## Saturday, 25 February 2012

### Reflections on Bruce Kovner

Bruce Kovner was interviewed towards the end of the 1980s.  This is a summary of what I found useful about how he traded.

Primary Market
Currencies and fixed income, commodities.

Risk Management

Early on in his career, he initiated a soy-bean futures spread which he, in retrospect, unwisely legged out of.  The implication being that the unhedged long futures position was "insanity" He also didn't handle the moment well as it all turned against him.   Describing himself as in "emotional shock" at his behaviour, in the face of all his years of cautious study, discipline, his lack of understanding of just where a market might go to - and how quickly - to inflict most damage.  This shock seems to have been a big deal for him - as well as having a sick feeling in his stomach, he didn't eat for days.  He viscerally felt that markets could burn you.  "I had lost a process of rationality that I thought I had...At that moment, I was confronted with the realisation that I had blown a great deal of what I thought I knew about discipline".  What I'm taking from this is that you do not always have to feel rational and disciplined in all interactions with the market, but you should behave rationally and with discipline (which mostly means sticking to your plan - and in particular to that side of the plan which deals with the scenario when the market moves against you).

Making regular trading mistakes is a good thing
He sites this as deeply important for a trader.  I guess he's emphasising the familiar point that failure to realise losses (small and early) results in bigger losses in the end.  I'm thinking about what he says about sizing down on losses.  If he finds losses affect his near term trading adversely, perhaps his emotional well-being too, then part of the strength he mentions is the strength to frequently put yourself through this wringer by following the rule of taking regular losses.  Clearly the pain is in proportion to the size of the loss, so smaller losses will do less behavioural/psychological damage than big ones, so taking regular losses could actually be a protection mechanism against those feelings.

Markets can really move to that level in that time frame
As well as sticking to the plan during moves against him, he attributes his success to his ability to understand that the markets can move quite a long way.  That big moves can happen.  And not just in markets.  This is perhaps a wider point - political systems can crumble (the book came out in 1989, around the time of the fall of the Soviet empire), commodities can go dramatically up or down.  Currencies can make enormous moves.

Strong, independent, contrary.  All three of these are related.  Core, I guess, is independence.  Since it in a way implies a certain strength of will.  As does holding a contrary view.  Notice, too, that it probably needs to fit in balance with his view that taking your mistakes in small and regular doses is key.  You might think that strong, independent and contrary thinkers would find it difficult to admit they're mistaken potentially most of the time.  Perhaps it is best to think of trading as a two phase operation - ideas generation and execution.  Maybe you get to firmly express your contrariness when generating the ideas, but need to switch to a more precautionary style when you're in the market.

Price action
When the market moves big on fundamentals, the initial move is an indicator/confirmation of direction.  In general though price action analysis ought to be an intellectual discovery concerning how some traders have just now behaved may or may not influence how other traders are about to behave.  This to me is the core of a fundamentalist approach to technical analysis. Kovner says it helped him form a hypothesis on this question: how is everybody voting?  He's not afraid to jump on breakouts even if a rational cause cannot easily be hypothesised.   The higher the speculator-to-hedger ratio in a market, the more likely you get 'false' signals.  He reckons bear markets have sharp down moves and quick retracements.  So if you're short the market and you get in too late in a down move, be prepared to be stopped out by the subsequent quick retracement.  His advice is to go short in bear markets only on the quick retracements.  This sounds like the bear market partner of the ancient 'buy on dips' advice.  In a bear market, buy on dips* in spades. [* where dip is a conter-trend movement].

He also reckons a trend following approach is more likely to be successful in an inflationary environment.  Equities price action versus commodities: the equities market has a lot more counter-trends. "After the market has gone up it wants to come down".  Whereas commodities supply and demand makes for a continuation of price action in those markets. This chimes with my understanding that equities fat tails are more in the left tail, whereas commodities are more in the right tail.

Markets, life even, undergoes structural change and that can break many quant or programmed systems, which can have, as an implicit support, an assumption that the future will remain sufficiently like the past to warrant encoding the set of rules.

