|
|
Introduction
Economists and the financial press have failed to sound
the alarm as a - so far unprecedented - curse hit the world: the rate of
interest in Japan started plunging to zero. While the public is kept in the
dark, authorities are trying desperate measures, including the encouragement
of and subsidy to the yen-carry trade, in the hope to relieve the downward
pressure on the rate of interest in Japan through the sale of Japanese bonds
- to no avail. The yen carry-trade is the very mechanism whereby the
deflationary tumor metastasises across the Pacific.
What it does is merely to shift the downward pressure on the rate of interest
from Japan to the United States. From this vantage point, the gold-carry
trade is also highly deflationary as it also contributes to the downward
pressure. The world has been hit by the greatest deflationary iceberg in
economic history, and the captains of the international monetary system are
busy rearranging the deck-chairs, telling passengers that the boat is
unsinkable.
A reliable indicator of deflation is the rate of
interest falling towards zero. It can be compared to a blackhole
the attraction of which is hard to escape and from where (just as from Hades)
no one has ever returned. Why is a zero-bound rate of interest such an
unmitigated disaster? Because, as we shall see, it makes the (present value
of) debt go to infinity - just like gravitation becomes infinitely large
inside of a blackhole of the physical universe.
There are, in fact, two blackholes
in the economic universe: blackhole #1 is the
infinite-interest blackhole (runaway inflation),
and blackhole #2 is the zero-interest blackhole (runaway deflation). When gold was put beyond
the pale just 25 years ago in 1976, little did the world realize that it was
firing the policeman in charge of cordoning off blackholes. By 1980 the
world had a close brush with runaway inflation as the dollar was tottering at
the brink of one of the blackholes. Many ad hoc
explanations were concocted, from the machinations of the gnomes of
Zürich to the malice of Arab sheiks. Nobody raised the question whether
the firing of the policeman may have had something to do with the disaster.
The scare passed when the Soviet Union showed signs of disintegration and the
successor states of the Evil Empire, like a dry sponge, started absorbing
dollars with abandon. The people of these countries were lapping up dollars
at the trough as if they were imbibing the elixir of life which had for so
many years been denied to them. This meant a stay of execution for the dollar
which appeared doomed only a few short years before. But neither the
economists' profession, nor the top brass at the Fed was interested in taking
advantage of the remission to initiate an inquiry into the wisdom of having
put into abeyance the monetary clauses of the United States Constitution half
a century earlier. Instead, they joined the enthusiastic crowd at the
Currency Circus watching with fascination Clown #1, the dollar doing fancy
footwork dancing away from blackhole #1, while
neglecting to pay attention to Clown #2, the yen waltzing around blackhole #2.
Of course, the rate of interest can never be exactly
zero. Likewise, it can never be infinity either. We are in the habit of
rounding figures. But just as we never round to infinity, we should never
round to zero either. For example, if the rate of interest is 0.01%, one must
not round it to zero. It could get halved a dozen times or even a hundred
times, and will still not be exactly zero. Yet each halving would mean that
the debt burden got doubled, causing ever more bankruptcies and punching ever
greater holes in the balance sheets of banks.
The question arises why it is that we never hear
about blackhole #2, even though the other blackhole is familiar to students of inflations. In fact,
the phrase 'runaway deflation' is not even in the dictionary despite great
formal similarities between the two blackholes. The
answer is that John Maynard Keynes, the godfather of contra-cyclical policy,
a seasoned speculator himself, made a serious blunder in ignoring the
consequences of bond speculation when he designed his Brave New World of
irredeemable debt and irredeemable currencies. Alan Greenspan's greatest
ambition appears to be that he be remembered as the most worthy henchman of
the godfather. He copies the original blueprint, warts, blunders and all. As
far as we know, he has never ordered a detached, critical analysis of the
principles underlying contracyclical policy. After
all, the policy worked in the 1930's, didn't it? Blackhole
#2 didn't devour the world, did it? So the Keynesian nostrum worked, didn't
it? Well, the truth is that we shall never know. The nostrum certainly did not
work up until 1940. Thereafter the interference of World War II confused the
issue. It seems to be a reasonable assumption that, without the war, the rate
of interest in America would have continued its plunge to zero. This is an
indication that a lot more research ought to be done on this question, but
nobody is doing it. The frightening possibility cannot be dismissed that
history will repeat itself and it will take another World War to escape from
the clutches of the blackhole once again.
