Tuesday 25 December 2018

crescet and peredit

It seems to me that the wealth process, which is largely fed by the income process, ought to be the primary process in economics for consumption.  That process, crescet, diminishes through consumption (and taxes) and grows through investment of some (high) fraction of one's wealth (or all of it, with a fraction assigned to safe cash) and a series of jumps on the wealth process through unexpected spikes and troughs on the wealth process.  Rather like Modigliani's lifetime consumption hypothesis, and like Friedman's refinement for non-windfall income, there needs to be a model of the fraction of wealth which is considered appropriate for investment.  There may be a much tighter constraint on the wealth process around leverage (and borrowing).  But there essentially needs also to be a fraction of wealth which would substitute the need for income borrowing.

As a matter of western world 2018 fact, we're mostly always in a position of borrowing, thanks to the desire embedded in most of us to own our own houses.  The next question which arises is, how to model the wealth process mathematically, and how the fraction given over to investment ought to evolve over the lifetime of the individual.

Sunday 23 December 2018

crescet: waste and pine

"..man in short that man in brief in spite of the strides of alimentation and defecation is seen to waste and pine waste and pine.."  Lucky





The relationship between wealth and income is clearly strong.  There's an argument in economics that we do in fact consume based on our expectation on our lifetime income (under either a simplifying assumption of untrammelled friction-less borrowing from one time period to another or given a borrowing constraint as a function of wealth).  A (real) wealth process is in general replenished by the remnants of income minus consumption, per unit time.  It is also replenished by a rate of return on savings, and there's probably a jump process (in the sense quants use when modelling price action in derivatives pricing) on wealth - unexpected increases and decreases in wealth due to either an inheritance, an unexpected and large cost, etc.

In practice, the relative proportion of wealth over income (or expected income, or average income, smoothed over a lifetime) determines how much one can borrow, and one's consumption response function during times of economic hardship.  In other words, crescet drives your current (and expected future personal credit spread). If the amount you can borrow is always expressed to you as a percentage of your wealth,  then the associated borrowing cap and associated credit spread for loans are both functions of crescet.  Clearly it works well in a theorist's modelling of crescet if one can reasonably assume that there's a stable pattern of consumption based on a lifetime income  model.  It takes the variance out of the expected income process and allows for the possibility for doing a simpler lifetime crescet model.

The natural initial assumption, I think, is that consumption is based on a lifetime wealth process (which has embedded in it a lifetime income model).  It is also natural to assume that lending to consumers will always have a budget constraint based on some simple measurable proxy for the lifetime wealth process.  Current income appears to be the provable, easy to calculate proxy of choice, certainly in the western world.  

In the western world, it is a decent first approximation certainly for the last 70 years, to assume that the average person's wealth is held jointly in the property they live in, and in their pension.  However, when you look at this closely in the context of consumption smoothing and lifetime income, it becomes harder to make a distinction between wealth and sensible foregone consumption for future times.

Needless to say, the general flow of theory on consumption modelling goes like this: classical period economists model consumption largely as a function of interest rates.  Fisher introduces inter-temporal consumption and a budget constraint. Keynes in effect operates on the assumption that the average consumer is very much bound by the limit to borrow (i.e. he theoretically honours a real constraint on borrowing not present in theoretically pure models with frictionless borrowing).  He also, like lenders, approximates income as this year's income, and has no model for income to be shifted from one time to another (saving).  Modigliani added this element in the 1950s.  By far the biggest discontinuity in the lifetime income model is the fact of retirement, so in a sense, Modigliani introduces retirement to the model, and brings with that an ability to save and borrow.  I think this probably is a sign of the times.  In the early 20th century of Keynes, only about 10% of homes were privately owned.  By the 1950s of Modigliani's time, the US already had 50% home ownership.  So having a model which accounted for this significant fact of lumpy consumption in a person's life was an advance.

Friedman further made a sub-distinction between that fraction of 'reliable' and 'windfall' income, and considered the former more determinant in the consumer's life.  Windfall income adjustments (a legacy from a rich relative, a lottery win, a sudden hospital bill, a windfall tax) ought to have a net effect on one's wealth, depending on how that is consumed or saved.

One final chronological piece of background - the 1930s of course foregrounded the problem of persistent unemployment - a phenomenon not 'solved' by adjusting the real interest rate.  This leads Keynes to pursue aggregate consumption, including the less controversial personal consumption function (my current focus) and the much more variable investment function.

Lastly the rational expectations mob arrive on the scene to point out that if the consumer is doing all this good modelling and smoothing of lifetime consumption, then any actual changes to consumption would therefore have to be random (meaning unpredictable).  There's something quite beautiful about that step, even though the assumption made in it is initially hard to believe.

Sunday 18 November 2018

crescet pool

The traditional 'asset allocation' industry typically makes 'investor risk appetite' your problem, not theirs.  They then perform this outdated pre-MPT analysis of the kinds of asset your risk profile might need.  In reality, you need them all, in toto, and your risk appetite only drives the degree of leverage on that total portfolio.  Secondly, using some nineteenth century maths on annuities and perpetuities, they take your requirement of needing a fixed amount at a date in the future, run the formula, and work out what your monthly premium ought to be to achieve that future cashflow.  Note this too works only by eliminating all asset types and strategies except relatively safe loans/bonds, together with a hope that inflation doesn't destroy the real future value.  However, if you're willing to accept uncertainty in the primary return stream (which becomes increasingly OK  the longer your relevant time horizon is), then you can replace a safe (close to risk free) return with increasingly risky returns.

But I think one should try to build a model of the risk appetite, which is to say  a model of the wealth process.  This would be a rather complex process.  Stochastic no doubt, and with feedback from the actual experienced output of your core investment model.  It is much grander (much more destined to failure too) than knocking off a perpetuity to pay for your children's university bill.

