2-19, Let A be the balance in a fund on January 1, 1999, B the balance on June 30, 1999, and C the balance on December 31, 1999. If there are no deposits or withdrawals, show that the dollar- (a) weighted and time-weighted rates of return for 1999 are both C-A A equal to (b) If there was a single deposit of W immediately after the June 30 balance was calculated, find expressions for the dollar- weighted and time-weighted rates of return for 1999 (Assume simple interest for periods of less than a year.) (c) If there was a single deposit of W immediately before the June 30 balance was calculated, find expressions for the dollar-weighted and time-weighted rates of return for 1999. (Assume simple interest for periods of less than a year.) (d) Give a verbal explanation for the fact that the dollar- weighted rates of return in parts (b) and (c) are equal. Show that the time-weighted rate of return in part (b) is (e) larger than the time-weighted rate of return in part (c)