## Friday, December 25, 2015

### Macro : The Year of Volatility (That Was Not)

2015 was supposedly the year of the comeback year of the volatility.

And no it did not, we had some occasional jitters, but mostly volatility in most asset classes, measured in standard manner, are in line with long dated averages, realized, as well as implied (say for example VIX for equity, Deutsche Bank CVIX for currencies and BoAML MOVE for treasuries).

But what we did see was a comeback of vol of vol instead. See the charts below. They plot rolling averages of number of weeks with 1.5x move (compared to recent weekly realized vols), on a longer range (2000 till date, left hand side) and a zoomed in version (2010 till date, right hand side).

What we have seen in 2015 is a definitive spike in days with large moves, in almost all asset classes, but with average volatility in line with long-run history.

Additionally there is a large divergence in this pattern across asset classes. Historically volatility tends to spike (following major events) across asset classes simultaneously. The current divergence we observed beginning in 2015 is unprecedented in terms of the degree.

This is quite different than the last hike cycle of 2004, perhaps driven by diverging monetary policies. Alternatively vol as an asset class no longer captures a single risk factor and relates to asset specific risks. Liquidity in rates? re-balancing in commodities? may be even HFTs in equities? - at this stage, we can only guess.

This means if this continues, those risk-factor based investors are going to find it difficult to generate consistent and commensurate returns as pricing gamma becomes more difficult. The upshot is for tactical and discretionary asset agnostic investors, who will have it easier to insure short gamma strategies (by selling the richest gamma and buying the cheapest in another asset class).

## Monday, December 14, 2015

### December Fed: A Study in Ornithology

That time when you can't really tell a dove from a hawk!

December Fed is mostly settled, with a good old 25bps rate hike priced in. However, the real question will be then what!

If it is just a 25bps hike, market will perhaps focus mostly on the subsequent press conference - for clues on future rate hike path. And I am certain the Fed chairman will make an extra effort to explain that bit. And if there is a hike, the press conference will be certainly dovish. I think the primary motivation for any hike at this point in time is more an effort to return to a resemblance of normality, than inflation worries. The Fed will definitely maintain that policy action works with a lag and this hike is just being on the cautious side. While that is true and the labor market is definitely showing early signs, the split in goods and services inflation across the globe makes it difficult for central banks, including Fed. And I doubt they have a very strong conviction on that.

We have no comparable history of this potential first rate hike in almost a decade. But we do have something close. The taper of QE program was announced in 2013, in a very similar manner. The market was mightily perturbed mid year, then the Fed backed off in September. Finally it went ahead with cautious taper in December and life went on (deja vu?). Below are some charts that capture how asset classes reacted during entire 2013 and especially around December 17-18 FOMC in 2013, compared to this year's move (click to enlarge).

The points to note here: the rates moved a lot more in 2013, and after taper tantrum, again sold off after the actual taper, till market realized around late January 2014 that taper is NOT tightening. Compared to that, we had quite feeble move in rates this year, despite it being possibly the actual tightening year! On equities again, the taper in December was quite a good risk-on move, but of course on P/E basis, S&P was much cheaper back then. And dollar, well, dollar sold off, mostly, in 2013 or in 2015. Note, this is Bloomberg DXY index, which given the correlations (CHF, SEK etc), is almost 65% Euro!

Now let's take a look at the same thing, in relative value terms (again, click to enlarge).

Well, now things do look a bit different. For one, in terms of excess yield (inverse of forward P/E less 5y yields), the S&P now and then are more or less in the same territory. Yes S&P was much cheaper in P/E terms then, but we also have to remember how much the rates rallied since. On the dollar chart, well it is more or less similar, except the recent sell-off looks a bit too sharp. This perhaps got little to do with dollar itself and mostly re-pricing of Euro after the Dec ECB. And finally, rates! Yes in general a start of the hiking cycle puts flattening pressure on the curve, but see the difference then and now. In fact the USD curve is so flat (despite the solid commodities thump) that some complain the risk premium (like calculated by the ACM method) is historically low compared to any other hiking cycle in history.

The key here is less about predicting what the FOMC will decide, and more about how the market will react to different outcomes.
• A no hike will be quite surprising and will almost surely have a negative outcome ("they know something we do not") - unless it is properly explained in the subsequent press conference. Risk-off rally in rates and sell-off in equities.
• The other outcome of 25bps hike will hardly be surprising. As mentioned earlier, then the entire focus will be on the press conference, which is most likely to be quite dovish on the future rate hike path. And overall this should mean a relief rally for equities. Mild risk-on for equities and rates will have limited response. Given the recent flattening, a very dovish press meeting will also mean some asymmetric positioning in steepener in USD rates makes sense.
• And another outcome, something in between - say a 15 bps hike. That will be really interesting. Given the charts above, it is highly likely the market will really understand and trust the Fed's point on slow pace of rate hike much easily with this outcome. And with a proper press conference it will be nowhere near "what do they know that we do not" risk-off argument. In a word, it will accentuate the risk asset reaction from the second point above. And a possible steepening in rates as well.
Trades Here: So overall looking from above, the probability weighted tactical positioning is:
• long equity (perhaps Dec expiry calls to leverage gamma to optimize payout ratio)
• convex steepening in USD rates in limited size (and/ or against Euro)
• long dollar to hedge above

## Friday, December 11, 2015

### Note to Self: December Fed

If you are looking for tactical positioning around Fed decision day, the major question you should ask yourself is what if they hike by less than 25bps which has been the standard. The market prices in a full hike. How it will react to a "half-hike" is what the most asymmetric positioning will be about.

More on it later.

## Monday, November 30, 2015

### Trade Idea: Positioning For ECB

The expectation for the next ECB is running high. Many from ECB, including Mr. Draghi, has already talked up further measures. There are already talks of telescopic monetary policy taxation in the air. And a lot of speculations to follow.

The ECB announced QE at the end of Jan this year. Then the market priced in QE aggressively as well, and yet ECB over-delivered. The continuation of the movement past the announcement and well past the start date is a proof of that. This time as well the market has priced in possibilities of further measures aggressively. The question is this time, will they deliver, or over-deliver?

