Technology and Operational Risk
– Purpose of technology is to lower operating costs, increase
profits and capture new markets for the FI.
– Economies of Scale: The degree to which an FI’s average
unit costs of producing financial services fall as its output of
services increase
– Economies of Scope:The degree to which an FI can
generate cost synergies by producing multiple financial
service products.
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– Technology Risk occurs when technological investments do
5
- Operational Risk : The risk that existing technology or support systems may
malfunction or break down.
•Foreign Exchange Risk: The risk that exchange rate changes can affect the
value of an FI’s assets and liabilities located abroad. If a U.S. FI is net long in
foreign currency denominated assets, any depreciation of the foreign currency
against the US dollar would lead to a loss for the U.S. FI . If a net short position
prevails, then an appreciation of the foreign currency would lead to a loss.
- Even if we match the amounts of the assets and liabilities, we would still
not be fully hedged if we have exposure to foreign interest rate risk from a
maturity mismatch (simple maturity matching does not lead to a good hedge
either, we need to match durations, but more on that later).
•Country or Sovereign Risk: The risk that repayments from foreign borrowers
may be interrupted because of interference from foreign governments.
•Liquidity Risk : The risk that a sudden surge in liability withdrawals may leave
an FI in a position of having to liquidate assets in a very short period of time and
at low prices. ( Fire-Sale ) (RUN!)
•Insolvency Risk: Not having enough capital to offset a decline in asset values.
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Interest Rate Risk
The Repricing Model
• Also called the funding gap model.
• A book value accounting cash flow analysis of the
repricing gap between the interest revenue earned on an
FI’s assets and the interest paid on its liabilities over
some particular period.
• Repricing Gap:The difference between those assets
whose interest rates will be repriced or changed over
some future period (Rate sensitive assets) and liabilities
whose interest rates will be repriced or changed over
some future period (Rate sensitive liabilities).
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Assets
Liabilities
Gaps
____________________________________________
__
1 day
$20
$30
$-10
1day-3mos
30
40
-10
3mos-6mos
70
85
-15
6mos-12mos
90
70
+20
1yr-5yrs
40
30
+10
over 5yrs
10
5
+5
_______
______
______
$260
$260
0
The above breakdown in maturities has been required
by the Fed from Copyright
all banks
in the form of repricing
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gaps.
8
• Bank calculates the gaps in each bucket, by looking
at rate sensitivity of each asset and liability (time to
repricing).
 DNIIi = (GAPi) DRi = (RSAi - RSLi ) DRi
• The above applies to any i bucket.
• This can also be extended to incorporate cumulative
gaps
• Look at cumulative gap for the one-year repricing gap
CGAP= -10 + -10 + -15 + 20 = -$15
If the interest rates that apply to this bucket rise by 1
percent
DNII 1-yr = (-$15million)(.01) = -$150,000
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Assets
Liabilities
____________________________________________
_____
ST consumer loans $50 Equity Capital
$20
(1-yr mat.)
LT consumer loans
25
Demand deposits
40
30
Passbook svngs
35 3 mos CDs
30
70
20
40
3 mos BA’s
6 mos Comm.P.
1yr Time deps
20
60
(2-yr mat.)
3 mos. T-bills
6 mos T notes
40
3yr T bonds
10yr,fixed-rt mtgs
30yr floating-rt mtgs
20
(rate adj. Every 9mos)
40
2yr time deps
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• Rate Sensitive Assets----One year
–
–
–
–
ST consumer loans
3 month T-bills
6 month T-notes
30 year floating mtgs
$50
30
35
40
$155
• Rate sensitive Liabilities----One year
–
–
–
–
3 month CDs
3 month BA’s
6 month Comm. Paper
1 year Time deps.
$40
20
60
20
$140
• CGAP =RSA - RSL = 155-140 =$15 million
• If the rates rise by 1 percent
>>>DNII=15(.01)=$150,000
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• Arguments against inclusion of DD:
– explicit interest rate on DD is zero
– transaction accounts (NOW), rates sticky
– Many DD are core deps, meaning longterm
• Arguments for inclusion of DD:
– implicit interest rates (not charging fully for
checks)
– if rates rise, deposits are drawn down,
forcing bank to replace them with higheryield rate-sensitive funds
• Similar arguments for passbook savings
accounts
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•Weaknesses of The Repricing Model
•Market Value effects (true exposure not captured)
•Overaggregation (mismatches within buckets) liabilities may
be repriced at different times than assets in the same bucket.
•Runoffs : are periodic cash flows of interest and principal
amortization payments on long-term assets such as
conventional mortgages that can be reinvested at market rates.
Assets Runoffs
Liabilities
Runoffs
_________________<1yr >1 yr______________________<1yr____>1yr______
ST consumer loans $50
Equity Capital
$20
LT consumer loans
5
20
Demand deposits
30
10
3 mos. T-bills
30
Passbook svngs
15
15
6 mos T notes
35
3 mos CDs
40
3yr T bonds