Before doing that, it might be worth thinking if there are any macro or qualitative insights which might be gleaned by thinking about a world where everyone, rich and poor, operated a wealth process.  Are there implicit biases in the behaviour of investors based on how wealthy they are?  Secondly, how distributed is wealth?  How does that matter?

Why the long face?


Why the long bias in strategy allocation?  Well, first of all, logically, you need to have a long  market first, and of a sufficient maturity and depth, before you can create institutions enabling shorting activity.  And you need to be able to short in order to perform many hedged and complex strategies.  A set of farmers needed to pre-exist before an institutional framework existed which allowed the possibility of shorting commodity prices.  The primordial investment activity is the handing over of capital for the purchase of goods and/or services that could result in the growth over time of that capital.  The primordial investment activity is the act of sowing seeds, tending to those seeds, managing their growth until a yield is achieved.  The primordial investment activity is in the act of acquiring a herd, having it propagate new members of the herd, while tending to to herd.  At its most basic, you can see that the investment that is being made is often in terms of the time, manual labour and thought, planning and management.  This is the initial investment.  And the outcome is a commodity which is valuable to humanity - wheat, milk, meat.  Here it is clear how work is seen as an investment.  It was originally performed as a machine which transformed effort, planning, time and intelligence (the labour factor, as economist rather plainly summarise it) into goods (commodities).  Insofar as cash is partly an agreed external measure of the value of a basket of goods, then so too can the primordial value be measured.  This is how François Quesnay would have launched the ideas of the tableau économique.


Apart from the logic of chronology, it is a long standing commonplace that investment capital is often used to build or make something.  In modern times, this is especially so, as capital pays for the location, the input materials, the tools and the skill-set and effort of a workforce to manufacture a desired output of production.  In the modern economy, this product usually has a price.  If the primordial investment activity can be thought of as human capital and effort manufacturing the means of immediate physical survival, then this widens to the manufacture of more sophisticated product, requiring more refined division of labour, more expensive tools, a more costly locale.  In short, as the product of economies become more sophisticated than immediate means of survival, those products became more expensive to make, and that made it harder to launch an entrepreneurial episode without having access to capital.  Secondly, there entered into the process an idea of economies of scale, and economies of scale allowed the unit price of goods, all other things being equal, to reduce, which in turn led to more people to be able to afford it.  But scaling an enterprise up in order to reduce unit prices was a costly operation.  Up front capital, in size, was also required.  By these means, the manufacture of goods for broad consumption came to require large sums of capital.

The insurance business was the first great short.  Certain businesses were happy for certain risks to be taken from them.  This act is a kind of hedge, a short.  For farmers, agreeing a pre-harvest price is an example.  For shippers, eliminating the potentially ruinous risk of a loss at sea is a kind of short.  Another early institution of hedging was the conglomerate.  The idea that a business might own divisions which contained businesses perhaps exposed to different parts of the business cycle.  In aggregate, an enterprise might survive better with this build-in diversification benefit.  In a similar vein, having sales distribution agents in different geographies minimised certain risks.  There is evidence of marine insurance in fourteenth century Pisa but primordial economies would also have contained various forms of mutual aid, institutionalised for example in community granaries.  The Greeks and Romans created benevolent societies, proto-guilds, which in effect provided life insurance.  Modern life insurance kicked off after Pascal and Fermat injected genius into the proto-subject of probability theory.  

Futures, and in particular, commodity futures, were the first great single asset of the shorting industry.  Just as life insurance as an industry is an idea based on one's understanding of annuity tables - based on probability theory and correlation of events, so too did the futures market evolve in the eighteen and nineteenth centuries.  If you understood your correlation well, you could make sophisticated estimated of how likely any given kind of person was to die, and how related or correlated that death was to others.  And if you ran your business to sufficient scale, then you could expect to make fairly expert predictions of the expected payout your company might have to make per year due to death.  With this expected cost, and with the knowledge of the size of your insured population of customers, then you could set a premium in a way which protected everyone and made your firm a profit.  With farmers and with properties, these rules were also true, but with agricultural insurance, there was a higher risk of 'correlation one' events - severe weather, for example.  One then needed to have an institution which was either geographically diverse, or which had reserve funds for the occasional severe weather event.  So, with human life and with weather, the institution was using statistics, large numbers, to allow them to arrive at expected yearly costs to run their business.  From this perspective, the futures exchanges were simply insurance hubs with very standardised products.  This allowed the insurers to diversify some of their risks when it suited them.  To do that, it required one further institutional category - the speculative investor.  If an insurance company was providing this kind of service to risk-adverse makers and producers, who was it on the receiving end of the futures contracts which the insurers wanted to use to lay off their risks.

First of all, it has to be remembered that it was (and is) possible to run an insurance business in the absence of a futures market.  The collection of skills which resided in understanding how the correlations worked in death, in accidents, in crop yields, were self sufficient, but that set of skills would spill out into the speculative community.  Imagine you and your work colleagues knew about as much as there was to know about managing the risks associated with running an insurance business.  And imagine you had the idea that you could exploit that knowledge but without having to insure the makers of industrial or agricultural goods.  If you understood the risks, the cycles, the opportunities, perhaps you too might be tempted to start to speculate with the new futures institutions of the nineteenth century. We have the price takers, the makers of the economy, and we had the price makers, the mathematical model builders of the financial and insurance industry, and soon thereafter this set of model builders began to separate into the market makers (who made a profit by offering insurance products to the real economy) and speculators, who felt they could trade those markets themselves to make larger profits.  This division, somewhat related to the 'buy' and 'sell' side of modern finance, somewhat related to the investment banking / hedge fund division of modern banking remains with us today.  Political regimes in the West have sometimes wanted these two sets of model builders to be separated, and at other times, they've conspired to stay close together.  Again, from the point of view of conglomerate-as-diversifier of risk, it is clear why the companies themselves want to stay together and given that there are more risks taken on the speculative side of the business and that model will have low correlation to the insurance business, since they're in aggregate on opposite sides of the trade.

