How to pinpoint risk

In the past, flawed statistical models have prevailed for reasons of mathematical expediency (see box: Bell tolls for Gaussian assumptions). Therefore, the most sensible way for many people to manage risk has been by mental accounting: they view equity holdings as separate from other assets and sources of income, so are happy to pick stocks on the individual investment cases, as this is 'the risky stuff' anyway.

Mental accounting is perfectly valid, especially if it ensures mistakes aren't costing people their homes or money they depend on to get by, although thinking more scientifically about portfolio construction can help achieve the most efficient trade-off between risk and potential returns. As a caveat, no risk management system can ever be perfect and there is still no substitute for common sense. The first rule of investing should be not to put money on the line if an adverse outcome would prove ruinous.

Making the best estimate of how much you could lose

When it comes to applying risk-management techniques to existing portfolios (the process for starting a portfolio from scratch will be revisited in a forthcoming feature), the starting point is quantifying the size of losses with a given likelihood. In a recent Investors Chronicle portfolio clinic ('Overseas equity and fixed income could help meet your target') we used the software of US firm PrairieSmarts to estimate that our reader's portfolio (primarily UK equities) was risking on average £83,000 in the worst 0.5 per cent of months. Algorithms behind this calculation are based on the understanding that asset returns have 'fat-tail' distributions (see the box for explanation) and they give a far more conservative appreciation of risk than more widespread measures such as Value at Risk (VaR).

How useful is this risk figure? Put another way, there are 200:1 odds of losing £83,000 (roughly 19 per cent of the portfolio value) in a month; unlikely but not a possibility remote enough to dismiss. Of course, this is an average figure and there is a chance losses could be worse in such a terrible month, but taking account of fat tails makes for more realistic appreciation of what might befall the portfolio. By comparison, the Gaussian VaR suggests that in the worst 0.5 per cent of months the loss would, on average, be £46,000 or just over 10 per cent of the portfolio.

An investor may have constructed this portfolio on the basis that they are comfortable with a 200:1 chance of losing £46,000, but they might be less sanguine about the prospect of an £83,000 reversal being this likely. What's more, it follows that a loss of £46,000 is much less of an outlier than the investor assumes from the VaR measure. According to PrairieSmarts, an average loss of that size might be experienced in the worst 1.2 per cent of months. So a 10 per cent portfolio drawdown could be more than twice as likely as the VaR prediction.

More granular portfolio management

Using the same fat-tail distribution model and probability algorithm, we can also look at the average size of the biggest rises and falls that each individual holding might suffer. Switching attention to the worst daily losses, our reader's portfolio could lose an average of £15,900 on the worst 0.5 per cent of days. The sum of the total potential daily losses for all the individual holdings at the same confidence level is £25,600. Therefore, a simple calculation shows 38 per cent of the individual holdings' combined risk has been diversified away within the portfolio.

This diversification score is a useful indicator of how well the portfolio has been constructed. The reader who sent in this portfolio is very experienced and the score is testament to his sensible decisions to spread investments across sectors, with the largest allocations made to defensive stocks. Holdings are heavily concentrated in UK shares, however, which begs the question: could the diversification figure be improved if exposure to other less correlated assets was added?

Returning to the figures for extreme daily losses and gains, the disparity between outlying positive and negative returns for each security is known as its 'range of motion'. In other words 99 per cent of all daily returns should fall between these two values. This is a good supplement to just considering the volatility of a security and it has a use in terms of managing the size of individual holdings.

Our reader's portfolio with daily risk analysis

Source: PrairieSmarts

At this juncture it is worth reminding ourselves of the different layers of risk investors need to consider. The risk we can do least about is systemic - this is the possibility that the whole financial system could collapse, which did not seem far-fetched during the perilous few weeks of autumn 2008. Broad asset class (especially through direct ownership) and income diversification are the best protection but, just looking at portfolios of securities, for now our focus is on systematic and idiosyncratic risks.

