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Topic 2. Measuring Interest Rate Risk

Topic 2. Measuring Interest Rate Risk. 2.1 Repricing model 2.2 Duration 2.3 Convexity. 2.1 Repricing model. Repricing model is based on the book (historic) value of assets and liabilities in the FI’s balance sheet.

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Topic 2. Measuring Interest Rate Risk

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  1. Topic 2. Measuring Interest Rate Risk 2.1 Repricing model 2.2 Duration 2.3 Convexity

  2. 2.1 Repricing model • Repricing model is based on the book (historic) value of assets and liabilities in the FI’s balance sheet. • Under the repricing model, FI is required to divide its asset and liability portfolio into different maturity intervals (maturity bucket). • Rate sensitive assets of the ith maturity bucket (RSAi) is defined as the total book value of all FI’s assets which the first repricing at current market interest rates is within ith maturity bucket. • Rate sensitive liabilities of the ith maturity bucket (RSLi) is defined in the same way as RSAi for the FI’s liabilities.

  3. 2.1 Repricing model • The repricing of an asset or a liability in ith maturity bucket can be the result of a rollover of an asset or a liability in that bucket or the corresponding asset or liability is variable-rate instrument in which the rate is reset within that bucket. • Change in net interest income in the ith bucket NIIi is defined as where Riis the change in the level of interest rates impacting assets and liabilities in the ith bucket. GAPi = RSAi – RSLi

  4. 2.1 Repricing model • Under Eq. (2.1), we assume the same interest rate Ri is applied to all assets and liabilities in the ith bucket. In reality, different interest rates should be used for assets and liabilities with different maturities within the bucket.

  5. 2.1 Repricing model Examples of RSAi and RSLi: Let the i-th maturity bucket be (3 months, 12 months]. Fixed rate 1-month loan is NOT RSAi or RSLi since the first repricing is not within i-th maturity bucket. Fixed rate 9-month loan is RSAi or RSLi since the first repricing is within i-th maturity bucket. Fixed rate 2-year loan is NOT RSAi or RSLi since the first repricing is not within i-th maturity bucket. Floating rate 10-year loan (interest rate reset every 1 month) is NOT RSAi or RSLi since the first repricing is not within i-th maturity bucket. Floating rate 30-year loan (interest rate reset every 6 months) is in RSAi or RSLi since the first repricing is within i-th maturity bucket. 5

  6. 2.1 Repricing model Table 2-1

  7. 2.1 Repricing model • In first bucket (1-day), GAP1 = $10 million If the 1-day interest rates (fed or overnight repo rate) rise 1% per annum, i.e. R1= 0.01, then from Eq. (2.1) That is, the negative gap (GAP1 < 0) resulted in a loss (gain) of $100,000 in net interest income in the first bucket for the FI when the 1-day interest rate increases (decreases) 1%.

  8. 2.1 Repricing model • Cumulative 1-year gap (CGAP1-year) where RSA1-yearand RSL1-yearare the total value of assets and liabilities in which they are repriced within 1 year. Taking [0, 1 year] as the first maturity bucket and from Eq. (2.1), we have where R1-year is the average interest rate change affecting assets and liabilities that can be repriced within a year.

  9. 2.1 Repricing model From Table 2-1 with R1-year= 0.01, we have For R1-year =1%, the 1-year net interest income drops by $150,000.

  10. 2.1 Repricing model where A is the total asset value. The gap ratio in (2.4) is useful to tell us that • the direction of interest rate risk exposure (+ve or –ve CGAPt) • the scale of that exposure relative to the total asset size of the financial institution.

