Interest rate risk and the repricing gap model

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Interest rate risk and the repricing gap model
Mafinrisk 2010Market risk

• Session 1
• Andrea Sironi

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Agenda

• Interest rate risk
• The repricing gap Model
• Marginal and cumulative gaps
• Problems of the repricing gap model
• The standardized gap

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Interest rate risk

• Assets maturity gt liabilities ?refinancing risk
• Assets maturity lt liabilities ?reinvestment risk.
• A change in the level of interest rates has a
  double economic effect
• Direct effect change in the market value of A/L
  and in the level of interests paid and received
• Indirect effect change in the amounts of
  financial activities

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The Repricing Gap Model

• Income oriented model ? target variable Net
  Interest Income (NII) Interest Revenues
  Interest Expenses
• Interest Rate Gap ? difference between assets and
  liabilities sensitive to interest rates changes
  in a predefined time period.
• An asset or a liability is sensitive if, in the
  relevant time period (gapping period), it
  reaches its maturity or there is a renegotiation
  of the interest rate.

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The repricing gap
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The model at work

• Starting point
• We can also write
• If the change is the same for assets and
  liabilities interest rates

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Follows
Gap Positive Negative
Increase of int. rates (?i gt 0) Increase of net interest income (NII?) Decrease of net interest income (NII?)
Decrease of int. rates (?i lt 0) Decrease of net interest income (NII?) Increase of net interest income (NII?)
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The model at work

• Some useful indicators
• ? impact on profitability of lending
  activity
• ? Impact on profitability (Return on
  fin. assets)
• ? scale independent

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The timing problem

• We have made the assumption that a change of the
  interest rate will produce the same effect for
  every sensitive asset or liability
• Under this assumption
• In the real world the effect is different for
  every A/L and is proportional to the time gap
  between the renegotiation time and the ending of
  the gapping period

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Examples
Gappping period 12 months
11 months
Case 1Interbank deposit with a residual life
of 1 month
Gappping period 12 months
today
1 year
p 1/12
time
fixed rate
new rate conditions
Case 2CCT with repricingin 6 months
today
1 year
p 6/12
6 months
time
Fixed rate
new rate conditions
For any sensitive asset
the same applies to sensitive liabilities
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The solution for the timing problem

• We can write
• ij current int. rate for the asset j-th
• interest rate after variation
• pj is the time (expressed as a fraction of the
  gapping period) from today to the next
  renegotiation of the int. rate

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The solution for the timing problem (follows)

• We can do the same for liabilities
• We can calculate the maturity adjusted gap
  (every A/L has a weight proportional to the
  distance from the renegotiation period to the end
  of the gapping period)

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Marginal and cumulative gap

• An alternative to Magap that can be used to
  estimate the true exposure of the bank to changes
  in interest rates is the one based on the use of
  gaps relative to different time periods.
• Marginal Gap the difference between assets and
  liabilities with renegotiation of the interest
  rate in a certain time period.
• Cumulative Gap difference between assets and
  liabilities with renegotiation of the interest
  rate before a certain date.

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An example
ASSETS m LIABILITIES m
Deposits with banks (1 month) BOT (3 months) CCT (5 years) (next rate revision 6 months) Short term loans (5 months) Floating rate mortgages (20 y) (next rate revision 1 year) BTP (5 years) Fixed rate mortgages (10 y) BTP (30 years) 200 30 120 80 70 170 200 130 Deposits with banks (1 month) Floating rate notes (next revision 3 months) Floating rate notes (next revision 6 months) Fixed rate notes (1 year) Fixed rate notes (5 years) Fixed rate notes (10 y) Junior debt (20 y) Shareholders Equity 60 200 80 160 180 120 80 120
Total 1000 Total 1000
1 month gap 140 3 months gap 30
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Marginal and cumulative gaps
Time Period Sensitive Assets Sensitive Liabilities Marginal GAP Cumulative GAP
0-1 months 200 60 140 140
1-3 months 30 200 -170 -30
3-6 months 200 80 120 90
6-12 months 70 160 -90 0
1-5 years 170 180 -10 -10
5-10 years 200 120 80 70
10-30 years 130 80 50 120
Total 1000 880 - -
The bank has a long net position for the first
month and for the period from 3 to 6 months and a
short net position for the period from 1 to 3
months and for the period from 6 to 12 months.
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Follows
Given the null 1 year gap if in every time
sub-period the interest rate change is adverse
the bank can experience a decrease in the net
interest income.
Time Period Marg. GAP ( mln) Int. rate Assets Int. rate Liabilities ?i with respect to T0 (basis points) Effect on the NII
T0 6.0 3
1 month 140 5.5 2.5 -50 ?
3 months -170 6.3 3.3 30 ?
6 months 120 5.6 2.6 -40 ?
12 months -90 6.6 3.6 60 ?
Total ?
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(No Transcript)
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The effect on Net Interest Income