Not to be confused with systemic, systematic risk is the market risk (or beta) of securities that cannot be diversified away. In other words, you can hold shares in two companies in different sectors, with unrelated operational concerns, but both will be affected by the market falling. This type of risk has in the past been mitigated by diversifying across asset classes - buying government and corporate bonds, overseas shares, private equity, real estate and holding cash.

Most basically, there is the idiosyncratic risk of holding an individual security. These include factors such as companies losing major contracts, poor management performance or accounting scandals. All of which will affect a company's share price but not necessarily other holdings in the portfolio. Returning to our portfolio risk analysis, the range of motion can be used to quantify the extent to which an individual holding risks overall objectives.

Concentration alerts

According to PrairieSmarts, our reader's portfolio could lose 3.6 per cent on average on the worst 0.5 per cent of days. As well as the overall risk, it is useful to look at how exposed the portfolio is to the idiosyncratic risk of particular shares and funds. This is done by setting a level of daily loss that is unacceptable and considering whether adverse moves in just a single holding render such a loss a) possible and b) likely given its range of motion. Assuming that our investor is prepared for the possibility of a 3.6 per cent loss but considers anything worse as unacceptable, then he should take corrective action if a security is likely to suffer a fall in value large enough to constitute a 3.6 per cent loss for the portfolio.

Take the largest holding, which is £45,000 of Unilever (ULVR) shares. Roughly 10 per cent of the portfolio, it is mathematically possible that a 36 per cent fall in the value of Unilever could cause an unacceptable loss. That said, based on the pattern of past returns and the probability of all the parameters which could have generated them, the worst expected loss for Unilever is much lower. On average, on the worst 0.5 per cent of days, little more than a 5 per cent reverse is expected from the defensive bellwether. As this would represent a portfolio loss of only around £2,250, or roughly half a per cent, there is no over-concentration in the stock from a risk perspective.

How does our reader perform versus the FTSE 100 on a risk-adjusted basis?

Benchmarking the portfolio against the FTSE 100, our reader appears to be living proof that it is possible to outperform the market and, crucially, take less risk in the process. PrairieSmarts' reward-to-risk ratio divides the average expected positive daily returns of a stock or portfolio by the expected average daily extreme tail risk. According to this measure, the portfolio has a higher risk to reward than the FTSE 100 when compared over a daily, weekly or monthly period.

Our reader's portfolio versus the FTSE 100

Source: PrairieSmarts

Interestingly, the figures tell us to expect more outperformance over one week and one month, than in a day. We know investing in defensive stocks works and these figures affirm the portfolio strategy. Versus the FTSE 100 on a monthly basis, the portfolio has a far superior reward-to-risk ratio (1.54 vs 1.09), which makes sense given the diminishing probability of the worst returns occurring on consecutive days.

We might even make a bolder hypothesis: the higher expected returns for the portfolio aren't rewards for taking on greater risk. Rather, the figures suggest constituents and weightings of this portfolio are just inherently more risk-efficient than the market capitalisation weighted benchmark.

How can the portfolio be improved?

For a UK equities portfolio, our reader has a well-diversified selection of shares with the proportionately bigger holdings in defensive stocks. The range of motion for individual holdings is such that it is unlikely that idiosyncratic risk will cause an unacceptable loss and the portfolio has a relatively low beta (market risk). So, how can we improve it?

Choosing risk-optimised strategic portfolio allocations is a big topic in its own right (to be tackled soon) but for now, looking at simple rule-of-thumb asset allocations makes for an interesting comparison. Theoretically, as it stands, the portfolio represents an aggressive allocation. Counting real estate investment trust (Reit) Primary Health Properties (PHP) as a property allocation, then roughly 83 per cent of the portfolio is in equities. Subtracting the overseas shares from the Scottish Mortgage (SMT), then UK shares make up 77 per cent.

