The Case of Misleading False-Negative Returns

Summary

Total returns since inception may be artificially low (or negative), due to early losses which have far reaching effects on total compounded returns over the life of a portfolio. For this reason trailing N year returns should always be considered, when looking at the "true" performance of a portfolio.

Background on Time Weighted Returns

Returns are the key indicator as to the performance of your portfolio, and the the investment decisions you have made. In the simple case, with no external cash flows, the return for any given period as a percentage is defined by:

Return=\left(\frac{Ending Portfolio Value}{Starting Portfolio Value} - 1\right) \times 100

When you introduce cash flows into the equation, you would measure the value of the portfolio immediately before the cash flow, and apply that to the numerator:

Return=\left(\frac{Ending Portfolio Value}{Starting Portfolio Value + External Cash Flow} - 1\right) \times 100

Your total return is then obtained by linking the individual period returns together. Others have written about this much better than I have, and two good articles may be found on Investopedia and Wikipedia. In summary though, the key formula to understand is the following, which measures the true time weighted return:

Return = \left(\left(\frac{M_1}{M_0+C_0}\times\frac{M_2}{M_1+C_1}\times\cdots\frac{M_n}{M_{n-1}+C_{n-1}}\times\right)-1\right) \times 100

Where in the above, M_n is the value of the portfolio at period n, and C_n are any cash flows occurring in period n (conventionally the cash flow is measured "immediately before" the valuation before period n+1).

False-Negative Returns

One key point to make is that when you are looking at returns, all things being equal, your return after a loss must be greater than the loss itself, to get back to where you started, on a percentage basis. For example, if your portfolio drops from $100 to $90, that is a 10% loss. However, to get from $90 to $100, you need an 11.1% gain. To further explore this example, consider a portfolio which is valued at $100 on Jan 1, $90 on July 1, and $100 on December 31. The return on the first period is -10%, the return on the second period is +11.1%, and the total return by linking those returns is 0%. So even though you did extremely well in the latter half of the year, your net return is still 0% overall (since you finished where you started).

So, for better or for worse, the TWR measures the true performance of your portfolio over time, reflecting every investing decision you have made. But when you are reviewing your total returns over time, you may have misleading results. Consider the following hypothetical portfolio from Jan 2000 to Dec 2002. In this portfolio, I have invested $100 on Jan 1, 2000, and the company I invested in tanked completely, losing me $98 on my $100 investment. Then on Jan 1, 2001, I heard of an even better opportunity, and invested another $1,000. Unlike the initial $100 investment, the second investment performed extremely well, more than doubling by the the end of 2002, after which point I closed off the portfolio. Overall, I have invested $1,100, and at the end of it all I have walked away with $2,260.33, a tidy profit of $1,160.33, or 105.48%; more than double my initial investment! But, if you look at my total time weighted return since inception (i.e. Jan 2000)—which includes every trade, both good and bad—my investment decisions show that I am actually down in excess of 95%!