On this reading of history, the futures and options markets play an essential and long running role in the management of risk in developed economies.  Out of these activities has come the concept of superior returns for investors.  The promise that your wealth process can be super-charged, leading to you being financially better off than otherwise.  And with it too has come the possibilities of the various forms of gambler's ruin.  With inflation and transaction costs nipping at the heels of your wealth process, yet with the promise of superior returns to fund your lifestyle needs - the purchase of all those expensive to make products - yet without falling for gambler's ruin, the strategy allocation industry was born, and the long bias in investing is the main reason why the industry sometimes goes by the name of the asset allocation industry.

Sunday 30 September 2018

returns, volatility of returns, correlation of returns

If all investment occurred via a single product, with a single pattern of returns, and no choice, and if this happened over a sufficiently long period that the short term swings of volatility become secondary when measured against the timeline of a typical investor's expected life, then the only one fact you can survive with is the (long term) expected return of that product.  I refer to it as a product and not an asset because I imagine it to be the offering of a company or set of companies which may have the freedom to manufacture this product.  

But reality isn't like that.  And as soon as a second product emerges as a choice (or even if you examine how the company manufactures this product), then correlation and (therefore volatility) enter into the frame.  

In the history of major assets, cash was invented first.  (Of course, loans existed before all that, and were a huge part of early human culture - the loans being loans of non-cash valuables for non-cash rewards e.g. slaves, food; these goods, like cash, may also have been understood to be fungible and tradeable).  Not surprisingly, the place which brought us writing also brought us the first bond.  The city state of Nippur in Sumeria offered one.  Italian city states pioneered state bonds as far back as the twelfth century, quite a while before the official story that Amsterdam and then the Bank of England invented them.  Certainly they set the modern pattern.   Shares were known certainly in Roman times, as was property, which had deep underpinnings as the earliest Greek and Roman religions were domestic hearth ancestor religions.  This simultaneously raised the cultural value of property but also introduced a whole bunch of restrictions, rules, taboos around selling property.  As the Roman republic evolved, and as class war between patricians and plebs loosened the grip of the old domestic gods, property as an asset class began to evolve too.

Inflation, of course, is not an asset.  But it is the force which makes cash experience volatility in real terms.  So these are the primary financial assets:  Cash (and loans), Property, Equities, Bonds.  And inflationary pressures contribute to the volatility of all four of these assets.  The primordial question is to work out how much each one will return to you, and how uncertain that return could be, and finally, to design of set of weightings which might exploit their time-evolving correlations.

By the time Markowitz came to develop the standard maths of modern portfolio theory, he addressed just two assets, equities and bonds.  Why?

Sunday 23 September 2018

crescet and titubit

The speed with which one's wealth grows, and its absolute level, are tied to one's life style (one's consumption of one's income).  A useful simplification is to assume one's income derives largely from one's wealth.  Economically, this is almost completely unreasonable, since it applies only to a vanishingly small fraction of humanity.  One then needs to spend to live from this wealth.  There are however minimal quality of life spends which may imply several modalities in the relation between the wealth growth process and the spend process.  I assume for simplicity that wealth is sufficiently large that the income spent can be made in a way which still leaves wealth growing.  Put another way, there is an assumption that the wealth process grows faster than both inflation and the daily consumption of your lifestyle.   A second critical threshold is for now also ignored - as with the case where the lifestyle spend significantly impacts the wealth process, transaction costs also can incur a third hurdle to overcome.  These assumptions clear away much of the thrust of the Darst book on asset allocation.

Next, an implicit starting assumption is that wealth at time $t$ may be considered as residing in one or more currency (short term fixed income) buckets.  One then imagines that the mean value theorem can be applied to the act of taking financial risk above this risk free (globalist) position.  That is to say, in equilibrium, the entirety of the job of strategy allocation and capital deployment can be waved away as solved for now, and modelled as a single 'bet' over an appropriate time frame, whose outcome can be a win or a loss.  One then determines the ideal bet size, per unit of time, based on the mathematics of Gambler's Ruin.  That is to say, that the average bet size can be no bigger than some fraction $\delta$ of wealth at point $t$ if volatility (and long term, ruin) is to be avoided.

Of course, in reality, the complete opposite applies with titubit.  Bet sizing is often ignored and instead one's lifestyle generates the major driving constraint to investment returns variance tolerance.  In short, our lack of funds makes us bet too big - this together with transaction costs, destroys our wealth.

Thursday 20 September 2018

Strategy allocation: a wealth process (crescet), a volatility constraint (titubit) and an expected life (fugit) and a cycle (circuit)

In chapter 2, Darst tries to carve up the space of approaches to 'asset allocation' through dimensions of style, then how strategic the approach is, and finally how quantitative the approach is.  As I mentioned in the last post, I think the 'style' dimension is bogus.  This in the limit can be replaced by owning the market of available strategies in toto), in their market weights, and then by implementing risk appetite purely through levering the in toto portfolio.  Next his seemingly clear quantitative versus qualitative  distinction breaks down too - for an ideal strategy allocation algorithm, the parameterisations are empirically calibrated and the discovery of new strategies are qualitative, whereas ideally the implementation, given a broad parameter set, ought to be quite algorithmic and computationally tractable.  Again ideally, the re-allocation decision might in theory be near-real time.
Finally, the dimension of 'strategic' v 'tactical' is the difference between Kant and Machiavelli. 

