Interest rate risk and the repricing gap model

Session 1Andrea Sironi

Mafinrisk – 2010Market risk

2

Agenda

Interest rate risk

The repricing gap Model

Marginal and cumulative gaps

Problems of the repricing gap model

The standardized gap

3

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

4

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

5

The repricing gap

SensitiveLiabilities

(SLt)

Not SensitiveLiabilities

(NSLt)

SensitiveAssets(SAt)

SensitiveAssets(SAt)

Not SensitiveAssets(NSAt)

Gapt (>0)

6

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

7

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)

8

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

9

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

10

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

14

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

25

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%