The recent moves in the market in many way resembles the time period before the last QE announcement. And at the same time differs in some crucial details. The major similarities: 1) selling off in Euro, 2) rally in the front-end and major differences are 1) remarkable steadiness of the long-end and 2) steepening of the curve. In the figure below we show the excess moves in rates and slopes (relative to the US). The Blue columns are the move in the last QE. Red one current moves and the green columns shows what the current move should be if we adjust for the move in Euro (that is we assume the euro move correctly prices expectation and compare rates move based on that). As we can see the move in the front end much stronger than before, and reverse for the long end. In fact corrected for euro, the move in 10y is about fair. While 8th euro futures rallied most and 30y did much less than expected.

The bone of contention here is of course what exactly will the ECB do. As clear from the picture above, the market is fully or to a large extent pricing in an action in the front end, that is, a significant depo cut. And with all the stories of -20/-50 tiers or -35 flat or all other possible combination, it is hard to say what happens if ECB does a significant deposit cut. There is no reason to believe they cannot exceed expectation. So perhaps short end move is justified.

But here is the key. Whatever be the depo cut, it is in itself not important. It is plausibly true that the point of this depo cut is simply to make the QE program more tenable. With 15% of euro area govies trading below current depo, the ECB has a strong incentive. The question is if they do deliver, what does that mean for long end. It does not mean we have an increased supply, nor it means depo is reflationary. All it does it to save the QE program by making more bonds eligible. I have not checked for euro area, but based on Germany distribution of yields and amount outstanding, roughly a move from -20 to -35bps makes 84b more available. With a capital key of 18% that is ball-park 470b more papers to buy for ECB, approx. 8 months worth of QE. This is significant. But we also have to count in the feed-back response, as the market may potentially push the curve further down, and thus neutralizing a part of the impact the ECB hoped to create. So if this depo rate comes with any significant expansion of QE in terms of time or size, the long end should be biased for rally. And if the depo rate cut does not match market expectation, the short end will sell off back to previous levels.

And while we have all these, another point to note is the levels of vols. The implieds are way to high compared to delivered. But if we adjust for the Fed hike expectation (by computing implied/realized premiums in EUR over USD), the front ends are still cheaper compared to long end on a realized basis, with 5y around fair.

I believe whatever ECB does, it will hardly be a lasting change. Europe needs fiscal stimulus now. Monetary policy is just a tool to avoid falling behind, but can hardly give a large push ahead. Whatever the move follow the momentum, and then position for a fall back. ECB claims the QE has "clearly" worked, but the real rates in euro area were back at the 2014 levels at end of August, before the new QE expectation kicked in.

The trade here: A convex flattening position in 5s30s or 5s10s. If not through spread options, given the vol richness and underlying directionality, buying the belly payer vs. long end looks better than the alternative in a risk-reward consideration.  Otherwise a Nothing. Wait till the announcement and no point trying to fade the market from here.

EDIT (3-Dec-2015:08:55 UTC): The hidden risk to this view is the ECB doing away with the yield-floor limit for QE eligibility. That may lead to a large upward correction in long term yield and a significant steepening.

## Sunday, November 29, 2015

### Time Series Momentum Strategies | The Spirits Within - Part V

This is part of a series on time series momentum. Previous post on this are:

1. Part-I: Time Series vs Cross-sectional momentum
2. Part-II: Nature of linear time series momentum filters
3. Part-III: Types of sizing function
4. Part-IV: Strategy characteristics for random walk with a trend

In the last post we analyzed characteristics of a generic momentum strategy in case of an underlying following a random walk with a known trend. In this post we look in to the returns characteristics in case when the underlying is an auto-regressive process, specifically AR(1).

Once again, we assume the sizing function $\Psi$ is linear. Asset return $r_t$ is given by $r_t = \phi.r_{t-1}+\sigma.\epsilon$, where $\epsilon\sim N(0,1)$. Remebering that for linear function the expected return of the strategy is expected value of the signal times return, we get
$$E(R_t)=E(S_tr_t)=E(\sum_{s=t-k}^{t-1}(w_sr_s)r_t)=\sum_{s=t-k}^{t-1} w_s E(r_sr_t)=\sum_{s=t-k}^{t-1} w_s \gamma_s=\Sigma^2\sum_{s=t-k}^{t-1}w_s.\phi^s$$
Here $\Sigma^2= \frac{\sigma^2}{1-\phi^2}$ is the unconditional variance of the process. This makes the expected return sensitive to both the weighing scheme, i.e. the signal, and also exponentially sensitive to the auto-correlation coefficient $\phi$. Notice how this differs from the previous case. The higher order moments like variance and skew also vary exponentially with $\phi$ as in the figure below (left-hand one).

Everything (expected return, strategy vol, skew) increases with $\phi$, however increase in return is more than vol hence Sharpe improves as $\phi$ increases. While comparing across different types of positioning function (right hand chart above), the change in the Sharpe ratio is not very significant at reasonable values of $\phi$. The major difference comes in terms of higher order, i.e. skew and excess kurtosis. Again, we see that double-step and sigmoid present a competing choice between Sharpe ratio and positive strategy skew, perhaps with a bias to double-step in this particular case.

The influence of the weighing scheme on the strategy performance of course will depend on the underlying process. Here for the AR(1) process, the shorter the MA lookbacks, better the performance, to the extent that for very large MA (like 50/200) the strategy skew even turns negative (not shown here). Similar to the results above for varying $\phi$, the expected return and the strategy skew is more sensitive to the weighing function than strategy volatility.

The optimization objective here again would be to estimate the process parameters, but perhaps hoping for higher accuracy than the case for a known drift random walk. This is not only because the performance sensitivity is an order higher than before, but we also need to optimize the weighing function (i.e. the signal) which depends on the underlying process parameters. In previous case, we would be happy to have confidence on the sign of the drift term, ignoring the accurate estimation of its value. But in this case, with a given risk/reward budget, we need much more accuracy in the estimated value of $\phi$. Nevertheless, as far as positioning is considered, we again see sigmoid and double-steps are good competing alternatives, with a favor for sigmoid for a implementation with linear instruments and double-steps for non-linear instruments.

## Monday, November 16, 2015

### Five things I do not believe in...

But have no evidence to the contrary. Yet.