The see-saw of interest rates and bond
prices
The fact that the yield and the price of a bond are
in inverse relation is well-known to security analysts, although it is often puzzling
to laymen. Even professionals may occasionally think that it is merely a
statistical law. Well, it is not: it is a mathematical law that does not bear
exceptions. A related common mistake is to think that the see-saw of the rate
of interest and the bond price is limited by the face value of the bond,
which cannot be exceeded by market value no matter how far the rate of
interest falls. The fact is that the face value of a bond does by no means or
manner limit its market value. Any given value, however
high, is theoretically possible, provided that the rate of interest is low
and maturity is far enough. The mathematical apparatus to prove this is given
in the Excursus. The mathematics becomes quite simple for perpetual bonds (consols in British parlance). A perpetual bond with face
value A has no maturity, but it pays interest in perpetuity at a fixed
rate a. If the market rate of interest is b, and the market value of
the perpetual bond is B, then the mathematical relationship between
the variables b and B is given by the attractive formula:
Aa = Bb = constant
For example, if the rate of interest b gets
halved, then the market value of the bond B must double (and
conversely). Moreover, if b plunges to zero, then B tends to
infinity. While there are no perpetual bonds around any more, the formula is
still relevant, because it provides a reasonable approximation for the
relation between the market value of long-term bonds and the rate of
interest. Furthermore, the longer the term the better the approximation (for
a 30-year bond the error in percentage terms would still be single-digit).
Wrong way to report debt
Another reason why the formula Aa
= Bb is relevant is the fact that it can be used to calculate the
present value B of the total debt A in a country as a function of the rate of
interest b (a > 0 is immaterial.) In this case it has the force of
an exact formula rather than an approximation, because the existing debt of
virtually every country today may be assumed to be perpetual debt. By no
rational calculus can it be expected that it will be retired through the
normal process of repayment (that is to say, without wiping it out either
through default of through currency depreciation). We may verbalize our
formula as follows: Every time the rate of interest in a country is cut
into half, the (present value of the) total debt in that country doubles. In
particular, as the rate of interest is plunging to zero, the (present value
of the) total debt is galloping to infinity.
It follows that our way of reporting total debt is
flawed. It reports only the principal sum due, overlooking the stream of
interest payments thereon that is also due. Under a gold standard, where the
rate of interest does not go up-and-down and cannot fall to the ground like a
Yo-Yo, the omission is not significant. When you discount an infinite stream
of payments by a stable rate of interest, the present value is finite. By
contrast, if we discount a stream of payments as small as one dollar per year
by zero, we shall find that the present value is infinite. This epitomizes
the plight of the world in view of the rate of interest plunging to zero. The
total debt in Japan (or in any other country) could double without approving
a single new loan, or without selling a single new bond, if the rate of
interest is halved.
The zero-bound rate of interest brought the United
States to its knees in the 20th century when it was the greatest creditor
nation on earth. It may do it again in the 21st for the stronger reason that
the U.S. is now the greatest debtor nation on earth.
Siren song from the blackhole
The blackhole threatening
Japan is particularly treacherous. From it comes the sweetest siren sound,
drawing unsuspecting seafarers to their destruction. But Odysseus had been
warned of the dangers. He was shrewd enough to have himself tied to the mast
and to order the ears of his crew members to be plugged, so that the siren
song could not deflect their journey. No such shrewdness is apparent in the
conduct of skipper Greenspan. The boat under his command is happily sailing
ever closer to the source of the inviting siren song, promising budgetary
surpluses as far as the eye can see, and abolishing inflation for good, quite
unaware of the dangers awaiting the people aboard.
