Interest Rate Risk Management Elias S. W. Shiu Department of Statistics & Actuarial Science

1. Interest Rate Risk Management Elias S. W. Shiu Department of Statistics & Actuarial Science The University of Iowa Iowa City, Iowa U.S.A.

2. Frank M. Redington, F.I.A. “Review of the Principles of Life-Office Valuations” Journal of the Institute of Actuaries Volume 78 (1952), 286-315

4. Last sentence in the first paragraph: The reader will perhaps be less disappointed if he is warned in advance that he is to be taken on a ramble through the actuarial countryside

5. Last sentence in the first paragraph: The reader will perhaps be less disappointed if he is warned in advance that he is to be taken on a ramble through the actuarial countryside and that any interest lies in the journey rather than the destination.

6. For a block of business and for t > 0, let At = asset cash flow to occur at time t (= investment income + capital maturities)

7. For a block of business and for t > 0, let At = asset cash flow to occur at time t (= investment income + capital maturities)For example, if the company’s assets consist of one 2-year bond with face value of $100 and semi-annual coupons at a nominal annual rate of 6%

8. For a block of business and for t > 0, let At = asset cash flow to occur at time t (= investment income + capital maturities)For example, if the company’s assets consist of one 2-year bond with face value of $100 and semi-annual coupons at a nominal annual rate of 6% and one 3-year bond at 8%

9. For a block of business and for t > 0, let At = asset cash flow to occur at time t (= investment income + capital maturities)For example, if the company’s assets consist of one 2-year bond with face value of $100 and semi-annual coupons at a nominal annual rate of 6% and one 3-year bond at 8%, then A0.5 = ½ (6+8) = 7

10. For a block of business and for t > 0, let At = asset cash flow to occur at time t (= investment income + capital maturities)For example, if the company’s assets consist of one 2-year bond with face value of $100 and semi-annual coupons at a nominal annual rate of 6% and one 3-year bond at 8%, then A0.5 = ½ (6+8) = 7, A1 = 7, A1.5 = 7

11. For a block of business and for t > 0, let At = asset cash flow to occur at time t (= investment income + capital maturities)For example, if the company’s assets consist of one 2-year bond with face value of $100 and semi-annual coupons at a nominal annual rate of 6% and one 3-year bond at 8%, then A0.5 = ½ (6+8) = 7, A1 = 7, A1.5 = 7, A2 = 103+4 = 107

12. For a block of business and for t > 0, let At = asset cash flow to occur at time t (= investment income + capital maturities)For example, if the company’s assets consist of one 2-year bond with face value of $100 and semi-annual coupons at a nominal annual rate of 6% and one 3-year bond at 8%, then A0.5 = ½ (6+8) = 7, A1 = 7, A1.5 = 7, A2 = 103+4 = 107, A2.5 = 4, A3 = 104

13. For a block of business and for t > 0, let At = asset cash flow to occur at time t (= investment income + capital maturities)Lt = liability cash flow to occur at time t (= policy claims + policy surrenders + expenses  premium income)

14. Let A = Asset Value at time 0. Then,

15. Let A = Asset Value at time 0. Then, But yield curves are not (necessarily) flat.

16. Let A = Asset Value at time 0. Then, But yield curves are not (necessarily) flat. Generalize: Then

17. Similarly, let L = Liability Value at time 0. Then,

18. Similarly, let L = Liability Value at time 0. Then, Surplus (Net Worth or Equity) = Asset Value - Liability Value = A - L

19. Similarly, let L = Liability Value at time 0. Then, Surplus (Net Worth or Equity) = Asset Value - Liability Value = A - L Instantaneous interest rate shock: How does the surplus change?

20. Instantaneous interest rate shock means

21. Instantaneous interest rate shock means Assume that the asset cash flows {At} and liability cash flows {Lt}do not change as interest rates fluctuate.

22. Instantaneous interest rate shock means Assume that the asset cash flows {At} and liability cash flows {Lt}do not change as interest rates fluctuate. That is, there are no embedded interest-sensitive options.

23. Instantaneous interest rate shock means Assume that the asset cash flows {At} and liability cash flows {Lt}do not change as interest rates fluctuate. That is, there are no embedded interest-sensitive options. The more general case of interest-sensitive cash flows is a much harder problem.

24. Changed asset value is

25. Changed asset value is Changed liability value is

26. Changed asset value is Changed liability value is Changed surplus is S* = A* - L*

27. Question: How will the surplus not decrease?

28. Question: How will the surplus not decrease? A - L  A* - L*?

29. Question: How will the surplus not decrease? A - L  A* - L*? Define two (discrete) random variables X and Y: Pr(X = t) =

30. Question: How will the surplus not decrease? A - L  A* - L*? Define two (discrete) random variables X and Y: Pr(X = t) = Pr(Y = t) = (The cash flowsare assumed to be non-negative.)

31. Define the function f(t) =

32. Define the function f(t) = Then

33. Define the function f(t) = Then

34. Define the function f(t) = Then

35. Define the function f(t) = Then

36. Similarly, L* = L E[f(Y)].

37. Similarly, L* = L E[f(Y)]. Original Surplus is S = A - L.

38. Similarly, L* = L E[f(Y)]. Original Surplus is S = A - L. Changed Surplus is S* = A* - L* = AE[f(X)] - LE[f(Y)].

39. Similarly, L* = L E[f(Y)]. Original Surplus is S = A - L. Changed Surplus is S* = A* - L* = AE[f(X)] - LE[f(Y)]. Now, assume A = L, i.e., assume S = 0.

40. Similarly, L* = L E[f(Y)]. Original Surplus is S = A - L. Changed Surplus is S* = A* - L* = AE[f(X)] - LE[f(Y)]. Now, assume A = L, i.e., assume S = 0. Then S* = A{E[f(X)] -E[f(Y)]}

41. Similarly, L* = L E[f(Y)]. Original Surplus is S = A - L. Changed Surplus is S* = A* - L* = AE[f(X)] - LE[f(Y)]. Now, assume A = L, i.e., assume S = 0. Then S* = A{E[f(X)] -E[f(Y)]}, and S*  0 if and only if

42. Similarly, L* = L E[f(Y)]. Original Surplus is S = A - L. Changed Surplus is S* = A* - L* = AE[f(X)] - LE[f(Y)]. Now, assume A = L, i.e., assume S = 0. Then S* = A{E[f(X)] -E[f(Y)]}, and S*  0 if and only ifE[f(X)] E[f(Y)].

43. f(t) =

44. f(t) = In the Redington (1952) model, it = i and = i + , where is a positive or negative constant.

45. f(t) = In the Redington (1952) model, it = i and = i + , where is a positive or negative constant. Thus f(t) is an exponential function, which is a convex function.

46. f(t) = In the Redington (1952) model, it = i and = i + , where is a positive or negative constant. Thus f(t) is an exponential function, which is a convex function. In the Fisher & Weil (J. of Business 1971) model, where c is a positive constant.

47. f(t) = In the Redington (1952) model, it = i and = i + , where is a positive or negative constant. Thus f(t) is an exponential function, which is a convex function. In the Fisher & Weil (J. of Business 1971) model, where c is a positive constant. That is, f(t) = ct

48. f(t) = In the Redington (1952) model, it = i and = i + , where is a positive or negative constant. Thus f(t) is an exponential function, which is a convex function. In the Fisher & Weil (J. of Business 1971) model, where c is a positive constant. That is, f(t) = ct, which is also a convex function.

49. Jensen’s Inequality

50. Jensen’s Inequality: E[f(X)]  f(E[X]) for all convex functions f.