  11. 2.1 Repricing model Example 2.1 11

  12. 2.1 Repricing model Determine CGAP1-year. RSA1-year consists: Short-term consumer loans: repriced at the end of one year. Three-month T-bills: repriced on maturity (rollover) every 3 months. Six-month T-notes: repriced on maturity (rollover) every 6 months. 30-year floating rate mortgages: repriced (mortgage rate is reset) every 9 months. So, RSA1-year = $155M. 12

  13. 2.1 Repricing model RSL1-year consists: Three-month CDs: repriced on maturity (rollover) every 3 months. Three-month bankers acceptances: repriced on maturity (rollover) every 3 months. Six-month commercial paper: repriced on maturity every 6 months. One-year time deposits: repriced at the end of the year. So, RSL1-year = $140M. CGAP1-year=$15M and From the calculated gap ratio, RSA1-year is more than RSL1-year by 5.6% of the total asset value. 13

  14. 2.1 Repricing model Under equal changes in interest rates on RSA and RSL, CGAP effects refer to the relation between changes in interest rates (R) and changes in net interest income (NII). 14

  15. 2.1 Repricing model CGAP > 0 ΔR and ΔNII are positively correlated. CGAP < 0 ΔR and ΔNII are negatively correlated. 15

  16. 2.1 Repricing model • Under unequal changes in interest rates on RSA and RSL, where RA and RL are the interest rates applying on asset and liability respectively. Spread effect: For all else being fixed, there is a positive relation between spread and NII.

  17. 2.1 Repricing model From Eq. (2.6), when both CGAP effect and spread effect play their role to ΔNII, we observe the following 17

  18. 2.1 Repricing model Weakness of repricing model • Ignores market value effect The repricing model does not consider the change of the present value of cash flows on asset and liability as interest rates change. (rates-asset has corr, duration profile) • Overaggregative The actual maturities distributions of assets and liabilities within individual maturity bucket is not considered. Mismatching of maturity could cause refinancing or reinvestment risk. (maturies not enough granularity)

  19. 2.1 Repricing model Although GAP3-mth to 6-mth = 0, the assets’ maturity < liabilities’ maturity (reinvestment risk). 19

  20. 2.1 Repricing model • Ignores effects of runoffs The periodic cash flow of interest and principal amortization payments on long-term asset being classified as rate insensitive and ignored under the repricing model. Actually, these cash flows can be reinvested at market interest rates. • Ignores off-balance-sheet activities

  21. 2.2 Duration Duration of a bond • Given a N-year coupon bond with annual coupon rate of c and principal (face) value of F. Suppose the coupon frequency is m and the bond yield is R (per annum) with the compounding frequency as the same as the coupon frequency. The price of the bond at the commencing date is given by

  22. 2.2 Duration • The duration of a bond D is defined as the weighted average of cash flow dates of a bond

  23. 2.2 Duration • D is also called Macaulay duration. • Example 2.2 Consider a 2-year coupon bond with coupon rate of 4% per annum and principal value of $100. The coupon frequency is 2 and the bond yield is 4%. m = 2; ti = i/2; R = 4% Let CFi be the cash flow at ti. Let DFi be the discount factor at ti.

  24. 2.2 Duration D = 194.19/100 = 1.9419

  25. 2.2 Duration • On the other hand, it can be shown that (refer to the lecture notes of SEEM 2520) MD is called the modified duration.

  26. 2.2 Duration model If the changes of R (R) is small, then Eq. (2.9) can be approximately written as • From Eq. (2.10), MD measures a bond’s price sensitivity as a percentage change of its current price to the small changes in its yield.

  27. 2.2 Duration Properties of duration • Duration of a zero coupon bond is equal to its maturity. •  coupon •  maturity •  yield   Duration (D)

  28. 2.2 Duration Duration of a portfolio of bond Consider a Portfolio, A, consists of N bonds. Notation: For i =1, …, N, Pi : Price of the bond i Fi : Face value of the bond i mi : Coupon frequency of bond i Ri : Bond yield of bond i Di : Duration of bond i MDi : Modified duration of bond i ni : Number of units of bond i in Portfolio A (ni > 0: Long position; ni < 0: short position)

  29. 2.2 Duration The value of Portfolio A is given by For small Ri, i=1, …, N, we have 29

  30. 2.2 Duration If R1= R2=…= RN= R (parallel yield shift), then Eq. (2.11) becomes The modified duration of Portfolio A, MDA, is then given by 30

  31. 2.2 Duration In particular, if R1=R2=…=RN (flat yield curve) and m1=m2=…=mN , the duration of Portfolio A, DA, is given by