The optical illusion that halving a small number is
inconsequential has desensitized the public to the danger. The rate of
interest declining from 0.2% to 0.1% is no big deal, people say. Not so! It
is doubling the debt burden, same as a decline from 16% to 8%. The role of
Cassandra is a thankless one. To begin with, there is euphoria that we have
"licked inflation". The doubling of the present value of the public
debt is a piece of intelligence that is easily hidden under the bushel by
treasury officials, for example, by phasing out the 30-year T-bond. All we
are told is that "the budget deficit will also be licked" as the
rate of interest keeps falling; but we are not told what is happening at the
other end of the see-saw. So when the public debt gets doubled, and doubled
again, nobody will care a hoot. On the contrary, people will be jubilant that
more public money can be raised at ever lower rates of interest.
At first there is little sympathy for firms going
bankrupt. People say: "serves them right, they have taken all the money
from the banks while the taking was good". The realization will come
later that even the most conservatively managed firms are in deep trouble on
account of the plunging rate of interest. While the low rate may appear as a
palliative, it will turn out to be a trap. Furthermore, those who take the
money on these spectacularly easy terms will be sorry later when the rate of
interest gets halved again and again. Prices decline together with the rate
of interest. Businesses scramble to liquidate inventory and debt. As they do,
a vicious circle sets in. Liquidation of debt becomes the cause of another
round of interest-rate cuts; and liquidation of inventory of another round of
price cuts. Lower prices squeeze profits. Receivables may become illusory or
may disappear altogether, and so may profits. Servicing debt contracted at
higher rates of interest earlier becomes excruciatingly painful. Those
businessmen are the lucky ones who have sold out their businesses at the
start of deflationary spiral and put the proceeds in bonds. They have no
payroll to meet, no risks or labor trouble to face,
no receivables or profits to worry about while they are enjoying not just a
steady fixed income from the coupons, but also a nice capital gain on the
principal. And, when the deflationary storm blows over, they could repurchase
their businesses at a fraction of the price.
What makes people walk away from their
dream-home?
Most of us have never had the opportunity to serve
as the CEO of an enterprise and have little appreciation of business problems
connected with falling interest rates. For those who had to face the problems
connected with the financing of the family home the following example may be
more meaningful in showing how deflation can make innocent people suffer.
You bought the house of your dreams, putting down
50% of the purchase price, your entire life-savings, which is now your
equity, financing the other 50% with a conventional mortgage at 6% interest.
Now deflation comes and the same mortgage is offered at 3%, half of what you
have to pay. The bank refuses to renegotiate the mortgage as it is flooded
with similar requests. You shrug and say: "So what? I like my home and
stay put. I don't care that my equity has been wiped out. Those are just
numbers."
Brave talk! But now you see "house for
sale" signs cropping up around you and hear talk that
houses don't move even at half of the appraised value. The situation
is now this: you carry twice the debt on a house the market says is worth
only half of what you have paid for it. For a moment you think that you want
to walk away from your dream home, because it makes no economic sense to pay
that much per month for shelter when you could rent a nice apartment for a
fraction of debt servicing. But throwing away your life-savings is just too
painful to contemplate.
However, you are a confirmed optimist and say:
"Let's just tough it out and wait for the turnaround that the President
says is just around the corner". Your house is now your coffin, and you
are lying in it. They threaten to drive in the nails when your salary is
unexpectedly reduced by half. Then you climb out of your coffin and,
broken-hearted, walk away from your dream-home, because you prefer life to
death.
Enter the bad guy
The script so far has followed that of Keynes. The
gold standard was blamed for everything. The "barbarous relic", as
he preferred to call it, was deflation-prone. Man has a pathological craving
for gold, making it responsible for the contractionist
tendencies of the capitalist economy. Gold must be taken away from
"man's greedy palms" and put beyond his reach so that he can start
spending his income rather than saving it. He likened craving for gold to
craving for the moon. Man must be told that he cannot have the moon but that
green cheese (sic) is just as good, and the Central Bank is tooling up
to produce lots of green cheese for his delight. Since the capitalist mode of
production is structurally flawed, Keynes went on saying, therefore the
government ought to step in and prime the pump by public work spending. He
had a pet project for public works in mind: the government should put
greenbacks in coke bottles, bury them underground, and then hire the
unemployed to dig them up again. That, he was fond of saying, would make more
sense than prospecting for gold. Finally, he suggested that the central bank
buy the bonds that finance public works in order to lubricate the economy, so
that the capitalist system of production will not come to a screeching halt -
as in his opinion otherwise it obviously must.