I think you want the algorithm to be as autonomous as possible, and to make a call on the strategic/tactical dimension based on the following inputs: where you are on your own expected wealth process and your expected lifespan.  Your spend process ought to follow from these two, and shouldn't count as an input.  Likewise this set of input parameters can be used in the determination of how much leverage to use (how long do we think it will take us to get there).  Your expected spend (and the lumpiness thereof) is really a (time-dependent) constraint on the volatility you desire on your wealth process.

The starting point (the long term equilibrium point) would be based on the maximum likelihood weightings, based on as much data as there is available for the strategies.  If one then subsequently had a model of strategy cycles, then that would be burned in too, to a degree proportional to one's confidence in the cycles model.  The mean value theorem guarantees that your long term equilibrium parameters are a good starting point, in the face of no certainty about cycles at all.

Crescet, titubit and fugit are facts about you.  Curcuit and the long term equilibrium weightings are parameters of the strategies.

Wednesday 19 September 2018

leverage and the universe of strategies (the strategy market)

In just the same way that prior to Markowitz investment advisers performed this charade of matching the riskiness of a set of single names with the risk appetite of the customer, safe stocks to widows, racy high growth stocks to adventurous greedy investors with long time horizons, so to does Darst expect his potential customers to be willing to accept this cumbersome and sub-optimal selection process to satisfy.  It doesn't.  Just as with CAPM, one can satisfy the customer's risk profile simply by leveraging up (or down) on the market portfolio, likewise one ought to be able to sample strategies in the same way - having the full universe and weighting them with leverage according to their risk/return profile.  This is indeed what the population of managers of 'fund of funds' do.  They take investor capital, then, depending on how risk hungry they are, they avail themselves of more or less PB leverage on their collection of strategies, and reap a hedge fund average return.  Strategies, like companies, can be born, can run to exhaustion, can be merged, delist, go bust.  And like stocks, the space of strategies can be partitioned into its own sector or industry - just as GICS partitions the universe of stocks.  And occasionally new 'strategy sectors' can be born (just as, eg real estate can become a new GICS sector).  

How would this index of all strategies arrive at weightings?  Probably by capital invested in the strategy.  If this drove your allocation sizing, then you'd always get 'the market'. And just like with ETF providers who model the whole market by targeting a subset only of representative stocks, accepting a degree of tracking error, so too in theory with strategies.  You could select representative institutions or canonical implementations of the main strategies, and this would serve you, to a sufficient degree of error, as a proxy for the market of strategies.  These days, factor models allow you to get a handle on the factor exposure, so I think it ought to be possible to apply the same technology, given the right data, to the issue of constructing a portfolio of names which, in toto, sufficiently closely tracks the full investible market. 

By sheer AUM allocated (the equivalent of market capital), I would suspect that long equities and long credit would simply dominate the weightings.  Also each strategy has its own internal (eg asset based) leverage, so the concept of a singular leverage value which can be tweaked up and down needs to be revised.  Some strategies are inherently more capital intensive, some less, and the meaning of 'increase leverage to get more risk' needs to be transformed into a series of leverage adjustments for each of the strategies.  A further point is that one can also have in the equity world the concept of an equal weighted index, which typically gives you more risk and more return.

In practical terms, find the set of n ETFs which best represent the universe of all strategies.  Find the weightings by invested market capital.  Buy the basket.  Lever up or down, depending on your risk profile.  Re-balance on a very slow timescale.  Unless of course you have a cycle based model for investment weightings.  Perhaps cyclicity would be one of the 'factors' in a strategy factor analysis.  What else might take the place of, say, fundamental factors?  

I wouldn't put the strategy equivalent of market cap in there, since this is expressed in the weightings.  Perhaps sensitivity to volatility, to equity, to credit, and to interest rates.  Perhaps asset class exposure, country exposure, sector exposure and inherent leverage of strategy (gross exposure over capital allocated) and liquidity.  These would be the areas I'd be looking to get strategy factor models out of.  They also provide more or less well known hooks into modelling cycles (credit conditions predict economic contractions,  the debt/equity relationship of corporates could see itself recapitulated in the relative allocation weights for debt and equity.

Sunday 16 September 2018

The anti-FOMO movement

There are n strategies, each with returns $r_i$.  Ranked top to bottom, so $r_1$ is the strategy with the highest return (long term).  Why not just put your wealth all in strategy 1?  Putting only a fraction of it in 1 and fractions in 2,3,...n leaves one with a feeling of missing out.  I suspect if you live to be 640, then this would be the effective result of the ideal allocation strategy.  Indeed if you have a 60 year perspective, this might also be the case.  But history doesn't always repeat itself.  So you can never be sure the future will continue sufficiently to be like the past.  Hence you'll want to diversify.  For example if you are Russian, living at the turn of the twentieth century and happened to note in 1901 that the St Petersburg stock exchange was your $r_1$, and decided to put all your wealth in there, then you'd be in for a shock when the Russian revolution came and wiped your wealth to zero.  If you were an ultra risk-adverse German post WW1 and thought you'd keep your money in nice liquid deutsche-marks, then the hyper-inflation would have likewise wiped you to effectively zero.
The degree to which you trust the institutions which underpin the strategy returns you feel you have access to is the degree to which larger and larger fractions of your wealth will go into strategies 1, 2 etc. rather than into tail end strategies.  Conversely, the degree to which you are uncertain of the future of those enabling institutions (and this, to be sure, is an uncertain act of political tea-leaf-reading) determines how distributed your wealth will be.  Your degree of confidence in strategies 1, 2 also grow to the extent that your future wealth-investment time horizon is long.

Besides the above unknown unknown, is the idea of correlation.  If all strategies 1,...,n are fully correlated with each other, then each of n is as good, in this one respect, as all the others.  But the degree to which any two (or more) strategies are uncorrelated or lowly correlated, opens the possibility that there was a combination of these strategies which was ideal, in some wider, as yet to be defined sense.  