• To quantify the effect of the various interest
  rate changes we have to keep track of the length
  of the time period on which every change has an
  effect.
• Even with a null annual gap we can have a non
  zero effect on the 1 year net interest income
  because every interest rate change has an effect
  on a different time period with a different
  marginal gap.

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Follows

• We can weight every marginal gap for the
  difference between the average renegotiation
  period inside the marginal gap and the end of the
  evaluation period (usually 1 year).
• T global gapping period (1 year)
• ti average renegotiation period inside the i-th
  gapping period
• n number of the time periods evaluated inside
  the global gapping period
• WGAPT sensitivity of NII to changes of interest
  rates ? duration of NII.

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Some numbers
Time period Marg. GAP ( mln) Asset Int. Rates Liab. Int. Rates ?i (b.p.) (GAPx?i) ( mln) T-ti (T-ti) x GAPi ( mln) (T-ti) x GAPi x ?i ()
T0 6.0 3.0
1 month 140 7.0 4.0 100 1.4 0.96 134.4 1,344,000
3 months -170 7.0 4.0 100 -1.7 0.83 -141.7 -1,416,667
6 months 120 7.0 4.0 100 1.2 0.63 75.6 756,000
12 months -90 7.0 4.0 100 -0.9 0.25 -22.5 -225,000
Total 0 0 46.4 464,000
Not weighted GAP
Time weighted GAP
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Conclusion

• Non zero marginal gaps can generate a non zero
  variation of the interest margin even with a null
  cumulative gap for two main reasons
• The changes of interest rates can be non uniform
  across different time sub-periods
• The effect on the net interest income of the
  change of interest rates is different across
  different time sub-periods
• To have a zero sensitivity of the NII we need
  zero marginal gap for every time sub period

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Maturity-adjusted gap versus time weighted
cumulative gap

• The maturity-adjusted gap is more precise, as it
  considers the actual maturity of each asset and
  liability
• The time weighted cumulative gap (based on
  marginal gaps) considers one virtual maturity,
  equal to the median value
• However, marginal gaps have an advantage they
  allow to estimate the impact on NII of different
  interest rate changes that may occur during the
  year

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Limits and problems

1. Assumption of a uniform change of assets and
   liabilities interest rates.
2. Assets Liabilities with no maturity (e.g. call
   deposits)
3. The model does not consider effects on the market
   value of A/L.
4. Assumption of a uniform change of interest rates
   for different maturities.
5. The model does not consider the effect of a
   variation of interest rates on the volume of
   financial assets and liabilities

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Answer to problem 1 Standardized Gap

• The first problem can be addressed with the
  following procedure
• We identify a reference market rate, for example
  a 3 months interbank rate
• We estimate the sensitivity of different assets
  and liabilities interest rates to the reference
  rate
• We can calculate the standardized gap to evaluate
  the sensitivity of the NII to a change of the
  reference rate

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Standardized Gap
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An example
ASSETS m ? LIABILITIES m ?
Deposits with banks (1m) BOT (3m) Floating rate loans (5y) Floating rate loans (on call) Variable rate mortgages (10y) (euribor 100 basis points) 80 60 120 460 280 1,10 1,05 0,9 0,95 1,00 Deposits with banks (1m) Deposits (on call) Floating rate notes (next revision 3m) Fixed rate notes (1y) Floating rate bonds (10y) (euribor 50 b.p.) Shareholders Equity 140 380 120 80 160 120 1,10 0,80 0,95 0,90 1,00
Total 1000 Total 1000

• GAP 120 vs Standardized GAP 172
• Higher average sensitivity of Assets
• We can also solve the problem of call deposits
  and loans