Stops  (a.k.a. Markets shouldn't really move to that level in that time frame)
Put stops at points where the market shouldn't move, beyond some technical barrier.  On the face of it, this is just generic stop placement advice.  But the implication is a hope, a degree of certainty that markets can't get there from the moment the trade goes on.  There's another slight dissonance with his advice, which he sees as essential, to understand that the market really can move to some level in your given time horizon.  If he rationally believed this for both sides of his bet, then any stop would be perhaps too close.  Saying 'the market really can get there' is a way of saying that the market can blast through the support and resistance of technical analysis.  Saying 'the market can't really get there, so I'll put a stop there' sounds contradictory.

Client money represents owning a call
I bet he regrets offering that rather honest view on client money.  It isn't just serving as leverage - it is a call.  He's getting leverage without exposing himself to downside vis-a-vis their money.  Clearly, it is a different story with his own money.  But I guess, to continue the analogy, if he fails to deliver, then it is a call with very  high implied vol.  Has he paid a fair price for the vol?  Plus, the call he owns can itself be called away by investors under a range of circumstances.

Approach to fundamentals
The market price is the correct one; find out what will happen economically, politically to change this price at the margin.  Wait for a market to confirm one of his several scenarios concerning what might happen.  He's trying to get into the heads of real people (central bankers, politicians) and I guess, perhaps of leviathans and other non-humans too (the state, the average oil purchaser, the average consumer).

## Thursday, 23 February 2012

### Mechanical Options Pricer

I have created a design (in my head) for the construction of a mechanical options pricer.  Searching on the web, I have never come across anyone else who's had this idea, nor have I ever read of such a thing.  If you'd like to know, or to help in its construction, get in contact.

## Tuesday, 21 February 2012

### If leviathan could speak, could we understand it?

Wittgenstein's famous 'if a lion could speak we couldn't understand it' provides a nice summarising metaphor when considering the pitfalls of generalising from the micro-economic to the macro-economic.  This is what Keynes was getting at with his paradox of thrift and fallacy of composition in the General theory.  The system of logic which works at a household level may not scale to the macroeconomic level.  Folksy home truths which work for a household could be catastrophic for the economy in general.

Keynes didn't invent macro, by any means,   Check out William Petty, John Law, Richard Cantillon, Thomas Mun, Dudley North, Henry Thornton, to name but a few.  They pre-date classical micro-economics.  And they truly invented a different language.  It is a language which essentially assumes that the actors are states.

Understand the historical context of this language, this logic, and you get closer to understanding macro-economic events, perhaps even of exploiting them in financial markets.  Petty was Hobbes's personal secretary, so probably got the leviathan mindset directly from his master.  These thoughts, these times were truly great.  They set in motion  - for the Western world, at least - a kind of alien logic which I don't think anyone has gotten anywhere near to the bottom of yet.

### Monadology : In the long run, we all share the same tombstone

Many of us have experienced the situation where we go to sign up for an email provider, a twitter account, whatever, only to find out somebody's taken our name.  So we get creative, and reach a somewhat acceptable compromise.  After all, there are probably hundreds of Joe Bloggs' our there and they can't all share the same service login.  But in the long run, this policy will have to change.   Assuming for a moment that services will survive long enough for this to be an issue, all the Joe Bloggs' out there will die, and therefore all immediate permutations of their names will be blocked.  If that policy remained, it would drive us into crazier and crazier circumlocutions.  So service providers will uniformly close dead accounts - I mean, accounts of dead people.  Perhaps park the dataset somewhere.  And open it up to any living Joe Bloggs out there.  Who owns all the closed accounts?  The estate of the deceased?  Do they revert to the service providers, like pension annuities?  If so, will they be made publicly available to researchers?  To anyone for a fee?

These service 'handles' operate as points into which humans momentarily breathe life, before passing out the other side.  Dimmed, the handle waits for the next human to manipulate it, or to write it, if you're a fan of Derrida.  This image is not unlike Leibniz's idea of a monad.  And Google and Facebook are in the business of differentiating and integrating humanity.  Funded by ourselves as consumers whilst paying in turn for the experience of being different and the experience of belonging.