10
60
3 mos BA’s
20
10yr,fixed-rt mtgs
2
18
6 mos Comm.P.
60
30yr floating-rt mtgs 40
1yr Time deps
20
2yr time deps
20
20
-------- -----------------$172
98$
$205
$65
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Interest Rate Risk
The Maturity Model
• Market Value Accounting: The assets and liabilities
of the FI are revalued according to the current level of
interest rates.
• Examples:
– How interest rate changes affect bond value:
• 1 year bond, 10% coupon, $100 face value, R=10%
– Sells at par, $100
– if interest rates go up, R=11%, sells at 99.10
– capital loss (DP1) = $0.90 per $100 value
– (DP /D R)< 0
– Rising interest rates generally lower the market values of
both assets and liabilities of an FI.
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• Show the effect of the same interest rate change if
the bond is a two-year bond, all else equal.
–
–
–
–
At R=10%, still sells at par
At R=11%, P2 = $98.29
But DP2 = 98.29 - 100 = -1.71%
Thus, the longer the maturity of a fixed-income asset or
liability, the greater its fall in price and market value for any
given increase in the level of market interest rates. But, this
increase in the fall of value happens at a diminishing
rate as time to maturity goes up.
• Maturity Model with a Portfolio of Assets &
Liabilities
– MA or ML designates the weighted average of assets and
liabilities.
– If bank has $100 in 3 year, 10% coupon bonds, and had
raised $90 with 1-year deposits paying 10%, Show effects of
a 1% rise in R.
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– Show effects of Copyright
a 7% rise
15
• Original B/S
___A________L_____
_
A=100
L=90 (1 year)
(3 year)
E=10
• 7% rise in Int. rates
___A________L_____
_
A=84.53 L=84.62
E=-0.09
D E = DA - DL
• 1% rise in Int. rates
-10.09 = -15.47 - (___A________L_____
5.38)
_
Bank is insolvent.
A=97.56 L=89.19
E=8.37
The situation is tragic
if bank has extreme
D E = DA - DL
Asset Liability
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-1.63 = (-2.44) - (mismatch
16
• In the case of Deep Discount (zero coupon)
Bonds, the problem is extreme, and the implications
are disastrous.
– Show the effect on the same balance sheet, if the assets
were 30-year deep-discount bonds.
• A 1% increase in interest rates, reduces the value of the
30-yr bond by -23.73% per $100. Thus the bank will
have net worth of -12.92 completely and massively
insolvent.
• Maturity matching, by setting MA = ML , and having
a maturity gap of 0, seems like might help. Let’s
see:
• Maturity Matching and Interest Rate Risk
Exposure
– Won’t work. Example:
– Bank issues a one-year CD to a depositor, with a face value
of $100, and 15%
interest. So, $115 is due the depositor at
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year 1.
17
• Maturities are matched, and if interest rates remain at
15% throughout the year:
– at half-year, bank receives $50 + $7.5 in interest (100 x .5 x
.15), $57.5
– at end-of-year, bank receives $50 + $3.75 in interest (50 x .5
x .15) plus the reinvestment income from the $57.5 received
at half-year, (57.5 x .5 x .15), $4.3125, for a total of $58.06.
– Bank pays off the CD at $115, and has made $0.5625
• BUT, if interest rates fell to 12% in the middle of the
year, this would not affect the 15% on the loan, nor
the 15% on the CD, but reinvestment of the $57.5 will
have to be at 12%, THUS:
– at half-year bank still gets $57.50
– at end of year, bank receives $53.75 from loan, but $3.45
from reinvestment of the $57.50 (57.5 x .5 x .12), a total of
$114.7.
– Bank pays off CD at $115, and loses $0.3, despite maturity
matching of assets and liabilities. DURATION next.
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Interest Rate Risk
The Duration Model
• Duration and duration gap are more
accurate measures of an FI’s interest rate
risk exposure
• Interest elasticity - Interest sensitivity of an
asset or liability’s value
• More complete measure as it takes into
account time of arrival of all cash flows as
well as maturity of asset or liability
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Same loan example as before: $57.5 at half-year, and $53.75 at 1-yr.
Taking present values at 15%:
PV at half-year = 57.5 / (1.075) = 53.49
PV at one-year = 53.75 / (1.075)2 = 46.51
Notice Present Values add up to $100.
Duration is the weighted-average time to maturity using the
relative present values of the cash flows as weights.
Relative present value at half year = 53.49 /100 = .5349
Relative present value at one-year = 46.51 /100 = .4651
DLoan = .5349 (1/2) + .4651 (1) = .7326
If financed by the one-year CD, DCD= 1, Negative Duration Gap!!!
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General Formula for Duration
N
D=
N
 CF  DF  t  PV  t
t =1
N
t
t
 CF  DF
t =1
t
=
t
t =1
N
t
 PV
t =1
t
Examples:
• Duration of a Six-Year Eurobond. Show 4.993 years
•Duration of a 2-year Treasury Bond. Show 1.88 years
•Duration of Zero-coupons. Always equal to maturity.
•Duration of a Perpetuity = 1 + (1/R)
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Features of Duration:
•Maturity
D
0
M
•Yield
D 2
0
2
 M
D
0
R
•Coupon Interest
D
0
C
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•The Economic Meaning of Duration
•Start with price of a coupon-bond:
C
C
CF
P=