Keynes assumed that no bad guy could wreck his plans
to save the world. But, as it turned out, there was a bad guy waiting in the
wings: the bond speculator. Keynes, of course, knew him well. They were
buddies in Germany in 1923, speculating in Reichmarks.
However, Keynes thought he was as harmless in the bond as in the foreign
exchange market. For every bull there must be a bear. The speculators'
activities cancelled out in the long run and could be safely ignored. Bond
speculation is a zero-sum game, he insisted, and that's that. With this
assumption Keynes made the greatest scientific blunder of the century, if not
in the entire history of science. Nowhere in his voluminous work did he
examine the consequences of bond speculation as related to his proposed
schemes of pump-priming, make-believe work, and debt-monetization in spite of
the fact that he treated speculation at length. It is our task here to show
that Keynes' assumption of a zero-sum game in the bond market does not hold
water.
Speculation versus gambling
The speculators' long- and short legs entering the
bond market do not alternate randomly. Bond speculators march in lockstep.
Their activities give rise to enormously destructive forces, threatening the
international financial system with a collapse. Bond speculation, just as
foreign exchange speculation, is a form of gambling. It has no social value
any more than shooting craps has. The risks in the bond and foreign exchange
markets are entirely artificial: they were created when governments first
sabotaged and then destroyed the gold standard. Neither form of speculation
had existed in countries adhering to the international gold standard. Foreign
exchange and interest rates were stable by nature, leaving no room for
speculative profits. This conclusively proves that foreign exchange and bond
speculation is the consequence of artificial risk-creation and it offers no
benefits to the public. On the contrary, it is a rip-off. The same is of
course not true for commodity speculation where the risks are nature-given
and the speculator's activity has great social value and public benefit in
alleviating gluts and shortages whenever they occur.
The bond market is huge. Initially it was about ten
times the size of the stock market but, by now, it has grown by leaps and
bounds thanks to the proliferation of derivatives such as bond futures,
options on bond futures, interest-rate swaps, repos, knockout puts, and so on
ad libitum. Not only does this activity require a large pool of
currency (tens of trillions of dollars in America, which is a high multiple
of the amount required to support productive activity) that otherwise would
have no reason for being in existence. It also draws capital and talent away
from productive enterprise. All this is bad enough, but the worst part is
that bond speculation exposes society to the gravest dangers of economic
collapse. It is drawing the economy into the blackhole
as speculation drives up bond prices, that is to say, pushes the rate of
interest to zero. This is the evil Keynes didn't see. Generations of
economists following him didn't see it either. They simply pretended that
bond speculation was irrelevant because it was a zero-sum game. They believed
that the government and the central bank can do as they please in the bond
market with impunity. Now it turns out that their actions, too, have
consequences.
It is still a sacrilege to criticize Keynes. The postwar generation of economists put Keynes on a pedestal
and worshiped him as savior. Greenspan is no
exception, his pretensions of being a free-market man notwithstanding.
"We are all Keynesians now", declared the President who defaulted
on America's foreign gold obligations. Keynes' recipes are followed blindly
and slavishly by Greenspan. Unlike Odysseus, he doesn't pay attention to the
warning that he should steer away from the blackhole.
It is not too soon to question Keynes' theories which have apparently led the
world into temptation - and to disaster.