## Tuesday, 14 February 2012

### Money market investment yield and bond equivalent yield

In a previous post I mentioned that the economic return, or internal rate or return, or yield to maturity (YTM) of a money market instrument (one with a year or less of a term to run) can be calculated.  But this can't be all there is to the bond equivalent yield.  After all, why the name bond equivalent when yield to maturity or internal rate of return would be more on the money?  The answer is that the bond equivalent yield of that money market instrument is a yield translated to facilitate a direct comparison with a (US) Treasury bond with less than a year until it matures.  It is assumed the bond will pay semi-annual coupon, so the bond equivalent yield ought to take into consideration the compounding effect at the 6-month horizon.  If the last but one coupon on a US treasury has been paid, then the treasury bond's YTM calculation is the bond equivalent yield as described in the earlier post.  If, however, there are two puffs left on the cigar, so to speak, then the YTM calculation is slightly more awkward.

Why would you want to prefer to work out the equivalent (final year US Treasury) bond yield, and not just the economic internal rate of return?    Well, you'd do both, depending on your purposes, but there's be many reasons why you'd want to compare the money market instrument with the equivalent maturity US Treasury.

So, if you really want to know what a money market instrument's economic yield is (YTM), you are best to just ignore the bond equivalent yield (which would bear a lower yield number, since there is an additional internal compounding for those instruments expiring in more than 6 months and the more frequently an instrument compounds the lower the rate can be to result in the same present value), but instead apply the $\frac{F-P}{P} \times \frac{365}{d}$ if you happen to have the money market instrument's price already to hand.

US Treasury (notes and) bonds have a semi-annual coupon and they have an ACT/ACT day count, which amounts to 365/365 three years out of four and 366/366 on the leap year.  The indicator 'ACT' implies that the convention doesn't bother about months and instead counts actual days, making sure that a full year of days makes up precisely 1.0 years.

All Yield to maturity calculations share this in common - namely that they relate the market price of a series of cash flows with its valuation.  It is that single yield which equates the market price and theoretical price.  So you can see that to translate a discount basis into a YTM money market basis you equate their prices.  By the way you can think of the terminology bond equivalent yield as a shorthand for: 'money market rate re-based into coupon bearing equivalent-riskiness bond yield'

Recall that discount yield is $y_d = \frac{F-P}{F} \times \frac{360}{d}$ and YTM (no intervening compounding) is $y_m = \frac{F-P}{P}\times \frac{365}{d}$.  Express these both in terms of $P$ and equate them, then solve for $y_m$ and you get  $\frac{365}{360} \times \frac{y_d}{1-\frac{d}{360}y_d}$. The best way to remember this formulation is as a pair of boosters applied to the smaller $y_d$ number.  The first booster corrects upwards for the 360 day basis of $y_d$ and the second corrects up for the economically irrelevant 'divide by face value' convenience of the definition of $y_d$.

There's a cleaner formula for this, which is $y_m = \frac{365 y_d}{360-y_d}$, assuming that you'd like your annualised real yield to be based on a 365 day year.  You'll sometimes see $y_m$ referred to as the money market investment yield, and perhaps it is quoted on a 360 day basis, which would be simply $y_m = \frac{360 y_d}{360-y_d}$

## Wednesday, 8 February 2012

### The archaic discount yield and the truer bond equivalent yield

In the world of fixed income there are lots of different kinds of yield.  It can, and does, get confusing.  I'm going to start simply and take it slowly.  First, imagine a simple interest calculation based on two cash values at two different points in time, $C_{t_1}$ and $C_{t_2}$.  This represents a return of $\frac{C_{t_2}-C_{t_1}}{C_{t_1}}$ for that time period $t_2-t_1$.

Now instead of considering these two points as equally important, lets emphasise one, then the other.  In other words, let us leave the realm of mathematics, which doesn't care too much about what these two points in time actually mean to us. First, imagine $C_{t_1}$ is a sum to be invested, your starting capital, as it were. Then we're likely to say we gain $C_{t_2}-C_{t_1}$ (or lose it if negative).  This is what you might call the common-or-garden interpretation of a return: simple interest (no intervening compounding dates internal to the end point dates $t_1$ and $t_2$, an asset starts with value $C_{t_1}$, and over period $t_2-t_1$ it grows to be worth $C_{t_2}$.  This is what the person on the street usually means by a return.
Next switch focus to $C_{t_2}$.  From the perspective of this point, then we might call  $C_{t_2}-C_{t_1}$    the discount we would get off the full price $C_{t_2}$ if we were to have owned it at $t_1$.  The rate $\frac{C_{t_2}-C_{t_1}}{C_{t_1}}$  is called the bond equivalent yield.