 ... 
2
(1  R ) (1  R )
(1  R ) N
We are after a measure of interest rate sensitivity, So:
dP
-C
- 2C
- N (C  F )
=

 ... 
2
3
dR (1  R )
(1  R )
(1  R ) N 1
We rearrange terms
dP
1  C
2C
N (C  F ) 
=
 ... 
2

dR
1  R  (1  R ) (1  R )
(1  R ) N 
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Remember the concept of elasticity from economics, such as
income elasticity of demand:
Dx / x
x =
DI / I
Remember also our definition of Duration:
1
D=
C
C
(C  F )
 2

...

N

(1  R )
(1  R ) 2
(1  R ) N
C
C
(C  F )

 ... 
2
(1  R ) (1  R )
(1  R ) N
Notice that the denominator is just the price of the bond,
C
C
(C  F )
1
 2
 ...  N 
2
(1  R )
(1  R )
(1  R) N
D=
P
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C
C
(C  F )
DP = 1 
 2
 ...  N 
2
(1  R)
(1  R)
(1  R) N
Notice that the right hand side is identical to the term in brackets
in the last equation on slide number 11. So substitute DP into
that equation, we get:
dP
1
= PD
dR
1 R
or ,
dP
P
= -D
dR
(1  R )
Interest elasticity of
price?
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A further rearrangement allows us to measure price changes as a
function of duration:
Applications:
dP
 dR 
= -D
P
1  R 
•The 6-year Eurobond with an 8% coupon and 8% yield, had a
duration of D = 4.99 years. If yields rose 1 basis point, then:
dP/P = -(4.99) [.0001/1.08] = -.000462 or -0.0462%
To calculate the dollar change in value, rewrite the equation above
 dR 
dP = ( P )( - D ) 
1  R 

= (1,000)(-4.99)(.0001/1.08)= $0.462
The bond price falls to $999.538 after a one basis point
increase in yields.
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Obviously the higher the duration the higher will be the
proportionate drop in prices as interest rates rise.
A note on semiannual coupon adjustment to the duration - price
relationship:
dP
 dR 
= -D
P
1  1 / 2 R 
Duration and Immunization
•FI needs to make a guaranteed payment to an investor in five
years (in 2004) an amount of $1,469. If It invests in the market
and hopes that the rates will not fall in the next five years it would
be very risky (and stupid), after all the payment is guaranteed!
What to do? Two alternatives:
•Buy five-year maturity Discount (Zero coupon) Bonds
•Buy five-year duration coupon bond
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•If interest rates are 8%, $1,000 would be worth $1,469 in five years.
Buy 1.469 five-year zeros at $680.58 for 1 bond, paying
$1,000, and you are guaranteed $1,469 in five years. Duration
and maturity are matched, no reinvestment risk. All OK.
•If on the other hand, FI buys the six-year maturity 8% coupon, 8%
yield Eurobond with duration of 4.99 years, AND:
•Interest rates remain at 8%: Cash Flows:
Coupons, 5x80
$400
Reinvestment (80xFVAF)-400
Proceeds from sale of bond, end of year 5
69
1,000
$1,469
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•If interest rates instantaneously fall to 7%
Coupons
Reinvestment Income
Proceeds from sale of bond
$400
60
1,009
$1,469
•If interest rates instantaneously rise to 9%
Coupons
Reinvestment
Bond sale
$400
78
991
$1,469
Matching the duration of any fixed income instrument to the FI’s
investment horizon immunizes it against instantaneous interest rate
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shocks.
•Duration Gap for a Financial Institution
Let DA be the weighted average duration of the asset portfolio of the FI,
and DL be the weighted average duration of the liabilities portfolio, Then
 DR 
DA = - D A  A  
1  R 