1. That the dealers are running zero corporate bond inventories
2. That China shorts is going to make money for investors (UPDATE: At least not in macro shorts. Possibly in selective equity shorts. There seems to be a fissure within the old and the new economy in China)
3. That the next crisis (whenever that happens) will mean a dollar rally (against euro) (UPDATE: See this, although I think it misses the point. It is about in what currencies global assets and liabilities are funded)
4. That migration crisis is just another one for Europe
5. That we have reached the peak Geo-political crisis (think about power balance in post-oil scarcity world)

## Wednesday, November 11, 2015

### Trade Idea | Long Euro anyone?

Forget the Fed, forget the parity. Euro is more Yen than anything else. If on Friday we have a rating downgrade on Portugal, then ECB suddenly has almost euro 200b less room in current QE. That is NOT good for the expectation of an expansion. PSI 20 has done approx 16% correction from recent peak already, approx 6% in latest bout.

We have a few Fedspeaks scheduled this week, but next ECB is sufficiently away to be wary of anyone from there to talk down euro anytime soon. Minimal down side on a tactical long euro trade given the current level, and given the market is now perhaps pricing fully a Dec Fed hike.

The aggressive trade here is long euro. A more balanced one is convex long euro with cheapening achieved by long Portuguese equities.

## Monday, November 9, 2015

### Time Series Momentum Strategies | The Spirits Within - Part IV

This is part of a series on time series momentum. Previous post on this are:

1. Part-I: Time Series vs Cross-sectional momentum
2. Part-II: Nature of linear time series momentum filters
3. Part-III: Types of sizing function

In this post we look in to the returns characteristics of a generic time series momentum (TSMOM) strategy. We have the expressions for the returns and moments for the previous post. We here consider two cases of the behaviors of the underlying asset - one where the asset behave like a Gaussian random walk, and in the second where the asset returns are autogressive (of the order 1).

Gaussian Random Walk: Let's assume our sizing function $\Psi$ is linear and the underlying asset is a random walk. That is asset return $r_t$ is given by $r_t=\mu + \sigma.\epsilon$, where $\epsilon\sim N(0,1)$ is Gaussian noise. In this case we can find the expected return from a TSMOM strategy as below
$$E(R_t)=E(S_tr_t)=E(\sum_{s=t-k}^{t-1}(w_sr_s)r_t)=\sum_{s=t-k}^{t-1} w_s E(r_sr_t)=\sum_{s=t-k}^{t-1} w_s (\gamma_s+\mu^2)=\mu^2\sum_{s=t-k}^{t-1}w_s=\mu^2$$
Here $\gamma_s$ is the autocovariance of underlying returns at a lag $s$. We obtain the results using the facts that $\gamma_s=0$ for $s\neq0$ in our particular case, and also that $\sum_{s=t-k}^{t-1}w_s=1$ by design. This is a case of strict TSMOM strategy in the sense all $w$ are positive. The result is intuitive. The position size is proportional to the expected return $\mu$ and so is the return on this size, hence the square of $\mu$ term. Note, this result does not depend on the exact type of signals, as long as the weights are positive and adds up to one. Similarly we can show the volatility (square root of variance) of this strategy is proportional to $\mu\sigma$. Figure below shows simulated results for different parameters

As we can observe, the expected returns and strategy volatility is as discussed above. The skew of the strategy is positive and increases with decreasing $\mu$ (till a certain threshold) and increasing $\sigma$. Excess kurtosis increases with decreasing $\mu$. The signal function $S$ is a 10 vs 50 period simple moving average cross-over signal. Since for this special case of random walk, all the individual terms under the summation evaluates to the same expression for all terms (this is true for all moments), the underlying signal function parameters (i.e. simple vs. exponential or 5/10 period vs 50/250 period) do not influence the performance.

However, the positioning function $\Psi$ will influence the performance. The above results are for a linear function $\Psi=S$. Below is the performance comparison for different types of $\Psi$.

As we can see, there are variations in statistical characteristics across different choices of $\Psi$. The sigmoid function behaves similar to the linear function we have already seen. This is expected, for example, sigmoid can be made to resemble a linear function (with position cut-off) with appropriate choice of parameters. In general for the random walk case, binary function will show similar expected returns and variance as the underlying itself and little skew or excess kurtosis, Compared to both, linear will have higher skew due to higher potential position on the extreme. Sigmoid usually will show a reduced expected return (but maintaining the Sharpe Ratio more or less).

However, the double-step shows markedly lower Sharpe and higher skew (in spite of the position limit). It has a lower vol but an even lower expected return makes the Sharpe lower overall (compared to the benchmark linear case).  The higher skew comes from the sharp increase in position at a relatively lower threshold of signal (compared to, again, a linear function). Also higher the threshold $\epsilon$, higher is the skew.

So in the case of random walk with deterministic drift, the optimization problem is rather trivial. The underlying signal function does not affect the strategy performance much. That includes the type of the signal and the parameter space of the signal function. The choice then reduces to finding appropriate positioning function $\Psi$. Usually the linear is NOT preferred because of potentially very large exposure. Sigmoid is a good choice for position limiting with a higher Sharpe. On the other hand double-step is a good choice for a high skew strategy. Depending on the trading style (confidence in underlying process estimates, along with risk management), instruments (linear or convex) and trading horizon (we will come back to trading horizon later in details), a combination of sigmoid and double-step can deliver the desired mix of Sharpe and positive skew.

