The second most important parameter to look at when considering an investment is the volatility (the most important one is the Return, see here). Volatility is important because it shows how varied the price is. In other words, how consistent the growth is.

Why the consistency of the growth is important? Take a look at this chart:

Low and High Volatility

All 4 lines starts from 100 and ends on 119. But how they arrive at 119 are very different. The volatility of these 4 lines are green: 8.93, red: 9.65, blue: 1.54, purple: 5.34.

Red is up and down like a yoyo. In Sep 2012 I would have no clue whatsoever whether in the next few months it would be going up or down. It goes from +15 and +17 in Oct 2011 and Apr 2012 to -13 in March 2012. Statistically, because it’s like up & down dramatically like a yoyo, the probability of it’s going up is almost the same as it’s going down. The future is completely unpredictable.

Green is shooting up very high in Oct to Dec 2011, then flat in Jan and Feb and down a lot in March. Then up very high again in April, then stepping down until Sep 2011. It goes from +13.5, +12.5 and +17 in Oct 2011, Dec 2011, Apr 2012 to -13 in Mar 2012. Again, we have no idea of where it’s going. But judging by “the near past is more influential than distant past”, in a few months from Oct 2012 I think the probability of it’s going down is high than going up.

Blue is pretty consistent with the growth. It goes from +4 in Jan 2012 to -1.5 in Mar 2012. It is almost a straight line. It is by far more resembling a straight line than red or green. From May to Sep it has been pretty consistent with growing +1 to +2 each month. It is much more predictable than Red or Green. In a few months from Oct 2012 I am pretty sure it will continue to go up at the same pace.

Purple was pretty much flat for 9 months from Nov 2011 to Aug 2012 then suddenly shoot up to reach 119. It is far better than Red or Green, but it is worse than Blue. It is better than Red or Green because staying flat is better than going up and down like a yoyo. But going up consistently +1.5 each month like Blue is better than staying flat for 9 months then shoot up at the end. Better means more predictable.

Yes you are right, I am assuming that if it has been rain, rain, rain in the past 6 days, then the chances of it going rain tomorrow is bigger than sunshine. And if it has been sunshine for the last 6 days, the changes of it’s going to be sunshine on tomorrow is bigger than raining.

If the growth has always been +2 in the past 6 months, the chances of the growth being +2 next month is a lot higher than if it has been yoyo-ing +2, -2, +2, -2, +2, -2.

These charts represents either a company, or a fund (or commodity, or FX). There are real things behind these charts. Assume we are talking equity. These 4 lines are 4 different companies in the stock market. Company red which is up and down like yoyo, is not trustable at all. There is something fundamentally wrong with this company. Why is the share price goes up and down like yoyo? Why did people rushing to buy (hence the price was up) then the following month people rushing to sell? Why in Feb people were buying, in March people were selling, in April buying, in May selling, in June buying, in July selling? We need to find out what’s going on inside this company. There must be something going on. There is no smoke without fire.

Company blue on the other hand is much more consistently growing compared to Red. It is a much more stable company. And because the company is more stable (based on the share price), it is more predictable.

If I have to choose between these 4 companies to buy, hands down I will choose Blue. It is buy far the most stable, the most predictable growth. Yes all for of them gives the same return (from 100 to 119 in a year = 19% return), but I have to predict the future movement, I say that Blue has the highest change of making good, consistent growth in the future.

There is another reason why I prefer the one most resembling straight line (Blue) than the yoyo ones like Red. If it is straight line, I can exit at any time without problem. If it is yoyo-ing, exiting could be a little problem. For example, March 2012 was a bad time to exit from Red because the price was down 12.

If a company made a big fall in the past it is more likely to make it again in the future. Red and Blue in March are both down 12. Whereas Blue is never down more than 1.5. Compare 12 and 1.5: the 12 are more scary! We know that if we buy Blue, if it goes down, it doesn’t go down much. Whereas if we buy Red or Green, we must be prepared to accept big falls.

The volatility of these 4 lines are green: 8.93, red: 9.65, blue: 1.54, purple: 5.34. Blue has the lowest volatility (“vol” for short). Low vol means “growth is consistent”, like it’s always +1 or +2 every month. Red and Green have the highest volatility. High vol means “up and down violently like a yoyo”, like Jan +10, Feb -13, March +15, … Purple’s vol is in the middle between Red/Green and Blue. It’s not “growing consistently” and it’s not “up and down like yoyo” either. It’s consistently flat then up.

So we prefer investment with low vol. Meaning that its growth is consistent.

Calculating Volatility

Calculating Volatility - Blue

These are the numbers for Blue. The growth is between +4 (Jan) and -1.5 (Mar). The average of growth is 1.58.

Delta means how far is the growth from the average. In Oct 2011, the growth is 3.5, so the Delta is 3.5-1.58 = 1.92. In Nov 2011, the delta is 1.5-1.58 = -0.08. Delta^2 means Delta Squared = Delta2. In Oct 2011 the Delta2 was 1.922 = 3.67. And in Nov 2011 the Delta2 was (-0.08)2 = 0.0069, rounded to 0.01. The reason we square the Delta is to avoid the negatives cancelling the positives.

