At the start of each year, an investor invests 3,000 into a pension fund. The annual effective investment return earned over year is denoted by the random variable , for = 1,2, ... It is assumed that {}, = 1,2, ... are independent and identically distributed. Let be the accumulated value of the pension fund at the end of year . Assume that 0 = 0.

(a) Derive a recursive formula for [ 2 ] in terms of moments of 1 and .

You do not need to derive a recursive formula for the first moment 1.

(b) If [] = 0.05 and the standard deviation of is 0.12, then [9 2 ] = 21,268,794,314 (you are not required to show this).

(i) Calculate [9 ].

(ii) Hence calculate [10 2 ].