So a world with a lot of serious unknown unknowns presents a difficult environment for the ideal strategy allocation algorithm, as does a world with cross strategy high correlation.  Thankfully so far the world we live in is somewhat known, somewhat predictable .  And this is the space that the theory of the ideal strategy allocation algorithm can work within, where the past can tell us something about the future, and where strategies have less than perfect correlation.


Floor entropy and ceiling noose

The whole space of strategy allocation is shaped by two massively important risks - inflation risk and gambler's ruin.  The first bites your wealth from below, when your allocation strategy overall is too focused on principal protection, where the return can be below the inflation rate, and the second bites your wealth from above, when your allocation strategy overall is focused on principal growth and your 'bet on green' at the roulette table of life stops you out and you go home early.  Each fate is ugly, you either dying in dog food penury or dying young in a bloody accident.  It ought to be the goal of an ideal strategy allocation approach to avoid both outcomes and instead enjoy healthy lunches - free and paid for - over as long a stretch of your life as you can.

This blog post is in general a post for everyone, but of course poor people first need to arrive somehow at a pool of investable capital (separate from their day to day living costs and the capital they have for investing in their business).  I think it is a fair statement, at this early stage, to suggest that younger folks with capital can afford to be closer to gambler's ruin than older folks, since they can trade their labour, brain, body, time for paying their today costs, whereas post-retirement oldies have less flexibility and hence have a big income draw-down demand.

How close a young investor gets, of course, to gambler's ruin, is a cultural question as well as an economic one.  Their appetite for risk ought to be higher, insofar as making the tilt for wealth growth over wealth protection is paid for by their greater expected lifespan. I've heard it said that private equity / startup investors like to hear from a founder that they failed once or twice in the past.  Secondly, bankruptcy law is all about buying back in gamblers who have reached ruin with their firm.  Our culture of long term GDP growth has some of this risk taking burnt in.  There's a sense that the fable of Icarus is seen not only as a warning but also as admirable somehow.  Back through human history, we have moved forwards in time by combining our prudence with our spirit of adventure.

And vigour, life, vitality, novelty, creativity, growth, these are all inter-connected concepts culturally.  As opposed to self-sustainability, entropy, predictability, familiarity, maintenance.  But the ideal strategy allocation algorithm must partake in all of those concepts.   Unfortunately, all too often, both of these existentially definitive risks are under-emphasised on behalf of investors.  But wealth generation is in the limit a lifetime activity (longer, for companies, or for aristocrats or for anyone who plans to leave an inheritance for their loved ones).  It is a common fate for us collectively to understand the importance of long term planning only as we get to be old.

Friday 24 August 2018

The Art (pah) of Asset (pah) Allocation

Of course calling a book The Art of Asset Allocation is just asking for trouble. Back in the olden days of investing, you bought assets, the primary uncertainty being what fraction of your investable wealth was to be allocated to which broad asset category.  These days this has been generalised to strategy allocation, for the financial industry (and for a growing number of individuals too).  You allocate to equity long short, to volatility arbitrage, to mergers and acquisitions, to capital structure arbitrage, to convertible arbitrage.  Each strategy, in other words, could contain long and also short positions, subject to financing costs and limited by a degree of leverage typically offered to hedge funds.

Hedge funds were created in 1949 by Alfred Jones (covering equity long short strategies); convertible arbitrage   was pioneered in the 1960s by Ed Thorpe, after the casinos banned him from his card counting and expectation-based betting; volatility arbitrage blossomed in the years after equity index option trading on exchange met the Black-Scholes calculator; merger (risk) arbitrage had already made it into the third edition of Benjamin's "Security Analysis", 1951.  Capital Structure is much more high powered than that, and had to wait until Merton's 1974 model of credit in terms of the set of assets and liabilities (including residual equity) of the firm.

A key fact about successful trading strategies is that, by definition, they become popular and 'over funded' (tragedy of the commons), leading to more money (and, on average more diluted talent) chasing the same market.   This fact ought to be written in stone on any 'guidelines for strategy allocation' work.  It is continually chipping away at the returns associated with these second generation strategies.

The first generation of strategies are the purchase of assets and liabilities on a buy-and-hold basis.  Here, the term would have been relevant.  The primary question facing first generation investors was: how much of which asset class to hold, and for how long until the next re-balance process.  This first generation of strategies of course is still around, and super slim, in the form of the burgeoning ETF markets.  For a modern take on the first generation investors, there were two paths you could go down.  The old (but still popular) and CAPM-ignorant (pre Markowitz) strategy of not just buying the market, but attempting to buy sectors (or themes) in ratios not related to their market cap ratios.  This shades into thematic investing.  The idea here is the investor knows something the market doesn't  The finance professors are usually not so keen on this strategy, which I'll call gen-1-slanted.

The second path a modern investor may take when it comes to their buy-and-hold allocation is to buy the market en toto  and to fine tune your risk appetite via using leverage to achieve whatever level of relative volatility (or beta) you'd like.  If you want to own the market, the path is clear, with ETFs and futures.  If you currently express a less fulsome risk appetite, you place some of your funds in cash or treasuries, then achieve your <1 beta.  Consequently if you want more risk, you lever up your ownership of the market (eg in futures, in leveraged ETFs).

Which brings me to Darst's use again of 'art' here.  If the real problem which he's expecting his investor clients to solve is one of understanding the relationship between all these strategies, then that's a big ask for many investors.  Especially the part where he asks investors to understand not only the returns, volatility of each of these scenarios, but to understand their fundamentals and valuations, together with technical and liquidity dimensions, plus market psychologies on top of all that.   In other words he's claiming it is an art then expects the truly hard part to be performed by the investor (or perhaps a further set of costly advisers).