## Friday, November 6, 2015

### Time Series Momentum Strategies | The Spirits Within - Part III

This is part of a series on time series momentum. Previous post on this are:

1. Part-I: Time Series vs Cross-sectional momentum
2. Part-II: Nature of linear time series momentum filters

The second phase of designing a momentum strategy is designing the positioning function $\Psi$. This is the function that converts the signal in to a positions. The common choices are:

1. Sign/ Binary function (i.e. maximum long position allowed if positive signal, maximum short otherwise): The simplest of the lot. Sharp change in positioning near 0 level of signal (ambiguous zone) which can lead to increasing turnover and related costs. $\Psi = Sign(S)$
2. Linear (including constant): Simple, but no limit on maximum position. $\Psi =c S$, $c$ is a constant scaling factor.
3. Step function: Sudden change in direction (although no longer around the ambiguous zone). $\Psi=+1|S>\epsilon, -1|S<-\epsilon$. Here $\epsilon$ is the threshold.
4. Sigmoid function (error function) or tangent hyperbolic filtering: Smooth combination of linear and binary, moving gradually from one to another depending on parameters. $\Psi=erf(S)$, or $\Psi=\frac{-e^{-S} + e^S}{e^{-S} + e^S}$
5. Reverse sigmoid function: Sigmoid with peak sizing in long or short zone. $\Psi=e^{1/2}S.e^{-S^2/2}$

Given this set up, now we are ready to look in to the performance of such a strategy. By definition, the one-period return of the strategy is $R_t=\Psi(S_t).r_{t}$. The expression for one-period mean is as below.
$$\mu = E\left(\sum{\Psi(S_t).r_t}\right)$$
The k-th moment is given by
$$M(k) = E\left(\sum{(\Psi(S_t).r_t)^k}\right)$$
As we have seen already, S can be (in case of linear filters) expressed as $S_t=\sum(w.r_t)$. Also for linear $\Psi$ we can take the coefficient outside the summation notation.

## Wednesday, November 4, 2015

### Time Series Momentum Strategies | The Spirits Within - Part II

This is part of a series on time series momentum. Look here for the previous post on this.

Here we focus on time series momentum strategies in a single underlying (as opposed to diversified momentum trading).

The typical time series momentum trading strategy has two distinct design phases. The first one is generating a trading signal based on some logic applied to the underlying price levels or price returns. This, therefore, can be thought of as a function $S$ converting the underlying prices or returns to a trading signal. The second phase is designing an appropriate positioning function $\Psi$. This accomplishes converting the output from $S$ in phase one in to a sizing or positioning. In many cases, we can have the third phase consisting of risk management. This phase includes putting different types of risk management logic, like stop losses or take profit or we can even club volatility filter under this category. For the sake of practicality and simplicity, we keep risk management out of scope and concentrate on a strategy involving the basic two steps as above - designing $S$ and $\Psi$. The schematic below shows how the price or returns (first terms) flows through these filters to generate a profit or loss number (last term)
$$logP_t \Rightarrow S(logP_t) \Rightarrow \Psi(S(logP_t)) \Rightarrow r_{t+1}\Psi$$
$$r_t \Rightarrow S(r_t) \Rightarrow \Psi(S(r_t)) \Rightarrow r_{t+1}\Psi$$
First set refers to price-based signals, and the second set refers to returns based signal. Here $r_t=(logP_t - logP_{t-1})$ is the one period return. Note we are using logarithms of the prices for filtering, while in most cases (like moving average) simple prices are used.  This for convenience so that we can write the percentage returns as a difference of logarithms of prices. The designing objective is to choose $S$ and $\Psi$ to optimize the performance.

The two most common types of signal designing is either a momentum signal on the returns (RMOM) or a moving average crossover signal (XMOV). An RMOM signal computes a weighted average of recent returns and goes long if they are positive. A simple strategy based on such a signal is to buy an asset if, say the recent monthly return has been positive. A simple TSMOM signal will be as below
$$S_t^{RMOM}=\sum_{s=1}^nw_sr_s=\sum_{s=1}^nw_s(logP_{t-s+1} - logP_{t-s})$$
Here $w$ are the weights and $n$ is the window of the applied filter. A buy signal is generated for $S_t^{RMOM} \ge 0$. Similarly, a moving average cross-over signal tracks two moving averages and signals a buy when the fast one crosses the slow one from below. The moving average signals will be as shown below
$$S_t^{XMOV}=MA_t^{fast} - MA_t^{slow}=\sum_{s=1}^nc_s^{fast}(logP_{t-s+1}) - \sum_{s=1}^nc_s^{slow}(logP_{t-s+1}) = \sum_{s=1}^n(c_s^{fast} - c_s^{slow})logP_{t-s+1}$$
Here $c$ are the weights and $n$ is the window of the applied filters. A buy signal is generated for $S_t^{XMOV} \ge 0$.

Pedersen and Levine (from AQR Capital) have shown that these two are in fact equivalent ways of expressing same filtering. They have even shown that in general all linear filters are equivalent. For example, the equivalent ways of expressing the XMOV signals in equivalent RMOM expression is to compute the equivalent weights as
$$w_s=\sum_{j=1}^s(c_j^{fast} - c_j^{slow})$$
Here are some examples of price level filters mapped back to returns space applying these results.

Here the simple MA crossover is based on a 50 period and 250 period (fast and slow respectively) moving average filters. The corresponding EWMA is designed to have similar filtering (in the sense that the net signal has similar center of mass). Also note that the net signal weights are both positive and negative for the price space, but strictly positive on the return space in these two cases.

So we see that in general, we can represent the signal function $S$ as weighted past returns, in the form of  $\sum{w_sr_s}$, at least for linear filtering and scaling. Next we look at the positioning function $\Psi$.

This covers a significant number of technical indicators (like ROC, MACD, or even normalized momentum filters like CCI, assuming a known and constant volatility - i.e. constant scaling). However, this will exclude non-linear indicators like Aroon oscillator.

## Tuesday, November 3, 2015

### Time Series Momentum Strategies | The Spirits Within - Part I

This is first part of a small series on some observations on general time series momentum strategies. In this I lay threadbare generic time series momentum strategies with the objective of establishing general theoretical underpinning on the strategy performance and optimization approach.

The class of "time series momentum" strategies are very common and popular among investors. At its simplest, it means buy high and sell low. Or more precisely, buy high (an asset that has recently appreciated) to sell at a even higher price. Similarly sell low (an asset that has recently sold off) to buy back at a even lower price. From this point of view, this is fundamentally different than the value investing paradigm. In other words, any returns generated from these class of strategies should be independent and hence should represent an independent risk factor to the investors.

Most of the theoretical interest in momentum investing started with the classic paper by Jegadeesh and Titman in 1993, where they found strong evidence of momentum profit. Further research followed these interesting observation since then. However, what is described as momentum in this case is usually termed as cross-sectional momentum (XSMOM). This is fundamentally different than what is usually understood to be time series momentum (TSMOM).

