One pitfall of looking at investment performance in the rear-view mirror, is a tendency to gloss over bad times and focus on total return over the holding period. Invariably, the distress felt when markets do plummet is not fully imagined from historic data, especially when (as is the case with shares) past gains are impressive over the long term.
Understanding capacity to absorb a big loss is one of the starting points in deciding investment strategy. A sensible way to look at this is in terms of the size and frequency of losses a portfolio can be potentially exposed to, given the timeframe the investor is hoping to be rewarded in. For example, a person in their 20s could, in a bad year, probably bear a 25 per cent drawdown in the value of their pension fund, knowing it has 40 years over which to meet its target annualised return. If the objective is something more immediate, like buying a house in five years, then such a loss is unacceptable.
Therefore, an investor needs to consider not only their required future return but also the level of risk they are prepared to take. When plotted on a graph, the line depicting the trade-off at different levels of absolute risk tends to slope upwards. In theory, an investor should be prepared to hold a portfolio represented at any point along this line, so it is known as their 'indifference curve'.
Indifference curves and Modern Portfolio Theory
Modern Portfolio Theory (MPT), as pioneered by Harry Markowitz in 1952, presumes that rational investors demand higher compensation for greater risk. Through diversification it is possible for a portfolio to suffer less volatility per unit of positive return and the goal of the investment manager is to select the most efficient portfolio, where the expected return per unit of risk is highest.
The trade-offs for all possible portfolios, when plotted, forms a parabola on a graph which is commonly referred to as the Markowitz bullet. The upward slope of the parabola represents the 'efficient frontier' of the investment opportunity set, where portfolios have a higher expected return than any other combination with the same level of risk. In MPT, the optimal portfolio for an investor exists where there is a point of tangency between their indifference curve and the efficient frontier.
Risk-free rate of return
Building on Markowitz's work, the capital asset pricing model (CAPM) designed by William Sharpe and others, introduced the concept of a risk-free rate of return (usually the yield on three-month treasury bills). Separation theory is concerned with combining the risk-free rate with the risky opportunity set to achieve financial targets.
On a graph, the Capital Market Line (CML) depicts how the availability of a risk-free rate increases possibilities for an investor. The ability to combine this asset with the riskier opportunity set means investors can invest along the CML, meaning more moderate financial objectives can be pursued with less overall risk. In effect, the point of optimal utility for the investor now exists along the CML, and their indifference curve moves upwards and to the left.
Drawbacks with traditional theory
Ultimately, all models suffer for relying on past data to estimate future returns. The demise of the defined-benefit pension scheme provides a cautionary tale. Up until the millennium, actuaries believed they could model based on annualised equity returns of 9 per cent plus. Sadly, the fallout from the subsequent disappointing performance of shares is still being felt, as employees face ever-dwindling pension entitlements.
Accepting that forecasts based on historical returns is an imperfect method, it is prudent to focus on the risk side of the investment trade-off. The importance of risk management highlights another problem with the CAPM, which is that it quantifies risk in terms of the standard deviation of asset returns from the mean. This causes it to understate the probability of serious falls in asset prices that are most unsettling to investors.
In spite of this, the CAPM is a useful starting point for choosing an asset allocation. There is a close relationship between the mathematics of standard deviation and the analysis of correlations between investments in the opportunity set. This means an asset allocation based on optimal co-movement between securities and standard deviation can be generated by one model. For a true appreciation of risk, however, it is necessary to use other, more powerful tools.
In March, Investors Chronicle used the CAPM to generate diversified portfolios from an opportunity set of 12 US-listed exchange traded funds (ETFs). The measure of portfolio efficiency derived from CAPM is the Sharpe Ratio, which is calculated by dividing a portfolio's excess return over the risk-free rate, by the standard deviation. Setting the model parameter to maximise the Sharpe Ratio, resulted in the selection of six ETFs (splits can be viewed in the table).
To test whether this portfolio really did offer superior risk-adjusted returns, it was assessed using software designed by PrairieSmarts, an American risk analytics firm. The PrairieSmarts algorithm looks at the non-normal distribution of asset returns and assigns probability to every parameter that could have generated them. While it cannot predict the magnitude of extreme losses in the future, it is designed to give a far more realistic appreciation of significant falls in asset prices, which are less unusual than a normal returns distribution would suggest.
Based on the probabilities assigned to returns on the worst 0.5 per cent of days, the PrairieSmarts software can extrapolate findings to estimate what the average portfolio loss would be in the worst of months. For our maximum Sharpe Ratio portfolio, it was predicted that the average return in a very bad month would be -5.07 per cent. To put this in perspective, using the more common Value at Risk (VaR) measure (which assumes a normal distribution of returns) the portfolio is predicted to lose -3.78 per cent, on average, in a bad month.
There is always the danger of a totally unforeseen occurrence, or 'Black Swan' event, altering statistical parameters. Still, it was reassuring that when using a more rigorous model, the portfolio allocations were still shown to pose only a moderate risk to capital. What the result did bring into question, however, is whether having a more accurate appreciation of risk can alter how we define an efficient portfolio.
