You are given the following information about 2 accounts: Account 1 Time Account Value before transactions Deposit Withdrawal 0 100 0.25 110 X 0.75 120 3X 1 82 Account 2 Time Account Value before transactions Deposit Withdrawal 0 100 0.5 120 2X 1 140 You are also told that the dollar weighted return over the year on account 1 is i. If the time weighted return over the year on account 2 is also i, what are X and i

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Explanation:

For account 1:

Dollar weighted investment = 100 for entire year + X for three fourth of the year - 3X for one fourth of the year = 100 + 3X/4 - 3X/4 = 100

Dollar return = Closing balance - opening balance - (Total deposit - total withdrawal) = 82 - 100 - (X - 3X) = 2X - 18

Hence, dollar weighted return = i = Dollar return / Dollar weighted investment = (2X - 18) / 100

Or, 100i = 2X - 18 Or, 50i = X - 9

For account 2:

Time weighted return: It has two components:

100 growing to 120 in 0.5 year

Immediately after deposit of 2X, the capital becomes 120 + 2X that grows to become 140 in the next 0.5 year

Hence time weighted return = 1 + i = 120 / 100 x 140 / (120 + 2X) = 168 / (120 + 2X) = 84 / (60 + X)

From the first equation, i = (X - 9) / 50

Hence, from second equation, 1 + i = 1 + (X - 9) / 50 = (41 + X) / 50 = 84 / (60 + X)

Hence, (60 + X).(41 + X) = 50 x 84

Hence, X2 + 101X + 2,460 = 4,200

Or, X2 + 101X - 1,740 = 0

It's a quadratic equation that can be factorized as:

(X - 15).(X + 116) = 0

Hence, X = 15

Hence, i = (X - 9) / 50 = (15 - 9) / 50 = 0.12 = 12%

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