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# The duration gap model and clumping

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1. Mafinrisk – 2010Market Risk The duration gap model and clumping Session 2 Andrea Sironi

2. Agenda • Market value versus historical cost accounting • The duration gap model • The Clumping Model

3. The Duration Gap Model • “Market Value” model  target variable = market value of shareholders’ equity • Focus on impact of interest rate changes on the market value of assets and liabilities • Gap = difference between the change in the market value of assets and the market value of liabilities

4. Market Value vs Historical Value Dec. 31, 2008 Dec. 31, 2009

5. follows Dec. 31, 2010 On the 1/1/2009 the ECB increase the interest rates of 100 bp Nothing changes in the FS of the bank

6. follows In 2011 the bank has to finance the 10Y Mortgages with a new fixed rate note issued at the new market rate: 4% The effect of the increase of the interest rates on the profitability of the bank appears only two years after the variation itself.

7. follows This problem can be solved using in the FS the market value of A/L instead of the historical value Dec. 31, 2009

8. follows Next Year (2010) Notes (maturity)  90 Dec. 31, 2010

9. Agenda • Market value versus historical cost accounting • The duration gap model • The Clumping Model

10. The Duration gap The same result could be obtained using the duration gap

11. The Duration gap • The change in the market value of Shareholders’ Equity is a function of three variables: • The difference between the modified duration of assets and the modified duration of the liabilities corrected for the bank’s leverage (“leverage adjusted duration gap”) duration gap (DG) • The size of the intermediation activity of the bank measured by the market value of total assets • The size of the interest rates change

12. Immunization • If MVA = MVL MVE is not sensitive to interest rates changes if MDA = MDL. • If MVA > MVL MVE is not sensitive to interest rates changes if DG=0, i.e. MDA < MDL. In this case the higher sensitivity of liabilities will compensate the initial lower market value and the change in the absolute value of assets and liabilities will be equal.

13. The example again Let’s go back to our bank

14. follows For an interest rates increase of 100 bp the market value of shareholders’ equity would decrease by 6 m€ (60% of the original value)

15. Some remarks The result (-6) is different form what we got before (-3.63) for three main reasons: • -6 m€ is an instantaneous decrease estimated at the time of the int. rates change (January 1st 2004); • In the – 3.63 m€ we also have 2.3 m€ of interest margin • The duration is just a first order approximation

16. Duration gap: problems and limits • Duration (and duration gap) changes every instant, when interest rate change, or simply because of the passage of time • Immunization policies based on duration gap should be updated continuously • Duration (and duration gap) is based on a linear approximation • Impact not estimated precisely • The model assumes uniform interest rate changes (Di) of assets and liabilities interest rates

17. Every time market interest rate change, duration needs to be computed again wuth new weights (PV of cash flows) Even if rates do not change, duration decreases: linearly with “jumps” related to coupon payments Problem 1:duration changes Duration Coupon payments t1 t2 t3 time

18. Answer to problem 2:convexity Rather than proxying % change in value with the first derivative only …we could add the second term in Taylor(or McLaurin) including second derivative: See following slides

19. Answer to problem 2:convexity Second derivative of VMA to i Modified convexity MC Dividing both terms by MVA : Convexity, C

20. Answer to problem 2:duration gap and convexity gap Substituting duration and convexity in the second order expansion Multiplying both terms by MVA: Same for liabilities: The change in market value of the bank’s equity can now be better estimated: duration gap convexity gap

21. Duration gap and convexity gap: our example First order proxy, -6.23 convexity gap,equal to 61.6 Very close to the true change (-5.94)

22. Answer to problem 3:beta-duration gap Similar to standardized repricing gap. For each asset (liability) estimate: Then substitute in the change of the value of the bank beta-duration gap The impact of an interest rate change depends on 4 factors: • average MD of assets and liabilities • average sensitivity of assets and liabilities interest rates to the base rate (beta) • financial leverageL • size of the bank (MVA)

23. Residual Problems • Assumption of a uniform change of assets and liabilities’ interest rates. • Assumption of a uniform change of interest rates for different maturities. • The model does not consider the effect of a variation of interest rates on the volume of financial assets and liabilities