Bond equivalent yields are often annualised - to facilitate comparisons.  You can do that in one of many ways, but in all cases, what you're trying to do is find out what the $t_2-t_1$ time period rate would mean if you could continue it for a whole year.  These different ways of scaling rates along the time axis are called day count conventions.

The history of lending naturally segments time into day sized chunks.  In the olden days, there would have been little practical point in getting any more fine grained than a day (except perhaps for periods of hyper-inflation).  And of course, human culture is permeated by the seasonality of the whole year.  So often you're flitting between a days-level view and a year-level view.  By many human calendars, there are also months as in-between  time periods.  So you'll find some day count conventions taking the month time period into consideration too.  For quite some centuries, loans have been made on a multi-year basis, on a month-by-month basis and even, for large amounts to large borrowers, perhaps even on a day by day basis (today's day by day lending capital markets have many overnight lending activities).  It almost always comes down to counting days - either directly, or in assumed 30 day or 360 day or 365 day chunks.

The time unit we typically all settle on to facilitate comparison is the good old fashioned calendar year.  And the most natural of all the ways of scale a $t_2-t_1$ time period rate into a corresponding annualised rate is to multiply by $\frac{365}{t_2-t_1}$.  This is natural in the sense that it quite closely corresponds to reality, since we usually have about 365 days in a year.  Even if you're comparing two loans which only last a few weeks apiece (say, a 2 week and a 3 week), you'd still look to annualise them, based on an agreed or common day count convention.

A second major alternative is to multiply the  $t_2-t_1$ time period rate by $\frac{360}{t_2-t_1}$.  Any why 360?  Well, this is an example of the influence of the month on interest calculations.  Even though there are 12 months in our calendar year, they're quite different in length, ranging at worst from 28 to 31 days.  That's a 10% difference right there.  Lenders don't like this complicated variability so they invented the concept of the 30 day month.  A notional period of time which has the desired advantage of all 12 of these 30 day months being equally 30 days long.  So formulae could be developed which treated any month as the same as any other.  But $12 \times 30 = 360$.  Hence the need, in some markets, for the day count convention which scales the $t_2-t_1$ time period rate to 360 days.  What you're doing is seeing what the corresponding rate would look like for 12 equi-length pseudo-months.

So the two major forms of annualised bond equivalent yield are  $\frac{C_{t_2}-C_{t_1}}{C_{t_1}} \times \frac{365}{t_2-t_1}$ and $\frac{C_{t_2}-C_{t_1}}{C_{t_1}} \times \frac{360}{t_2-t_1}$.  Remember, the fixed income market has been around for a long time, and the shortest time period is the day.  Continuous compounding came much later.  So why don't I just replace time period $t_2-t_1$ with $d$ days.  And, to make the terminology slightly more bond-familiar, lets call $C_{t_2}$ the final or face value $F$ and $C_{t_1}$ the initial price paid, $P$, $F-P$ being my discount.

In other words,  two important varieties of annualised bond equivalent yield are $\frac{F-P}{P} \times \frac{365}{d}$ and $\frac{F-P}{P} \times \frac{360}{d}$.  The measure closest to the true economic return is clearly the 365 day based one.

So what about the discount yield?  Well, it is an inferior measure, dating back to a time when people were doing a lot of these calculations per day, by hand.  Replace the economic return $\frac{F-P}{P}$ at the heart of the bond equivalent yield with a more convenient denominator, but one which makes less economic sense: $\frac{F-P}{F}$.  Usually $F$ is a face value, for example 100 USD.  Clearly dividing by a face value is so much more convenient than dividing by, say 98.34, a current price.  The discount yield, too, could be annualised, and again you could annualise to a real year or to a fake, homogeneous $12 \times 30 = 360$ year.  In the world of short term fixed income, with simple interest, it turns out that the 360 day based annualisation was preferred, certainly in the US market.  This means that the annualised discount yield is often expressed as the formula  $\frac{F-P}{F} \times \frac{360}{d}$.

