0. We let () (where 0) be accumulation function of an investment fund launched at time 0. Questions

   1. (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.

   2. (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.)