If we average all the Delta2, we get 2.37. This is called the Variance. Variance is a measure of how far a set of numbers are spread out from the mean.

If we take the square root of the variance, we get Standard Deviation (SD). Its matematical symbol is Greek alphabet sigma: σ. It also shows how spread out the values are from the mean. In investment, the values we are comparing is the growth. Hence it is called volatility, meaning how volatile the price is, how different the growth is from one day to the next.

In the growth is distributed normally, 1 sigma = 68%, 2 sigma = 95%, 3 sigma = 99.7%. In the above case of Blue, the sigma is 1.54. Hence 2 sigma = 3.08 so 1.58 ± 3.08 covers 95% of the population.

Below are the calculations for Red, Green and Purple:


Calculating Volatility - Red


Calculating Volatility - Green


Calculating Volatility - Purple



Return is how much money the investment has grown. In other words, it is the value of it today, compared to its value in the past.

As always it is better to use an example when explaining. Consider the following investment, Red and Blue:


During the 3 years period (Sep 2009 to Sep 2012), Red grew from £100 to £142, whereas Blue grew from £100 to £184. So Red made £42 and Blue made £84. Red’s return is 42% for 3 years, or 14% per year. Blue’s return is 84% for 3 years, or 28% per year. This 14% and 28% are not accurate, because the return is compounded. The accurate numbers are 12.4% (cube root of 1.42 minus 1) and 22.5% (cube root of 1.84 minus 1).

Return is the most important parameter when considering an investment (the other being the risk). We simply pick the one with the highest return. This sounds very simple, but many people don’t always see it clearly.

That is why it is called “Return on Investment”. It is how much money the investment is making. We have an investment £100 in Red and we get £42. We invest £100 in Blue and get £84.

Let do another example so we really understand the formula. This time the prices don’t start at 100:

Not from 100

Red starts from 140 and grew to 182. Blue starts from 120 and grew to 204. The gain is 42 for Red and 84 for Blue, for 3 years. The return for Red is 42/140 = 30%, and for Blue 84/120 = 70%. These 30% and 70% are for 3 years, so per year the return is 10% for Red and 23.3% for Blue. This 10% and 23.3% are not accurate, because the return is compounded, so it is not linear. The accurate numbers are 9.14% (cube root of 1.3 minus 1) and 19.3% (cube root of 1.7 minus 1).

What if it doesn’t cover 3 years duration? Like this:

Diff duration

Red covers 3 years (Sep 2009 to Sep 2012) but Blue is only from March 2010 (30 months = 2.5 years).

Return for Red = (180-140)/140 = 40/140 = 28.6% for 3 years. So per year the return is cube root of 1.286 minus 1 = 8.75%.

Return for Blue = (180-120)/120 = 60/120 = 50% for 2.5 years. So per year the return is the 2.5th root of 1.5 minus 1 = 1.5 ^ (1/2.5) – 1 = 17.6%.

So that is how we calculate the return on an investment. This is the number one consideration when investing (in my opinion) so we need to understand it well.

I believe we should learn from actual numbers, not theoretical cases. This is Apple shares, from 18 Sep 2009 to 21 Sep 2012: (source: Google Finance)


Apple share price grew from $185 on 18/9/2009 to $700 on 21/9/2012.
Return = ($700-$185)/$185 = $515/$185 = 278.38% for 3 years = 55.83% per year.

On 23 Sep 2011 the share price was $404.
Return = ($700-$404)/$404 = $296/$404 = 73.3% for 3 years = 20.1% per year.

So the return based on 3 years data is not the same as the return based on 1 year data.

Is 20.1% a good return or not? Yes it is a good return. Generally speaking, for the period Sep 2011 to Sep 2012, return above 10% is a good. Above 20% is very good. How do I know this? By looking at major indexes. An index like FTSE 100 or S&P100.

This is S&P 500 Jun 2011 to Sep 2012:


On 21 Sep 2009, S&P 500 was 1460. On 25 Sep 2009, S&P 500 was 1044. The 3Y return therefore is (1460-1044)/1044/3 = 13.3%

On 23 Sep 2011, S&P 500 was 1136. The 1Y return is therefore (1460-1136)/1460/3 = 9.5%


FTSE 100 (above) was 5853 on 21/9/12, 5067 on 23/9/11 and 5082 on 25/9/09. In case you are wondering why not all 3 years on 23 Sep, it’s because Google Finance data is on Fridays (weekly data). So the 1Y return is 5.2% and 3Y return is 5.1%.

S&P500 is reflects the growth of the biggest 500 companies in the US. FTSE100 reflects the growth of the biggest 100 companies in the UK. Which is why Apple’s 20.1% return is good. Because it’s higher than the FTSE’s 5% and S&P’s 9%-13%.