The Ideal Portfolio: managing risk

If you have a goal - typically required real total return within a specified timeframe - then as a rational investor you want to take the least risk achieving it. While UK shares may not yet be overpriced, they remain a volatile asset class that could fall sharply in value following a shock to the market. If such an event were to occur (and events in Europe or the Chinese sell-off could provide the detonator) and there was a long period until prices recovered, buying in at today's theoretically fair value levels would still mean getting caught out at the top with all the distress that entails. While this possibility cannot be eliminated, investors can mitigate the impact to their objectives by understanding how each of their holdings exposes them to potential losses.

Understanding risk models

With a theoretical basis in the work of the great German mathematician Carl Friedrich Gauss, the common assumption has been that investment returns are normally distributed as a bell curve of observations around the mean. This is why in modern portfolio theory volatility, as measured by standard deviation, is used as the expression of risk. The problem, however, is that markets are driven by human behaviour, which is influenced by a multitude of factors - so large movements in asset prices occur more often than a normal distribution would suggest. The obvious flaw of the Gaussian model is that, as very little probability is assigned to outcomes of a magnitude past three standard deviations from the mean (see diagram below), 'impossibly large' changes in asset prices are predicted far less frequently than they actually happen.

Black Monday in October 1987, the dot-com crash and the post-Lehman crisis, to name but three memorable periods of market turmoil, caused severe losses and portfolios heavily invested in shares would have taken a long time to recover their value. The most damning indictment of standard risk models is that they would have assigned so low a probability to each of these events that theoretically they shouldn't have all happened within a 21-year period. Despite this, due to a historical lack of computation power and for reasons of mathematical expediency, flawed models persisted. Nowadays, with the computational software to work with 'fat-tail' return distributions (ie where outcomes of extreme magnitude occur more regularly than a normal bell curve would suggest - this is known as leptokurtic distribution in statistical jargon) more widely available, the financial industry is moving towards increasingly sophisticated methods to try to quantify the potential impact when serious events occur in future.

The potential for single events that invalidate previous statistical understanding - or 'black swans' as they are popularly referred to - can never be removed. Despite this, using a tool that takes account of fat-tail distributions aids portfolio management by reflecting empirical evidence of the size and frequency of losses. By better allowing for levels of price volatility that are probable in real life, the number of genuine black swans (which by definition cannot be predicted from past data) is reduced.

Understanding how different asset classes and individual securities have behaved and been correlated in the past gives investors a better idea of the potential for losses the next time markets face challenging times. To demonstrate, I asked the US creator of a proprietary risk solution to put strategies based on Investors Chronicle's Ideal Portfolios through their paces. PrairieSmarts is a start-up company that has developed a software solution called RealRisk™ to put a monetary figure on the 'worst case' potential losses of a portfolio and flag which holdings pose the greatest risk to long-term goals.

To recap, the three hypothetical UK Ideal Portfolios have different mixes of major asset classes. The cautious strategy holds 20 per cent shares, 60 per cent gilts, 5 per cent gold and 15 per cent cash. For the balanced portfolio the ratio of shares to gilts to gold to cash is 50:40:5:5 and for the adventurous strategy the allocation ratio is 70:20:5:5. PrairieSmarts ran US equivalents, using ETFs that track the S&P, US treasuries and gold. So, while different to our portfolios, they are certainly useful for demonstration purposes. An additional adventurous portfolio was also run using individual stocks rather than an index tracker for the domestic (US) equity allocation and using ETFs to internationally diversify some of the equities exposure in other developed and emerging markets. An ETF was also used to invest in US real estate.

Reporting risk in our ideal portfolios

For each portfolio style, a starting pot of $400,796 was assumed (this was equivalent to £250,000 when the reports were run). The key metrics that the reports delivered were the risk of loss on a really bad day as a dollar figure and as a percentage of overall holdings; the expected loss on an average 'down day'; and the diversification index of the portfolio, ie how the co-movement of securities reduces the sum of individual risks of all holdings. Before looking at how this information can be used to inform asset allocation decisions it is worth examining what each figure means in more detail.

