2-19. Let A be the balance in a fund on January 1, 1999, B the balance on June 30, 1999, and the balance on December 31, 1999. (a) If there are no deposits or withdrawals, show that the dollar- weighted and time-weighted rates of return for 1999 are both equal to C=4 (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.) 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.) Give a verbal explanation for the fact that the dollar- weighted rates of retum in parts (b) and (c) are equal. (e) Show that the time-weighted rate of return in part (b) is larger than the time-weighted rate of return in part (c). (c) 2-19. Let A be the balance in a fund on January 1, 1999, B the balance on June 30, 1999, and the balance on December 31, 1999. (a) If there are no deposits or withdrawals, show that the dollar- weighted and time-weighted rates of return for 1999 are both equal to C=4 (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.) 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.) Give a verbal explanation for the fact that the dollar- weighted rates of retum in parts (b) and (c) are equal. (e) Show that the time-weighted rate of return in part (b) is larger than the time-weighted rate of return in part (c). (c)