Keeping score

For once – and possibly the only time – Dickens wins it on brevity. In this feature we are dealing with nothing but facts; facts as they apply to assessing the performance of your investment portfolio, a matter that is, as Warren Buffett acknowledges, of enormous dollars-and-cents importance.

Knowing how well – or badly – your portfolio is doing is vital. Without those facts you are floundering. And, in assessing performance, there is no point in applying a flattering gloss. This isn’t a marketing exercise. This isn’t an investment manager or your stockbroker reporting to you. Nothing is gained if you fool yourself. The aim is to quantify portfolio performance. Without that information you can’t begin to understand what you have done well, what you have done badly; whether you are on the right lines, whether you are off track. Here are the simple mathematical tools – and more besides – to show how to assess the facts of your portfolio’s performance.

First, you need raw materials. Any statistical exercise – primarily what this is – needs stats. And the basic raw material is the value of your portfolio updated regularly. How often depends on your investment style and/or the nature of your portfolio. If the portfolio is full of illiquid assets – property, shares in private companies, say – then values may change only slowly and the valuation exercise will contain a wodge of subjectivity. In that case, valuations need doing probably only twice a year.

However, if your capital is exposed to lots of derivatives, stuff whose values are leveraged to underlying assets, then values will swing wildly and you need to know what’s happening day by day. In that case, this feature isn’t for you. That said, we might add the rider that if you don’t already understand the simple ideas we’ll discuss shortly, you probably shouldn’t be in derivatives in the first place.

As you will see, the data we are collecting come in monthly instalments, although the picture only becomes really interesting when we aggregate that into annual performance. If you’re starting out and a bunch of annual data seems remote so why obsess over monthly reckonings, don’t think like that. The seasons, they go round and round and soon the months become several years.

Monthly valuations are most suitable for the conventional sorts of portfolios that we generally deal with in the pages of this magazine – mostly quoted securities, although that can include bonds as well as equities; ditto collective funds, which are likely to have accessible daily valuations even if they are unquoted open-ended funds.

The other basic raw material that’s needed is a benchmark, which you will use as the touchstone to compare with your portfolio’s performance. Without that, assessments will linger in a confusing twilight. It’s all very well to know that your portfolio’s value rose, say, 10 per cent in 2017, but is that good or bad? It may feel good, but then you discover that the average performance of all the portfolios that were invested in the sort of assets that yours held rose 20 per cent. Suddenly you are a laggard. No fun, but it’s best you know that.

Hence the usefulness of a benchmark, which should – as implied – be a widely available index of the sort of assets that a portfolio is invested in. In the case of the Bearbull income portfolio – an in-house ‘shadow’ portfolio run by Investors Chronicle about which we have plentiful data – the benchmark is the FTSE All-Share index, a widely used and long-established index of the value of listed shares on the London Stock Exchange. That is appropriate because most of the Bearbull portfolio’s investments are – and have been – components of the All-Share index. The City of London Investment Trust (CTY), a £1.4bn fund whose accessible and abundant data are also used here, also benchmarks its performance to the All-Share. In addition, it monitors itself against the progress of similar investment trusts which are grouped into the UK Equity Income sector of the Association of Investment Companies, the trade body for investment trusts.

True, you could go with an absolute value as your benchmark – say, 8 per cent a year – which is what many hedge funds do. If so, you have to know why you are choosing a particular number and that it’s sensible. For example, on average the total return (changes in capital values plus income received) from the All-Share index has been 6 per cent a year since 2000. Would that be an acceptable benchmark? Maybe. But, if so, when the market was dicey you would need the mechanisms in place to prevent the value of your portfolio dropping too far below 6 per cent and the cost of those mechanisms – essentially, pricey insurance achieved via derivatives – would drag on upside performance, perhaps substantially. So, over the long term, you would be sacrificing extra investment returns for less volatility. Yet it is debatable whether there would be any point in doing that while you still have a long-term investment horizon. In the end you would arrive somewhere behind where you would have been except that you would have travelled by a smoother path.