Why bond speculation is not a zero-sum
game
Speculators don't buy the bond because they want to
earn the interest income. They buy it because they want to ride the uptrend
in bond prices. In view of the see-saw we can also express this by saying
that they want to ride the downtrend in the rate of interest. They don't sell
the bond because they want to invest the proceeds in the real economy. They
sell it because they want to ride the yield curve (selling the short bond and
buying the long), or they want to hitch a ride on the bandwagon of the
carry-trade (selling the bonds of a country with a lower, and buying the
bonds of a country with a higher interest rate). It is clear that bond
speculators act in concert, and not at random as Keynes thought. They put
themselves in position so that they can enjoy the tail-wind provided by the contracyclical policy of the central bank. They all want
to buy the bond just before the central bank enters the open market with its
own shopping list, and take profits after. Collectively, they represent a
formidable financial force. It does not matter whether they act as part of a
conspiracy, or they are acting on their own. The effect is the same: theirs
is not a zero-sum game. They are on the winning side virtually all the time.
It is simply not true that one speculator's gain is another's loss. The bond
speculators profit together, at the expense of the productive segment of
society. The public is losing whenever there is a protracted fall in interest
rates, but it is also losing when there is a protracted rise. To understand
this, we have to remember the see-saw. In the first instance, when interest
rates fall, there is a prolonged rise in the present value of debt and
productive enterprise is forced into bankruptcy. In the second, when interest
rates rise, the value of bonds falls and with it falls
the value of productive capital.
Bond speculation is a parasitic activity on the body
economic. Of course, this is not meant as a smear on the character of any
individual. Speculators acting on their own can be, and we may assume that they
mostly are, upright people. Like everybody else, they are trying to eke out a
living. They are certainly not responsible for the establishment of this
vicious system which, by staying the "invisible hand", victimizes
the majority. The blame is entirely on the government which is responsible
for the institution of this iniquitous system which, rather than promoting
social cooperation, pits one citizen against the other. It is another matter
if bond speculators act in collusion with one another which would make the
majority the victim of vampirism by a small minority. In that case we are
facing a conspiracy which is purposely trying to milk society dry of its
substance. It is not suggested here that there is evidence of such a
conspiracy, only that the existing system is conducive to such conspiratorial
activity on the part of bond speculators, hiding behind the façade of
a hedge fund, or a gold mining concern with a hedge book, for example. Be
that as it may, bond speculation represents an enormous financial force which
one can ignore only at one's peril. It controls financial resources that in
all likelihood eclipse those of the government. And, above all, bond
speculators have a superb strategy which makes the central bank, slave to
Keynes's mindless contracyclical policy, a sitting
duck.
Linkage
In my talk Kondratieff Revisited given at Sf.
Gheorghe, Romania, I discussed at length the phenomenon of linkage which
causes the price level and the rate of interest to move, apart from leads and
lags, in the same direction. Just as when a man is walking his dog on a
leash: while it is possible for either one to get ahead of the other by a few
steps from time to time, it is not possible for the two of them to move in
opposite directions for any great length of time. Linkage has been recognized
and treated by several distinguished economists such as Knut Wicksell, Wilhelm Röpke,
Gottfried Haberler, Irving Fisher and many others
(internet authors, following Keynes, refer to this phenomenon as the
"Gibson-paradox"). In terms of blackholes,
linkage means that the zero-bound rate of interest is followed by a weakening
or collapsing price structure and profit-squeeze. The financial system may
snap as receivables become uncollectible, and the domino-effect sets in
wiping out all but a handful of well-heeled enterprises.
Contra-cyclical or counter-productive?
The contracyclical policy
mandate of the central bank calls for open market intervention in the bond
market. Every time the price structure is weakening (or the stock-market is
in danger of collapsing) the central bank enters the open market as a buyer
of bonds, in order to replenish bank reserves. By the see-saw effect, the
central bank drives down the rate of interest. This is just fine, as far as
bond speculators are concerned. They welcome the tail-wind making their sails
bulge. Central bank intervention is a virtual guarantee that they will be
able to dump their bonds at a hefty profit. At the same time the linkage
guarantees that lower interest rates are translated into weaker prices, so
that the contra-cyclical policy should more appropriately be called
the counter-productive policy of central banking.
