Question

You are given the following information on an investment account: 1/1/2017 5/1/2017 9/1/2017 1/1/2018 Account value before deposit or withdrawal 50,000 75,000 90,000 67,000 Deposit 15,000 Withdrawal 25,000 Calculate the absolute value of the difference in the yield rates by the dollar-weighted and time-weighted methods.

Answer 1

Answer:

The answer is "2.33%"

Explanation:

A = pre-payment or withdrawal account value= 50,000

B =end-of-year balance = 67,000

Deposit = 15000;

Withdrawal = -(1000)

   

Return rate annual dollar-weighted:

 $= \frac{(67000 - 50000 - 15000 + 25000)}{(67000)( \frac{12}{12}) + (15000)(\frac{8}{12}) - 25000(\frac{4}{12})}\\\\= \frac{(27000)}{(67000)( 1) + (15000)(\frac{2}{3}) - 25000(\frac{1}{3})}\\\\= \frac{(27000)}{(67000 + 5000 \times 2 - 8333.3)}\\\\= \frac{(27000)}{(67000 + 10000 - 8333.3)}\\\\ = \frac{27000}{68666.67} \\\\= 3.63 \%$

Method of timing: the time frame is 1 year from the table provided  balance at the beginning

:

$B_0 = 50000 \\\\ B_1= 75000$

return rate=$(1 + i) \frac{12}{12}$

$= (\frac{B_1}{B_0}) \times ( \frac{B_2}{B_1+W_1}) \times ( \frac{B_3}{B_2+W_2}) \times (\frac{B_4}{ B_3+W_3})$

$= (\frac{75000}{50000}) \times (\frac{90000}{75000+15000}) \times (\frac{67000}{90000 - 25000})\\\\= (\frac{3}{2}) \times (\frac{90000}{90000}) \times (\frac{67000}{65000})\\\\= (\frac{3}{2}) \times (1 ) \times (\frac{67}{65})\\\\= (\frac{3}{2}) \times (1 ) \times 1.03\\\\= \frac{3.09}{2}\\\\=1.5\\\\\to 1 + i = (1.5) \times 1 \times 1.03 \\\\= 2.307%$

$\therefore \\ i = 2.307 - 1 = 1.307$

Therefore, by dollar-weighted and timeweighted approaches the gap in involves collecting:

$=3.63 - 1.307 \\\\ =2.33 \%$