Once you know the price, the face value and the term, you can calculate either yield directly.  Notice that the discount yield will always result in a smaller yield than the economic return of the bond, since the price of these zero coupon or discount bonds is (almost) always less than the face value.

But why bother with the clearly inferior annualised discount yield when you can have a 365 day based economic return calculation, the annualised bond equivalent yield?  Well, because markets have a history which can't easily be eradicated.  Once certain markets started producing quotes in annualised (360) discount yields, there was no going back.  The convention of quoting a rate in a 360 day annualised discount yield basis pops up in many markets, but none so large and so important as the US Treasury bill market.  But before going on to look at the T bill market, there's one more wrinkle in the bond equivalent yield which needs to be laid out.  That'll be the subject of my next blog post.

## Saturday, 4 February 2012

### The only option - chasing joyriders

Equity derivatives is a complex place.  Microsoft shares have options (calls and puts) on them across a whole range of maturities and strikes.  Like a pair of tables.  Think of excel. Say we'll create a separate excel spreadsheet to document all of the calls and puts you can have on Microsoft.  In that sheet we'll have two tabs, one for all the calls currently in play, one for all the puts in play.  In each tab, imagine expiry dates running across column-wise at the top, and strikes running down the tab.  Now that excel sheet gets replicated, one for every equity name out there which also has options on it.  There'll be an Apple spreadsheet, one for Wal-Mart too.  Now, the set of options currently in play for Microsoft are different from the set of options which were in play last year.  Or a decade ago.  Or which will be in play a decade from now.  Likewise for all the names.  That's quite some variety.  Perhaps we can impose some organisational structure on these sheets - let's imagine storing each Microsoft sheet for a given day in a different sub-folder on our computer.  Think of all those folders.

Now for something surprising.  This can all be replaced by a single excel sheet, with a single excel tab.  This is because the theory behind the option pricing model defines an option's value in terms of a couple of parameters of the underlying share - namely its price, expected volatility and dividend payout structure over the life of the option.  Stripped down like this, all shares look identical to the option pricing model.  It doesn't care which particular share it is a derivative on.  Only that there's always a price, expected volatility and dividend payment structure available to it for any moment it is asked for a valuation.  The option also needs to know the current values of a couple of low risk interest rates, and finally it needs to know four definitional constants - start up parameters, if you like - Is it a call or a put?  Can the current holder get out of the deal during the life of the option, or only on the final day?  Just when is that final day? And what's the line-in-the-sand strike price of the option, the price point around which it operates?

This single model can actually be implemented in a couple of ways, but they'll essentially give the same result.  And I'm deliberately ignoring the existence of so-called exotic options for now.

What about history?  Microsoft is now trading at 30.24.  Way back in 2005, Apple was also trading at 30.24.  If they had the same dividend payment structure and expected volatility, and if interest rates were the same at those points in time too, then all the calls and puts you could imagine on these two names ought to be valued at the same levels.    It doesn't matter one bit to this insight that in reality interest rates are never exactly the same, or that volatility expectations are never exactly the same.  The insight still holds - there is only one option.

The reason for this has to do with how the recipe for the option model is defined.  Imagine a car filled with joyriders barrelling down a quiet highway.  The driver's been drinking, he's playing loud music.  his drunk friends are urging him to go faster, his cute girlfriend urging him to go slower.  That is the model for the underlying stock price.  Now the option represents a second car whose job it is to follow the first car.  It accelerates and decelerates based on what the first car is actually doing.  The interest rate could be something like the gradient of the road.

The option car doesn't need a corporate valuation model - which in this case would be a guess as to which person the driver cares to listen to at any moment, or a prediction of what his right foot will do next - the option car simply tracks the car as best it can.  The car contents can be a black box to the option car.  Once the option car learns this trick, it can track any driver on any road anywhere, for any time.

Even more dramatically, you could pre-calculate all these numbers in an excel sheet and store them down.  From that point on, the valuation of any stock's option would be a straight table lookup in excel.  People would never do this because the lookup table would be too large; running the recipe is a better time-space trade-off proposition.