The answer depends upon just three things. One is the volatility of the relative performance of the stocks you hold: this determines how likely they are to out-perform a lot. The second is the correlations between your shares. If all your stocks do well at the same time, you are more likely to beat the market than if some do badly when others do well. The third is the number of stocks you hold. This matters because the volatility of your relative returns, and hence your chances of beating the market by a particular amount, is a weighted average of the volatility of your shares and the covariances between them. The more shares you own, the less important will be their volatility and the more important will be the covariances between them.
Let’s put some numbers on this.
To calculate the relative volatility of your individual stocks, you simply take its returns, deduct returns on the All-Share index and measure the standard deviation of what’s left. For big household name stocks, annualised relative volatility will often by under 20 percentage points, which means it has a one-in-six chance of beating the market by 20 percentage points or more over a 12-month period. For more speculative shares, it can be well over 50 percentage points; Falkland Oil & Gas’s annualised relative volatility – to take an example of a popular Aim stock – has been 56.8 percentage points since the start of 2005.
Our second variable is the covariance between relative returns: the covariance is simply the standard deviation of two stocks multiplied together and then multiplied by the correlation between their relative performance.
If we take two shares at random, the covariance of relative returns will often be zero, because if one beats the market there is a 50-50 chance the other will underperform. However, in practice, the stocks we own might well be positively correlated with each other because of one or more of the five Ss:
■ Sentiment. If investors’ sentiment improves many risky speculative shares will beat the market at the same time. And if sentiment deteriorates, lower-risk shares will outperform together.
■ Size. Market indices are weighted by shares' capitalisation. This means that if big stocks do well, the market will rise even though most stocks will underperform. And if big stocks do badly, most stocks will outperform. This causes the relative performance of smaller stocks to be positively correlated.
■ Skill. Whereas a bad golfer sprays his shots all over the fairway, a good one is more consistent. The same’s true for stock-pickers. A good one should see most of his holdings do well.
■ Sector. Stocks from the same sector are more likely – though not certain – to rise and fall together.
■ Style. Stocks will similar characteristics – value, momentum, low volatility and so on – will be correlated. This means that insofar as investors favour a particular style, their holdings will be correlated.
My first table gives an idea of how these factors generate an annualised volatility of relative performance for different portfolios of 20 assets. By 'funds' I mean those investment or unit trusts which have an annualised relative volatility of 10 percentage points; small cap funds will be more volatile than this, but general UK equity funds will be less so. By 'blue chip', I mean shares with a volatility of relative performance of 17 percentage points; this is the case for Glaxo and National Grid since 2005 but it also a reasonable description of general international equity funds. By 'middling', I mean stocks with a volatility of relative performance of 30; this describes Antofagasta, for example, and many second-liners. By 'speculative', I mean shares with a relative volatility of 60 percentage points; this applies to Aim stocks such as Falkland Oil & Gas and the more cyclical main market stocks.
| Annualised relative performance volatility for 20-stock portfolios | ||||
| Correlation:- | 0.0 | 0.1 | 0.2 | 0.3 |
| Funds | 2.2 | 3.8 | 4.9 | 5.8 |
| Blue chips | 3.8 | 6.5 | 8.3 | 9.8 |
| Middling | 6.7 | 11.4 | 14.7 | 17.4 |
| Speculative | 13.4 | 22.8 | 29.4 | 34.7 |
The numbers in the table have an easy interpretation. Assuming you have no stock-picking skill, you have a one-in-six chance of beating the market by this amount, and of underperforming it by this amount. So if you have 20 stocks of middling relative volatility and a zero correlation between them, you have a one-in-six chance of beating the market by 6.7 percentage points over a 12 month period.
I stress that I’m speaking here of the average relative volatility of your shares. This could be around 30 if you hold lots of second-liners or if you mix blue chips with speculative ones.
Note the first line in this table. This tells us that if you hold 20 zero-correlated funds, you have a two-thirds chance of seeing annual returns within 2.2 percentage points of the market’s returns. This is why I say portfolios of funds are often like closet trackers. There’s nothing wrong with tracker funds – and plenty right – but there is something wrong with them if you are paying higher charges.
My second table shows how your portfolio’s relative volatility declines as you add more stocks to it.
| How performance volatity falls with number of holdings | ||||
| No of holdings: | Funds | Blue chips | Middling | Speculative |
| 5 | 5.3 | 9.0 | 15.9 | 31.7 |
| 10 | 4.4 | 7.4 | 13.1 | 26.1 |
| 20 | 3.8 | 6.5 | 11.4 | 22.8 |
| 30 | 3.6 | 6.1 | 10.8 | 21.6 |
| 40 | 3.5 | 6.0 | 10.5 | 21.0 |
| Based on correlations between holdings of 0.1 | ||||
Whether all these numbers are big or small is to a large extent a matter of taste. The question of how much you want to take on the chance of beating the market – or of underperforming it – is a matter of personal choice. The point is to ensure that the composition of your portfolio fits the amount of relative performance risk you are willing to bear. If you’re not willing to risk underperforming the market, for example, you shouldn’t be holding small portfolios of second-line or speculative stocks. Maybe you can beat the market – but you can’t beat the maths.
