Question

A stock initially trades for $50 per share. An investor decides to purchase 1,500 of shares. After 5 years, the portfolio is worth $120,240.00. At that time, the investor decides to purchase an additional 120 shares. At the end of year 7, the portfolio is now worth $140,178.60. The investor then decides to sell 60 shares. At the end of year 11, the portfolio is now worth $129,667.20.

Find the dollar weighted rate of return and the time rated rate of return?

Answer 1

Lets divide the portfolio value/investments on a timeline:

1. Investment made today being Inv(0) = $50 x 1,500 = $75,000

2. Portfolio value after 5 years being Port(5) = $120,240 (so one share is worth $120,240/1,500 = $80.16)

So, purchase of additional 120 shares then being Inv(5) = $80.16 x 120 = $9,619.20 (total shares now stand at 1,620)

Portfolio value after 5 years and additional purchase being PortFinal(5) = $120,240 + $9,619.20 = $129,859.20

3. Portfolio value after 7 years being Port(7) = $140,178.60 (so one share is worth $140,178.60/1,620 = $86.53)

So, sale of 60 shares then being Sale(7) = $86.53 x 60 = $5,191.80 (total shares now stand at 1,560)

Portfolio value after 7 years and sale being PortFinal(7) = $140,178.60 - $5,191.80 = $134,986.80

4. Portfolio value after 11 years being Port(11) = $129,667.20 (here, we need to assume that the portfolio is sold at this value in order to be able to calculate the Returns required in the question)

Dollar Weighted rate of return:

For this, we need to calculate the discount rate (R) which would equate the Present Value of Inflows and the Present Value of Outflows and this discount rate is our Dollar Weighted rate of return. There are two outflows i.e. Inv(0) and Inv(5) and two inflows i.e. Sale(7) and Port(11)

So the equation goes like this:

Inv(0) + [Inv(5) / (1+R)^5] = [Sale(7) / (1+R)^7] + [Port(11) / (1+R)^11]

$75,000 + $9,619.20 / (1+R)^5 = $5,191.80 / (1+R)^7 + $129,667.20 / (1+R)^11

Solving this, we get R = Dollar Weighted rate of return = 4.62%

Time Weighted rate of return:

In case of Time Weighted rate of return, we primarily deal with change in portfolio values for the same number of shares and calculated holding period returns for those same number of shares and then multiple these holding period returns to calculate the Time Weighted rate of return.

Return for 1st holding period of 5 years = R(0-5) = ($120,240 - $75,000) / $75,000 = 60.32%

Return for 2nd holding period of 2 years = R(5-7) = ($140,178.60 - $129,859.20) / $129,859.20 = 7.95%

Return for last holding period of 4 years = R(7-11) = ($129,667.20 - $134,986.80) / $134,986.80 = - 3.94% (negative return during last holding period)

So, Time Weighted rate of return = [ ( 1 + R(0-5) ) x ( 1 + R(5-7) ) x ( 1 + R(7-11) ) ] - 1

= [ ( 1 + 60.32% ) x ( 1 + 7.95% ) x ( 1 - 3.94% ) ] - 1 = 66.25%

Time weighted rate of return is much higher because it effectively gives us the holding period return over 11 years!

(Note: If we wanted to know effective annual rate of return from above, we could just do [(1 + 66.25%)^(1/11) - 1] and arrive at an effective annual rate of ~4.73%)