In case of a serious decline in commodity prices which
by the linkage may indeed be brought about by a zero-bound rate of interest,
the central bank is desperately trying to reflate the economy. But pumping
more money into the economy is just pouring oil on the fire. The new money is
not flowing to the commodity market (nor to the
stock market) as intended. Why should it? The fun is elsewhere. The central
bank can put all the money in the world into circulation, but cannot make it
flow uphill. The new money is going to flow to the bond market, where the fun
is.
Avalanche alert!
Deflation is a term loosely applied to the
phenomenon of general money scarcity in the economy. Hardly ever is the
question asked just what it is that makes money scarce. Here is the answer to
that crucial question. Money is made scarce by bond speculation. The bond
market acts like a gigantic vacuum cleaner running amok. It siphons money
away from every nook and cranny of the economy. The money is channeled into bonds serving as chips in the hands of the
speculator with which they make their bets.
A vicious circle is hereby set into motion. The
scarcity of money calls for central bank intervention. The open market
purchases of bonds by the central bank drive up bond prices making the
profits of bond speculators soar. Thereupon bond speculators get bolder and
start pyramiding, using easy money made available by the banking system,
thanks to the intervention of the central bank. Money becomes scarce again,
and the central bank is called upon to act once more. And so on it goes.
We use the term "bond market" and
"bond speculation" in the broadest sense, as a generic term
embracing not just bond trading and speculating in the narrow sense of the
word, but also trading and speculating in derivatives based on bond and interest
rate futures, options and swaps. Even gold leasing comes under this heading,
as the short position in the gold market is offset with a long position in
the bond market. It is an extremely dangerous development that the bond
market in the broadest sense is snowballing exponentially, as new players
jump on the bandwagon and the old ones pyramid. Even the blind can see the
explosion of derivatives. It is an avalanche that will ultimately kill the
innocent and unsuspecting people who live down there in the valley.
The central bank is helpless
It is impossible to isolate a single economic force
that is capable of resisting the coming avalanche. Would foreigners blow the
whistle and withdraw their funds from the casino gone mad ?
Far from it: they will send in more to partake in the fun, so irresistible is
the attraction of the short road to riches. Would the central bank step in,
raise and hold up the rate of interest long enough to put an end to the mad
party, and bring speculators to their senses? Not likely. The central bank is
helpless. A higher rate of interest would be lethal to the banking system. It
would punch enormous holes in the balance sheets of the banks as the volume
of non-performing loans exploded. This is the medicine that was tried in
Japan. Predictably the patient, the Japanese banking system, went into a coma
where it still lies. Nobody dares to shut off the patient's life-support
system in fear of the consequences that are incalculable.
Murdering savers in their sleep
It is difficult to put ourselves in the long-forgotten
land of the gold standard where interest and foreign exchange rates just do
not Yo-Yo, blackholes are off limits, and the rate
of interest could not fall to zero. Long before that happened
bondholders - who, incidentally, were all genuine savers in that strange land
- would step in and sell their overpriced bonds at a profit. They would wait
until sanity prevailed once again and bond prices returned from outer space;
only then did they buy back their bonds. Keynes was totally insensitive to the
evil consequences of abolishing the gold standard. He was content to see
bondholders of old, among them widows and orphans, be murdered in their
sleep. The new bondholders, the speculators, trampling through the dead
bodies of the old, are no longer inclined to sell the bond after the rate of
interest fell. They want to go the whole hog. They rather see the rate of
interest go to zero, so firmly are they in the saddle at the other end of the
see-saw.
But let's give credit where credit is due. Keynes
was a great trickster. He bequeathed to Greenspan the magic word how to
increase green-cheese production, but took with him the secret of the other
magic word, how to decrease it, to the grave. Greenspan in charge of
green-cheese production is now hostage to the speculators who have cleaned up
in Japan, and are now after the United States. This year Greenspan is cutting
interest rates for the seventh time in a row, each time announcing his
intention in advance, causing great rational exuberance among bond
speculators.
Conclusion
I am not predicting that the world will plunge into
the black hole of zero interest. I am trying to sound the alarm that it
might. There is no scientifically credible way to predict the future when it
comes to imponderables. Another possible scenario is that speculators make a
complete volte-face and, instead of accumulating, they start dumping the
bonds. This, however, would bring other dangers in its train. When all the
people aboard congregate on the port side and then all of a sudden switch to
the starboard side, the boat is in danger of capsizing. A third possible
scenario is that countries without a bond market - which makes them immune to
the attraction of blackhole #2 - stumble over blackhole #1, pulling other countries along.
