Personal Finance
[ December 9, 1999 ]
Financial Deviance
By James Carlisle (TMFJimmyC)
On Tuesday, I explained how you get the average of a range of numbers. I discussed a game of lucky dip in which we draw one envelope from five containing £0, £5, £10, £20 and £50. The average value of an envelope was found to be £17. This is a useful number since it tells us the "expected value" of an envelope. From this, we can determine how much we should be prepared to pay for the privilege of picking an envelope. The answer is anything up to £16.99. However, to understand the possible range of outcomes properly, we need to know another number. That is, we need to know the amount by which we should expect the envelope that we pick to differ from the average. This is important in assessing and controlling our risk. To illustrate this, we need to up the stakes a little.
Imagine the same game, but this time, the envelopes contain £0, £50,000, £100,000, £200,000 and £500,000. Imagine that we only have £200,000 to start with. The expected value of an envelope is £170,000. So, should we be prepared to pay up to £169,999 to pick an envelope. No way! Not unless you're very rich indeed or you are brave to a point that would make Eddie the Eagle look like a pansy. The reason for this is that the range of outcomes is too wide. There is too big a chance of losing too much money.
If we played for £169,999, only two of the five (that is, 40%) of the envelopes would leave us with enough money to play the game a second time. For us to have an acceptable likelihood of profiting by paying £169,999, we'd need to play the game an awful lot of times. For us to be certain of profiting, we would need to play an infinite number of times. Playing for £169,999 in the above circumstances is plain daft. This is because our expected deviation from the expected value of an envelope is too high. This is often described as "volatility".
So, I'd play the game for £169,999 only if I had a banker standing behind me who had very deep pockets indeed and who had undertaken to back me through thick and thin. Have you heard of Long Term Capital Management? It's the hedge fund that went bust last year and nearly caused global financial meltdown (or so they say). Their situation can be compared with this. Essentially what they were doing was playing this game for the equivalent of £169,999. In other words, their "positions" were essentially profitable. The problem they had was that their backers deserted them after a long run of very bad luck. Hard cheese, I say. They should have thought harder about their volatility and how long they could stay in the game.
Anyway, I digress. Think of another game. This time, the amounts in the envelopes are £168,000, £169,000, £170,000, £171,000 and £172,000. Again, let's say that we have £200,000 to start with. The average envelope still contains £170,000. How much would you pay to play this game? Well, perhaps not quite £169,999, but I'd happily pay £169,900. On average, I'd be winning £100 each time. However, just as importantly, I'd be able to sustain a very bad run of luck indeed, before I had to retire from the game (that is, when I no longer had the money to buy an envelope). Even if this run of bad luck occurred, I'd still have at least £168,000. In other words, my worst case scenario (and very unlikely it is too) would give me a return of -16%. A nasty hit, but by no means a disaster.
The Risk Premium
So, we'd pay very much more to play the second game than we would to play the first. This is because, in the first case, there is far more risk involved. We therefore need a bit of a cushion in our calculations to account for this. This cushion is often described as a "risk premium". Imagine that you could trade the right to play both these games: you might end up with the first game trading in the market at, say, £140,000 and the second game trading at, say, £169,900. The difference of £29,900 is the risk premium and is entirely due to the greater risk of the first game.
However, remember that there are different people in the market. If everyone in the market was a billionaire with a strong bank behind them, you might expect the risk premium to fall massively. This type of investor may pay very nearly the same to play both games, because they have deep enough pockets to keep going on and on until the odds come down in their favour: as they eventually must (if they are playing for less than £170,000). Now imagine that a market which is half made up of these people and half made up of people with only £200,000. Here, market price for playing the game might, perhaps, be about £154,950. In this case, the people with deep pockets and who don't need the risk premium, can clean up. This is because they effectively get the risk premium for free.
This is like a long-term investor in equities. Having a long time to invest is like having deep pockets in the lucky dip game. You can sustain a period of very bad luck, because you can just keep on going. However the equity market contains people who have shorter time horizons and who therefore demand a bit of a risk premium. Those who are able, and prepared, to invest for the very long term should therefore be able to take advantage of this.
Measuring Volatility
The most normal measure of volatility that people use is the "standard deviation". Plenty of people disagree with its use. However, I think that this generally stems from people not attributing a time period to it. This would be like specifying our standard deviation for the lucky dip game without saying how many times we are playing it (that is, how much money we have). The standard deviation of the return from equities (relative to its expected value) is more over one year than it is over 30 years. In the same way, the standard deviation of our return from playing the lucky dip game is greater (again, relative to its expected value) if we only play it once than if we play it 30 times. Anyway, I'm getting ahead of myself again. Let's see how we calculate the standard deviation.
Going back to the low stakes version of the lucky dip game. Our range of outcomes and how they differ from the average looks like this.
Outcome Average Difference
0 - 17 = -17
5 - 17 = -12
10 - 17 = -7
20 - 17 = +3
50 - 17 = +33
Now you might have thought that the most sensible thing to do would be to take the average difference. You'd get this by dropping the minus sign from the negative differences, adding them all together and dividing by 5. So, 17+12+7+3+33=72 and 72 divided by 5 is 14.4. The average difference is therefore 14.4. This is therefore the amount by which we expect the value of the envelope that we pick to deviate from its expected value. Unfortunately, the mathematicians don't like doing it like this. They can't stand it when you say things like "just drop the minus sign". Frankly, it plays havoc with their clever formulae. So what do they do instead?
Well, the way that mathematicians get rid of a minus sign is to "square" the number (that is, multiply it by itself). Because multiplying two negative numbers gives you a positive number, this therefore gets rid of all negative numbers. Then, they take the square root of the final number to get back to normality.
So, to get the "standard deviation", they do this. They take the all the differences and square them. Like this:
Envelope Average Difference Difference
Squared
0 - 17 = -17 289
5 - 17 = -12 144
10 - 17 = -7 49
20 - 17 = +3 9
50 - 17 = +33 1,089
-----
Total 1,580
Then you divide the 1,580 by 5, to get 316. Finally, since we squared in the beginning, we have to take the square root to get us back to normality. The square root of 316 is 17.78 (that is, 17.78 squared equals 316). So, the standard deviation in our return from playing our lucky dip game once is 17.78.
In finance then, when people talk about risk, they generally mean the volatility of returns from a particular investment over a particular time scale. The most common measure of this volatility is standard deviation.
The standard deviation is a fairly odd number, but it does at least describe the sort of range to expect from a collection of possible returns. In fact, it can help you be quite a lot more precise than that. For instance, if you have what is known as a "normal" distribution of outcomes (the heights of a collection of people would be distributed like this), you can say that there is a probability of 68.27% that any particular outcome would be within 1 standard deviation of the average. In other words, take a group of 100 people. You know that their average height is 5 ft 8 inches, with a standard deviation of 3 inches. You can say that, if you pick a person randomly from that group, then there would be a 68.27% chance of that person having a height between 5 ft 5 inches and 5 ft 11 inches. I will go into more detail on this on another day. If you can't wait that long, then Paul Marshall, TMF JonnyT, has written an excellent article on risk which touches on normal distributions.
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