Pray Tell Me, Are Derivatives Good or Are They Bad?

For the past 30 odd years, derivatives have acted as the barometer of popular sentiment toward markets. In good times, they are praised as ingenious inventions that allow companies to save money in their finances. In bad times, they are scorned as the mysterious devices created to game the system. At all times, they are not understood. The level of discussion never goes beyond the derivatives-are-good/derivatives-are-bad platitudes.

This appalling insubstantiality is in full display again in the latest round of talks about the regulation of markets that, naturally, involves derivatives. In its latest research paper, the Bank of International Settlement likens markets to “pharmaceuticals” and argues that more potent drugs – that would be derivatives – should be made available only by prescription. (The analogy cannot be carried any further because that would imply that only very sick companies could use complex derivatives.) Sage of Omaha is on the record with this gem of a thought that derivatives are the “financial weapons of mass destruction”. George Soros thinks that some derivatives are good and some are bad and should be outlawed, exactly the sort of penetrating analysis one would expect from a money manager who wants to be known as an intellectual and a philosopher.

This past Friday, Floyd Norris of the New York Times picked up the subject. “In the world of derivatives, profit for dealers comes from complexity and secrecy.” After that dramatic lead sentence, he went on to assert that “even when derivatives do allow financial risks to be transferred, that is not always a good thing” because, he quoted a finance professor, derivatives in fact “shift risks from those who understand them a little to those who do not understand them at all.” Norris is among the more perceptive of the business reporters.

Speaking of those who do not understand derivatives at all, here is what I wrote in the opening paragraph of the Foreword to Vol. 2 of Speculative Capital:

Derivatives are the functional form that speculative capital assumes in the market. This form is fundamentally a bet. But like the bodies of the damned in Inferno whose deformity corresponds to the sort of sin they have committed, the particular composition of each derivative corresponds to the sort of opportunities that speculative capital intends to exploit. Arbitrage opportunities are many and varied; hence the confusing array of derivatives and the tortuous legal documentation that must accompany each.

I then added:

The subject of risks of derivatives is a virgin territory. That is not due to the dearth of attempts to exploit it. Quite the contrary; the mountain of material on the subject has few rivals in any discipline. But the territory of concern to us is in the realm of theory and thought, into which unthinking material cannot penetrate no matter what its size.

These lines were written over a decade ago. They remain as valid now as they were then.

Meanwhile, the crisis goes on.

An Excerpt from Vol. 4: A Primer on Bond Mathematics (2 of 2)

In its original form, the equation FV = PV (1 + i) assumes nothing. It merely takes the facts of the transaction – $100 lent at 4% for one year, in our example – and calculates how much money is due to the end of the term. Of course, the lender, like all lenders, assumed that he would be paid back in full, otherwise he would not grant the loan. But that is the lender’s assumption and does not affect the equation. Even if the borrower defaults, the validity of the relation would stand, as it only calculates what should be due to the lender.

Things change when we solve the equation for the present value, PV:

PV = FV/(1 + i)

Mathematically, all we did here was to rearrange the equation, a simple operation familiar to 6th graders. But that technical operation shifted the focus to the present value.

This emphasis and the name “present value” would confuse our borrower and lender. “What do you mean by the ‘present value’ of the loan?,” they would demand to know.

– “The present value is the current value of the loan: how much it is worth today.”

– “What do you mean by the ‘worth’ of the loan”?

–“That is how much the loan is worth! How can we say it? How much it cost the lender to finance the loan”

–“That is $100. It is the loan’s principal. Why do you call it the present value?”

But the present value is not the same as the principal. The new vocabulary is telling us that we are no longer in the private world of borrowing and lending between two individuals. Rather, we have entered the world of securities and markets. The relation PV = FV/(1+ i) expresses relations in the markets and presupposes them. Only then the concept of present value becomes meaningful.

To presuppose markets is to presuppose buyers and sellers. The new form of Eq. (1) assumes that when the creditor takes his IOU to markets, he will find Moneybag waiting for him, ready to buy the note at its “fair value”.

What happens if there are no buyers?

To answer that question, we must ask why and under what conditions would there be no buyers?

There could be two reasons; one particular, the other, general.

The particular reason has to do with the perceived “credit risk” of the individual security, credit risk being the risk that the borrower will fail to pay back the loan and interest on the due date. If there are concerns about our borrower’s ability to pay back the loan, Moneybag, like other potential buyers, will stay away. No one comes to market to suffer a loss, and with many securities to choose from, there is no reason to risk one’s capital on a risky bet.

Our creditor would react to this situation the only way sellers all over the world react when the demand for what they are selling softens: by discounting the price. Instead of asking $100, which is the note’s original “fair price”, he marks it down and offers it at say, $98.