  32. 2.2 Duration From Eq. (2.13) and Eq. (2.14), we observe that the modified duration and also the duration of a portfolio are the weighted average of the modified duration and duration of its components respectively. Portfolio A is called duration neutral if MDA =0. The value of a duration neutral portfolio is insensitive to the small change of the interest rates. 32

  33. 2.2 Duration Example 2.3 Suppose Portfolio A consists of 3 bonds: B1, B2 and B3. Here, 1 unit of bond corresponds to $100 in bond’s face value. 33

  34. 2.2 Duration Duration model Let MDA be the duration of asset portfolio of a FI which consists of NA bonds. Let MDL be the duration of liability portfolio of a FI which consists of NL bonds. Assume parallel yield shift. From Eq. (2.13), we have

  35. 2.2 Duration Let A and L be the value to asset and liability portfolio of the FI respectively. Assume R to be small and the same for both asset and liability portfolio. From Eq. (2.12) and Eq. (2.13), we have

  36. 2.2 Duration Let E be the equity value of the FI. k is a measure of the FI’s financial leverage. 36

  37. 2.2 Duration (MDA – kMDL) is the leverage adjusted modified duration gap (modified duration gap for short) which reflect the degree of duration mismatch in an FI’s balance sheet. To immunize the equity or net worth (E) from interest rate shocks, FI should set the modified duration gap to 0. Immunization: The procedure to protect against the interest rate risk. 37

  38. 2.2 Duration If all the bonds in the asset and liability portfolio have the same coupon frequency and equal to 1 and also the yield curve is flat, then Eq. (2.15) becomes In Eq. (2.16), (DA – kDL) is called the leverage adjusted duration gap (duration gap for short). 38

  39. 2.2 Duration Example 2.4 The FI’s initial balance is assumed to be *bond’s price A: 5-year zero coupon bond L: 3-year zero coupon bond Suppose R = 10%, R = 1%, DA= 5 years and DL = 3 years.

  40. 2.2 Duration For 1% increase in interest rate, the equity value drops $2.09 millions. The insolvency occurs when E  $10 million. This corresponds to R  4.783%.

  41. 2.2 Duration • Financial leverage (gearing) of a firm is the extent to which debts are used to finance a firm’s assets. If a high percentage of a firm’s assets are financed by debts, then the firm is said to have a high degree of financial leverage. • The higher the financial leverage of a firm, the higher the chance for it to bankrupt. 41

  42. 2.2 Duration • The following are common ratios which are used to measure the financial leverage: • Ratio (1) and (2) are increasing as the level of leverage increases, while the ratio (3) is decreasing as the level of leverage decreases. 42

  43. 2.2 Duration Limitations of duration model • It is costly and time consuming for the FI to restructure the balance sheet to achieve the immunization against the interest rate shock. The growth of asset securitization has eased the speed and lowered the transaction costs of the balance sheet restructuring. • Immunization is a dynamic process since duration depends on instantaneousR.

  44. 2.2 Duration Duration model performs poorly under non-parallel shift of yield curve. In reality, points in a yield curve do not often shift by same amount and they sometimes do not even move in same direction. Large interest rate change effects are not accurately captured. This can be improved by introducing the convexity. 44

  45. 2.2 Convexity Convexity • The duration measure is a linear approximation of a nonlinear function. If there is a large changes in R, the approximation is much less accurate.

  46. 2.2 Convexity P(R) Duration model Error R* ΔR R* R* +ΔR R (%) 46

  47. 2.2 Convexity P(R+R) – P(R) can be better approximated by using Taylor Expansion with 2nd order as follows 47

  48. 2.2 Convexity In Eq. (2.18), CX is defined as CX is called the convexity of a bond. 48

  49. 2.2 Convexity Example 2.5 Consider a 6-year maturity coupon par bond with annual coupon of 8% and principal value of $1,000. The coupon is paid annually. Since the bond is par, the annual yield is also 8%. From Eq.(2.8), D = 4.9927. From Eq. (2.19), 49

  50. 2.2 Convexity Suppose R = 2%. P(10%) = $912.8948 (Exact value) Without convexity adjustment (Eq. (2.10)): With convexity adjustment (Eq. (2.18)): 50

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