1. Interest rate risk and the repricing gap model Session 1 Andrea Sironi Mafinrisk – 2010 Market risk

2. 2 Agenda  Interest rate risk  The repricing gap Model  Marginal and cumulative gaps  Problems of the repricing gap model  The standardized gap

3. 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. 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.

5. 5 The repricing gap

6. 6 The model at work  Starting point:  We can also write  If the change is the same for assets and liabilities’ interest rates:

7. 7 Follows GapPositiveNegative 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. 8 The model at work  Some useful indicators:  impact on profitability of lending activity  Impact on profitability (Return on fin. assets)  scale independent

9. 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

10. 10 Examples: today 1 year p =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 year p =6/12 Fixed ratenew rate conditions 6 months Gappping period: 12 months time Case 2: CCT with repricing in 6 months For any sensitive asset: the same applies to sensitive liabilities

11. 11 The solution for the timing problem  We can write  i j = current int. rate for the asset j-th  = interest rate after variation  p j = is the time (expressed as a fraction of the gapping period) from today to the next renegotiation of the int. rate

12. 12 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)

13. 13 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.

14. 14 An example 1 month gap = 140 3 months gap = –30 ASSETS€ mLIABILITIES€ 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 Total1000Total1000

15. 15 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 PeriodSensitive Assets Sensitive Liabilities Marginal GAP Cumulative GAP 0-1 months20060140 1-3 months30200-170-30 3-6 months2008012090 6-12 months70160-900 1-5 years170180-10 5-10 years2001208070 10-30 years1308050120 Total1000880--

16. 16 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 T 0 (basis points) Effect on the NII T0T0 6.0%3% 1 month1405.5%2.5%-50  3 months-1706.3%3.3%+30  6 months1205.6%2.6%-40  12 months-906.6%3.6%+60  Total 

17. 17

18. 18 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.

19. 19 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)  t i = 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.

20. 20 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 (€) T0T0 6.03.0 1 month 1407.04.01001.40.96134.41,344,000 3 months -1707.04.0100-1.70.83-141.7-1,416,667 6 months 1207.04.0100+1.20.6375.6756,000 12 months -907.04.0100-0.90.25-22.5-225,000 Total 0046.4464,000 Not weighted GAP Time weighted GAP

21. 21 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

22. 22 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

23. 23 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

24. 24 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

25. 25 Standardized Gap

26. 26 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) 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 Total1000Total1000

27. 27 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

28. 28 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 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) Second step: allocate deposits to different corresponding maturity buckets

29. 29 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 Never Beyond 6 m Within 6 m Within 3 m Same month The sensitivity coefficients may change depending on the sign of the interest rate change Ex. Interest rate on deposits

30. 30 Maturity adjusted Gap standardized and non-standardized Non standardized MaGap: 638.3 – 678.3 = -40 Standardizzato MaGap: 618.9 – 610.7 = 8,2

31. 31 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

32. 32 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

33. 33 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

34. 34 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.).

35. 35 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 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

36. 36 Questions & Exercises 5. 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.