Portfolio optimisation based on tail risk
Normal statistical distributions appear as a bell curve about the mean value on a histogram. Real life asset price returns display characteristics of being skewed and events of a significant magnitude occur more frequently than the normal distribution implies. On a histogram, this appears as a fat-tail of observations. Therefore, when trying to build an efficient portfolio, it is surely more prescient to think in terms of an optimal return relative to tail risk, rather than simple standard deviation.
In a tail-risk analysis the ideal portfolio allocations, in terms of the co-movement between asset prices, cannot be arrived at from the same model used to determine average probable losses. Therefore, a different approach is required in order to test which splits can deliver the highest overall return relative to tail risk.
The co-founder and creator of PrairieSmarts, Dr Ron Piccinini, has suggested a methodology for systematically testing combinations of investments. Just looking at the six ETFs used in the maximum Sharpe Ratio portfolio, Dr Piccinini selected thousands of random portfolios from possible weightings of the ETFs. The algorithm he used first appeared in a paper by Rowland R Hill in 1976, and it enables determination of the number of possible portfolios from an opportunity set of investments, provided there is a restriction on the size of incremental changes in allocation towards individual assets.
Dr Piccinini assumed a starting pot of $100,000 (£69,408) and that investments in each of the six ETFs would vary by $5,000 increments. (This approach also has the advantage of more practically sized allocations.) Having generated random portfolios, with different combinations of the six ETFs, on this basis Dr Piccinini tested the selections with the PrairieSmarts risk software. The results were very interesting, with five portfolio combinations available that, with a similar level of risk, offered higher returns than our initial maximum Sharpe Ratio portfolio.
A starting investment, in 2006, of $100 in the maximum Sharpe Ratio portfolio (risk in worst months -5.07 per cent) would now be worth $171. There is, however, a less risky and more rewarding opportunity using the same six ETFs. One combination with a monthly risk of -4.95 per cent would now be worth $180.
Other opportunities included a portfolio that would have grown the principle investment to $192 with a monthly risk of -5.43 per cent. To put that in perspective, assuming a capital value of $100, this is a loss of only 36¢ more than the Sharpe Ratio portfolio is expected to suffer, on average.
It is highly likely that a rational investor would consider risking 36¢ more in a really bad month, if the trade-off was a 92 per cent overall return instead of 71 per cent. This introduces a new question; how to systemically express that investors consider some monthly tail risks to be equivalent and they could show a preference for higher returns in such instances?
More work is required to come up with an answer to this quandary but, for now, it is useful to examine some other metrics for assessing the merits of different splits. The reward-to-risk ratio that PrairieSmarts uses is a simple ratio of the average expected daily return, compared with the average expected daily losses based on tail-risk analysis. Thinking about portfolio efficiency in terms of the highest reward per unit of risk, then there is not a great deal to choose between the splits analysed in the table, so again an investor may simply choose to go with the highest expected return.
The power of diversification is a cornerstone of Markowitz's work. The amount of tail risk reduced thanks to co-movement of assets in the portfolio can be assessed by dividing the expected tail loss (ETL) of the portfolio by the sum of ETLs for all individual holdings, and subtracting the result from one. In the case of the four optimal portfolio allocations suggested by Dr Piccinini's analysis, around half of the risk has been diversified away.
The methodology for calculating risk is cutting edge but, having confirmed that none of the portfolio choices display significantly different risk or diversification characteristics, a simple normative decision can be made on which selection to go for. Many investors will, in this case, probably go for the best return but crucially the starting point for their allocation decision will not have been chasing that higher rate; but rather a scientific assessment of what they actually might lose.
Portfolio allocations
| Portfolio | S&P 500 | Russell 2000 | 3-7 yr US Treasury | 7-10 yr US Treasury | Gold | US Real estate | Expected worst monthly loss | Growth of $100 since 2006 | Reward-to-risk | Diversification index |
| IC Max Sharpe | 15.41% | 1.07% | 53.31% | 21.99% | 1.81% | 6.41 % | -5.07% | $171 | 2.37 | 48% |
| ETL 1 | 10% | 5% | 40% | 30% | 10% | 5% | -4.95% | $180 | 2.39 | 54% |
| ETL 2 | 5% | 5% | 45% | 25% | 15% | 5% | -5.03% | $180 | 2.31 | 52% |
| ETL 3 | 15% | 5% | 15% | 55% | 5% | 5% | -5.09% | $184 | 2.33 | 49% |
| ETL 4 | 5% | 5% | 50% | 10% | 25% | 5% | -5.20% | $186 | 2.08 | 49% |
| ETL 5 | 15% | 5% | 15% | 45% | 15% | 5% | -5.43% | $192 | 2.28 | 51% |
Readers will note the apparent anomaly where, compared with the ETL 2 portfolio, the IC Max Sharpe Ratio allocation has a worse expected monthly tail loss and less impressive past growth, yet still has a higher reward-to-risk ratio. This is because the reward-to-risk ratio is based on expected gains and losses (in the statistical sense of the term), so it won't be a ratio of Historical Returns/ETL. Actually, this makes the figure more robust, because the historical return number can vary very rapidly depending on a few weeks of market action, but the average gains and losses, and the probabilities of gains and losses, would not change very drastically in weeks.