Granted, those approaching the time when they need to start turning their portfolio into cash may want the reduction in volatility that training on an absolute target can bring. In which case there are low-volatility absolute-return funds available. However, matching fund performance against a benchmark index is right for most investors.

We are also assuming that investors have in place the means to calculate the monthly performance of their portfolio. For those wanting help, the Bearbull Portfolio Management Tool, a workbook of spreadsheets, is still available on the Investors Chronicle website as is a detailed article (29 May 2015) that explains how to use them.

And there is no doubt that the fun starts when there is lots of data to play with. The first task is to quantify how a portfolio is actually performing. Let’s take Table 1, which compares the monthly performance of the Bearbull Income Portfolio with the All-Share index over the 18 years 2000-2017.

The table shows three key pieces of information. First is the average return, which, for example, shows that over the 18 years the Bearbull portfolio has generated an average yearly capital gain of 7.1 per cent compared with 2.7 per cent from the All-Share. ‘Average’ here is just what the man in the street thinks of as ‘average’. It is the arithmetical mean – the sum of a series of numbers divided by – as it were – the number of numbers. For the Bearbull portfolio, the sum of the numbers is 127 and it’s divided by 18 (the number of numbers, or of years) which comes to almost 7.1. It is significant because it addresses the question, what is the typical return that a fund produces? And from that it peers into the future by answering the really important question, what is the fund’s expected return?

Still, expectations come in various shades, so how likely is that 7.1 per cent capital return or, say, the portfolio’s 11.5 per cent average total return? This is answered by the second key piece of information derived from the table, the standard deviation, a statistical tool that measures the average distance of all the numbers in a series of data from their average. The smaller the gap between the average and the standard deviation, the more useful the average as an indicator of expected returns for the coming year. Clearly, therefore, 7.1 per cent is a more useful predictor of the expected capital return from the Bearbull portfolio, where the standard deviation is 12.4 per cent, than is 2.7 per cent for the All-Share index, where the standard deviation is 15.6 per cent.

Statistical laws also tell us that there is a 0.7 probability that any year’s return will be within plus or minus one standard deviation of the mean. The box below deals with that in more detail, but, in practical terms, that means there is almost a 70 per cent chance that the Bearbull portfolio will return something between a 19.5 per cent gain and a 5.3 per cent loss this year. Or, taking the data for the Bearbull portfolio’s total returns, the range becomes plus 23.9 per cent down to minus 0.9 per cent. Put more bluntly, this also tells us that the chance the fund will return worse than nominal losses (after dividends) is slim (less than one in three); as for serious losses, say, worse than 14 per cent, don’t get alarmed – that will happen once every 50 years.

The third key piece of information is the growth rate. This is the compound rate of interest that took the Bearbull portfolio’s capital value from about £103,500 at the start of 2000 to £316,500 at the end of 2017. In other words, it is the smoothed interest rate that joins together two values separated by time and it assumes that the interest from one period is added to the capital value before calculating the return for the next period (that’s what ‘compounding’ is). If there were absolutely no variation between the rate of growth from one period to the next then the growth rate would be the same as the average return. But because in reality there will always be some variation then the growth rate will be lower than the average return and the more that the returns vary from period to period then the greater the gap between the average rate and the growth rate. Understand, however, that it is the average rate that changes, not the growth rate.

To sum up, the average rate is the statistic to use when estimating the most likely return for a forthcoming period, factoring in standard deviation to get a feel for the reliability of the estimate. The growth rate is the stat that describes the smoothed pace of growth over a number of periods. As such, it is backward looking. It tells us how we have done, but it says almost nothing about the likely return in one period.

Tables 2 and 3 – and Charts 1 and 2 – make explicit a fact of investing that was implicit in Table 1 – the importance of interest in investment returns and, specifically, the importance of dividends to equity returns. Knowing how much value comes from capital gains and how much from income received is a core part of portfolio assessment. Take the data in Table 2, which assumes that the Bearbull portfolio’s value and the dividends it had received (and distributed onwards) has been unitised in pence. Starting in 1 January 2000 with a unit value of 103.5p (top left of the table), the fund ended 2017 with a total unit value of 486.4p (bottom right), thus adding 382.9p of value. The table shows that 212p of the value added – or 55 per cent – came from growth in capital and 171p came from dividends received.

