Sunday, June 14, 2009

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?


Anonymous said...

That i>0?

Nasser Saber said...

Dear Anonymous,

Yes, the interest rate is always positive. The so-called "negative" interest rate is the result of adjusting for inflation. If you lend at 3% and inflation is 4%, presumably you will be losing 1%. That is a "negative" rate of 1%. In our discussion, we ignore inflation (temporarily) so the interest rate, as always, is positive.

Thomas GER said...

There is a distinction between "nominal interest rate", which should always be larger than 0, and the "real interest rate", which can be negative, if the rate of inflation is larger than the nominal interest rate.