An XSMOM strategy looks at the relative performance of a basket of assets, and invests in the winners and shorts the losers. The buy/sell signal is generated from relative performance of different assets (cross-section) within a given time interval. On the other hand, a TSMOM strategy looks at past performance of an asset, and buys if has been a winner, or sell otherwise. A TSMOM strategy may or may not involve a basket of assets, it does not depend on a basket crucially for the strategy implementation (unlike the XSMOM which is meaningless without the context of a basket). When a basket is used for a TSMOM it is for diversification and risk management.

This fundamental difference in construction shows how the performances of these two strategies can be similar and different. For sake of comparison, let's assume in both cases we have a basket of two assets. As it is evident if we indeed have strong correlated changes in asset prices (past performance predicts future), then both strategies will perform, as we will be buying and selling the right assets by construction. However, even if we have a change in the momentum, if there is an increase in dispersion of the asset performance (e.g. winner becomes losers, but losers become even more so - not a complete reversal, as in winners become losers and losers become winners) then the XSMOM will still perform. Similarly, even if we do not have strong auto-correlation, but persistent trends the TSMOM will perform better.

It seemed to me the first major theoretical insights in to TSMOM strategy was presented by Moskowitz, Ooi and Pedersen in 2012. The paper also captures the essence of the above paragraph, by breaking down the sources of profits in XSMOM and TSMOM strategies, as follows (in terms of one period expected returns):

$$E[r_{t,t+1}^{XSMOM}] = \frac{N-1}{N^2}tr(\Omega) - \frac{1}{N^2}[l^T\Omega l - tr(\Omega)] + 12\sigma_{\mu}^{2}$$
$$E[r_{t,t+1}^{TSMOM}] = \frac{tr(\Omega)}{N} + 12\frac{\mu^T\mu}{N}$$

Here $\Omega$ is the covariance matrix, $\mu$ is the mean returns vector, $\sigma_{\mu}^{2}$ is the cross-sectional variance of the means, $N$ is the number of assets in the basket and $tr()$ is sum of the diagonals. The above expression clearly shows the points made in the previous paragraphs. The TSMOM returns is driven by auto-covariance and strength of the drift terms. Where as XSMOM, in addition to auto-covariance, also depends on cross-variance (dispersion) and cross-variance of the mean returns (dispersion, again), but not particularly on the strength of the mean returns. These results are valid for linearly weighted basket (linear in returns), but in general give good guidance.

## Wednesday, October 28, 2015

### Inflation Puzzle: Back In Time

"Inflation typically rises during an economic expansion, peaks slightly after the onset of recession, and then continues to decline through the first year or two of recovery. During the present U.S. expansion, however, inflation has taken a markedly different path. Although more than six years have passed since ____, inflation in the core CPI (the consumer price index excluding its volatile food and energy components) has yet to accelerate."
Guess the missing time line in the above paragraph. It is from a paper from NY Fed, dated 1997. It refers to the 1990-91 recession. But it could easily apply to the 2008 GFC. With the benefit of hindsight, it is now questionable if the Fed should have been more hawkish during the 90s and 2000s to avoid the great financial meltdown. But in 1997, with all fairness, there was little support for more hawkishness in data.

A recent Bloomberg article talks about six million reasons for Yellen to think before raising rates. But I wonder if we will ever have those kind of jobs back in listing which the 60 somethings like Mr Elanko are skilled for. 20 somethings are working overtime and designing apps against Miss Yellen.

In a case of structural unemployment, the numbers matter less than the rate of increase in wage and employment cost. The Fed knows it. So for me, a December pause is more worrying than a hike.

## Saturday, October 17, 2015

### Inflation: Think Global (In Chart)

A few observation on inflation from a global perspective

#1: Global inflation has been weak, but core has been steady. Here the global data points (like headline or core inflation) are calculated using the GDP weighted national measures of the top 20 countries in terms of GDP in current dollars (representing 79.9% of world GDP. Pareto!!).

The difference between the core measure and the headline is even more important as the wedge between them is currently driven mostly by a single factor - energy prices. The concept of inflation is an overall price rise. A change in a particular component is mostly a relative price change, not an overall price change. Central banks have little controls over production of individual goods and services. If there is a large relative price rise for doughnuts for some reason, a hike in policy rates most probably is not going to help it (unless this relative price rise permeates through the economy and finally in wage expectation through second round effect).

#2: The smack-down of inflation in commodity exporting countries is most prominent for the ones with fixed exchange rate regimes. With a few exceptions, most of the metal and energy exporters are not suffering any great dis-inflationary pressure in core measures otherwise.

#3: In terms of professional forecasts, inflation expectation remains steady, but the market based measures for the US are not so.

#4: The consumer demand is weak. Especially if we measure in dollar terms. We have a scenario of low rates, a strong dollar, very weak commodity prices and weak global demand. Commodity prices respond a lot to investments expenditure globally. However, the consumption expenditure has been a relatively stable component of economies across countries and time historically. Since mid of last year the consumption expenditure globally in dollar term has been in a strong contraction phase, approx 6% from peak till Q2 2015. This is only matched by an approx 8% drop during the GFC. And this is not driven by US or China much, rather rest of the world, including Euro area and Japan. It is hard to say if this has bottomed out and we will see the savings from drop in energy prices being channelized to recover consumer demand.

Nonetheless, the possibility of a wage driven inflationary pressure cannot be dismissed. The chart on the left shows scatter plot of job opening rate (JOLT), Employment Cost Index and PCE core inflation against headline unemployment rate on x-axis, since 1980. The starting points are marked in red and end points in green. As we see in case of job opening, there has been some significant hysterisis (unemployment rate higher, given the job opening, if we measure the slope from the earlier part of the curve). This may points to a case of structural problem in unemployment. That will put forth a case against a downward revision of Fed's unemployment target (NAIRU). On the other hand the wage inflation (here ECI) and broader inflation (PCE) still shows inverse relationship. The Phillips curve is still alive (esp. for wage inflation), although flatter in recent times. Given the fact that Fed action has always a lag before it affects the real economy, this will keep the case for a early hike on the table.

