
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.)