# Calculating Momentum From Returns

I am doing some modelling and have some data:

Index         XYZ                1
1994-02-01  -0.005128205          0.994871
1994-03-01  0.013089005           1.007893
1994-04-01  0.012224939           1.020215
1994-05-01  0.018518519           1.039107
1994-06-01  0.017817372           1.057622
1994-07-01  0.019438445           1.078180
1994-08-01  0.006741573           1.085449
1994-09-01  0.017582418           1.104534
1994-10-01  -0.004319654          1.099762
1994-11-01  0.004310345           1.104503
1994-12-01  0.002150538           1.106878
1995-01-01  0.006382979           1.113943

I am trying to calculate the total return of the above column vector without transforming it to an equity curve first. Is it correct to just add it up?

The sum of the return = 0.1088808, while the discrete and continuous returns are 0.113944 and 0.107907. Why is there this difference?

Thanks,

EDIT: For the calculation, i simply used @Richard's equation.
Discrete = (1.113943 - 1) / 1 = 0.113944 Continuous = ln(1.113943/1) = 0.107907

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How do you calculate the above returns? discrete, log? Can post the prices too, so that we can replicate the results? Then it should become clear what the "correct" return for total is. –  Richard Mar 8 '13 at 12:44
Ok see above for the new modification. The price series, is just a equity series through cumprod. –  user1234440 Mar 8 '13 at 17:22
The answer on why there is a difference can be found by doing the algebra. Apply Chris' answer and the formulas. Adding up ratios is just the same as adding logarithms. If you want the log return between some day $t_1$ and some $t_2$ is found by adding logreturns. for discrete returns you need Chris' formula. –  Richard Mar 8 '13 at 20:27
Ooops typo, I didn't mean 'just the same' I meant 'not the same'... –  Richard Mar 9 '13 at 10:10

It depends on what you do with your returns. If your returns directly affect your capital base, regardless of positive or negative returns, and if you employ all the generated returns in new trades on which you subsequently calculate returns then you should use compounded returns. Else your returns should be treated as additive and simply aggregated through addition.

Please keep in mind that most buy-side portfolio managers and prop groups on the sell-side and hedge fund side do not re-employ returns, at least not right away. It is not a hard science in that all returns, gains and losses are moved into a segregated account but traders and portfolio managers generally do not look at each day's capital base, including all generated returns, when they make decisions of position size, profit targets and when they cut losing positions. Especially loss limits and position limits on the sell-side are re-evaluated very infrequently by superiors. Of course losses are immediately reflected in aggregate net profitability but just because you start off a 100 unit capital base and generated a 10 unit profit today does not mean your boss lets you get off the hook easily if your single asset position limit was 5% of the capital base at your last meeting with your boss and you took the liberty to put on a position of 5.5 units tomorrow. Buy-side portfolio manager committees generally sit down quarterly or so and re-evaluate re-investment decisions that are based on the profits/losses generated.

At the least, I would not blindly assume a constant re-investment on daily, weekly, even monthly basis unless there is a clear intention to re-invest behind it.

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I am just trying to measure asset class momentum. Would I then just add them? –  user1234440 Mar 8 '13 at 4:59
if you trade off the same base regardless of the profits/losses generated in the interim then yes you should just add them. –  Matt Wolf Mar 8 '13 at 5:32

Returns are supposed to be compounded. For example, if I make 10% today and another 10% on top of that tomorrow, then I will have made 21%. Addition would only make sense if I had taken my profits out at the end of the first day.

So no, you can't add returns like this. Instead, you must multiply the returns:

$$\prod_{i=1}^{n} (x_i + 1) - 1$$

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Nice answer and for other readers. Returns are no black boxes. They are derived from prices if $$r_t = \frac{P_t-P_{t-1}}{P_{t-1}}$$ then it is clear from algebra that adding up just does not make sense. With log returns defined as $$r_t^{log} = \log(\frac{P_t}{P_{t-1}})$$ it does. –  Richard Mar 8 '13 at 8:09
@Richard, so you are saying its inaccurate to state that if you started off with 100, made 10 yesterday and 5 today that you so far generated a return of (+10%, + 5%) 15% return? –  Matt Wolf Mar 8 '13 at 12:08
No, I am not commenting on your answer. I just thought it is sometimes useful to recall how those returns are defined. I think the original question is rather basic. In general log returns can be summed up over time and discrete returns not (due to the algebra that I wanted to show). If we turn to derivative markets and theresuch things get complicated (e.g. the return in a futures position if you have margin $x$ and such issues). My comment should just reveal basic connections. –  Richard Mar 8 '13 at 12:31
@Richard, well the simple arithmetic example above sums returns and those are not log returns and should not be calculated as log returns. –  Matt Wolf Mar 8 '13 at 12:33
@Freddy but the topic is "momentum". Thus I assume that the returns are calculated from equity prices. Then how are these returns calculated? I will ask the "asker". –  Richard Mar 8 '13 at 12:43