A key philosophical question which comes up, and for which the Markowitz approach may not be sufficient, is how many different kinds of strategy could there be, and how does one allocate between them.  How stable can they be?  What is the evolution of their life-cycle returns?

There's a great confluence of fairly simple mathematics here - gambler's ruin, covariance matrices, regression / series analysis - which will provide the intellectual backbone to a proper look at modelling the act of optimising the spread of your investment wealth across an unknown number of life-cycle-sensitive strategies in the face of uncertainty.  This book doesn't go anywhere near this, but I shall carry on reading it.

Wednesday 22 August 2018

Will you still need me, will you still feed me, when I'm 640

Imagine a world where humans lived much longer than their current 70-80 year range (for Westerners).  Imagine they lived 640 years.  Earning just a single 1% above the prevailing inflation rate would transform one unit of capital into 601.  That's surely going to be enough to retire on.  One presumption here is nothing about the economy changes, but in a sense this could change everything - for a start there'd be a lot more capital seeking a return.  Also, at what point in those 640 years would we decide to stop working?  Nevertheless, that's an assumption of this post.

Would it be enough to retire on?  To answer that, I'd first like to know how a typical salaried person's salary growth would slow down over the centuries.  We work from 20 to 60, approximately, and we see over that period a growth in salary.  This (again in the Western world) represents a career trajectory which, I think, we can't extend onward for centuries. The pattern of real  lifetime wage growth, I strongly suspect, would flatten out after a while and we'd have a more or less inflation-stable income.  Of course, so much is uncertain here.  Would we be as productive or less so, aged 200?  We're in the realm of science fiction, for sure but that's useful to imagine a flat-lining, since one could then conceive some parameter, Ï‰, ranging typically between 0 and 1, which, when achieved might lead us to retire.  The parameter represents the fraction of our mature stable salary S such that we'd be happy to retire on Ï‰S for our remaining time alive. By 'retire' of course, I don't mean become inactive, I mean having in essence the ability to self-fund a liveable income.

To translate this into capital terms, how much capital would one need to accumulate so that it earned us a real return of Ï‰S indefinitely.  Let's further assume for simplicity that we immediately start earning S at the beginning of our working career.    In other words, how much capital would you need to accumulate in order to be able to pay for a perpetuity worth, in real terms, Ï‰S paid to you yearly foreverGiven the length of time here, it is fine to approximate the fixed term annuity with a perpetuity, since they'll both amount to a similar value, and the maths for a perpetuity is simpler.

This capital amount R would be our retirement trigger such that  $R=\omega S/r$.  With $r$ the real rate of return in the above example set at 1%, $R=100 \omega S$.  A general guideline of 67% is often given for the expected final pension of retiring Westerners.  This means on the ultra conservative estimate, you'd better have 67 times your salary before you can retire.  That's a lot.

How long would you have to work when you could put some savings fraction $\delta$ of your salary away every year and until you reached  $R=67 S$? I.e. how many annual payments of $\delta S$, growing each year in a retirement pot for you at a real rate of return again of 1% would result in a pot of size  $R$?  This second problem isn't a simple annuity problem, since even though you're paying a fixed amount each year for $n$ years, the real point is that each year your pool of retirement capital grows, and it is this larger pool which is subject to the following year's growth of 1% real return.  This compounding element will mean many fewer years to wait for freedom from wage slavery.  But how many years?  This structure isn't a plain annuity but more like a sinking fund, whose formula is $\frac{Kr}{(1+r)^n-1}$ where $K$ is the target amount you're planning to need in $n$ years.  Assuming annual compounding and real growth of $r$ which you can consistently receive on your growing fund.

For my current needs, I'm saying that $\delta S = \frac{r\omega S/r}{(1+r)^n-1}$.  I now want to rearrange this to solve for $n$.  First of all I notice that on the top line the rates cancel, so I can write
$(1+r)^n-1= \frac{\omega S}{\delta S}$ and rather conveniently the capital amounts cancel,  $(1+r)^n-1= \frac{\omega}{\delta}$,  The capital amounts cancelling merely reminds me that this simplistic analysis would hold, given the same simplifying assumptions, for any wage slave, regardless of their actual income level.  Moving on, $(1+r)^n= \frac{\omega}{\delta}+1$ and if I take logs on both sides $n \ln(1+r)= \ln(\frac{\omega}{\delta}+1)$ before finally arriving at $n = \frac{\ln(1+\frac{\omega}{\delta})}{\ln(1+r)}$.

Let's plug some sample values in.  Stick with $\omega=\frac{2}{3}$.  Now, we all try to save 5% of our salary at least into the pension pot each year with our current life timeline.  Let's assume this doesn't change.  $\delta = \frac{1}{20}$.  Again let us make the real return 1%.  That's $\frac{1.1583}{0.0043}$ or 268 years (or 41% of your extended life of 640.  For reference purposes, 41% of 60 working life years is about 25 years.  So if you start at 20 and die at 80, saving 5% a year, on the expectation of two thirds final salary means you can retire at 80-25 or 55.

What if you earned a real 2% on your annual saving, all else staying the same?  You get 134 years of saving.  And if you were prepared to forgo 10% of your salary each year for pension saving, all other things the same?  You'd work for 205 years.  Next, if you got a 2% real rate and saved 10% of your salary, you'd take 103 years (16% of your potential working life) before you could retire.

According to the Fed, 5.89% is the Western world's long term current real rate of return.  So, unless you were unlucky enough to hit a world war, this rate of return on 10% pension contributions would have you working for only 35 years, out of your 640 years of living.