Slightly increasing the cash allocation to 10 per cent, we can experiment with flexing the rest of the portfolio to invest a less aggressive 60 per cent in shares (three quarters in the direct UK holdings and a quarter targeting overseas equities); 20 per cent bonds (half in the SLXX exchange traded fund and half in a sovereign debt ETF); 5 per cent in property, through PHP, and adding a 5 per cent allocation to a gold ETF.

This is back of a napkin stuff - and is certainly not intended as advice - but does this intuitively more diverse allocation reduce risk? Running the modified portfolio through the PrairieSmarts software suggests that it does. In the worst 0.5 per cent of months, the system predicts losses would be just shy of £61,000, on average (the Gaussian VaR makes an estimate of about £33,500). This is a big reduction in risk (by over a quarter) compared with our reader's original portfolio. Although the trade-off is lower predicted returns our new selection is more efficient, with a reward-to-risk ratio of 1.59 also an improvement.

According to PrairieSmarts' diversification score, 47 per cent of the sum risk of holdings is eliminated, thanks to the lower degree of coincidence between security price movements in the portfolio as a whole. In keeping with much of the theory, we can ascribe the reduced risk and greater efficiency to diversification.

Our flexed diverse portfolio

Source: Investors Chronicle, PrairieSmarts

So, should our reader now go and flex his portfolio to match our new allocations? The short answer is no. Firstly, these allocations are based more on intuition and textbook theory than any current realities in the markets. Secondly, the type of risk we are focusing on here is peak-to-trough drawdown in the value of capital. This reader stipulated a need for income and the danger is that by adding exposure to assets like gold (which offers no income) and sovereign debt (where yields are tumbling) we would diminish the portfolio's capacity to deliver on this requirement.

The detailed risk analysis does, however, provide a more accurate sense check of a strategy's downside. It should also be possible to come up with more risk-efficient portfolios that are tailored to individuals' specifications. What is clear, is that in a low-return world, investors need to keep a very careful eye on more accurate risk measurements, as they are forced into more volatile asset classes to chase target returns.

Bell tolls for Gaussian assumptions in finance

German genius Carl Friedrich Gauss was responsible for much of the work on standard deviation and distributions that forms the basis for modern statistics. Gauss found that in a 'normal' distribution observations occur evenly, either side of the mean value. This plots a bell curve on a histogram, with the most common results within one standard deviation above or below the mean; more extreme statistics appearing within two, three or more standard deviations. This relationship has been shown to be empirically correct when it comes to the size of hills or human height but there are drawbacks in using Gaussian distributions to analyse security returns.

In practice, large drops in asset prices happen more frequently than a Gaussian distribution would suggest, yet most financial models still assume a bell curve for returns. This means that far too low a probability is assigned to major stress events in markets, a failing brutally exposed in the 2007-9 financial crisis. In models like Value at Risk (VaR), sizeable negative returns are assigned a probability of less than 16 per cent, or even less than 0.5 per cent, on the basis they are two or three standard deviations below the mean value.

A histogram of actual asset returns, however, is more likely to display a 'fat-tail' - with falls of a significant magnitude not uncommon and some serious drops proving not to be outliers. Recognising this failing, several academics have modified VaR to take account of leptokurtic (fat-tail) distributions. The problem is that these models, such as Modified Value at Risk (MVaR) or Conditional Value at Risk (CVaR), still rely on observing patterns from past results.

The PrairieSmarts system we have used to analyse this portfolio also has the unavoidable limitation of relying on past data. Unlike MVaR or CVaR models, however, PrairieSmarts uses a unique probability algorithm that estimates the likelihood of each parameter that could have generated those past returns. This innovation makes it accurate to the 0.5 per cent confidence level when back-tested, a significant improvement on MVaR which is accurate to roughly 4.5 per cent. So, while no method can predict the size of market falls perfectly, we can make estimates more confidently than ever before.