Question

An investor buys three shares of XYZ at the beginning of 2002 for $100 apiece. After one year, the share price has increased to $110 and he receives a dividend per share of $4. Right after receiving the dividend, he buys two additional shares at $110. After another year, the share price has dropped to $90, but the investor still receives a dividend per share of $4. Right after receiving the dividend, he sells one share at $90. After another year, the share price has gone up to $95, the investor receives a dividend per share of $4 and sells all shares at $95 immediately after receiving dividends.

3. What are the arithmetic and geometric average time-weighted rates of return and what is the dollar-weighted rate of return of the investor in the above example (for the dollar-weighted return assume that (i) the cash flows from dividends received at the end of a given year are based on the number of shares held at the beginning of that year, and (ii) cash flows from dividends occur on the same day as the cash flows from buying and selling shares)?

4. Why is the dollar-weighted average rate of return in the above example lower than the geometric average rate of return?

Answer 1

3.

Dollar weighted return is the return at which PV of cash outflows = PV of cash inflows from an investment. For calculating Dollar weighted return we first need to calculate the periodic cash flows:

Year	Period	Cash Flow on stock	Dividend cash flow	Total cash flow
2002 beg	0	-300.00	0.00	-300.00
2002 end	1	-220.00	12.00	-208.00
2003 end	2	90.00	20.00	110.00
2004 end	3	380.00	16.00	396.00

Using the property of Dollar weighted return:

r = -0.166%

Time weighted return: Return on an investment which removes the effect of cash contributions or withdrawls

Time weighted return for period n = ((MV_n - MV_n-1 + D₁ - CF₁)/MV_n-1)

Where:

• MV_n-1 = beginning market value,
• MV_n = ending market value,
• D_n = dividend/interest inflows,
• CF_n = cash flow received at period end (deposits subtracted, withdrawals added back)

So we first need to calculate each of these factors for all of the holding periods:

Period	MVn-1	MVn	Dn	CFn
1.00	300.00	550.00	12.00	-220.00
2.00	550.00	360.00	20.00	90.00
3.00	360.00	0.00	16.00	380.00

• TVR1 = (550-300+12-220)/300 = 14.00%
• TVR2 = (360-550+20+90)/550 = -14.55%
• TVR3 = (0-360+16+380)/360 = 10.00%

Arithmetic average = (14.00-14.55+10)/3 = 3.15%

Geometric average = (((1+0.14)*(1-0.1455)*(1+0.10))^(1/3))-1 = 2.33%

4. The dollar weighted return is affected by the period of cash contribution and cash withdrwals. contributing cash in periods when the cost of share was high and withdrawing them in the periods of low stock prices resulted in decrease in the dollar weighted return.

In time-weighted return, this effect of cash contribution/withdrawl is removed and only the actual return on the stock is calculted, resulting in a higher return than dollar weighted return.

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