#5: And related to above point of global consumption, the global imbalance in excess savings seem to be heading towards a forced reduction. The left chart shows excess savings (or equivalently current account balance) in nominal dollar terms. As we can see the large CA deficit of US has historically been balanced by large surplus of Japan and lately China. The EM had a spike just after the late 90s Asian Crisis. But that is mostly negated now. The recent cause of concern (Euro Glut) was a large and ballooning surplus of Euro Area. With the fall in oil prices, the Petrodollar balance is now going the other way to counter it. These low commodity prices may play a crucial role in re-balancing the flow of trades and capital across the globe. ( it is evident from the chart that trade volume has come down significantly.) It is not clear to what extent this balancing act will help consumption and through what channel, but it is definitely better than exploding imbalances in the medium to long term.

Also since the financial crisis, after the very initial period, it has been mostly a battle fought by central bankers, with fiscal stimulus sitting mostly on the sideline. In fact the withdrawal of high fiscal stimulus just after the collapse might as well have countered central bank efforts. We are politically getting in a better position to consider and use fiscal stimulus than the height of European Crisis and talks of austerity. The global budget balance is in fact back to the pre-crisis average level. And if the economy is not, there is a good reason and scope for fiscal stimulus in coming years.

The key takeaways: Despite the weak global demands and large savings imbalance (which are related), there is a case that the commodity prices has done some corrections, and a persistent weak demand/ high global excess savings may not be realized. And we still have the upside of fiscal stimulus in case consumer demand needed a booster does. At a global level, most measures of core inflation, and non-market based inflation expectation remains robust. However, the market seems to be pricing a very pessimistic outlook for inflation globally. And also as mentioned earlier the inflation skew pricings are improving on the upside surprise.

Is this a case of peak (dis-)inflation worries and significant consolidation and upside from here. Hard to be sure, but I would say chances are good than they were before. Of course inflation can go either way from here, but in most scenarios they have a better chance of ending up higher than current levels. And a reasonable dollar weakness from here can tilt the balance in its favor further.

1) In case Fed is on time (which we will only know with the benefit of hind sight): long inflation upside and nominal rates sell-off with short dollar for cheapening.
2) In case Fed is delayed: long vol - a sharper rate of hike will catch many unsuspecting asset classes on the wrong foot.

## Saturday, October 10, 2015

### Upside Surprise: What The Inflation Skew Prices In

Inflation is low, has been so, and is forecast that way for a while. I mostly agree. Nevertheless, let's look at the other possibilities. Below is a picture that captures what is driving the headline inflation in the US. Yes, no surprise here, it is the oil price.

If we take out that part though, things looks interesting. As we can see, a large contribution comes from housing. Most will perhaps agree the oil related part is transitory. So the real question is what about the housing. At this point it seems within the dual mandate of the Fed, the unemployment part as just a target number becomes less relevant. It is at 5.1%, can go down to 5.0% or go up to 5.2%, or there can be serious debate about what is the level of NAIRU. But the fact of the matter is over the past few decades the revered Phillips curve has weakened a lot. With a drop in productivity, and labor share of GDP, the unemployment number is a lot less related to inflation now. So for policy makers the real points are two-folds - look out for signs of wage inflation, and also look out for heating up of the housing markets (or risk premia compression in general). And so far we do not have any strong confirmed signs of any. Unfortunately the action of the Fed has a certain lag on the policy targets. So to move in after seeing confirmed signs of these may be too late. Which means a slow and steady path of hikes replaced by a delayed and steep one.

The market seems to seriously be thinking over it. These is based on the volatility markets of the US inflation. Which trades typically as zero coupon swaps on the CPI (urban) index and options on that. The chart below shows the implied vol of a 5y options at a 2% strike (blue), the vol premium (green, implied vol over the actual realized vol of the index) and the skew (red, difference in vols between a 2% and a -1% strike on right hand axis, note: these vols are approximate and implied from Bloomberg quoted price)

The inflation vols has picked up a bit recently (but nowhere near the paranoid inflation phobia in 2009 - 2010 following the QE), but the vol premium has shrunk has well. The strong negative skew since 2013 correctly predicted/ coincided with increasingly softening inflation expectation, and maintained the negative skew till very recently. (Apart from a short flare up of headline inflation in Q2 2014, which the Fed rightly called the bluff). But this is now almost reversed and poised for a come back on the other side. It seems although market is not much upbeat about the breakeven inflation forwards, it is increasingly suspicious of a further downside surprise. Contrast this with the vol markets nominal rates (USD swaptions), where the skew is now flipped on the receiver side for a while.

## Thursday, September 17, 2015

### Cross Asset Correlation

Cross asset correlation update. Note this is correlation at daily frequency (unlike weekly in previous cases) to capture the large moves seen in August across asset classes.

The vol correlation dynamics is interesting (the last column). This vol is usually dominated by equities, and naturally equities have a strong negative correlation. However, the spike in rates and vols correlation 3 months back has given away to commodities and more lately FX. Also it appears rates and inflation simply have diminished significantly.

While rates and inflation correlation has gone down, USD and GBP 5y and 10y breakeven swaps correlation to rates (belly and 10y) are still significant (60%+ ). Between rates and equities, US rates are still major drivers (correlation ~ 50% to equities across G3). However, GBP rates and EUR 5s30s have picked up correlation to European equities. However G3 rates has little correlation to EM equities (including China).

Also we have seen a general increase in correlation for USD rates. That includes a pick up of USD rates vs. equity vol, oil and CRB commodities Index and Yen and Euro FX. In addition US 5y/ 10y breakeven has been highly correlated to oil (Brent) and CRB commodities index (CRY) and Yen. In general break-even across G3 has picked up correlation to equities.

On the FX side, the major correlation maker has been JPY, with a strong correlation to equities across geographies - capturing the risk-off mood back in August. In fact there has been a substantial increase in correlation between equities and FX across the board. Separately, EM vols has in general been more correlated to commodities move than DM vols.

## Friday, September 11, 2015

### Economics: The Myth of "Quantitative Tightening"

It is the latest populist theory doing the rounds in the financial media. Even the mainstream media is now flooded with this now. See here and here

To see why it does not make much sense, we need to understand what quantitative easing actually is, in terms of Economic models.

