this is a problem from actuary class, please give brief reasoning and complete steps, thank you

"mathematics of investment and credit"

Unless otherwise specified, questions involving duration assume a flat term structure. I 7.1.2 Suppose that the yield rate and coupon rate on an n-coupon bond are the same. Show that the Macaulay duration is n valued at the yield rate. Find the Macaulay duration of a 6-coupon bond with coupon rate 10% per coupon period and yield rate 10% per coupon period. 01 Rathematics of Investment and Credit 7th Edition a = Not Authorized for Resale 46 CHAPTER 7 Eru-Lle" 20). then S-S 2 0 and the assets immunize the liabilities. This analysis is valid even if the term structure is not flat 7.4 DEFINITIONS AND FORMULAS Macaulay Duration The Macaulay duration, denoted D. is usually referred to simply as "duration." ExK,(-7) E-K (+4)* P ,04) Modified Duration The modified duration, denoted MD sometimes referred to as "volatility of the set of cashflows K.K. is Day (20.4 5/6,047 rk, (+7) K, -.72 Modified duration and Macaulay duration of an w-year rere coupon bond D () = D.)(1+1) = (1+i) = .. D () . . . (1.4) First-order approximate change in the present value of a cashflows Macaulay Pli+h) = PO (14 Modified Pli+h)-PO) = x P) = -AxPxD.. (75) 448 01 Rathematics of Investment and Credit 7th Edition a = Not Authorized for Resale CASHFLOW DURATION AND IMMUNIZATION 417 Macaulay duration of an year coupon bond ErxFrxw/+wxFxv Macaulay duration of a portfolio LYS-(1+r) 4 XSDX Redington Immunization If asset cashflows are for 1 01, and liability cashflows are for 1 -0,1..., then the liability cashflows are Redington immunized by the asset cash flows at valuation rate i, if the following conditions are met PY.CO. - PV 4.0 - rv2.0 7.12) Convexity Modified Convexity Pro Ex +140+4)** ( PV.08 4 0-) 449 ne 01:PMathematics of Investment and o Credit 7th Edition a = Not Authorized for Resale 418 CHAPTER 2 Macaulay Convexity EF41+6) Relationship Between Modified and Macaulay Convexity COSD+D_00 Full Immunization A portfolio is fully immunized if 4. EL-V for any i > 7.5 NOTES AND REFERENCES The concept of duration was introduced by Frederick Macaulay in the book titled S Theatrical Problem Sweeted by the Movements of Interest Rates, Band Yields and Stock Prices in the United States since 1856, published by the National Bureau of Economic Rescarch in 1938 The concept of duration was not widely used in practice until the 1970s. I would like extend my gratitude to Robert Alps for providing detailed comments on duration and in particular for enlightening me on the first and second-order Macaulay approximations. In 2017, the Society of Actuaries incorporated a study mote for the professional FM examination The study note is written by Mr. Alps. The early development of immunization theory can be found in the paper Review of the Principles of Life Office Palations, by F. M. Redington published in the Journal of the Institute of Actuaries in 1952 A discussion of full immunization can be found in Chapter 10 of Introduction to the Mathematics of Finance, by J. J. McCuncheon and W.F. Scott, published in 1986 by Oxford: Heinemann Professional Publishing. In a presentation to the 1993 AFIR (Actuarial Approach to Financial Risks) Colloquium, S. Coutts pronounced that "Imm ation is Dead." The paper outlined a number of problems with Redington immunization, including its difficulty in dealing with a stochastic interest rate environment. 450