Kinky coins are the norm

 

Books on probability, and all student lessons on probability have relied on the good old thought experiment of tossing a coin.  The so-called Bernoulli trial.  A single toss, $P(X=k|\theta)=\theta^k(1-\theta)^{1-k}$ where $\mathbb{E}[X]=\theta$ and ${\sigma}^2=\theta(1-\theta)$, $k$ being either a 1 or a 0.  We all no doubt got to think of 1 being head, semantically being associated with something of value, a king or a human head or a male, and the 0 as something not valuable, a body or a tail, or a female.  Maths, of course, doesn't care what you map the numerical outcome to.

Jacob Bernoulli married Judith Stupanus and they had six children, three boys and three girls ($\theta=0.5$ one might speculate).  In repeating the sexual act at least six times with her husband Jacob, this 17 year old daughter of theology professor Christoph enacted in the flesh of her own uterus that even more useful probability distribution, the Binomial $P(X=k|\theta) = {n \choose k} \theta^k(1-\theta)^{n-k}$, which was also first documented by the owner of the penis which impregnated her so frequently, in the book Ars Conjectandi (1713).  She may have contented herself that she delivered the maximum likelihood number of boys and girls to Jacob before he died ($P(X=3|0.5) = {6 \choose 3} 0.5^3(1-0.5)^3$) and this happy balancing act means I don't need to state whether we interpret $X$ as the number of boys she delivered or the number of girls.

It is noteworthy however that three of these children went on to be mathematicians in their own right, and quite unlike the situation with their gender, the distribution of mathematicians across his offspring's genders is as far away from the maximum likelihood as it is possible to get - all three mathematicians, not terribly surprisingly, were male.

However, in the paper "Measuring paternal discrepancy and its public health consequences" Volume 59 Issue 9, Bellis et.al. we find that these days, certainly for Swiss neighbours Germany, DNA tests show that between 12% and 21%, on average 16.8% of children were not the issue of the sperm of the pater familias. Let us engage in the art of conjecture, in honour of Jacob.  Let us ask, what was the probability that all 6 of the little humans which emerged from uterus of Judith were all Jacob's.  

It is with a certain deviant pleasure mixed with some anxiety that I say, let $X$ be the number of times a wife delivers a child who isn't the genetic descendant of her husband. And let us assume she issues six of her own children in this lifelong experiment of licentious lust.  Let us first, respectfully, work out the probability that Jacob was father to all 6.

$P(X=0|0.168) = {6 \choose 0} 0.168^0(1-0.168)^6 = 33%$.  And now let us turn to the event of $X>0$, which has a cumulative probability of 66.8%.  Jacob might turn in his grave, like a logarithmic spiral but just like the Archimedean spiral which actually ended up on his tombstone by mistake, so too are the figures in this blog post.  He only had two children.  So he likely had a 70% chance of being the daddy.

 All this slight of hand is by way of introduction to the idea that it is quite likely that no coin which has ever been made has the probability of heads and tails be equal at 0.5.  Nor has any population of coins ever had a population mean, or even a sample mean of 0.5.  We humans just don't care to look or measure hard enough.  This is a function of the finitude of each of our times here on earth, a function too of the very approximate and crude tools available to us to measure things and of our desire to live in the dream of mathematical perfection.  No coin has ever been fair or ever will be fair.  As with love and death.

In the great Kantian divide of noumenal / phenomenal, we see the Bayesians happily reside in the phenomenal corner but my point here is that there may well be nothing we can truly (or probabilistically) say about  the so called "real world".  

I could also imagine a coin which had a tiny hollowed out channel in its body with a little droplet of mercury in the channel, which made the concept of a singular objective probability of heads even more difficult to imagine.  Bayesians can chose to imagine the subjective probability they're measuring is a measure of their own state of relative ignorance of a radically real reality, but they can just as easily remain indifferent to any ontological commitment to the 'heads-ness' of any coin.  Perhaps the heads-ness of every coin is forever unknowable and all we can do is present credibility intervals.  But what a useful fable Jacob Bernoulli showed us how to tell ourselves.  (I note in passing that his brother Johann had a further four children on top of Jacob's two, making six Bernoulli trials).

Anatomy of a convert - dates of interest

There are so many things I'd like to say about the interest calculations whose inter-translation I covered.  First up is the step up in complexity when moving from simple interest to compounded interest (discrete or continuous).  I would imagine the maths for simple interest has probably been understood and practised for at least a couple of millennia.  And while compounded loans are surely not much younger, their mathematics, that is, showing what the fair price ought to be, is quite recent.  The definitive book about discrete compounding came out in 1613, by Richard Witt.  No doubt Indian mathematicians probably cracked it 600 years earlier, but in our Western dominated tradition, we like to 'reset the clock' on important intellectual discoveries like this, unfortunately.  There's a nice temporal recapitulation here - the mathematics for fairly valuing certain future cash flows was first published in a Western book a mere 41 years before Pascal and Fermat opened up the way for estimating the fair value of uncertain future events.  Likewise we're spending time on understanding the fixed income side of convertibles before looking at their optionality, which requires more probability theory to understand.  Also, whilst Napier first talked about the exponential constant in 1618 - a mere 5 years after the Witt book -  Jacob Bernoulli, in working on the compound interest problem, identified that $e^x = \lim_{x}(1+{\frac{1}{x})}^x$, namely that if someone came to you and offered you a 100% annualised, continuously compounded rate of return for a year, if you lend them £1, then you'd get back £2.72 approximately.

I've come to realise how Christian, Islamic and Jewish arguments against money generally and the practice of usury in particular (which to many an ancient mind was strongly associated with compound interest, often regarded as grossly unfair) tainted - and still does taint - the Western world's view, so perhaps it is no wonder that we have to wait until 1613 for a full book on the subject. Compare that with a modern definition of capital as a property which creates other properties.  All you need to do is realise the recursive nature of this definition and you have a compelling need to assume compounding as the basis for understanding how capital works.  We've broken through with the mathematics, but we retain much of the moral disgust which accompanies lending and interest generally.  Even those ancient loan makers who only lend out on a simple interest basis, assuming that when they get their payback, they lend it out again (namely lend out their repaid capital).  This practice of a sequence of simple interest based loans it itself a compounding operation when viewed from the perspective of the loan maker's business over time.  So any attempt to distinguish on moral grounds simple versus compound interest must surely be bogus.  It isn't the compounding frequency that's the problem in usury, it is the rate of return.  Any fair simple interest rate has a corresponding fair compounded rate.

There's too great a temptation to rush forwards in my overview of the anatomy of a convert, but I'll hang around a while on the subject of yields.  Remember where we are right now.  I'm seeing how to model the value of cash accruing to us in a future date so that we get a handle on the value now. In this world of rates, I started with so-called risk-free rates.  This allows me to ignore how to model credit, for now.

I'd like to spend some time on the general concept of a yield curve.  But even when I restrict for now my attention to maximum-creditworthiness borrowers, there can still be a confusing jungle of rate forms  (often called rate bases in the financial jargon).  The reason for this is we get those rates from several disparate actual markets.  And each of those markets has its own culture - its own quoting convention, time horizon.  If we ever want to imply anything from real rate market data, we'll need to understand each of those markets' quoting conventions.When you can do that, you can feed the rates into a homogeneous view, the yield curve.  And just as there are multiple conventions for quoting bonds or money market rates, or swap rates for market data quote interpretation, so too there are multiple ways of expressing the output yield curves. 

In the next posting I'd like to develop a general purpose and fairly simple framework for placing rate formalisms in a context which makes the operations seem totally sensible.