Friday, January 8, 2010 6 comments ++[ CLICK TO COMMENT ]++

A newbie question for you: can you show me how to compute some returns?

This is going to show my total newbieness but if you, as the reader, have some time to kill, can you compute the returns outlined below and leave a comment with your answer or e-mail me. As usual, I am trying to compute my performance for the year—all I know is, it's not good this year :|—and am getting confused. I'm starting to doubt the calculation I have performed in the past.


If you have time, can you compute the returns for the following scenario:


Start: You start with $100
Year 1: At the end of year 1, you have $150.

You add $100 so the starting value for year 2 is $250.
Year 2: At the end of year 2, you have $200.

You remove $50 from your portfolio so the starting value for year 3 is $150.
Year 3: At the end of year 3, you have $100.


What is/are the:

(i) Yearly (annual) returns (for year1, year 2, and year3)?
(ii) Overall total (aggregate) return over the 3 years?
(iii) Annualized return over the 3 years?



This is elemental stuff and I can't believe I am so confused by it. Anyway, let's see what you guys & gals think. I'll post my computation later, maybe tomorrow or on the weekend.

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6 Response to A newbie question for you: can you show me how to compute some returns?

JB
January 8, 2010 at 8:49 AM

I.

Year 1 Return = (150-100)/100 = 50%
Year 2 Return = (200-250)/250 = -20%
Year 3 Return = (100-150)/150 = -33.3%

II.

I calculated a Time-Weighted-Return of -33.3%.  I give your intial starting amount -- $100 -- a 1/3 weighting; then I add $100 to that $100 and give a 1/3 weighting to the resulting $200; then I subtract $50 from the $200 and give a 1/3 weighting to the resulting $150.

The weighted-average beginning amount (the denominator) comes out to $150.

Then I do a similar procedure for your ending amount:  You finished with $100 and I give a 1/3 weigthing to that; I added $50 to that for your Beginning Year 3 withdrawal and gave a 1/3 weighting to the resulting $150; and I subtract $100 for your beginning Year 2 contribution and give a 1/3rd weighting to the resulting $50.

The resulting number is 100.

In other words, your overall time-weighted return is (100-150)/150 = -33.3%

Alternatively, you could calculate your overall rate of return via geometric linking (which I think would be less accurate in the case of an individual, but might be more instructive to someone who potentially wants to invest with you, since it would give them an idea of what their returns would be if they gave you a lump sum in the beginning of Year 1 to manage for them):

(1+.5)*(1-.2)*(1-.333)-1 = -46.7%

III.

To arrive at the Annualized Return, you would perfrom the following calculation:

(1+r)^(1/n)-1

Where:

r = total return
n = number of years
^ = to the power of

Using the overall time-weighted return, your annualized return is -12.6%; using the geometrically linked overall return, your annulaized return is -18.9%.

MrParkerBohn
January 8, 2010 at 10:05 AM

Here's basically how I think about tracking returns.  This is certainly not the only way to track returns, but it is probably the most useful way to communicate your returns to other people.

Pretend that you are a mutual fund, and I gave you a dollar to invest.  What return would you have given me (compared to a benchmark) over whatever time period you are looking at?

If you only deposit/withdraw money at the very start/end of a year, then it should be pretty easy to calculate your annual returns.  Just divide the starting and ending amounts.

Based on your example:

Year One - 150/100 = 1.50 (+50% return)
Year Two - 200/250 = 0.80 (-20% return)
Year Three -  100/150 = .67 (-33% return)

Total return = 1.5 x .8 x .67 = .80 (-20% return)

Average geometric return = the cube root of .80, which is about .93, or -7% per year compounding (ie .93 x .93 x .93 = .80)

It gets a bit tricky if, like me, you sometimes make deposits in the middle of the year, but it is possible to follow the same basic method using an Excel spreadsheet and monthly rather than annual brokerage statements and account totals.

MrParkerBohn
January 8, 2010 at 11:01 AM

JB - you posted while I <span>was typing up my post, so I didn't see your post until after I posted.  I'm not 100% sure how your first method works.  The method I use is the same as what you called "geometric linking".
</span>

Your equation for geometric linking looks currect, but I think you hit the wrong button on your calculator or something, because

<span>(1+.5)*(1-.2)*(1-.333)-1 = -46.7%</span>
is wrong...
the proper answer is about -20%, which is the answer I got.

JB
January 8, 2010 at 12:37 PM

MrParkerBohn

Yeah, my geometric link number is wrong.

As for the method I called "time-weighted:"  I investigated further and see that it's actually dollar-weighted.  To understand the dollar-weighted mthodology and implications, think of it as follows:

1.  If you start with 100 and contribute 100 on Day 1, your assumed starting amount for the year will be almost 200;

2.  If you start with 100 and you contribute 100 on Day 364, then your assumed starting amount for the year will be almost 100;

3.  If you start with 100 and contribute 100 on Day 183, then your assumed starting amount for the year will be 150.

This methodology is more appropriate than geometric linking when you are managing your own account and deciding on the timing of contributions and withdrawals.  It is less appropriate when you are managing a comingled account where you do not have control over the timing of contributions and withdrawals.  if your situation lies somewhere in the middle, then I'm not so sure. 

For instance, you could be managing someone else's account, where control the timing of contributions and withdrawals and geometric linking might show them with a profit or loss, but the economic reality is that they have less or more money in the account than they've contributed, respectively.

From their perspective, geometric linking would give them a distorted picture of how their account has performed.  However, if the manager were to report the results to a third-party or a potential client, geometric linking might give that client a better picture of his track record for the given period.

But in Sivaram's case, it looks to me as if he is talking about his own account and that he has controlled the timing of contributions and withdrawals.  So I would say that for him to accurately assess how he has done, he ought dollar weight his returns based on the timing and size of contributions and withdrawals, as opposed to geometrically linking return periods.

Sivaram Velauthapillai
January 9, 2010 at 2:25 AM

Thanks for all the input. I really appreciate your comments. It clarifies my calculations.

I have actually been using the geometric computation method as outlined above but was doubting myself because I was trying to re-concile some numbers are they didn't match. The issue is the following:

Your invested capital is $150 ($100+$100-$50) while you end up with a final value of $100. So the loss is $50. I was getting confused trying to reconcile this with the numbers that were computed in (i), (ii), and (iii). How do you get this $50 loss from the -7.17% annualized number we compute? Or is there no way to arrive at the $50 using the annualized (or cumulative or yearly) number?

Sivaram Velauthapillai
January 9, 2010 at 2:53 AM

RE: JB's time-weighted computation

Regarding JB's note about time-weighted computation, I agree that the true cost of money (or the true gains) are not captured properly with the simple geometric multiplcation.

In light of your comments, I think I'm going to list the geometric computation numbers (which is what I have been doing in the past), along with what my personal finance software (Microsoft Money) computes. All my confusion started with problems I had reconciling the MS Money numbers (MS Money is being discontinued and I was trying to figure out how to do it by hand) and I'm still not sure how Money computes the returns. My guess is that it is some weighted average that, although not the same as your time-weighted computation, should capture the concept somewhat.

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