The standard Keynesian model is the famous IS-LM model. This captures the goods and money markets equilibrium simultaneously in an economy. The IS curve of the model, derived from the equilibrium of output and aggregate demand, captures the goods market equilibrium. It outlines the combinations of interest rates and economic output for which such equilibrium is possible. It is a downward sloping curve, as for a given level of external factors, a higher interest reduces the investment spending and hence output. The second part of the model is the LM curve. Derived from the demand of money, it captures the combination of interest rates and output for which the money market is in equilibrium. This is an upward sloping curve, as for a given amount of money stock, the demand for money goes up with higher income and lower interest. For more on this look here for a quick introduction. The entire economy is at equilibrium at the intersection of these two curves, which implies simultaneous equilibrium in goods and money markets.

However, I think to analyze QE, it is better to switch from IS-LM model to IS-MP. It is a variation of the IS-LM model which retains the same IS curve, but replaces the LM curve, by an MP curve (MP stands for Monetary Policy). The advantage is primarily two-folds. Firstly, unlike the implicit assumption in IS-LM model, most modern central banks do not target money stock, but rather a policy rate - which is explicit in the MP model. Secondly, the IS-LM is a bit ambiguous. Ideally the relevant interest rates for IS curve is the real interest rate, and nominal interest rates for the LM curve. So effectively it is a bit round-about to incorporate inflation directly in IS-LM. And as we will see QE is largely about (expected) inflation. For more details on IS-MP, look here (opens PDF and a bit wonkish)

Figure below shows a typical IS-MP curve. As mentioned before, the IS remains as it is. The MP is upward sloping. Which makes sense as most central banks uses a Taylor Rule approach to determine the appropriate level of real rate to target, balancing output and inflation. For a central bank targeting purely a real rate (i.e. inflation targeting), the MP curve will be horizontal.

In the IS-MP model, the economic shocks can be analyzed in a manner very similar to the IS-LM model. Suppose the economy is initially at equilibrium E0 with output at potential output of y0. If there is an external negative shock to aggregate demand (like the 2008 crisis), the IS curve shifts to the left (IS' in the plot), along with a drop in output y1 (which is below the potential output) at a new equilibrium of E1. The response of the monetary authority is to shift the MP curve towards right sufficiently (expansionary policy) so that the equilibrium point E1 shifts to E2, which brings the output back to potential, but at a lower real rate (r'). How the shifting of MP to right is actually achieved depends on many things. For a normal economy with sufficiently high nominal interest rates and stable inflation expectation, manipulating the nominal rate (setting fed funds etc) can achieve it. In case of a positive shock the dynamics works in the reverse. This is what central banks do in a nutshell.

The question is what happens if the nominal rates are not high enough (the so called liquidity trap). Or the initial shock is so large that to change real rate enough to reach the equilibrium E2, the nominal interest rate has to become negative (with a given inflation expectation). Obviously, this is not likely to work. Here the interest rate implies the general level of rates. Forcing the general level of nominal rates to negative territory is quite a challenge (if desirable at all), as people can just hold cash instead of bank deposits (thus avoiding negative interest rates, i.e. paying fees to park cash at banks).

The way out is to tweak the other component of the real rate. That is inflation expectation. If the demand is lower than potential, the inflation and inflation expectation has already started creeping towards a lower base. If the central bank can convince people that it is not going to stay low for long, and jack up the expectation, that can reduce the real rate, even at a zero nominal bound. Which in turn spark real activities. Quantitative easing is a tool to achieve just that. In fact we can express real interest rates as below (as a matter of definition):

Long term real rates = average path over expected future nominal rates + term premium - expected inflation.

Even at zero lower bound, the central banks can use tools to manipulate any of the three terms to achieve its objective. For example, the "forward guidance", adopted by Fed, is a tool to manipulate the first term. General asset purchase influence the second term. And depending on how the QE is planned and communicated it can influence the inflation expectation. In fact the standard way how QE works is mainly two channels - a) the portfolio re-balancing channel, which compresses the term premium, and b) the inflation expectation channel. And together they can work exactly like the expansionary monetary policy in the diagram above. Even at the zero nominal bound. That is pretty much what quantitative easing is. So by definition, "Quantitative Tightening" will work in reverse.

But, we are not talking about quantitative tightening by the domestic central bank here (i.e. the Fed), but rather foreign central banks. To analyze that, we need to extend out model to an open economy.

Much of the things remain the same. The stuffs that change are two-folds. Firstly, the IS curve is now influenced by the real exchange rate (opens PDF, a brief primer). An appreciation of dollar in real term will make imports attractive for domestic consumers and export costly for overseas consumers. So this works like a negative shock to the IS curve (domestic output), a shift to the left. Secondly, we also need to incorporate the foreign exchange market equilibrium, captured in the line BP (abbreviation for Balance of Payment). This equates the demand for foreign exchange (import over export) and supply (net FX inflows, ignoring central bank reserve changes, which is only applicable for pegged currencies or managed floats). For perfect capital mobility, this will be a horizontal line, as we can have only one interest rate at which we can have equilibrium. At every other rate, large inflows or outflows will overwhelm and restore balance. For general capital mobility, we have an upward sloping curve. The equilibrium for an open economy is achieved in the intersection of all three curves - IS, LM and BP
In such a scenario, negative demand shock can be countered as before. Assuming a floating exchange rate regime, an expansionary monetary policy, reducing fed fund target or QE as the case may be, pushes the MP curve towards the right to MP'. Given the lower rates, the new point is below the BP curve, which implies an imbalance in the FX markets. In this case, the dollar becomes cheaper in real terms, leading to simultaneous increase in net export (IS shifts right to IS') as well as improvement in current account (BP shifts right to BP'). This changes the output from y0 to y1 at a lower interest rate levels. The equilibrium changes from E0 to E2. Notice the change in real interest rate is less than the previous case. A tightening works in the reverse.

Now "Quantitative Tightening" by PBoC or other central banks, (i.e. selling of treasuries) is a totally different beast. PBoC has NOT decided overnight that it is the monetary authority for United States, and is NOT trying INDEPENDENTLY to influence the monetary policy for dollars. Nor it can change the total dollar money stocks. It is selling treasury because of its own monetary policy aim, which is to maintain the Yuan trading range.

