Download - Repricing Gap Session 1
Interest rate risk and the repricing gap model
Session 1Andrea Sironi
Mafinrisk – 2010Market risk
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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 > liabilities refinancing risk
Assets maturity < 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.
SLSAG
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The repricing gap
SensitiveLiabilities
(SLt)
Not SensitiveLiabilities
(NSLt)
SensitiveAssets(SAt)
SensitiveAssets(SAt)
Not SensitiveAssets(NSAt)
Gapt (>0)
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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:
NSLSLiNSASAiLiFAiIEIINII papa
SLiSAiNII pa
iii pa
iGiSLSANII
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Follows
iGiSLSANII
Gap Positive Negative
Increase of int. rates (i > 0)
Increase of net interest income
(NII)
Decrease of net interest income (NII)
Decrease of int. rates (i < 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
iE
G
E
NII
iFA
G
FA
NII
SL
SAGapRatio
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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
iGNII
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Examples:
today 1 yearp =1/12
fixed rate
new rate conditions
11 months
Gappping period: 12 months
time
Case 1:Interbank
deposit with a residual life of 1 month
Case 1:Interbank
deposit with a residual life of 1 month
today 1 yearp =6/12
Fixed rate new rate conditions
6 months
Gappping period: 12 months
time
Case 2:CCT
with repricingin 6 months
Case 2:CCT
with repricingin 6 months
iasipasia jjjj )1(For anysensitiveasset:
the same appliesto 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
jjjjjjjj piiSApiSAII 1
jj ii
n
jjjj piSAII
1
1 jjjj piSAII 1
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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)
kkkk piSLIE 1
m
kkkk piSLIE
1
1
iEMAGAPiEpSLpSAIEIINIIEn
jk
m
kkjj
)(11)(1 1
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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.
n
iiMGCG
1
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An example
1 month gap = 140
3 months gap = –30
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)
20030
120
8070
170200130
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
60200
80
160180120
80120
Total 1000 Total 1000
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Marginal and cumulative gaps
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.
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 - -
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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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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.
iEWGAPTiEtTGAPNIIE i
n
ii
1
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Some numbers
Time periodMarg. GAP
(€ mln)
Asset Int.
Rates
Liab. Int.
Rates
i (b.p.)
(GAPxi)(€ 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
n
i
m
jjjii SLSASG
1 1
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Standardized GapFigure 4: example of an estimate of the beta of a rate-sensitive asset
rrr jj 95.0
rj(rate
variations on on-
demand loans)
r(three-month Euribor variations )
95.0tan j
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An example
GAP = 120 vs Standardized GAP = 172
Higher average sensitivity of Assets
We can also solve the problem of call deposits and loans
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)
8060
120460280
1,101,05
0,90,951,00
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
140380120
80160
120
1,100,800,95
0,901,00
Total 1000 Total 1000
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Answer to problem 2: how to treat call deposits and other “no maturity” A&Ls
3 steps:1. Analyse how much and after how long, on
average, historically a market interest rate change gets reflected in call deposits rates
2. Divide SA and SL in coherent manner, based on the historical empirical evidence.
3. 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)
5%
27%
10%
8%
50%
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 Never
beyond 6 m
Within 6 m
Within 3 m
Same month
First step: estimate sensitivity to interest
rate changes
Ex. Interest rate on depositsGiven 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)
Second step: allocate deposits to different
corresponding maturity buckets
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One problem: sensitivity may be asymmetric
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Increase Reduction
NeverBeyond 6 mWithin 6 mWithin 3 mSame month
The sensitivity coefficients may change depending on the sign of
the interest rate change
Ex. Interest rate on deposits
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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
Attività as j s j as j ×(1-s j ) j as j ×(1-s j ) jPrestiti a tasso variabile (aperture di credito a vista) 460 0 460,0 95% 437,0 Depositi interb. attivi a 1 m 80 1/12 73,3 110% 80,7 BOT a 3 mesi 60 3/12 45,0 105% 47,3 Crediti al consumo a tasso variabile a 5 anni (revisione tra 6 mesi) 120 6/12 60,0 90% 54,0 Mutui a tasso var. a 10 anni (euribor + 100 basis points, repricing tra 1 anno) 280 1 - 100% - Totale 638,3 618,9
Passività ps j s j ps j ×(1-s j ) j ps j ×(1-s j ) j
Depositi in c/c da clientela 380 0 380,0 80% 304,0 Depositi interbancari a 1 m 140 1/12 128,3 110% 141,2 CD a tasso variabile (prossima revisione a 3 mesi) 120 3/12 90,0 95% 85,5 Obbligazioni a tasso var. a 10 anni (euribor + 50 bp, repricing a 6 m.) 160 6/12 80,0 100% 80,0 CD a tasso fisso a 1 anno 80 1 - 90% - Totale 678,3 610,7
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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 positiveB) The gap is positive, the standardised gap is negativeC) The gap is negative, the standardised gap is negativeD) 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 maturitiesB) 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.).
Sensitive assets and liabilities for Bank Omega (data in million euros) 1 month 6
months 1 year 2 years 5 years 10 years Beyond1
0 years Total sensitive assets 5 15 20 40 55 85 100 Total sensitive liabilities 15 40 60 80 90 95 100
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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 bank’s 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 ratesIV) A change in long term rates greater than the one
experienced by short-term rates
A) I and IIIB) I, III and IVC) I, II and IIID) All of the above
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Questions & Exercises5. Using the data in the Table below (and assuming, for simplicity, a 360-day year
made of 12 30-day months):
i) compute the one-year repricing gap and use it to estimate the effect on NII of a 0.5% increase in rates;
ii) compute the one-year magap and use it to estimate the effect on NII of a 0.5% increase in rates;
iii) compute the one-year standardised magap and use it to estimate the effect on NII of a 0.5% increase in rates;
iv) compare the differences among the results under i), ii) and iii) and provide an explanation.
Assets
Amount
Days to maturity/ repricing
Demand loans 1000 0 90% Floating rate securities 600 90 100% Fixed-rate instalment loans 800 270 80% Fixed-rate mortgages 1200 720 100%
Liabilities
Amount
Days to maturity/ repricing
Demand deposits 2000 0 60% Fixed-rate certificates of deposit 600 180 90% Floating-rate bonds 1000 360 100%