Money, Banking, and Financial Markets

1
Money, Banking, and Financial Markets

• Professor A. Sinan Cebenoyan
• Stern School of Business - NYU
• Set 2

2
Interest Rate RiskThe 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 FIs assets and the interest paid on its
  liabilities over some particular period.
• Repricing GapThe 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).

3

• 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 all banks in the form of
  repricing gaps.

4

• 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

5

• Assets Liabilities
• _________________________________________________
  
• ST consumer loans 50 Equity Capital 20
• (1-yr mat.)
• LT consumer loans 25 Demand deposits 40
• (2-yr mat.)
• 3 mos. T-bills 30 Passbook svngs 30
• 6 mos T notes 35 3 mos CDs 40
• 3yr T bonds 70 3 mos BAs 20
• 10yr,fixed-rt mtgs 20 6 mos Comm.P. 60
• 30yr floating-rt mtgs 40 1yr Time deps 20
• (rate adj. Every 9mos) 2yr time deps 40
• -------- ------
• 270 270

6

• Rate Sensitive Assets----One year
• ST consumer loans 50
• 3 month T-bills 30
• 6 month T-notes 35
• 30 year floating mtgs 40
• 155
• Rate sensitive Liabilities----One year
• 3 month CDs 40
• 3 month BAs 20
• 6 month Comm. Paper 60
• 1 year Time deps. 20
• 140
• CGAP RSA - RSL 155-140 15 million
• If the rates rise by 1 percent DNII15(.01)15
  0,000

7

• 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 higher-yield
  rate-sensitive funds
  
• Similar arguments for passbook savings accounts
• Gap Ratio (CGAP / A) 15/270 .056 5.6
• Tells us the direction of interest rate exposure
  ( or -)
  
• the scale of the exposure

8

• 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
_________________1 yr____________________
__1yr______ ST consumer loans 50 Equ
ity Capital 20 LT consumer loans 5 20 Dema
nd deposits 30 10 3 mos. T-bills 30 Pass
book 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 Ti
me deps 20 2yr time deps 20 20 --
------ -------- ------ ------
172 98 205 65
9
Interest Rate RiskThe 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, R10
• Sells at par, 100
• if interest rates go up, R11, sells at 99.10
• capital loss (DP1) 0.90 per 100 value
• (DP /D R)
• Rising interest rates generally lower the market
  values of both assets and liabilities of an FI.
  

10

• Show the effect of the same interest rate change
  if the bond is a two-year bond, all else equal.
  
• At R10, still sells at par
• At R11, 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.
  
• Show effects of a 7 rise

11

• Original B/S
• ___A________L______
• A100 L90 (1 year)
• (3 year) E10
• 1 rise in Int. rates
• ___A________L______
• A97.56 L89.19
• E8.37
• D E DA - DL
• -1.63 (-2.44) - (-0.81)

• 7 rise in Int. rates
• ___A________L______
• A84.53 L84.62
• E-0.09
• D E DA - DL
• -10.09 -15.47 - (-5.38)
• Bank is insolvent.
• The situation is tragic if bank has extreme Asset
  Liability mismatch

12

• 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. Lets see
  
• Maturity Matching and Interest Rate Risk
  Exposure
  
• Wont 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 year 1.
  
• Same bank lends to borrower 100 for one year at
  15, But requires half to be repaid in six
  months, the other half at end of year (plus
  interest, of course).

13

• 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.
  

14
Interest Rate RiskThe Duration Model

• Duration and duration gap are more accurate
  measures of an FIs interest rate risk exposure
  
• Interest elasticity - Interest sensitivity of an
  asset or liabilitys value
  
• More complete measure as it takes into account
  time of arrival of all cash flows as well as
  maturity of asset or liability

15
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!!!
16
General Formula for Duration

• 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)

17

• Features of Duration
• Maturity

• Yield

• Coupon Interest

18

• The Economic Meaning of Duration
• Start with price of a coupon-bond

We are after a measure of interest rate
sensitivity, So
19
Remember the concept of elasticity from
economics, such as income elasticity of demand
Remember also our definition of Duration
Notice that the denominator is just the price of
the bond,
20
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
Interest elasticity of price?
21
A further rearrangement allows us to measure
price changes as a function of duration

• Applications
• 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

(1,000)(-4.99)(.0001/1.08) 0.462
The bond price falls to 999.538 after a one
basis point increase in yields.
22
Obviously the higher the duration the higher will
be the proportionate drop in prices as interest
rates rise. A note on semiannual coupon adjustmen
t to the duration - price relationship

• 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

23

• 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 69
• Proceeds from sale of bond, end of year 5
  1,000
  
• 1,469

24

• If interest rates instantaneously fall to 7
• Coupons 400
• Reinvestment Income 60
• Proceeds from sale of bond 1,009
• 1,469
• If interest rates instantaneously rise to 9
• Coupons 400
• Reinvestment 78
• Bond sale 991
• 1,469
• Matching the duration of any fixed income
  instrument to the FIs investment horizon
  immunizes it against instantaneous interest rate
  shocks.

25

• 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

And since D E D A - D L ,
Multiply both sides with 1/A, we get
26
Where k L / A ,a measure of FIs leverage.
The above equation gives us the effect of
interest rate changes on the market value of an
FIs 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 t
erms, 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.
27
Example Suppose DA 5 years, and DL 3 years. F
or 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 a
djusted duration gap as close to 0 (zero) as
possible.
28

• 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.

29

• The actual price-yield relationship is nonlinear.
  See graph 7-6.
  
