# Abnormal Returns

If you thought this was a post about how to generate outsize returns you are partially correct. We show a way of splitting a return distribution into a well-behaved bell curve component and a nasty outlier component where the money gets made or lost (and the Efficient Market Hypothesis gets kicked on the backside).

We use the S&P500 Index and work on daily log returns. Using the closing prices of the index, we generate the time series {log[P(t)/P(t-1)]} where {P(t)} is the time series of closing prices. A histogram of the returns is plotted in Fig1 on which we superpose a histogram of “normal” returns – i.e. returns generated from a Gaussian distribution with the same mean and standard deviation as the actual returns data.

Fig 1: S&P 500 Returns – No Truncation (Source: Bloomberg)

As one would expect, the actual returns distribution is more concentrated in the middle (“leptokurtic”) and has more bars sticking out at the extremes (“fat tails”) than the Gaussian returns distribution. This makes it difficult to risk-manage mean reverting trading strategies (e.g. the theta-burn variety) constructed on the assumption of Gaussian returns: the probability of returns remaining within a given number of standard deviations for a given holding period is no longer as predicted by the Gaussian model. Similarly, break-out strategies are difficult to structure and risk-manage, given that large moves occur with a higher frequency than suggested by the Gaussian assumption.

It would be useful to be able to truncate the returns distribution – systematically take out outlier moves – so as to get a normal distribution. We could then view the actual returns distribution as a mixture of a normal distribution which holds x% of the time and another distribution which holds (100-x)% of the time. Using statistical hypothesis testing, we find x=90% for the S&P500 data used. This means that we expect the actual returns distribution to be Gaussian 90% of the time. We can then assert with a fair degree of confidence that 1 standard deviation of the truncated returns distribution represents 68% of the area under the (truncated) Gaussian curve. See Fig 2. Note that the 1sd of the truncated distribution is almost 40% lower than the 1sd of the actual distribution. This means that mean reversion strategies can be risk-managed more aggressively with distributional risk under tighter control.

Fig 2: S&P500 Returns – Truncated (10%) (Source: Bloomberg)

We plot the outlier distribution in Fig 3. These are the “breakout” days – expected to occur 10% of the time for the S&P 500 for the period analyzed. This means we would expect a breakout move with a median absolute return of 2.53% once every fortnight.  We thus have the ingredients in place to risk-manage a two week theta burn strategy.

Fig3 S&P500 Returns – Outliers (10%) (Source: Bloomberg)

We have chosen to fit a gamma distribution but other distributions could be deployed to tease out the key statistical characteristics of the breakout days.  Higher frequency trading strategies would seek to identify the microstructure of the outlier days (opening location, type etc.) so as to be able to ride the wave or step aside in time.

# US Treasuries – Comeback Kid

US Treasuries have been on a tear throughout 2014. Medium term bonds (proxied by IEF) have gained 7% in price terms and the long term bonds (proxied by TLT) have gained 24%. TLT in particular has entirely recovered from the steep decline in the second half of 2013, breezed past key resistance at 122 and is within sights of the post crisis high. See Fig 1. Is there any more juice left in the move?

Fig 1: TLT on a tear (Source: Bloomberg)

We would like to think so. First off, scaremongers notwithstanding, the US never came anywhere close to bankruptcy. The \$18tn Federal Debt is but a fleabite when one considers the oil and gas reserves owned by the US Government (in excess of \$200tn), quite apart from its ability to tax the most productive economy in the world.

Secondly, there is the US dollar. Fig 2 shows a graph of UUP – an ETF which replicates being long the US Dollar against a basket of liquid currencies. In the last quarter of 2014, the dollar broke out of a tight three year trading range and is powering towards the high from mid-2010. This is only partially as a result of the Federal Reserve’s indications that the tightening cycle will begin sooner rather than later. The main reason is that the other major Central Banks (Japan, Europe) are having to adopt very accommodative monetary policies in a bid to resuscitate their moribund economies. If the pace of QE accelerates in these areas and China follows suit as its economy weakens, the feedback loop is likely to be exacerbated and the dollar’s strength likely to continue well into 2015.

Fig2: US Dollar powering up (Source: Bloomberg)

Thirdly, dollar strength has unleashed powerful global deflationary forces, given that the greenback still remains the world’s preferred payments and funding currency.  Emerging markets with large dollar-denominated debts are feeling the pinch both as debtors and commodity producers. Fig 3 shows a graph of VWO – an ETF that tracks the performance of the FTSE Emerging Markets Index. There is price consolidation but no break-down as yet. However, the fact that the market has failed to take out the key 45 area despite several attempts, given that the higher time frame bias is up, suggests that we could be setting up for a fairly big move to the downside in 2015.

Fig3: Emerging Markets going sideways (Source: Bloomberg)

Fourthly, global deflationary forces are likely to ease inflationary pressures in the US and hence the Fed’s urgency to hike.  These forces may also have a negative impact on the profitability of US multinationals with knock-on effects on the US equity market, as investors shift to USTs.

This last point is particularly interesting. As Fig 4 shows, SPY – the ETF which tracks the S&P 500 has seen no significant (10% or more) decline over the last three years – every dip has been bought by investors believing that the QE-driven party is likely to last forever. It seems, given the strength of the US economy vis-à-vis the rest of the world, a bull market in equities is compatible with a bull market in long dated treasuries. However, one of them has got to give – and in the context of dollar strength, global slowdown and the fact that US firms are probably scraping the bottom of the efficiency barrel, our money is on the USTs reaching for the stars and the SPY breaking down for one heck of a ride.

Fig 4: SPY vs TLT – which will yield? (Source: Bloomberg)