In itself, this act of discounting is unremarkable. But something interesting happens on the technical side, when we substitute the new price in the PV equation. With the future value unchanged at $104 – it is the defining characteristic of the bond and cannot be altered – the one variable that must change is the rate i:

PV = FV/(1 +i)

98 = 104/(1+i), or:

i = 6.1%

Technically, this was expected. In equation FV = PV (1 + i), the bond price (PV) and interest rate i are inversely related. If rate increases, the price will decrease. Changing the order, it stands to reason that if price decreases, the rate will increase. This is no different than saying that if the area of the rectangle remains constant and its length decreases, then its width must increase.

Meanwhile, we have said nothing about interest rates in general. That is another way of saying that we assumed they remained unchanged. The increase in rate is therefore something specific to our particular security. That something is the borrower’s potentially deteriorating finances.

What is this rate? That is, what does 6.1% a year correspond to, or represent?

In the nomenclature of modern finance it means that if the borrower were to ask for additional loans, he would not be able to secure it at 4% and would have to pay 6.1%.

This conclusion is counter intuitive. Even primitive societies treat their vulnerable members with extra care. Certainly in the liberal democratic societies we are expected to see senior citizens, for example, on account of their reduced income, receive special discounts for a wide range of social and private services from transportation to movies. “Give me a break” is the cry of the hurt and vulnerable that demands a more lenient treatment. Obama administration’s Helping Families Save Their Homes Act fell squarely into this category before it was gutted out by the lobbyists.

In the case of the financially vulnerable, though, the fundamental relation of the bond mathematics dictates the opposite: if a borrower has trouble meeting his obligation, then interest rates on him must rise.

Such a “remedy” at once reveals the viewpoint of Eq. (1), which is that of finance capital. Far from being an abstract, neutral relation expressing a general truth, Eq. (1) expresses the power relation in a social structure in which finance capital dominates.

Under these conditions, the well being or survival of individual borrowers is not of concern – not because finance capital has “no heart” but because such matters are irrelevant. “So says the bond,” finance capital’s spokesman, Shylock, declares, further demanding that the judge second his view: “Doth it not, noble judge?” And when the judge asks him to bring a surgeon to attend Antonio's wounds lest he bleed to death in consequence of giving a pound of flesh, Shylock inquires: “Is it so nominated in the bond?” Therein lies the basis of the contract law which is the centerpiece of the Anglo-Saxon jurisprudence. All the learned erudition the law scholars at Yale and Harvard and the pompous musings of the U.S. Supreme Court judges on the subject of contract law never go beyond the self-serving utterances of this usurer.

As finance capital tightens its grip on the economy, its viewpoint is presented as a universal truth: the less credit-worthy the borrower, the higher the interest rate he must pay. Michael Milken’s junk bond operation in the 1980s was based on this idea. A straight line connects him to the recent sub-prime mortgage fiasco.

(A reader in the comment section asked whether interest rate is always positive. That, too, is the question that finance capital floats. Setting aside some technical exceptions such as when a security “goes special” in the repo market, the interest rate is always positive in the sense that the lender will always charge the borrower; a negative interest rate means that a lender will pay you interest to borrow his money. This is prima facie absurd. However, if the rate is 4% and inflation is running at 5%, the lender presumably loses 1% when lending. It is in that sense that the interest rate for him is negative. That, needless to say, is also the viewpoint of finance capital.)

The subject of the borrower’s deteriorating finances has been extensively researched in finance. The borrower’s probability of default (PD) and the lender’s exposure at default (EAD) and his loss given default (LGD) are extensively studied. Google these terms to see what I mean.

But then there is the general case of why there might not be any buyers in the market. In the aftermath of the Lehman bankruptcy in September 2008, the commercial paper market completely shut down.

Under this condition, no security, regardless of the financial health of the issuer or borrower, would find a buyer. The creditors holding the IOUs would cry in frustration: “But this security of mine is absolutely guaranteed to pay back $104 at maturity.” To which the market would reply by quoting the oft quoted line from The Godfather: It’s not personal Sonny!

Modern finance has nothing to offer on this topic. You will sooner find Taleban mullahs writing about distilling single malt whiskey than finance scholars of the Western liberal democracies writing about this subject – and for a good reason. The main pillar of modern finance, Equation (1), does not lend itself to even considering the question of the absence of buyers. Why there would be no buyers, i.e., why markets would break down, is something completely outside the realm of its consideration. In fact, the entire theoretical edifice of modern finance and economics is based on the assumption that every seller brings his own buyer to the market, a preposterous assumption that is refuted thousands of times a day every time a commercial is aired, an advertisement is posted, a sales call is made and a price is discounted. So in the aftermath of Lehman bankruptcy, when the CP market shut down, all the best and the brightest of finance in business and academia could offer was that buyers had “gone on strike”!