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Answer to problem 2 how to treat call deposits
and other no maturity ALs

• 3 steps
• Analyse how much and after how long, on average,
  historically a market interest rate change gets
  reflected in call deposits rates
• Divide SA and SL in coherent manner, based on the
  historical empirical evidence.
• Compute the repricing gap based on the new values
  of SA and SL

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Asset liabilities with no maturity (e.g.
current account deposits)
First step estimate sensitivity to interest rate
changes
1.0
50
0.9
Second step allocate deposits to different
corresponding maturity buckets
0.8
0.7
0.6
Ex. Interest rate on deposits Given a 1 increase
of the interbank rate, the interest rate on
Italian banks deposits increases by 5 bp
immediately, 27 bp the following month, other 10
bp in the following 2 months The total increase
is 50 bp(deposits have a 0.5 beta)
0.5
8
10
0.4
0.3
27
0.2
0.1
5
0.0
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One problem sensitivity may be asymmetric
Ex. Interest rate on deposits
The sensitivity coefficients may change depending
on the sign of the interest rate change
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Maturity adjusted Gapstandardized and
non-standardized
Non standardized MaGap 638.3 678.3
-40 Standardizzato MaGap 618.9 610.7 8,2
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Residual problems

1. The model does not consider effects on the market
   value of A/L.
2. Assumption of a uniform change of interest rates
   for different maturities.
3. The model does not consider the effect of a
   change of interest rates on the volume of
   financial assets and liabilities

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Questions Exercises

• 1. What is a sensitive asset in the repricing
  gap model?
• A) An asset maturing within one year (or
  renegotiating its rate within one year)
• B) An asset updating its rate immediately when
  market rates change
• C) It depends on the time horizon used as gapping
  period
• D) An asset the value of which is sensitive to
  changes in market interest rates

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Questions Exercises

• 2. The assets of a bank consist of 500 of
  floating-rate securities, repriced quarterly (and
  repriced for the last time 3 months before), and
  of 1,500 of fixed-rate, newly issued two-year
  securities its liabilities consist of 1,000 of
  demand deposits and of 400 of three-year
  certificates of deposit, issued 2.5 years before.
  Given a gapping period of one year, and assuming
  that the four items mentioned above have a
  sensitivity (beta) to market rates (e.g, to
  3-month interbank rates) of 100, 20, 30 and
  110 respectively, identify which of the
  following statements is correct
• A) The gap is negative, the standardised gap is
  positive
• B) The gap is positive, the standardised gap is
  negative
• C) The gap is negative, the standardised gap is
  negative
• D) The gap is positive, the standardised gap is
  positive

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Questions Exercises

• 3. Bank Omega has a maturity structure of its
  assets and liabilities like the one shown in the
  Table below.
• Find
• A) Cumulated gaps of different maturities
• B) Marginal (periodic) gaps relative to the
  following maturity buckets (i) 0-1 month, (ii)
  1-6 months, (iii) 6 months-1 year, (iv) 1-2
  years, (v) 2-5 years, (vi) 5-10 years, (vii)
  beyond 10 years
• C) The change experienced by NII next year if
  lending and borrowing rates increase, for all
  maturities, by 50 basis points, assuming that the
  rate repricing will occur exactly in the middle
  of each time band (e.g., after 15 days for the
  band between 0 and 1 month, 3.5 months for the
  band 1-6 months, etc.).

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Questions Exercises

• 4. The interest risk management scheme followed
  by Bank Lambda requires it to keep all marginal
  (periodic) gaps at zero, for any maturity band.
  The Chief Financial Officer states that,
  accordingly, the banks net interest income (NII)
  is immune from any possible change in market
  rates. Which among the following events could
  prove him wrong?
• I) A change in interest rates not uniform for
  lending and borrowing rates
• II) A change in long term rates which affects the
  market value of items such as fixed-rate
  mortgages and bonds
• III) The fact that borrowing rates are stickier
  than lending rates
• IV) A change in long term rates greater than the
  one experienced by short-term rates
• A) I and III
• B) I, III and IV
• C) I, II and III
• D) All of the above

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Questions Exercises

• 5. Using the data in the Table below (and
  assuming, for simplicity, a 360-day year made of
  12 30-day months)