# solution

1. 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

• ? 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.)

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