■ 1. Risk of loss

In planning a portfolio strategy the likelihood of sustaining large losses is an obvious consideration. The value at risk (VAR) is an estimate of losses that are equal to or above a certain amount at a given confidence level. Taking our more complex adventurous portfolio here as an example, at the 99.5 per cent confidence level the VAR is -1.61 per cent. In other words, there is a 0.5 per cent estimated probability of a daily return being -1.61 per cent or worse. Based on our starting portfolio value of $400,796 this implies a loss of $6,467 if such a bad day occurred right away.

VAR is, however, an inadequate measure as it is based on the standard Gaussian model and therefore severely underestimates tail risk. Furthermore, there is no indication of the severity of losses past the VAR number; in our example no daily loss worse than -1.61 per cent is quantified. The theoretical solution to this second problem is to use the expected tail loss (ETL), also known as the conditional value at risk (CVAR), which is the average of returns that exceed VAR. This would be the mean of all returns worse than -1.61 per cent in the example.

To attempt a better appreciation of tail risk, ETL can be calculated from the average negative returns beyond a figure known as the modified value at risk (MVAR). The purpose of MVAR is to "correct" the ordinary VAR by including skewness and kurtosis of the returns distribution in its calculation (see 'Magic Moments' below).There are, however, drawbacks to this method too; academic studies have questioned both the popular implementation of MVAR (Maillard, 2014) and its accuracy below a confidence level of 95.84 per cent (Cavenaile & Lejeune, 2010).

The PrairieSmarts model is different in that, rather than using past fat-tail distributions to predict volatility, it works on the premise that the instant volatility of asset prices is random. So, instead of looking to model trends, the PrairieSmarts approach is to assign a probability that volatility on any given day is of a certain value. This takes into account that returns are not normally distributed, using PrairieSmarts' own fat-tail distribution model which makes explicit adjustments for sampling errors, but avoids the inherent (and wrong) assumption of trending models that markets don't suddenly move from high to low volatility or vice-a-versa.

While no method is perfect, the PrairieSmarts system has matched the empirical returns distribution of the S&P 500 index more closely than other models. Estimating the most extreme parameters that could have generated the returns distribution, the system gives a more realistic appreciation of risk. For the complex adventurous portfolio, PrairieSmarts calculates the tail risk at 99.5 per cent confidence as 3.05 per cent. So, if that was to occur right away from the starting point of the $400,796 invested, the monetary value would be $12,225 - a truer reflection of the magnitude of losses investors might face than the lower figure given by the standard Gaussian VAR measure.

■ 2. Expected loss

The expected loss is simply the average of all the down days the portfolio has experienced since January 2006 times the probability of having a loss day. In the case of the complex advanced portfolio, the model predicts an average loss of 0.21 per cent of total funds on days the stock market falls.

Compared with the average gain on good or 'up' days, we can see that the overall reward-to-risk ratio of the complex advanced system is 1.169, which is better than for the S&P 500 or a benchmark of 60 per cent S&P 500 and 40 per cent US treasuries.

■ 3. Diversification index

The PrairieSmarts model computes hypothetical returns, based on every data point available back to January 2006, and analyses them as a single time series to estimate risk as a monetary figure. The same is done for the individual securities held. The diversification index is:

1 - (Overall portfolio risk/Sum of risks for the individual securities)

The diversification index for the complex adventurous example is 0.4, which means that the co-movement between securities is decreasing risk by 40 per cent overall.

Portfolio Management: matching risk to objectives

We've already seen that on a really bad day (probability 0.5 per cent), the complex risk portfolio might lose 3.05 per cent of its value. The worst drawdown (maximum drawdown refers to peak to trough falls in value) since January 2006, occurring between May 2008 and March 2009, was -24.8 per cent. The trade-off for this risk was an annualised total return of 8.17 per cent for the portfolio. This absolute performance was not only better than a 60:40 benchmark of the S&P 500 and US treasuries (which would have achieved 7.38 per cent annualised returns), crucially we can see from the table below that the more diverse portfolio takes lower risk in terms of downside on bad days for the S&P 500 (see RealRisk figure), has a better reward-to-risk ratio, and suffered significantly less drawdown (-24.32 versus -34.17 per cent in 2008-09).