• 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 Taylors series expansion and dropping
  the terms with third and higher order, we get

Where, MD is modified duration, D/(1R). CX
reflects the degree of curvature in the
price-yield curve at the current yield level.
30
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 Euro
bond CX 108 (999.53785-1000)/1000 (1000.
46243-1000)/1000 28
For a 2 rise in R, from 8 to 10
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
31
Market Risk

• Market Risk (Value at Risk, VAR) dollar exposure
  amount (uncertainty in earnings) resulting from
  changes in market conditions such as the price of
  an asset, interest rates, market volatility, and
  market liquidity.
• The five reasons for market risk management
• Management information (senior management sees
  exposure)
  
• Setting Limits(limits per trader)
• Resource Allocation (identify greatest potential
  returns per risk)
  
• Performance Evaluation (return-risk per trader

Bonus)
- Regulation (provide private sector benchmarks)
32
JPMs RiskMetrics Model

• Large commercial banks, investment banks,
  insurance companies, and mutual funds have all
  developed market risk models (internal models).
  Three major approaches to these internal models
• JPM Riskmetrics
• Historic or back-simulation
• Monte Carlo simulation
• We focus on JPM Riskmetrics to measure the market
  risk exposure on a daily basis for a major FI.
  
• How much the FI can potentially lose should
  market conditions move adversely
  
• Market Risk Estimated potential loss under
  adverse
  
• circumstances

33
Daily earnings at risk ( market value of
position) x (Price volatility)
where, Price volatility (Price Sensitivity) x (
Adverse daily yield move)
We next look at how JPM Riskmetrics model
calculates DEaR in three trading areas Fixed in
come, Foreign exchange, and Equities,
and how the aggregate risk is estimated.
Market Risk of Fixed Income Securities
Suppose FI has a 1 million market value position
in 7-yr zero coupons with a face
value of 1,631,483.00 and current annual yield
is 7.243 .
Daily Price volatility
The modified duration for this bond
34
If we make the (strong and unrealistic)
assumption of normality in yield changes, and we
wish to focus on bad outcomes, i.e., not just
any change in yields, BUT an increase in yields
that will only be possible with a probability, i.
e., a yield increase that has a chance
of 5, or 10, or 1(We decide how likely an
increase we wish to be worried about). Suppose we
pick 5 , i.e., there is 1 in 20
chance that the next days yield change will
exceed this adverse move. If we can fit a normal
distribution to recent yield changes and get a
mean of 0 and standard deviation of 10 basis poi
nts (0.001), and we remember that 90 of the
area under the normal distribution is found withi
n /- 1.65 standard deviations, then
we are looking at 1.65s as 16.5 basis points.
Our adverse yield move. Price Volatility -MD
(DR) (-6.527) (.00165) -.01077
DEaR DEAR ( market value of position) (Price
Volatility) (1,000,000) (.01077)
dropping the minus sign
10,770 The potential
daily loss with 5 chance For multiple N days, DE
AR should be treated like s, and VAR computed
as
35

• Foreign Exchange
• Suppose FI has SWF 1.6 million trading position
  in spot Swiss francs. What is the
  
• DEAR from this?
• First calculate the amount of the position
• amount of position (FX position) x (/SWF)
• (SWF 1.6million) x (.625) 1 million
• If the standard deviation (s) in the recent past
  was 56.5 basis points, AND
  
• we are interested in adverse moves that will not
  be exceeded more than 5
  
• of the time, or 1.65s
• FX volatility 1.65(56.5) 93.2 basis points
  THUS,
  
• DEAR ( amount of position) x (FX volatility)
• ( 1million) x (.00932)
• 9,320

36
Equities Remember your CAPM Total Risk System
atic risk Unsystematic risk

If the FIs trading portfolio is well
diversified, then its beta will be close to 1,
and the unsystematic risk will be diversified awa
y.leaving behind the market risk.
Suppose the FI holds 1million in stocks that
reflect a US market index, Then
DEAR ( value of position) x (Stock market
return volatility) (1,000,000) (1.65 s m) I
f the standard deviation of daily stock returns
on the market in the recent past
was 2 percent, then 1.65(s m) 3.3 percent
DEAR (1,000,000) (0.033) 33,000
37
Portfolio Aggregation We need to figure out the a
ggregate DEAR, summing up wont do, REMEMBER
If the correlations between the 3 assets are
Bond SWF/ US Stock Index
Bond -.2 .4
SWF/ .1
38
Then the risk of the whole portfolio, DEAR
treated like s, will be
Substituting the values we have
39,969
39

• BIS Standardized Framework for Market Risk
• Applicable to smaller banks.
• Fixed Income
• Specific Risk charge (for liquidity or credit
  risk quality)
  
• General Market Risk charge
• Vertical and horizontal offsets
• Foreign Exchange
• Shorthand method (8 of the maximum of the
  aggregate net long or net short positions)
  
• Longhand method Net position, Simulation, worst
  case scenario amount is charged 2

40

• Equities
• Unsystematic risk charge (x-factor) 4 against
  the gross position
  
• Systematic risk charge (y-factor) 8 against the
  net position
  
• Large Bank Internal models
• BIS standardized framework was criticized for
  crude risk measurements lack of correlations
  incompatability with internal systems.
  
• BIS in 1995 allowed internal model usage by large
  banks with conditions
  
• Adverse change is defined as 99th percentile -
  Minimum holding period is 10 days - correlations
  allowed broadly
  
• Proposed capital charge will be the higher of the
  previous days VAR, or the average daily VAR over
  the last 60 days times a factor (at least 3).
  Tier 2 and 3 allowed up to 250 of Tier 1.