Why and under what condition buyers would disappear en masse from a market is the subject of Vol. 4 of Speculative Capital. The condition is the prerequisite for the realization of systemic risk, its trigger point.

It is under these conditions that the role and activities of the Federal Reserve call for a brief comment.

For the past several months, the Fed has been trying to reduce the interest rates and particularly, the mortgage rates. To that end, it has been buying Treasuries – that is called “quantitative easing” – and the agencies, the latter being the IOUs of Fannie Mae and Freddie Mac. The logic is that buying the securities would increase the demand for them and thus, their price. (Remember supply and demand! If the price of IOUs increases, “their” interest rates would decrease.) The understanding of the members of the Board of Governors of the Federal Reserve of the nature and role of interest rate in the country – and hundreds of analysts, economists, policy makers, ex-banker, MBAs and quants that they employ – boils down to this embarrassingly crude logic. Therefore, in the face of rates behaving in the most unexpected manner, they have nothing to say. “Learned helplessness” has become de rigeur.


The Fed, at the same time, is being assigned the role of supervising the financial system with the purpose of preventing a systemic collapse. But with its mechanical view of how markets work, it would not know why buyers might suddenly vanish. The subject of finance is not the psychology of buyers and sellers but the laws of movement of finance capital. This subject is not simply within the Fed’s theoretical ken.

The setup reminds me of an Iranian poem on the eve of the Mongolian attack on Iran. The poet wrote:

The king is drunk, the world is in chaos and the enemies are front and back;

It is well too obvious what will come out of this.

An Excerpt from Vol. 4: A Primer on Bond Mathematics (1 of 2)

I am busy with Vol. 4 of Speculative Capital. Its subject keeps expanding because I digress. Each digression then proves to be the main subject. Here is a short excerpt on “bond mathematics” from the manuscript, with only minor editing for the blog, so you would see what I mean.

***

Consider a borrower who borrows $100 at the going rate of 4% a year for one year. As the evidence of his obligation to pay, he gives the creditor an IOU, a promissory note saying that, at the end of the year, he, the borrower, will pay back the original sum plus the accrued interest, for a total of $104. The calculus of the note is as follows:

100 + 100 x .04 = $104, or:

100 (1 + .04) = $104

The amount presently borrowed, $100, is the present value of the loan. The amount to be paid back in one year, $104, is its future value. If we designate these values by PV and FV, respectively, and let i stand for interest rate, we can generalize this relation as Eq. (1):

PV (1 + i) = FV

Eq. (1) is the fundamental relation of fixed income mathematics. It contains three parameters that uniquely define a debt instrument: principal, interest and maturity. In any lending and borrowing, you have to know how much you are lending or borrowing (principal), at what rate (interest) and for how long (maturity). (Maturity is hidden in Eq. (1) because we assumed it to be one year. This assumption has no bearing on our discussion.)

If, after lending the money, the creditor has a change of heart or suddenly needs $100, he cannot go to the borrower and demand the money. The term of the loan is one year. The borrower will not return it before the designated maturity date, before he had the full use of it, as contractually agreed. So the creditor’s $100 is “locked”, meaning that he has to wait one year before he could get back the principal and interest of his investment. His note, in financial jargon, must be “held to maturity”.

Thanks to the existence of capital markets, though, there is a way out for our creditor. He could sell his note there. What takes place in capital markets is the conversion of securities form of finance capital into money form. But these abstract concepts have as yet no meaning for us. For the time, simple buying and selling would do. So the creditor takes his IOU to market and presents it to a potential buyer, Moneybag.

– “How much are you asking for your note?”, asks Moneybag.

– “Well, the total amount due is $104”.

– “You have to stop trying to put one over us, my boy,” says Moneybag. “We the bond people are math savvy. Your note promises $104 1 year from now. Now is not "one year from now", if you know what I mean! You are selling your note today. The question before us is how much is the note worth today.”

We already know the answer. We only need to solve Eq. (1) for PV:

PV = FV/ (1+ i)

Substituting FV = $104 and i = .04, the PV of the promissory note is $100:

PV = $104/(1 + .04) = $100

We were expecting this result. In the absence of any change in the future cash flow, the term or the interest rate, the present value of the loan had to be what the creditor originally lent to the borrower.

Now, if Moneybag buys the note for $100, he would in fact be paying back the creditor and replacing him as the lender. The borrower need not even be aware of this change in his note's ownership. That is the critical function of financial and capital markets. They are the central pooling places for finance capital. In that regard, they provide capital at a scale beyond the reach of any single individual.