By the way, 67% salary as an annuity, discounted at 5.89% real, costs you about 11.4 times your salary.  The major element I leave out of the above is the fact that the annuity your retirement pot buys you is not going to grow with inflation.

UK working age income is currently (2018) 18k p.a.  So you'd need at least 204k in your pot.  For richer folks, say on 100k, you'd need more than 1.1 million in your pension pot to get you a 67k lifestyle (less, assuming the power of inflation).  I am, of course, ignoring the UK government state pension.

Sunday 12 August 2018

The art of asset allocation - poor figure 1.5

Darst ends his bombastic preface with a trite lesson on the etymology of the word 'art', being an expression of something beautifully put together, with skill and in adherence to a craft's skill base.  He adds, pompously and wholly inappropriately, "In addition to these senses of the term 'art', an important reason for naming this book .. relates to the use of more than 130 illustrations and charts intended to help investors to quickly grasp and retain important asset allocation and investment concepts".  Big self-praise indeed.  I've already indicated how strongly I disagree in my first blog on this book.

Let's take one of those early charts and dis-articulate it.  Figure 1.5 purports to show something simple and important - namely the effect of inflation on an asset, over various ranges of time and inflation rates.  How does one construct a chart like this.  Step 1 is go into excel, add a formula to a rectangle of cells, then take it into powerpoint, add crude arrows over the headings and hey presto.  This isn't art.  At all.

First of all, look how he's aligned the arrows (the only possible act of  creativity here).  He wants the downward facing arrows to indicate depreciation in real value as a result of inflationary erosion, so he overlays downward facing block arrows to semantically flag to the reader 'going down'.  However, when her comes to represent the effect of inflation, he clearly intends to have this go in the opposite direction (higher inflation after all erodes faster).  But putting the arrow the other way around (his claimed art-innovation here) merely shows an inflation rate reading pointing 'up' but with numbers decreasing.  This is a visualisation mismatch - a semantically jarring chart which, far from adding to clarity, pointlessly detracts away from clarity.

Second, I hear you say, '"but the guy's a finance guy, what matters is the rigour and discipline he applies to the numbers".  Well, wrong again.  I ran 3 versions of this simple table in a spreadsheet, first of all the correct way (with geometric inflation, since the effect of inflation is geometric) and secondly, using annual compounding.  In neither case did I replicate his numbers.  To get his numbers I have to apply simple interest adjustments, a process which at one point intensifies inaccurately the degree of erosion (helping him make his point, but via a mechanism which is unwarranted) and fails to represent any reality for how inflation as an economic phenomenon occurs.  

Here's the chart showing the erosion with geometric compounding $e^{-it}$
years
151020
inflation0.010.990.950.900.82
0.020.980.900.820.67
0.030.970.860.740.55
0.040.960.820.670.45
0.050.950.780.610.37
0.060.940.740.550.30
0.070.930.700.500.25
0.080.920.670.450.20
0.090.910.640.410.17
0.10.900.610.370.14
0.120.890.550.300.09
0.150.860.470.220.05

Taking as a representative point, the 10 year, 15% point, and applying the annualised formula I get 0.25 instead of 0.22.  I.e. you lose less.  Yet for this point he reports 0.20.  The only way to get there is to apply the following algorithm: $0.85^{10}$, which of course doesn't handle compounding at all.

In conclusion, aesthetically and numerically, I am not a fan of figure 1.5.   Also, I note that the western economies will try to position themselves at 2-3% inflation.  Let's assume this will continue to be the case, more or less, until one dies.  This will guarantee a 50% loss in real terms over the first half of the average working person's life.  Over the full 40 years, two thirds of your purchasing power would be eroded by putting your capital under a mattress in these circumstances.

The Art of Asset Allocation

I'm starting to read "The Art of Asset Allocation" by David Darst.  It already possesses the fly cover and typography of a mostly-empty finance book, and I have decided to go hard on it.

I've read the preface and chapter 1 and I can see that he does two things with diagrams.  One, he utterly recapitulates precisely the same message in his largely textual figures in the textual body of the book, in essence doubling up the message, flabbing the book contents out.  Two, he sees it as some form of art, whereas in reality it is largely a power-point mockery of art.

Here's his chapter 1 message.  The first enemy of the capital owner is the oft-present influence of inflation, eating away at capital's purchase power.  This, of course, is a message of returns (real returns being greater than 0 in fact) and not at all specifically directly related to asset allocation.  But it is fine nonetheless to borrow this out of kilter core concern of finance.

A point I really don't like about chapter one, but one which is partly true, is the way the author tries to make too many points of decision in asset allocation to be driven by investor preference.  For example inter-asset correlation isn't really just an investor decision.  There's of course a choice to be made in working out this window upon which you base your correlations, but this isn't simply a function of investor preference.  This,  ideally, is either something the asset allocation adviser can do for you, or if not, then provide guesses which are going to be at least as informed as your own.  

Also, what's going on with figure 1.2, which seemingly gives four fundamental meanings of asset allocation.  If you read these meanings closely, you'll find that they're largely repeats of each other - blending trade offs is really just the same as balancing characteristics and setting constraints on representation is precisely just a re-description of the very act of diversification.  Perhaps an unacknowledged goal for the author is to have a chunky tome. 

In terms of ideas from the book, chapter 1 introduces us to the following: first, there's a sequence of six steps in asset allocation (the diagrammatic 'art' here is a series of six boxes with a horizontal arrow running across the top - so, artless, really would be a better adjective.  He also attributes most of these plodding Powerpoint/Excel efforts as "source: the author").  Second, there's a Maslowian foundations pyramid.  Also, already at this point one asks what the relationship is between the steps and the pyramid, and to what degree these also are overlaps.  Third is a kind of meta-analysis - pros and cons of engaging in asset analysis.  Fourth, Darst reminds us of a pedestrian and widely recognised distinction between asset categories which protect capital and those which grow it.  Lastly, and to continue situating this quotidian distinction, he reminds us of the two elements of the entropic bite of financial reality - inflationary erosion and short term volatility - financially dying of old age and dying young in an accident.