Period Closing Portfolio Value Cash Flows during Period Period Return Return since Inception Total Invested Total $ Gain
Jan 2000 $0.00 $0.00
Feb 2000 $100.00 $100.00 $100.00
Mar 2000 $93.00 $0.00 (7.00%) (7.00%) $100.00 ($7.00)
Apr 2000 $81.00 $0.00 (12.90%) (19.00%) $100.00 ($19.00)
May 2000 $69.00 $0.00 (14.81%) (31.00%) $100.00 ($31.00)
Jun 2000 $57.00 $0.00 (17.39%) (43.00%) $100.00 ($43.00)
Jul 2000 $48.00 $0.00 (15.79%) (52.00%) $100.00 ($52.00)
Aug 2000 $39.00 $0.00 (18.75%) (61.00%) $100.00 ($61.00)
Sep 2000 $30.00 $0.00 (23.08%) (70.00%) $100.00 ($70.00)
Oct 2000 $19.00 $0.00 (36.67%) (81.00%) $100.00 ($81.00)
Nov 2000 $9.00 $0.00 (52.63%) (91.00%) $100.00 ($91.00)
Dec 2000 $2.00 $0.00 (77.78%) (98.00%) $100.00 ($98.00)
Jan 2001 $1002.00 $1000.00 (98.00%) $1100.00 ($98.00)
Feb 2001 $1057.00 $0.00 5.49% (97.89%) $1100.00 ($43.00)
Mar 2001 $1117.14 $0.00 5.69% (97.77%) $1100.00 $17.14
Apr 2001 $1176.24 $0.00 5.29% (97.65%) $1100.00 $76.24
May 2001 $1207.94 $0.00 2.70% (97.59%) $1100.00 $107.94
Jun 2001 $1195.88 $0.00 (1.00%) (97.61%) $1100.00 $95.88
Jul 2001 $1194.69 $0.00 (0.10%) (97.62%) $1100.00 $94.69
Aug 2001 $1276.98 $0.00 6.89% (97.45%) $1100.00 $176.98
Sep 2001 $1322.88 $0.00 3.59% (97.36%) $1100.00 $222.88
Oct 2001 $1355.91 $0.00 2.50% (97.29%) $1100.00 $255.91
Nov 2001 $1434.43 $0.00 5.79% (97.14%) $1100.00 $334.43
Dec 2001 $1516.08 $0.00 5.69% (96.97%) $1100.00 $416.08
Jan 2002 $1503.97 $0.00 (0.80%) (97.00%) $1100.00 $403.97
Feb 2002 $1585.07 $0.00 5.39% (96.84%) $1100.00 $485.07
Mar 2002 $1654.73 $0.00 4.39% (96.70%) $1100.00 $554.73
Apr 2002 $1735.71 $0.00 4.89% (96.54%) $1100.00 $635.71
May 2002 $1838.00 $0.00 5.89% (96.33%) $1100.00 $738.00
Jun 2002 $1894.92 $0.00 3.10% (96.22%) $1100.00 $794.92
Jul 2002 $1980.10 $0.00 4.50% (96.05%) $1100.00 $880.10
Aug 2002 $1980.10 $0.00 (96.05%) $1100.00 $880.10
Sep 2002 $2073.07 $0.00 4.70% (95.86%) $1100.00 $973.07
Oct 2002 $2104.14 $0.00 1.50% (95.80%) $1100.00 $1004.14
Nov 2002 $2192.43 $0.00 4.20% (95.62%) $1100.00 $1092.43
Dec 2002 $2260.33 $0.00 3.10% (95.49%) $1100.00 $1160.33

Clearly, this is misleading: I have walked away with more than double what I invested in overall, however due to a single bad decision at the outset, my total returns are dragged down completely. In fact, to gain ground from a 98% loss requires a staggering 4,900% return over time!!! The nuances of this are tied to the fact that time weighted returns take the geometric average of your historical returns: they multiply everything together, and due to the way the math works itself out, you are virtually never able to get back to where you started after a devastating failure.

Fortunately, there are ways to paint this picture in a different light. Observe what happens if we position the portfolio this way:

1 year return 2 year return 3 year return
49.1% 125.6% -95.5%

By breaking up the returns into tranches, the results look dramatically different.

The point of this is to illustrate that a single bad period can drag down the total return of a portfolio virtually forever. Similar to how when I review a firm I am typically only looking at 10 years of historical data, there is value in truncating the window at when one evaluates their returns as well; we learn as we move forward, and as long as on a historical basis we are increasing our returns, then we are doing relatively well. The final point is that while the "true time weighted return" is "true", one must review it with a grain of salt. True, on a percentage basis, the above example is still down 98%. However, overall the total return based on absolute dollar terms is in excess of 100%.

For the record, the above is a simplified version of my own portfolio. When I started investing in 2005 I followed “hot tips” from my coworkers who were “good friends” with traders on the trading floor, and I invested in two firms, First Calgary Petroleum (FCP.TO), an Alberta based refinery, and Paincare Holdings (PRZ.N), a US healthcare provider. Luckily I had only invested ~$1250 at the time: 50 shares of FCP.TO at $18.40/share, and 50 shares of PRZ.N at U$5.00/share. FCP.TO tanked, no pun intended, when one of their exploration operations did not pan out as expected. FCP.TO was eventually bought out by a foreign interest at $3.60/share, netting me a loss of 80%. Paincare Holdings was victim to a lawsuit, and the company eventually de-listed except for over the counter pink slips, and is now a private holding. The shares of PRZ.N virtually went to $0.00; they are actually still “in” my portfolio at BMO InvestorLine with a market value of $0.005, or U$0.0001/share, netting me a loss of 99.998% on that trade. As these are the first trades in my portfolio, they have been a constant drag on my true TWR. Since then, I definitely regained the original $1,250 investment in dividend income alone from other investments, so on an absolute (i.e. actual dollars profit) I am well ahead. So, if we carve out those outliers, every other investment I have completed has performed relatively well, something I hope to continue doing as time moves forward.



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