Study Table 3, which does the same exercise for City of London investment trust, and the added value is even more biased towards dividends. They accounted for almost 208p of the 357p added to total value per share in the period, relegating added capital value to the role of minority contributor.

As to why dividends are so important, it’s a bit like the tortoise and the hare. Dividends are the tortoise, trundling along at a slow pace, but varying little and becoming a significant contributor. Capital values are like the hare, bounding ahead some years but getting badly waylaid during others. True, the two funds under review here are both orientated to receiving and distributing dividends, so dividend income could be expected to be a bigger part of the whole and capital gains to be smaller. Clearly also, if a portfolio is wholly invested in equities that pay almost no dividends – a biotech fund, say – then capital gains must play the prime role. However, for all portfolios that chiefly hold shares in dividend-paying companies then the observations shown in Tables 1 and 2 will by and large hold good.

The connected point in this bit of portfolio assessment is the importance of minimising losses. That our data starts on 1 January 2000 illustrates this nicely, even though that particular start was miserable for the returns of the City of London trust. That was the year that UK equities, having suffered just two losing years in the previous 20, proceeded to fall three years running for the first time since the 1950s. Given its broad portfolio of UK shares, there was little that the investment trust’s managers could do to escape this trend so, by the end of 2003, its net asset value (NAV) was 31 per cent lower than it had been three years earlier and it was not until the end of 2006 that NAV rose above its starting value (see Table 3). Meanwhile, by the end of that year dividends paid by the trust had added 57p – 20 per cent – to its starting value.

Over all, however, the effect of falling UK stock markets was too much for the investment trust. Some of its losing years were steep (see Chart 2) and over the whole 18 years under review dividends paid contributed almost 60 per cent of its total value added, almost half as much again as that provided by capital gains.

Notice that in the previous few paragraphs we have studiously avoided using the term ‘total return’ when discussing the combined effect of capital gains and dividends distributed. That’s because what the fund management industry means by total return is different from how the average private investor would treat it. The essential difference is that, with retail funds, total returns are calculated as though dividends distributed were actually reinvested in shares or units of the fund. The arithmetic of such an exercise is shown in Box 2 below, so let’s confine ourselves here to a few comments.

Calculating total returns has meaning for the fund management industry because it is a practical proposition for their clients – the retail investors – to reinvest dividends into the funds they hold. Almost all funds offer reinvestment plans where dividends are used to buy second-hand shares in the market (in the case of investment trusts) and to buy new units issued at NAV (unit trusts). So some investors really do achieve the total returns that fund managers claim.

Besides, there are marketing considerations. As Box 2 shows, the effect of calculating total returns is to raise returns when there is a rise in share prices (for investment trusts) or in NAVs (for open-ended funds). Conversely, total returns fall when prices or NAVs fall. To the extent that the performance of investment funds tends to track their financial market and because markets rise more than they fall, then benchmarking by total return flatters performance. It makes returns higher than they would have been if capital gains and the yield on income paid out simply been added together. On that level, total returns may be a self-serving contrivance, but if one fund uses them, all the others must follow.

Meanwhile, for private investors assessing the performance of their own portfolio by calculating a total return along the fund management industry’s lines would be a waste of time. Sure, it could be done. Technically, a privately-run portfolio could reinvest the dividends received into new units that were issued at the portfolio’s prevailing asset value so as to avoid distortions. But what would be the point? Over the years lots of meaningless units would be created, but not a penny of value would be created or destroyed.

And there are more important matters to focus on, as Table 5 indicates. It is vital to know where your portfolio’s returns come from. That can have a profound influence on how you manage your capital. The point is that, almost certainly, returns will follow a so-called ‘Pareto distribution’. This is named after an Italian economist, Vilfredo Pareto, who gave the world what became known as the 80/20 rule. Basically, this says that 80 per cent of the effect of something will be generated by 20 per cent of the causes. Apply that rule of thumb to your portfolio and it tells you that 80 per cent of profits will come from one in five of your investments. So the other four-fifths will produce just 20 per cent of the profits between them.

