One common finding in behavioural finance research is that investors under-diversify by holding only one or two shares. However, looking at our readers' portfolios, the opposite is often the case. It's common for older investors to hold 30 or more shares. Which poses the question: how many stocks should one hold?

Ultimately, this is a matter of taste. But it can be informed by the maths of diversification. This requires just three inputs.

The first is the probability that any individual stock will under-perform the market by a given amount. This can be derived from the share's tracking error, which is the standard deviation of annualized returns relative to the All-share index. For large blue chip defensives such as Glaxo or BAT, this tracking error is under 20. For more volatile FTSE 100 stocks or typical mid-caps – such as Arm Holdings or Bellway – it is around 30. And for more speculative shares it can be over 50; Circle Oil's tracking error (based on relative monthly returns in the last seven years) has been 59.4.

These numbers have a simple interpretation - the one-in-six principle. They imply that there is a one-in-six chance of the stock under-performing reasonable expectations by its tracking error. So, for example, if we expect Glaxo to perform in line with the market over the next 12 months, then we should expect that there's a one-in-six chance that it'll under-perform by 20 percentage points or more. Some simple maths can derive probabilities of other sizes of under-performance over other periods frm these tracking errors.

Our second input is the correlation between relative returns: if one of our shares beats the market, what are the chances that another will do so?

For randomly chosen pairs, the correlation is around zero; often, roughly as many shares out-perform the market as under-perform. Many portfolios, though, have a tilt towards a positive correlation. This can be because they are overweight in particular sectors. Or it can be because there's a bias towards a particular strategy - be it buying value, defensives, free cash flow or whatever.

The third input is simply the number of stocks you own.

The matters for a simple reason. The more shares you hold, the smaller is the contribution of any particular stock to your portfolio's chances of under-performing the market, and the larger is the contribution of the correlation element. If you hold one stock, you have a big chance of under-performing the market by a long way. If you hold all stocks, you're likely to perform roughly in line with the market, give or take different weightings.

My table summarizes the maths for three different types of portfolio. One is a portfolio of blue-chips, comprising stocks with a tracking error of 20 and a correlation of relative returns of 0.3. I've chosen a positive correlation because defensive stocks are likely to mostly under-perform in good times and out-perform in good. The second is a value portfolio with stocks on a tracking error of 30 and correlation of 0.1. The third is a speculative portfolio of stocks with a tracking error of 60 and zero correlation. I've chosen zero because the chances of (say) a small oil company striking oil should be unrelated to the chances of another small company winning a big order or being taken over.

**Calculating a portfolio's tracking error**

No of stocks | "Blue chip" | "Speculative" | "Value" |

1 | 20.0 | 60.0 | 30.0 |

5 | 13.3 | 26.8 | 15.9 |

10 | 12.2 | 19.0 | 13.1 |

20 | 11.6 | 13.4 | 11.4 |

30 | 11.4 | 11.0 | 10.8 |

40 | 11.3 | 9.5 | 10.5 |

You can see that, in all cases, the portfolios' tracking error falls sharply as we move from one to 10 stocks, but falls less sharply thereafter. You can also see that a portfolio of 20 uncorrelated speculative stocks has about the same tracking error as five defensives which are slightly correlated with each other.

To interpret these numbers, let's assume you expect your portfolio to beat the market by five percentage points in the next 12 months; much more than this and you're probably over-confident, much less and there'd be little point holding the stocks in the first place. If your portfolio has a tracking error of 12, then the chances of under-performing the market is a 5/12 standard deviation event. Statistical tables tell us that this translates into a 34 per cent chance. The chances of it under-performing by 10 percentage points are a 15/12 standard deviation event, which translates into a roughly 10 per cent probability.

These numbers are symmetric. So you have a probability of 34 per cent of beating the market by 10 percentage points and a 10 per cent chance of beating it by 20 percentage points.

Whether these probabilities appeal to you is a matter of taste. If you think the probabilities of out-performance are too low, then you've probably over-diversified and should hold fewer stocks. If you think they are too high, then you should add more stocks – either ones with low tracking error and/or ones that are uncorrelated to your existing holdings.

The point is, though, to be aware that adding stocks to your portfolio tends to dilute relative returns. And this is especially true the safer is the share and the lower the correlation of shares in your portfolio.

#### THE MATHS

To calculate your portfolio's tracking error, it helps to work in variances, which are simply the squares of standard deviations.

You then perform the following steps:

1. Estimate the weighted average variance of your relative returns. Then divide this number by the number of shares you own.

2. Estimate the average correlation between stocks. Then multiply this by the average variance of your stocks. You then multiply this number by a fraction whose denominator is the number of stocks you hold and whose numerator is one less than that number. So if you have 10 stocks, the fraction is 9/10. If you have 20 it is 19/20.

3. Add the number from step 1 to the number from step 2. Then take the square root of this and you have the tracking error.

For example, say you have 10 stocks with an average tracking error of 25. Their average variance is therefore 625. Dividing this by 10 gives us 62.5. This is step one.

If the average correlation between stocks is 0.2, then we multiply this by the variance to get 0.2 x 625 = 125. We then multiply this by nine-tenths. Step two thus gives us 112.5.

Adding step one and step two's results gives us 62.5 + 112.5 = 175. The square root of this is 13.2. Your portfolio's tracking error is therefore almost half of the tracking error of the average stock in it.

**MORE FROM CHRIS DILLOW:**

Read more of Chris's comment pieces.

Chris blogs at http://stumblingandmumbling.typepad.com

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