One common mistake investors make is to over-diversify. They hold so many stocks and funds that they end up with what is in effect a tracker fund, except that they pay more in management fees and dealing charges than they would on such a fund – and spend more time on it too. This mistake is easy to make. To see why, let’s look at the maths of diversification.
Imagine you own just two stocks. Your chances of beating the market then depend upon two things. One, obviously, is the chance of each stock doing so. The other is the covariance of relative returns between the two stocks; if one underperforms when the other beats the market, the chances of your portfolio beating the market by a lot will be smaller than they would be if both stocks move in the same direction.
This is obvious. What should also be obvious is that as you add more stocks, the contribution of the individual shares to your portfolio’s performance diminishes, while that of the covariances increases.
We have an equation to describe this. It says that if you hold (say) 20 stocks, then the variance of the relative return on your portfolio is equal to one-twentieth of the average variance of your shares’ relative returns, plus nineteen-twentieths of the average covariance of relative returns.
An example will clarify this. Let’s say the average variance of relative returns of each of your shares is 900. This corresponds to a tracking error of 30 percentage points, which means there’s a one-in-six chance of the stock beating the market by 30 percentage points or more over 12 months. This sort of variance is typical of the smaller FTSE 100 or more secure mid-cap stocks. And let’s say the average covariance of relative returns is zero. This means that if you take each pair of stocks in your portfolio then if one stock in that pair beats the market, there’s a 50:50 chance the other will too.
Our equation tells us that the variance of your portfolio’s returns relative to the market is 45: one-twentieth of 900 plus nineteen-twentieths of zero.
It’s easier to see what this means if we take the square root of this, which is 6.7. This is your portfolio’s tracking error. It tells us you have a one-in-six chance of beating the market by 6.7 percentage points or more over 12 months, and a one-in-six chance of underperforming by that amount. To put this another way, there’s a two-thirds chance of your annual returns being within 6.7 percentage points of the market’s.
It's for this reason that some funds are often accused of being closet trackers, with returns similar to the market. This can happen not (just) because the fund manager is a coward, but simply because of the maths of diversification: holding lots of stocks diversifies away their relative performance and so produces returns that are similar to the market’s.
Now, here’s the thing. Let’s call this portfolio a fund. And let’s say you have 20 such funds, each of which has a 50:50 chance of beating the market if another does so – that is, your funds have an average covariance of relative returns of zero. The maths tells us that your portfolio of funds has a tracking error of 1.5 percentage points: this is the square root of one-twentieth of 45.
Your portfolio of funds therefore has a two-thirds chance of having annual returns within 1.5 percentage points of the market’s returns.
You have a tracker fund. Except that you’re paying 20 fund managers for the privilege. And those fees compound over time. Over 10 years, an extra percentage point of fees costs you around £150 for every £1,000 you’ve invested.
Brute maths, therefore, can easily force investors into holding what are in effect overpriced tracker funds.
My table shows calculations for other portfolios of 20 stocks or funds. It shows that if you have 20 funds assuming them each to have a tracking error of 10 percentage points then if these have an average correlation of zero, your portfolio will have a tracking error of 2.2 per cent; they’ll be a roughly two-thirds chance of your annual returns being within 2.2 percentage points of the market. If you have 20 blue chip stocks (each with a tracking error of 20 percentage points) then zero correlations among them deliver a tracking error of 4.5 percentage points. More speculative portfolios (with an average tracking error of the stocks of 50 percentage points) would have bigger tracking errors.
| Tracking error of 20-asset portfolios | |||
| Correlation | Funds | Blue chips | Speculative |
| 0 | 2.2 | 4.5 | 11.2 |
| 0.1 | 3.8 | 7.6 | 19.0 |
| 0.2 | 4.9 | 9.8 | 24.5 |
This table gives us some clue on how we might avoid having low tracking errors.
One way is to hold more speculative stocks – those with a high variance of relative returns. This is dangerous, though. Highly speculative stocks tend to do badly on average: Aim stocks, for example, have badly underperformed the All-Share index over the last 20 years.
A second possibility is to hold fewer assets. The fewer you hold, the less contribution low correlations make to your portfolio’s tracking error. This means placing more faith in your ability to pick good stocks or funds.
A third possibility is to ditch some funds while holding stocks. Doing so raises the average variance of relative returns of your portfolio, without forcing you into more speculative shares. It has the added advantage of saving you management fees.
A fourth option is to raise the covariance of relative returns above zero – to move to the second and third rows of my table. This means picking assets that are likely to do well at the same time.
Of course, everybody likes to think they can do this by picking good stocks. But there’s another way to do so – to back factors that pay off well on average.
Many of the better fund managers do this. They hold defensive stocks which tend to outperform over the long run at the expense of underperforming if the market rises sharply. Momentum stocks also tend to beat the market together – at least in sufficient quantities to raise the average covariance of relative returns. The same is true for cyclical stocks such as housebuilders and many retailers, which tend to rise together in good times and fall together in bad.
We can, therefore, avoid ending up with an overpriced tracker fund. But doing so requires care. In particular, we must judge stocks and funds not in isolation but also consider what they add to our existing portfolio. Covariances matter a lot, but it’s easy to neglect them.