Analysis of the Bearbull portfolio illustrates this neatly. From its start in September 1998 to January 2018, the fund generated £222,478 of capital gains on top of its starting capital. Table 5 shows just how concentrated those gains were. To be precise, rather than follow an 80/20 rule, the fund plotted an 86/13 course, with 86 per cent of its gains coming from just eight of the 64 holdings it bought. Not just that, but two holdings – Carr’s (CARR) and Henry Boot (BOOT) – account for 43 per cent of its gains.

The investment lesson this serves is that it is vital to run winning selections. Granted, we can’t know in advance which ones these will be so running them – leaving the gains to mount to something really significant – is about the good use of stop-loss tactics, which is outside the scope of this article. But it is only by the careful analysis of a portfolio, the accumulation and marshalling of facts that the big winners can be seen coming in the first place. As we said earlier, knowing the facts of your portfolio is of enormous dollars-and-cents importance.

How the laws of statistics help

How do we know there is a 70 per cent chance that the Bearbull income portfolio will return something between a 19.5 per cent gain and a 5.3 per cent loss this year? The short answer is because that’s the way it is; that’s what we can infer from the patterns that statistics throw up time and again. When a set of data is ‘normally distributed’ around their mean then 68 per cent of the data will be within one standard deviation of the mean and 95 per cent will be within two standard deviations.

‘Normally distributed’ means that the data points cluster evenly on either side of their average and then fade away – but still evenly – the further their values get from the mean. This is caricatured as the bell-shaped distribution curve that occurs in all sorts of sets of data, although technically only when each data point is independent from all the others – so ‘yes’ for sequences of tossing a coin, but ‘no’ for stock market returns because each is affected by what came before it.

Most of the time, however, that does not even apply to stock market returns, they look pretty normally distributed so expected returns derived from this branch of statistics work up to a point. The trouble is stored in the outliers, the data that occurs far from the mean. But how do we measure what is distant from the mean? For that, we need Z scores and – let’s stress – these are not the Z scores that forecast whether or not a company may go bust.

A Z score of 1.0 is one standard deviation from the average. That is calculated by taking the difference between an observation and the average of all observations then dividing the answer by the standard deviation. So the best total return that the Bearbull income portfolio has posted is 32.8 per cent back in 2004. That has a Z score of 1.72 – 32.8 minus 11.5 divided by 12.4. How likely are we to see that again? Not too often. The probability factor is 0.043; in other words, about once every 25 years.

The arithmetic of total returns

Total returns are calculated on the assumption that dividend income is used to buy more shares/units in the fund in question. The effect is that, when the share/unit price rises, returns are greater than they would have been if the dividend was taken as cash, but they are less when prices fall. Table 4 shows how.

Note first that the table does not reflect what actually happens because it assumes that the new shares, which contribute to the total return, are bought at the start of a year. In reality, shares are bought with declared dividends so they affect the subsequent accounting period not the current one. For illustrative purposes, however, the effect is the same.

So if an investment trust whose shares trade at 200p pays a 10p per share dividend, the proceeds would buy 500 shares for an investor holding 10,000 shares. That raises the shareholding to 10,500 whose value would be £24,150 by the end of the year if the share price rose to 230p, giving a total return of 20.75 per cent. Meanwhile, an investor who takes the dividend as cash generates added value of £4,000 – £3,000 in capital appreciation plus £1,000 in income – for a total return of 20 per cent. Of course the investor would also have the use of the cash, which could generate a return outside the fund.

True, the difference in returns does not seem that much. When, however, returns are compounded over many years in a rising stock market the effect is noticeable. Just suppose a share price rises by 20 per cent five years running. With dividends reinvested, the total return would be 51.3 per cent, but with dividends taken as cash the return would 49.8 per cent. Not much, but in marketing every little helps.

In a falling market, the reverse applies – reinvesting income penalises rather than rewards, as the table demonstrates. That said, the marketing people would point out that it means a given dividend buys more shares and therefore more exposure to the next upturn; that’s the virtue of what’s called ‘pound-cost averaging’.