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

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

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**Interest Rate Risk Management**Elias S. W. Shiu Department of Statistics & Actuarial Science The University of Iowa Iowa City, Iowa U.S.A.**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**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**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.**For a block of business and for t > 0, let**At = asset cash flow to occur at time t (= investment income + capital maturities)**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%**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%**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**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**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**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**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)**Let A = Asset Value at time 0. Then,**But yield curves are not (necessarily) flat.**Let A = Asset Value at time 0. Then,**But yield curves are not (necessarily) flat. Generalize: Then**Similarly, let L = Liability Value at time 0. Then,**Surplus (Net Worth or Equity) = Asset Value - Liability Value = A - L**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?**Instantaneous interest rate shock**means Assume that the asset cash flows {At} and liability cash flows {Lt}do not change as interest rates fluctuate.**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.**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.**Changed asset value is**Changed liability value is**Changed asset value is**Changed liability value is Changed surplus is S* = A* - L***Question: How will the surplus not decrease?**A - L A* - L*?**Question: How will the surplus not decrease?**A - L A* - L*? Define two (discrete) random variables X and Y: Pr(X = t) =**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.)**Define the function**f(t) =**Define the function**f(t) = Then**Define the function**f(t) = Then**Define the function**f(t) = Then**Define the function**f(t) = Then**Similarly,**L* = L E[f(Y)].**Similarly,**L* = L E[f(Y)]. Original Surplus is S = A - L.**Similarly,**L* = L E[f(Y)]. Original Surplus is S = A - L. Changed Surplus is S* = A* - L* = AE[f(X)] - LE[f(Y)].**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.**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)]}**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**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)].**f(t) =**In the Redington (1952) model, it = i and = i + , where is a positive or negative constant.**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.**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.**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**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.**Jensen’s Inequality:**E[f(X)] f(E[X]) for all convex functions f.