I'm keen to re-describe this more concisely and in an order which makes more logical sense, but I'm sticking with the criticism of chapter 1 as I see it.

So, these steps of his.  First, specify assumptions about future expected behaviour of asset classes.  This pretty much is a task he assigns jointly to the capital owner and the allocation specialist.  I challenge this.  And will challenge it at later points in this book - he's making this a joint responsibility of the capital owner.  This ought to be the job of the allocation specialist.  Making it the capital owner's job feels like pre-preemptive blame sharing.  By this step, Darst doesn't mean the classic portfolio optimisation step of deciding where the capital owner wants to be on the efficient frontier - this happens in step two.  No, by step one he means listing the future expected returns, associated risk (vol) and inter-asset correlations.  This is a largely empirical exercise.  Yes, there's implicitly a model behind it and yes there needs to be parameter selection (which to repeat comes on step two).  Step one, as far as I can see, is the running of a mean-variance-correlation analysis on asset categorisations observed universally.  This could be a singular input data set for all capital owners.  Nor does Darst hint at the monumental and incomplete effort this entails for the whole of humanity.    Describing this as the capital owner spelling out his assumptions on future returns, volatility and correlation is akin to the maths teacher asking his pupils to explain calculus to him before he starts that lesson.  Part of the motivation of this blame-sharing move is that the models which are used are rear-view mirror models masquerading as future-seeing  machines.  But the financial world of tomorrow is always somewhat surprising.  The best these models can do is to empirically adopt some form of maximum likelihood estimation principle or, through sheer random luck or through prescient and incredibly rare analysis, make statements about the financial future not observable in the empirical data.  Claiming that instead it ought to come printed on a page under the arm of a capital owner in that first series of meetings with his well-paid allocation analyst is quite improper.

I notice in passing that books like this, and certainly this book, relies heavily on the adjectival space of 'discipline'.  This is probably for two reasons.  First, the finance industry takes so much money off capital owners that it must be repeatedly made clear to them that nothing is being wasted here.  Second, books like this are a form of management consultancy brochure, exploiting and dumbing down academic research yet also papering over the found reality of what way money managers actually work.  There's frequently little concern for asset allocation precision and, believe it or not, for empirical analysis.  Thus 'discipline' is a marketing utopia.

Step two.  The selection of the right set of assets which "match the investor's profile and objectives", and pick the appropriate point on the risk/return profile.  I imagine that, in the limit, this is largely answering the same question for everyone everywhere.  Imagine a book which had a chapter for each of the currencies of the world.  In each chapter, a section for the capital owner's age, and within each section, some relevant data.  This singular book should answer most of these questions for most investors.  Also I think the idea of hiring an allocation analyst to deal with only a subset of your capital, and perhaps for a specific objective, is less optimal (though of course it happens) that a singular view of the person and their hopes for their capital through their whole life, and permeating across all levels of capital from a single dollar to many billions.  You'd then only need to pick a new chapter or section if your base currency jurisdiction or capital level changes significantly.  I'd also assume that all assets would be owned, even in fractional weightings.  That way, the act is not one of selection but one of allocating a weight - of deciding where to slice the pie up (or more generally, you're re-slicing an already sliced pie).

Another slight digression.  A capital owner already has implicitly or explicitly made an allocation decision.  Even if they hold their capital in US dollars under a bed, this is an allocation decision.  So the asset allocation industry is always in the game of making a series of re-allocations  across time.  The act of reallocating, however, in any discussion I've seen about it, is implicitly described as a singular, complete act of re-slicing a pie of capital.  However, there's a way of adjusting the slices which may be more efficient, and more in tune with how capital arrives with capital owners.  And that is to allocate any new incremental capital in such a way that the slices move to the weights you desire, without touching the current set of capital allocations.  This would work if the re-balancing occurred in line with the arrival of new capital.  For the sake of giving this a name, I will call it marginal re-balancing.  In the limit, ignoring costs, the marginal re-balancing process is continuous.  Related to marginal re-balancing is the idea that, for all reasonable sets of allocation decisions given the multifarious behaviours of world economies, it might be that there are certain low or high values for these re-balancing weights such that one can say that each asset class can have a fixed core allocation.  This is, of course, a popular and well understood allocation idea (core allocations and peripheral adjustments).   An advantage of recognising this point lies in the likely reduced transaction costs associated with permanently holding a large fraction (in aggregate more than e.g. 50%) of one's capital in the respective asset classes.  I will refer to this as the core allocation stability thesis.  It is either true at meaningful allocation levels or it isn't.  It is mathematically true that you're always going to get some non-0 threshold weighting for each asset.  The empirical question is whether these cores are in fact large enough in magnitude.  In answering this question, we need first to answer a different question, which is how dynamic is the theoretically perfect allocation algorithm likely to be?  A highly dynamic algorithm might require 0% in US equities at some point.  A conservative one perhaps looks for a set of long term fixed (or, in the limit, actually fixed) set of allocations.

Going back to the idea of continuous, theoretically ideal re-balancing.  The other element of a general modelling of this is a description of how the marginal fractional unit of capital arrives at the pool of extant capital which the capital owner possesses, and at what point in the capital owner's existence.  If capital arrives steadily (net capital grows steadily), this lines up well with the theoretical idea of continuous allocation decisions.  If net capital grows in a more volatile way over a person's life, then that phenomenon too might itself be an input to the ideal capital allocation process.  This process can be referred to as the net capital growth process, and it can have a (stochastic) volatility.