. Explain the attached questions below.

2.2. ** The continuously compounded interest rate formula can be derived by (a) splitting the time interval [0, ] into subintervals [0, 8t], [8t, 28t], .... [(L - 1) 8t, Lot], where at = t/L, and 2.8 Program of Chapter 2 and walkthrough 17 (b) assuming that the value of the investment increases by a relative amount pro- portional to rot over each subinterval. Letting fi = ist, this means D(ti+1) = (1 + rot) D(t;), (2.7 ) and hence D(1 = tz) = (1 +rot) Do. By writing this as D(t) = el log(1+ri/2)Do and using log(1 + e) = e + O(2) as e - 0, show that this model reproduces the formula (2.1) in the limit L - co (i.e. 8t - 0). Show that the models D(;+1) = (1+r81) D(1;) (2.8) and D(ti+1 ) = (1 + r (81)=) D(ti) (2.9) are not consistent with continuous compounding in the limit L - co. 2.3. ** Give an argument based on the no arbitrage assumption that justifies (2.3). 2.4. ** Establish (2.6) (a) by setting up suitable portfolios and applying the argu- ments used to get (2.4)-(2.5), and (b) separately, by using (2.4)-(2.5) plus put-call parity (2.2). 2.5. * * * Show that a butterfly spread with exactly the same payoff as that in Exercise 1.3 can be obtained using only a combination of European put op- tions. Use put-call parity (2.2) to confirm that the two spreads have the same set-up cost. 2.6. A forward contract, which is similar to a futures contract, operates as follows. Now, at time / = 0. Party A agrees to purchase an asset from Party B at a specified delivery time / = 7 for a specified price F. (Note that Party A is committed to the future purchase - by contrast, with a European call option the holder has the right, but not the obligation, to buy at the prescribed price.) Appealing to the no arbitrage assumption, show that a fair value for F is S(0)er.An institution has a liability to pay $15,000 per annum, half-yearly in arrears, forever. (i) Calculate the present value and volatility of the liability at 8% pa effective. [6] (ii) Calculate the duration of the liability at 8% pa effective. [1] The following two stocks are available for investment: (A) A special 5-year stock, redeemable at par, that pays a coupon of 5 per 100 nominal at the end of the first year rising, by 2% pa compound, to 5 x1.02# at the end of the fifth year. (B) An n-year zero-coupon bond, redeemable at par. The institution chooses to invest equal amounts of cash in Stock A and Stock B. (iii) If the institution requires that the duration of the assets must equal the duration of the liabilities, show that n, the term of the zero-coupon bond, must equal 22 years if interest rates are 8% pa effective. [8] (iv) Without doing any further calculations, explain whether the institution has managed to implement an immunisation strategy. [2] [Total 17]An institution has a liability to pay $15,000 per annum, half-yearly in arrears, forever. (i) Calculate the present value and volatility of the liability at 8% pa effective. [6] (ii) Calculate the duration of the liability at 8% pa effective. [1] The following two stocks are available for investment: (A) A special 5-year stock, redeemable at par, that pays a coupon of 5 per 100 nominal at the end of the first year rising, by 2% pa compound, to 5 x1.02# at the end of the fifth year. (B) An n-year zero-coupon bond, redeemable at par. The institution chooses to invest equal amounts of cash in Stock A and Stock B. (iii) If the institution requires that the duration of the assets must equal the duration of the liabilities, show that n, the term of the zero-coupon bond, must equal 22 years if interest rates are 8% pa effective. [8] (iv) Without doing any further calculations, explain whether the institution has managed to implement an immunisation strategy. [2] [Total 17]A whole life annuity is payable continuously to a life now aged 60. The rate of payment at time / is: p() = 10,000(1.02)' (1 >0) (i) Write down an expression for the present value of the annuity in terms of annuities-certain. [2] (1i) Write down expressions for the expected present value and variance of the present value of the annuity. [2] (iii) Calculate the expected value and the variance of the annuity assuming AM92 Ultimate mortality and 6.08% pa interest. [4] (iv) Simplify your expressions for the present value and its expectation assuming that i = 0.02. [2] (v) Calculate the expected present value of the annuity assuming AM92 Ultimate mortality and 2% pa interest. [1] [Total 11]A whole life annuity is payable continuously to a life now aged 60. The rate of payment at time / is: p() = 10,000(1.02)' (1 >0) (i) Write down an expression for the present value of the annuity in terms of annuities-certain. [2] (1i) Write down expressions for the expected present value and variance of the present value of the annuity. [2] (iii) Calculate the expected value and the variance of the annuity assuming AM92 Ultimate mortality and 6.08% pa interest. [4] (iv) Simplify your expressions for the present value and its expectation assuming that i = 0.02. [2] (v) Calculate the expected present value of the annuity assuming AM92 Ultimate mortality and 2% pa interest. [1] [Total 11]Question 2 (a) Briey explain what are human resource management activities and by relating to a company that known to you or a hypothetical company describe how human resource management activities should be related to a company strategic planning and justify the important of those linkages. (18 marks) (b) What is a performance appraisal? Based on your company/company of your choice/hypothetical company give an example how does right performance appraisal can inuence organizational success? Explain why to do performance appraisal effectively poses a particular challenge to manager. (16 marks) (0) Explain by giving examples why every manager (not only HR manager) needs to have some degree of knowledge in managing human resources. Briey justify why Human Resource management in your company should react or respond to the current era of digitization and how digitization has or could have inuenced its activities. (1 6 marks)