At TrinityCapM Ltd., a fixed interest liquidity and reserve manager, such issues are fundamentally addressed and answered by a unique investment process developed over many years. The process has its roots in ideas generated and fostered at JP Morgan in the early 1990s by Peter Rappoport, Victor Filatov, and Alan Cubbon. Victor is now Chairman of TrinityCapM and Alan is Head of Research.

The initial stage of the process involves framing our investment teams' views of where markets are headed over a short-term horizon of typically three months. There may come a time when all our strategists agree on such things, but it does not happen often! How best then to reconcile the differing forecasts and confidences of such opinionated people? We have adopted a heuristic "Bayesian" approach whereby the disparate views (how strategists think) are moulded into return distributions (how statisticians think) in a way which seeks to utilise as much of each of our strategists' points of view as possible.

One illustration of this flexibility is what we call the "low-best-high" method. For example we may feel that our best guess for where 10 year US yields will be in three months' time is 4.5%. However, we may think it unlikely that yields would widen out much further, say to 4.6% at most. On the other hand, maybe the oil price and exported euro-gloom will begin to impact the US economy and its growth prospects, so that yields might even fall slightly to 4%, or lower. And let's say that we put a 90% confidence rating on yields, finally ending up somewhere within that 4 – 4.6% range. From just these four numbers we construct a distribution of yields and returns which we can then work with. We may additionally incorporate "fat tails" if we have explicit views on volatility or construct "bi-modal" distributions if we expect event risk; a real consideration with emerging markets, for example.

Nothing in our investment process assumes or requires that returns be normally distributed. This means that we can include options in our universe as easily as any other asset, which has powerful consequences, as we shall see.

From our return distributions, we typically generate 1,000 equally-likely scenarios in a way that preserves the historical correlation of returns between assets. Each scenario then gives us the return for all assets in our investable universe. Hence we can calculate the return of any portfolio derived from the universe in any scenario and, likewise, any benchmark portfolio. Thus we have a thousand pairs of returns which we can plot on an x-y chart.

Drawing the line y = x across this chart, we can see that in any scenario above the line the portfolio has out-performed the benchmark. In the scenarios below the line, the portfolio has under-performed: the size of the under-performance, or "shortfall", is exactly the vertical distance of the scenario's point below the line. The probability of shortfall is given by the proportion of points below the line and the "average shortfall" is calculated as the average of all those vertical distances, including zeroes in cases where the portfolio out-performed.

Our investment process is geared towards finding portfolios whose average shortfall relative to that target is as low as possible for any given level of expected return. These portfolios are calculated by an optimiser that draws out an efficient frontier in classical Markowitzian style except that our chosen measure of risk is different: we use average shortfall.

We then can analyse the behaviour of these portfolios in the individual scenarios as well as having their summary risk and return statistics. This is useful as it serves to highlight the dangers both on the upside – if we should be over-reliant on a particular asset or market – and on the downside. For example, we wouldn't want to adopt a strategy, no matter how impressive its expected return or shortfall, if there were "rogue" scenarios out there that, even at a thousand to one shot each, which would close the fund down if one of them came about.

This is where the ability to include options in the asset mix comes into its own. Our optimisation model can use these to ensure that in no scenarios does the shortfall exceed a limit of our choosing. At the cost of a small amount of option premium we can set a floor on certain asset returns and eliminate the threat from the rogue scenarios.

Options can also be important in boosting expected return. Our optimisation constraints allow the writing of covered options so we can generate extra basis points of return through selling calls on assets we hold where the premium we earn we believe outweighs the extra upside of the underlying asset return. Option strategies are most prevalent when the "low-best-high" distribution is skewed strongly in one direction.

Which brings us to the idea of risk-budgeting. Our optimisation model budgets the risks allocated to each part of our eventual strategy all in one go. That's exactly what an "efficient" portfolio entails. This may sound like 'old hat', and indeed this principle has a long and venerable history, but its application in the hedge-fund world is crucial. In some parts of the world hedge funds are known as "speculative" funds and for good reason; it is impossible to generate returns of 10% or so per year without speculative trades involving risky assets and/or appreciable amounts of leverage.

At TrinityCapM we invest in the conventional, most liquid markets, but we use futures, swaps, forwards, swaptions, and options on both bonds and foreign exchange thereon to achieve our target returns. In traditional global macro hedge funds the head trader or Chief Investment Officer instinctively knows what the right allocation looks like and allocates risks across his team of traders responsible for key strategies. We believe that in these days of derivatives and leverage it is impossible to judge the optimal amounts of exposure to individual assets or strategies without some financial engineering and help from technology.

To illustrate, consider the extract in Table 1from a Euro-only frontier of a low-volatility hedge fund with a target return of Libor + 3% per annum. Each column in the table corresponds to a portfolio starting with Libor and then moving from the low-risk end of the frontier on the left to the higher-return end on the right. The numbers in the upper rows are summary statistics for each portfolio and are quarterly percentages with the exception of the value-at-risk, which is a daily 95% threshold number. The numbers on each of the lower rows correspond to percentage weights allocated to each asset.

The lowest risk portfolio is marked "A" where for each notional €100 million cash this would be overlaid with a €483m short two-year swap position, a long €359m three-year swap, etc. To expect even the best trader to come up with such allocations without an optimisation model would be completely impractical, if not impossible. But the model tells us "this is how fast the Red Queen should drive her Porsche."

You can see that portfolio A also has a very low value-at-risk at 10 bp of daily VAR. Downside-risk measures don't supplant volatility-based measures such as VaR and volatility, they supplement them. Indeed, there is a mathematical relationship between average shortfall and volatility in the case where returns are assumed normally distributed, such that the same portfolios constitute the frontiers based on those risk measures in both cases, though the efficient parts may differ. Therefore, unless we are dealing with portfolios that include large holdings in options, or other assets whose return distributions are far away from normal, we tend to see portfolios on the frontier that are simultaneously low-average shortfall, low-volatility, and low-value-at-risk.

Suppose we are considering introducing a second component into our strategy: a relative value trade where we short the SEK 5 year swap and go long the EUR 5 year swap. This is a relative flattener trade versus the Euro area where the driver is a belief of no further monetary easing in Sweden over our horizon, but a decent chance of a rate cut by the ECB. Is this trade worth putting on? How much of it should we do? To answer these questions we re-optimise using three new "assets": portfolios A and B as above, and portfolio C consisting solely of cash plus the relative value trade. The new efficient frontier is shown in Table 2 with the minimum risk being achieved through a combination of portfolios A and C in the proportion 81% to 19%.

Such an approach is no Holy Grail in boosting performance. However, it demonstrates that a talented team of portfolio managers that is good at making macro bets on markets and can devise creative relative value trades needs to be supported by a quantitative framework that efficiently splits their overall risk budget between strategies and trade ideas. Relying on arbitrary risk allocations to smart traders is not necessarily the most sensible approach. Running the portfolio at maximum VAR entails obvious dangers.

Whilst we are not recommending that other funds adopt our investment process, this is our way to mitigate unhealthy performance drawdowns when achieving client return objectives. Nevertheless we do believe that some kind of quantitative risk budgeting model is an essential component of any hedge fund investment process that should be found somewhere in the fund's high-performance toolkit.

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## Commentary

## Issue 132

## The Convexity of Trend Following

## Protecting your assets but perhaps not as much as you would like!