24. Questions & Exercises 1. Which of the following does not represent a limitation of the repricing gap model which is overcome by the duration gap model? A) Not taking into account the impact of interest rates changes on the market value of non sensitive assets and liabilities B) Delay in recognizing the impact of interest rates changes on the economic results of the bank C) Not taking into account the impact on profit and loss that will emerge after the gapping period D) Not taking into account the consequences of interest rate changes on current account deposits

25. Questions & Exercises 2. A bank’s assets have a market value of 100 million euro and a modified duration of 5.5 years. Its liabilities have a market value of 94 million euro and a modified duration of 2.3 years. Calculate the bank’s duration gap and estimate which would be the impact of a 75 basis points interest rate increase on the bank’s equity (market value).

26. Questions & Exercises 3. Which of the following statements is NOT correct? A. The convexity gap makes it possible to improve the precision of an interest-rate risk measure based on duration gap B. The convexity gap is a second-order effect C. The convexity gap is an adjustment needed because the relationship between the interest rate and the value of a bond portfolio is linear D. The convexity gap is the second derivative of the value function with respect to the interest rate, divided by a constant which expresses the bond portfolio’s current value.

27. Questions & Exercises 4. Using the data in the table below i) compute the bank’s net equity value, duration gap and convexity gap; ii) based on the duration gap only, estimate the impact of a 50 basis points increase in the yield curve on the bank’s net value; iii) based on both duration and convexity gap together, estimate the impact of a 50 basis points increase in the yield curve on the bank’s net value; iv) briefly comment the results

28. Agenda • Market value versus historical cost accounting • The duration gap model • The Clumping Model

29. A common problem and a possible solution • Repricing gap and duration gap  assumption of uniform change of interest rates for different maturities • The Clumping o cash-bucketing model a model with independent changes of interest rates at different maturities • The model is built upon the zero-coupon curve (both the repricing gap and the duration gap model were focused on the yield curve). • The model works trough the mapping of single cash flows on a predetermined number of nodes (or maturities) on the term structure.

30. How to estimate zero coupon rates: bootstrapping • For longer maturities we typically have no zero coupon bonds • We need to extract them from coupon bonds • One possibility is through bootstrapping • Assume we want to estimate the 2.5 (r2,5) zero-coupon rate • For this maturity we only have a 4.5% (semi-annual) coupon paying bond with a price of 100. • For the preceding maturities (t = 0.5; 1; 1.5; 2) we have zero coupon bonds (from their prices we can get their yield to maturity (rt))

31. How to estimate zero coupon rates: bootstrapping 1. From prices of zcb we extract the corresponding rt Ex. 2. We use these zero-coupon rates to estimate the present value of the first four cash flows (coupons) of the 4.5% coupon paying bond 8.55 102.25 2.25 2.25 2.25 2.25 0 0.5 1 1,5 2 2.5 Es. 2.21 2.16 2.12 2.07

32. How to estimate zero coupon rates: bootstrapping 3. Find the rate that equates the present value of 102.5 to the residual value of the bond which has not been explained by the PV of the four coupons 8.55 102.25 2.25 2.25 2.25 2.25 0 0.5 1 1.5 2 2.5 2.21 2.16 2.12 2.07 100 - = 91.45

33. What is the mapping for? • The mapping is a procedure to simplify the representation of the financial position of the bank. • Mapping is used to transform a portfolio with real cash flows, associated to an excessive number of p dates, into a simplified portfolio, based on a limited number q (<p) of maturity nodes (standard dates). • After mapping, it’s easier to implement effective risk management policies • Goal: reduce all the banks’ cash flows to a small number of significant nodes (maturities).