Note also the role of interest rate. If the rates rise to 5%, the creditor will not be able to get $100 for his note. Moneybag would pointedly remind him that he, Moneybag, could lend $100 with 5%, so he would be a fool to replace the creditor in a loan that only pays 4%. Under the new conditions, then, the creditor would have to accept less than $100 for his note. (Eq. 1) gives us the exact amount. We only have to remember that the future payment, $104, remains unchanged as that is all the borrower has agreed to pay. The overall rate, however, is now 5%. Substituting these into Eq. (1), we get:

PV = $104/(1 + .05) = $99.05

If the rates increase by 1%, the creditor will lose about 95 cents.

If the rate drops to 3%, the promissory note will be more valuable, as it pays 4% interest where others could only get 3%. The creditor will demand more for what, under the new circumstances, is a more profitable investment. The “extra” profit is 97 cents that we can calculate using Eq. (1):

PV = $104 /(1 + .03) = $100.97

This relation holds generally: Interest rates up, bond prices down, and vice versa. We see it in Eq. (1) as well. As interest rate i in the denominator of Eq. (1) increases, the present value of all promissory notes would decrease, and vice versa.

Eq. (1) is the fundamental relation of fixed-income mathematics, “fixed-income” being the universe of all the bills, notes, bonds, swaps, mortgages, accounts receivable, annuities – in short, any stream of future cash flows. The “mathematics” part is finding their present value , which should be the price at which the fixed income instruments is bought and sold.

You can take Eq. (1) and run amok. You could, for example, observe that a 1% increase in rates resulted in 95 cents fall in price while 1% decrease in rates resulted in 97 cents rise in price. So the price change of notes in response to a change in interest rates is not symmetric. You could spent a few years of your life studying the non-linearity of price-yield relations in bonds and then branch out and focus on the “convexity” issue, which is a second-derivative of sorts, dealing with the sensitivity of the sensitivity of price-yield relations in bonds.

Or, you could try to determine what happens if the borrower’s finances deteriorate. That, presumably, will increase the likelihood of the borrower's default, which should adversely affect the bond price.

Or, you could consider what would happen if the borrower could pay back his debt early. This “option” should obviously impact the bond price. That is a promising area of research worth a few hundred PhD dissertations on the subject of options adjusted bonds spreads/prices.

If you could do one or all these things, you would become a “quant”, a “rocket scientist”, a math wizard responsible for creating complex new products that would spearhead the globalization of finance. You could become a respected professor of finance at an Ivy League school of your choice. With a little luck, you might even receive a Nobel Prize in economics or become a policy maker at the Federal Reserve Board.

In short, in the realm of “mathematical finance”, you could be all you can be, and still understand absolutely nothing about finance, including its most fundamental relation in Eq. (1).

Let us look at it closely.

Eq. (1) belongs to a large class of physical, social and natural relations in the form of A = mB. These relations, without exceptions, have limits beyond which they are not valid. That is another way of saying that they are based on certain assumptions that limit their applicability. There is no ultimate equation of everything that is unconditionally valid across time and space.

Take, for example, Newton’s relation between force (F), mass (m) and acceleration (a), that is arguably the most profound relation in the universe. It states that

F = ma

The relation applies to all forces – gravity, electro-magnetic and weak and strong nuclear forces – and to all masses. In focusing on the seeming multiplicity of forces in nature and relating them to mass (matter), the equation defines the very discipline of physics which seeks to determine how the natural forces are related to one another and what is the nature of the matter. The equation is valid across the known universe, and helps plot the trajectory of satellites even outside the solar system.

Yet, it has limits. If force (F) increases, the acceleration (a) and, with it, the speed, will increase. But that is true only within “ordinary” speeds. As the speed approaches the speed of light, the mass also increases, countering the acceleration. At 300,000km/s, the relation is no longer valid. A different kind of physics governs.

What is the limit of PV = FV/ (1 + i)? That is, what are the assumptions and suppositions behind it?

First and foremost, this relation expresses a social relation, as evidenced by the presence of interest, i. That limits the applicability of the relation. Charging interest, for example, is forbidden in Islam. So in the Taleban controlled areas of Afghanistan and Pakistan, for example, Eq. (1) is not valid. If you try to enforce it, you would jeopardize your long term business prospects. Short term business prospects, too.

Shylock of The Merchant of Venice, by contrast, insists on interest. He lives by it. That is how relation (1) is a social relation, a product of historical development.

“That is an interesting observation, Mr. Saber. Very intellectual! But surely you realize that we do not live under the Taleban rule. We are citizens of Western liberal democracies where markets rule – the recent black eye they have gotten notwithstanding. So let us please focus on practical matters and leave the intellectual parts of finance to ivory tower academics.”

What else does Eq. (1) presuppose?