Let me give you three prophecies:

• The Sun will rise tomorrow
• It will rain tomorrow
• ​Aliens will land in Parliament Square tomorrow

Which do you think is most likely, and why?

You expect the Sun will rise tomorrow, right? If you live in the UK it's plausible that it will rain tomorrow, but we're not as sure about that as we were about the Sun rising. And not many people would think aliens will land in Parliament Square tomorrow.

Boris Johnson - not an alien

None of those likelihood judgements are based on logic. They're based on experience. The Sun​ has risen every day of your life. It has rained many times in your life. Aliens have never landed in Parliament Square (no, Boris Johnson doesn't count).

​The best we can ever do when making statements about the future is assign probabilities to different outcomes. The probability of the Sun rising is a tiny bit below 100%, rainy days happen 29% of the time in London and the chance of aliens landing in Parliament Square is a tiny bit above 0%.

If you invest you should think about the future value of your investments in exactly this way, in terms of probability.​ The probabilities will be based on what has happened to the historical value of investments. Of course we would like to always have positive returns, but in reality the return for any stock, bond or commodity on any day will be negative with almost exactly a 50% probability. If we count the number of times the FTSE 100 return was between 0 and 0.1%, and between 0.1% and 0.2%... imagine these were little sorting buckets. We would take the return on the FTSE 100 for every day since 1982 (there are 8,435 days) and put it into each bucket according to its size. The result looks like this:

This is the fingerprint of the FTSE 100 and it gives us a huge amount of information about the index. You can see that most of the time the return is close to zero. The FTSE rises (positive return) half the time and falls (negative return) half the time. More accurately the FTSE 100 was positive a little bit more than it was negative, it was positive on 51.6% of the days since 1984 which was enough of a nudge to generate a respectable 5.6% return over that 33 year period.

If we worry about risk, which we should, then its the dribs and drabs on the left hand side of the plot that concern us. This is the downside tail and that is our dreaded tail risk. On some days in which the market was in turmoil the FTSE fell by more than 5%. You can see this is very unusual because the number of days with a return below -5% are tiny. It occurred four times during the market crash in October 1987, eleven times during the Financial Crisis in 2008/2009 and a couple of times around 2001/2. That's 17 days out of 8435, and as a percentage that's less than 0.2% of the time. But the awful thing about those tail risk days is that they produce a cumulative loss of 70%!!! That's why we try to avoid and measure tail risk, and risk in general.

There are several risk measures which are used to assess and compare risks in the investment industry and we consider three: volatility, Value at Risk and drawdown. Each is a summary of the full return distribution and is therefore incomplete. It's always a good idea to take a look at the full return distribution rather than the risk and return summaries that are available in the media. That's really the only way to get to know an asset intimately.

Volatility

The odd thing about volatility is it ignores the sign of historical returns. This seems crazy because we would never think of a positive return as being risky. It's losses we worry about not gains. But volatility makes sense because it provides a very useful "surprise" measure. If you hear the FTSE 100 has risen by 1% should you be surprised? Is this even news? No! Here's why.

Volatility is the typical daily percentage price move

Here are some volatilities for frequently quoted stock, commodity and currency indices:

​Looking at the table you can see that the typical daily move of the FTSE 100 is 1.1%, so we would not be at all surprised to see the FTSE move up or down by 1%. If it were to move 2% that would be a bit surprising, 3% would be surprising and 4% or more would tell us some big news was driving markets.

The reason why volatility tells us about tail risk is that assets with higher volatility, or higher typical daily moves, also tend to have larger tails. But this need not always be true. For example you could have an asset that crashes spectacularly but day to day the typical daily move is small. Corporate bonds tend to behave this way, for example, and for these assets we need to focus on the tail. And that leads us to our other risk measures.​

Value at Risk

We can phrase our market worries in another way. Instead of typical up or down move we can focus on the amount that we might expect to lose in one year out of twenty (or one day out of twenty, or one month out of twenty...). The one-in-twenty means that our probability of taking that loss is 5%. If we invest £10,000 pounds then we might say:

Expect a loss of £2000 or more on average in one year out of twenty.

The £2,000 would be our "Value at Risk" (VaR). Volatility and VaR are related by a rough rule of thumb. One in twenty year annual VaR is about 1.64 times bigger than annualised volatility. Here are the daily and annual Value at Risk for the five indices based on an investment of £10,000.

Drawdown​

If we focus even more deeply on fear then we can dispense with the idea of "typical" losses completely. Drawdown is probably the simplest measure of risk and it addresses the question:

What's the worst peak-to-trough loss over the last year?

The drawback with this measure is that it focusses on the extremes of performance. It is very unlikely that this most extreme loss will be repeated, but for those of a nervous disposition this drawdown measure is of interest.

We can illustrate drawdown with this example for sterling for the time period from March 31st 2016 to March 31st 2017. This period is dominated by the effect of the EU referendum which massively weakened sterling. To calculate the drawdown we find the peak value over this period which was 1.479, then the following minimum, which was 1.204. This corresponds to a fall of 22.8%, and that's the drawdown value. The drawback of drawdown is obvious from this example: the effect of Brexit is already priced in and it is unlikely that sterling will take another 22.8% fall, but focussing on drawdown risk highlights such a possibility. If we look at the volatility of sterling it steadied quite quickly after the initial shock.

Which is the best measure?

​The three risk measures are quite different. Volatility is the most widely understood and quoted measure but it  treats upside and downside the same. Also if  an asset tends to crash volatility and Value at Risk may miss this. For example sterling fell sharply after the EU Referendum but the volatility and VaR of Sterling vs the US dollar does not fully reflect this risk because the currency suffered some short, sharp falls. Only the drawdown measure, which is based on cumulative return, picks up on this.

Before you dismiss volatility and VaR as being too insensitive, however​, remember that being too cautious could be costly. After a market crash there is often a strong rally, and religious devotion to drawdown will keep you out of the market for too long and you will miss the rally.

The truth is that there is no "best" measure. You should consider all three. Each tells you something different about the character of returns for an asset or a portfolio. Each is a summary of the full distribution of returns, and that should be your focus. A histogram of the daily returns is the richest source of information as this tells the most complete picture: upside, downside and all.

Try calculating these measures for your favourite asset. It may be you'll get to know it better and avoid big losses in future.