So in effect, in the above diagram, nothing changes. No dollar monetary base, nor real exchange rate, nor inflation expectation to move any of the curves. There is a potential of changing the term premium. But assuming it is selling foreign reserves for the purpose of exchange rate targeting, it must be selling not only treasury but all other reserve currencies as well. That means it will require a huge selling by PBoC to achieve a modest increase in the term premium. Which is unlikely.

Also a QE or reverse for a large bond markets like US treasury (approx USD 16 trillion outstanding) primarily works through inflation expectation than portfolio re-balancing channel. For example, the episodes of previous QEs by the Fed actually saw a modest increase in treasury yields, but an overall reduction in real yields (as computed through breakevens). In addition, the Chinese FX reserve can be around USD 3.6 trillions on paper, but given the size of the economy and exports and imports, China must maintain a part of it as a safe guard as per IMF recommendation (opens PDF). So effectively a much less amount is available for this so called Quantitative Tightening.

And lastly, the entire point of treasury selling of China is maintaining the FX policy. The recent capital outflows increased the devaluation pressure on China, and PBoC is selling dollars and buying Yuan to protect the range. So effectively it is keeping Yuan artificially overvalued, one can argue. And that means, if they do not do that, i.e. stops selling treasuries, that will actually have an worsening impact on the US, as USD real exchange rate appreciates and shifts the IS curve towards left.

Now enough of theories. Let's look at some hard data. How much net selling is happening anyways in treasuries - based on TIC data as of end of June 2015.

Hardly anything that suggests "Quantitative Tightening"!

Although official ownership of long term treasuries has gone down, this is more than compensated by increase in private ownership. The only countries where we have seen total treasury ownership going down is Japan and the Switzerland + Benelux block. And on overall basis foreign ownership of treasuries is on a steady upward path, after a sizable reduction for a brief period of Taper Tantrum back in 2013.

Only Fed can do a real quantitative tightening. "Quantitative Tightening" by PBoC is mostly a nonsense.

Nevertheless, what is interesting in this entire model thingy is the dynamics. You might have noticed how the entire thing works. Any monetary policy changes in response to a negative shock in demand lowers the real rate. Similarly a positive demand shock will increase the real rate for the same potential output. Interestingly in recent times, the demand shock distribution has been highly negatively skewed (you can have a look at the real GDP distribution since 80). It is hardly a surprise ever since we have a constant downward drifts in general rates levels. Forget about secular stagnation and other interesting theories. Even in a perfectly normal economy, a negatively skewed demand shock distribution, along with Keynesian central bank, implies rates will have a tendency to drift down and eventually hit the zero lower bound and get stuck there. There are only two ways out. Either reigniting the animal spirits and optimisms of the industrial revolutions or the post-war period. Or a higher inflation target. Else downward yields are far more likely than a sharp sell-off in rates. No matter which foreign central banks are re-adjusting their FX reserve.

## Monday, September 7, 2015

### Volatility: A Cheat-Sheet for Traders and Investors in Rates

Volatility in rates shows one of the most interesting dynamics compared to other asset classes. Unlike FX and equities, the skew shows considerable variation, in magnitude and even in sign, with changing macro-economic conditions. On top, a comparatively large share of derivatives markets influence the vol and the underlying in feed-back look.

Usually at any given point in time, the rates market is in any of the following three regimes
• Sticky-strike Regime: Very tight range trading of forwards. Realized and implied vols are range bound, no significant dvega/drate changes on the street, overall uncertainties low/stable
• Sticky-moneyness Regime: Trending rates regime, stable realized/implied vols, stable risk aversions/ risk appetite
• Uncertain Regime: Everything else. Usually accompanied by high vol, high policy and macro uncertainties.
Here is a quick cheat-sheet for traders and investors to identify which dynamics at play, based on the behaviors of traded prices.

 Negative Skew (Current Skew Negative) Positive Skew (Current Skew Positive) Sticky-Strike Regime Expected Behavior 1.       Fixed Strike Vols : Independent of level of swap rate (or forward swap rate) 2.       ATM Vol: decreases as the swap rate increases Sticky-Strike Regime Expected Behavior 1.       Fixed Strike Vols : Independent of level of swap rate (or forward swap rate) 2.       ATM Vol: increases  as the swap rate level increases Sticky-Moneyness Regime Expected Behavior 3.       Fixed Strike Vols : increases as the swap rate increases 4.       ATM Vol: independent of swap rate level Sticky-Moneyness Regime Expected Behavior 3.       Fixed Strike Vols : decreases as the swap rate level increases 4.       ATM Vol: independent of swap rate level Classical Uncertain Regime Expected Behavior 5.       Fixed Strike Vols : decreases as swap rate increases 6.       ATM Vol: decreases Classical Uncertain Regime Expected Behavior 5.       Fixed Strike Vols : increases as the swap rate level increases 6.       ATM Vol: increases

Speaking of volatility, here is an update of the cross asset volatility chart (see here for details, click to enlarge).
This chart presents volatilities across different asset classes in a comparable manner. Since the spurt of vol in rates since May this year, the volatility has switched back in to risk assets (commodities and equities) on the back of the Chinese Yuan devaluation. This supports the case that while positioning is important for rates, ultimately it is less about foreign central banks selling reserves, and more about which way the risk assets are heading. Historically that has been a better yardstick for directional calls on rates.

## Monday, August 24, 2015

### Equity Sell-Off : Looking Back

Histogram of S&P 500 one-month (20 days) move following a 3% or more sell-off, since 1960s. This shows historical probabilities of a recovery or a worsening sell-off during the next month, conditional on a 3% or worse sell-off today. Index is scaled to start at 100 after the first down move on day 1.

The worst 5 recoveries are all in 2008 Sep to Oct, except once in 1987 flash crash. The best 5 recoveries are mixed across time.
The cumulative probability of continued sell-off is 37.5%, Cumulative probability of 5% or more recovery is approx 40%. Median move is 1.9% rally. On cases we had further sell-off, the median sell-off is 6.6%, The median for rally case is 5.6% (so the distribution is skewed, as obvious from above). However, this skew disappear if you take out the tumultuous period of 2008 following the Lehman Brother failure. In that case the moves are slightly positively skewed, 4.6% sell-off vs. 5.4% rally. Also the cumulative sell-off probability drop is around 30%

But with all hard numbers, it is hard to keep calm when everyone else is in panic mode.