Clearly, for an investor who had medium to long-term horizons (ie somebody who was happy to stay invested and ride out the period of heavy drawdown in 2008-09), the complex adventurous portfolio was a good strategic asset allocation. As well as beating its benchmark and the S&P 500 index, we can see from the table that this particular portfolio has done better than either the standard adventurous, or the balanced portfolio in terms of performance and with less risk. The cautious portfolio is less risky but returned a lower 5.22 per cent. If one had a shorter time horizon, the conventional wisdom goes that the cautious strategy is best. However, drawdown for cautious was still -12.12 per cent in the financial crisis, so actually it might have been better to avoid strategies including equities altogether, if not investing for a longer period of time.

As well as the risk of losses in nominal terms, the other fundamental risk for investors is that of inflation eroding the value of money. With a target amount to achieve by the end of a longer period of time, allowing for inflation, a cautious portfolio might fail to deliver the necessary returns. Assuming target annual returns of 4 per cent, the Sortino ratio, which measures positive performance relative to historic below-target returns, is highest (best) for the complex adventurous portfolio at 0.040 versus 0.025 for ordinary adventurous, 0.028 for balanced and 0.020 for cautious.

Providing a sense check for holdings

By looking at a few key metrics, we can make informed decisions on portfolio strategy to take the least risk in pursuit of objectives. It is also possible to see how individual holdings contribute to overall risk. Returning to the complex adventurous portfolio as an example, the table below shows the range of motion of the holdings and the amounts of downside and upside that each investment exposes the portfolio to.

Range of motion is shown by the upper and lower parameters for expected percentage returns at the 99.5 per cent confidence level. When looking at individual portfolio holdings, it is useful in conjunction with a measure called the 'point of ruin'. The POR figure is defined as the percentage portfolio loss that is 'unacceptable', meaning the level of negative return that would be seriously detrimental to investment objectives. In the example, a -5 per cent loss for the whole portfolio has been set as unacceptable.

The point of ruin for individual holdings is the fall in value that a security would have to experience (assuming the prices of all other holdings stay the same) to make the portfolio lose 5 per cent. The value of stock in Apple (AAPL) for example would have to fall by 97 per cent to inflict a 5 per cent loss on the complex adventurous portfolio. So how likely is this? Well, as the range of motion shows, even allowing for fat tails, the size of daily movements in Apple's share price has been between -9.07 per cent and 10.21 per cent but as a price drop that could cause a 5 per cent loss for the portfolio is mathematically possible, it is highlighted.

Of course, in reality if the biggest company in the world were to take a 97 per cent tumble in value it is almost inconceivable that the rest of the portfolio would stay the same. The POR measure is purely theoretical but can be a useful flag if it lies within the range of motion (ie between -9.07 and 10.21 per cent). Were a portfolio to hold enough Apple stock that a change of say -8 per cent would lead to a 5 per cent overall loss, then the PrairieSmarts model would flag a 'concentration alert' against the holding. As the -8 per cent loss falls within the stock's range of motion based on fat-tail analysis, it would be a significant risk to the goals of the portfolio and action would be needed to reduce it proportionately.

At the tactical level, being able to visualise how adding or removing holdings affects risk provides a useful sense check. Returning to the last Market Tactics report, an investor may want to adjust his or her allocation to UK equities. UK shares may be fairly valued, but investors looking for greater returns might choose to proportionately increase exposure to overseas markets if these are looking cheaper. A model that gives a reliable appreciation of how portfolio adjustments affect the risks taken in pursuit of returns can be a powerful tool in the hunt for true alpha and we will be looking to do just that with the UK-based ideal portfolios soon.

Portfolio risk comparisons

Source: PrairieSmarts 2015

Complex adventurous concentration alerts - portfolio run 24 February 2015

The table shows that the points of ruin are well outside of the range of motions for the respective holdings; this is a sign that the portfolio is well diversified