34. Cash-flow mapping • We can get an interest rate curve with different rates for every individual maturity • Do I really need to consider MxN nodes? • No, cash-flow mapping allows to map a portfolio of assets and liabilities (with a large number of cash flows associated to a large number of maturities) to a limited number of maturity nodes • It represents a special case of mapping • A methodology to map a portfolio to a limited number of risk factors: e.g. international equity portfolio to S&P500, Dax and MIB 30

35. Analytical principal Given M securities, “maps” each of them to the “principal” maturity node Synthetic principal Given M securities, it only considers the maturity of principal (computes an average) Analytical duration Given M securities, it maps each of them to its duration Synthetic duration Given M securities, it only considers the duration (computes an average) Some simplifying cash-flow mapping techniques Modified analytical principal method Requires M nodes Extremely simplified Does not consider coupons reinvestment risk

36. Computing analytic duration for each asset and liability might be complex Using principal is not precise as it does not consider the coupons However, given the level of interest rates (e.g. 5% in the chart), there exists a relationship between principal and duration for bonds with different coupon level An hybrid technique: modified principal

37. To simplify the step from principal to duration consider only two cases e.g., < o > 3%) Divide principal values in few large maturity buckets Assign an average duration to each maturity (“modified principal”) Modified principal

38. The objective is the same: link real cash flows to a number q (<p) of “nodes” What changes? Rather than compacting flows into a single one at a unique date, each cash flow gets divided into more nodes How to map cash flows? Building a new security, identical to the real cash flow in terms of market value and riskiness A more refined technique: clumping dates 0,75 1,25 1,75 2,25 2,75 nodes 0,5 1 2,5 Clumping: dates 0,75 1,25 1,75 2,25 2,75 nodes 0,5 1 2,5

39. Clumping • In the clumping model a large number of cash flows, maturing in p different dates are reduced to q (with q<p) virtual cash flows on q different dates called “nodes” on the curve. • In order to choose the number and the position of the nodes we have to remember that: • Changes in short term interest rate are more frequent and larger than changes in long term interest rates. • The relationship between volatility and maturity of interest rates is negative. • Usually cash flows with short maturities are more frequent that cash flows with long maturities • It’s better to have a larger number of nodes on the short term part of the zero coupon curve

40. The nodes • The choice of the node is also influenced by the availability of hedging instruments: FRA, futures, swaps, etc. • When we divide a real cash flow with maturity in date t into two virtual cash flows with maturities on the nodes n and n+1 (with n<t<n+1), we must have: • The same market value • The same modified duration

41. Mapping in practice • We have two unknowns and two equations

42. An example • A cash flow with a nominal value of 50,000 € and maturity 3y and 3m. • Zero-coupon IR: 3.55%

43. follows Market Value and Modified Duration for the real cash flows Modified Duration for the two virtual cash flows

44. follows Market value for the two virtual cash flows Nominal value for the two virtual cash flows

45. follows • The sum of the market values of the two virtual flows is equal to the market value of the real cash flow. • The market value of the 3Y cash flow is greater than the MV of the 4Y cash flow. This happens because the real flow maturity is nearer to 3 than to 4

46. Clumping on the basis of price volatility • Another form of clumping centers on the equivalence between price volatility of the initial flow and the total price volatility of the two new virtual positions • This is calculated by taking into account also the correlations between volatilities associated with price changes for different maturities. VMt e VMt+1 are chosen in such a way that: • Since this is a quadratic equation, we get two solutions for  we need to assume that the original position and the two new virtual positions have the same sign 

47. Clumping • After the mapping of all the bank positions on the nodes it’s possible to: • Evaluate the effect on the market value of the shareholders’ equity of a change of the interest rates for certain maturities • Implement interest risk management activities • Implement hedging activities

48. Residual Problems • Assumption of a uniform change of assets and liabilities’ interest rates. • The model does not consider the effect of a variation of interest rates on the volume of financial assets and liabilities

49. The Basel Committee Approach • Banks are required to allocate their assets and liabilities to 14 maturity buckets based on their residual maturity • For each bucket, they estimate the difference between assets and liabilities (long and short positions, i.e. net position) • The net position is weighted by a coefficient that proxies the potential change in value • The product between the average modified duration and a 2% change in the interest rate (parallel shift of the yield curve)

50. The Basel Committee Approach • Banks are required to allocate their assets and liabilities to 14 different maturity bands • For each maturity bucket, the net position must be calculated (difference assets and liabilities) • Net position, NPi