 DR 
DL = - DL  L  
1  R 

And since D E = D A - D L ,
 DR 
DE = -[ D A A - DL L ]
1  R 

Multiply both sides with 1/A, we get
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 DR 
DE = -[ D A - DL k ]  A  
1  R 

Where k = L / A ,a measure of FI’s leverage.
The above equation gives us the effect of interest rate changes on
the market value of an FI’s equity or net worth, and it breaks down
into three effects:
1.
The leverage adjusted duration gap = [DA-DLk]
the larger this gap in absolute terms, the more exposure
2.
The size of the FI. The larger the scale of the FI the larger
the dollar size of net worth exposure
3.
Size of the interest rate shock. The larger the shock, the
greater the exposure.
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31
Example:
Suppose DA= 5 years, and DL= 3 years. For an FI with $100 million in
assets and $90 million in liabilities (with a net worth of $10 million),
the impact of an immediate 1 percent ( DR = .01) increase in interest
rates from a base of 10% on the equity of the FI would be:
DE = -(5-(.9)(3)) x $100 million x .01/1.1 = -$2.09 million
This is the reduction in equity : from $10 million to $7.91 million.
Obviously assets and liabilities go down according to the duration
formula (check the numbers please). As you can see the lower the
leverage ratio, and/or the lower the duration of the liabilities, and/or the
higher the duration of assets the higher the impact on equity.
What to do?
Get the leverage adjusted duration gap as close to 0 (zero) as possible.
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•Some Difficulties in the Application of Duration Models
•Immunization is a dynamic problem
Over time even if interest rates do not change, duration changes and
not at the same rate as calendar time.
The 6 year eurobond with 4.99 year duration (about the same as the
investment horizon -five years), a year later will have a duration of
4.31 years. Remember you were only immunized for immediate
interest rate changes. Now, a year later, you are facing a duration of
4.31 years with a 4 year horizon. Any interest rate changes now will
no longer be applying to an immunized portfolio. Need to rebalance
the portfolio ideally continuously, frequently in practice.
•Convexity
What if interest rate changes are large? Duration assumes a linear
relationship between bond price changes and interest rate changes.
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The actual price-yield relationship is nonlinear.
Convexity is the degree of curvature of the price-yield curve around
some interest rate level.
A nice feature of convexity is that for rate increases the capital loss
effect is smaller than the capital gain effect for rate decreases. Higher
convexity generates a higher insurance effect against interest rate risk.
•Measuring convexity and offsetting errors in duration model
After a Taylor’s series expansion and dropping the terms with third and
higher order, we get:
DP
1
= -( MD ) DR  CX ( DR ) 2
P
2
Where, MD is modified duration, D/(1+R). CX reflects the
degree of curvature in the price-yield curve at the current yield
level.
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The capital loss The capital gain 
CX = Scaling Factor from a 1 - basis  from a 1 - basis 


point fall
point rise

The sum of the terms in the brackets gives us the degree to which the
positive effect dominates the negative effect. The scaling factor
normalizes this difference. A commonly used scaling factor is 108.
Example: Convexity of the 8%, 6-year Eurobond:
CX = 108 [{(999.53785-1000)/1000} + {(1000.46243-1000)/1000}] = 28
DP
1
2 For a 2% rise in R, from 8% to 10%
= -( MD )DR  28( DR )
P
2
The relative change in price will be:
DP/P = - [4.99/1.08] .02 + (1/2)(28)(.02)2 = -.0924 + .0056 =-.0868 or
8.68%. Notice how convexity corrects for the overestimation of duration
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