Bonus Problem 1 (Optional, 20 marks) *Note: In this problem, you are going to explore some techniques in evaluating effective interest rate in an investment fund. We let () (where 0) be accumulation function of an investment fund launched at time 0. Questions (a) We let [1,2 ] be the effective interest rate of the investment fund, where 1 < 2. Using the definition of effective interest rate and the accumulation function of the investment fund, show that for any 0 1 < 0 < 2, [1,2 ] = (1 + [1,0 ])(1 + [0,2 ]) 1. (b) Another investor (investor B) has invested some amount into this investment fund at time 0. In addition, he invests an addition amount of $500 at time 1.5. It is given that The amount value of the investment account at time 1 is $2090; The amount value of the investment account at time 1.5 (after the deposit is made) is $2631.8. The amount value of the investment account at time 3 is $2763.39. The annual effective interest rate over 2nd year (i.e. [1,2]) and 3rd year (i.e. [2,3]) are both Using the information given, calculate the value of . (Hint: You can first consider [1,3] . One needs to be careful that the effective interest rate over [1,2 ] measures the amount of interest rate earned over the period [1,2 ] if $1 is invested at time 1 and no additional deposit/withdrawal are made within (1,2 ]. To find [1,3] for this case, you need to reduce the problem into the cases when there is only a single deposit made at the beginning of the period. Think about the result derived in (a) with suitable choice of 0.)

Bonus Problem 1 (Optional, 20 marks) *Note: In this problem, you are going to explore some techniques in evaluating effective interest rate in an investment fund. We let a(t) (where t > 0) be accumulation function of an investment fund launched at time 0. Questions (a) We let itt,[2] be the effective interest rate of the investment fund, where tz 0) be accumulation function of an investment fund launched at time 0. Questions (a) We let itt,[2] be the effective interest rate of the investment fund, where tz