Last but not least, let us not forget that we could
force a frank public debate on the errors of commission and omission of the
economists' profession. Such a debate might lead to a public inquiry into the
wisdom of having put into abeyance the monetary clauses of the United States
Constitution at the instigation of a foreign upstart. The U.S. House of
Representatives could by simple majority vote reinstate and uphold those
clauses, the repeal of which neither Greenspan nor his predecessors have had
the moral courage to initiate. It is true that such a development would
create a Constitutional crisis, but it would focus Greenspan's mind on the
danger facing the nation and the world like nothing else could.
And it is possible, if only barely, that the crisis
would be resolved by the Senate and the Supreme Court concurring that those
clauses must be upheld. This would indeed be a happy ending to a sad saga.
Thereafter the global regime of irredeemable currency would be remembered as
a brief reactionary episode in the history of mankind.
Excursus:
The mathematics governing bond values and
the present value of debt
Let the price of a bond with face value A and
coupon rate a maturing in T years be denoted by BT . There is a mathematical
relationship between the variable BT and the prevailing rate of
interest b, called the Bond Equation of which we present two
equivalent versions (a third version will be added later):
(First
Version) 
(Second
Version) 
where e denotes the basis of natural logarithm: e = 2.71...
We prove it in its First Version. On the right-hand side the first term Ae-bT is the present value of the lump
sum A payable T years from now. To this we must add the present
value of the income stream derived from the coupons. The coupon payable at
time t has value Aa. We have to
discount it by the prevailing rate of interest b to get the present
value Aae-bt of that coupon.
Therefore the present value of the income stream can be calculated as the
integral
This completes the proof of the Bond Equation. From
the Second Version it follows that
BT < A for b > a,BT =A for b
= a, BT > A for b < a
Because
 for all
b > 0 and T > 0. In particular, the market price does
indeed exceed face value of the bond whenever the rate of interest falls
below the coupon rate.
To get the value of consols
or perpetual bonds, we have to take the limit B of BT
as . Since in this case , from the First Version it follows that
 or
Aa = Bb = constant
where B is the price and Aa is the
yearly payout of the consol. It is clear from the Bond Equation that the rate
of interest b and the bond price BT are inversely
related (for consols, they are inversely
proportional).
The British government started selling 3% gold consols in 1751. The rate of interest on consols was lowered in 1889 to 2.75%, and again in 1903
to 2.5%. The issue was discontinued during World War II when the British
government defaulted on its domestic and international gold obligations. Even
though consols disappeared together with the gold
standard, the formula Aa = Bb is
still relevant today. First, B can be used as an approximation for BT . The parameters of this
approximation may be gleaned from another version of the Bond Equation:
(Third
Version) 
where and BT
A + aAT as b 0. The latter is
immediate from the Second Version, since
as b 0. This means that the present value of the
bond maturing in T years, when the rate of interest tends to 0, is
approaching A + aAT. This is
plausible, because aA is the face value of
every one of the T coupons so that this sum represents all the
payments to the bondholder. The present value is the same since it is
discounted by 0. From this we may conclude that the price of the bond BT
can be greater than any given value, however large, provided that the rate of
interest b is low and the maturity T is large enough. It also
follows that the approximation of BT by B gets
better as T gets larger.
The formula Aa = Bb
can also be used to calculate the present value B of total debt A
of a country if the prevailing rate of interest is b. Under the
assumption that the total debt is perpetual, this is an exact formula, not an
approximation, and it shows that if b is divided by n, then B
is multiplied by n. So as the rate of interest starts falling from 5%
to 0.5%, 0.05%, 0.005%,... then the total debt
increases by a factor of 10, 100, 1000,... For example, if the going rate of
interest 5% is cut into half every year for a period of ten years, then by
the end of the decade the rate of interest will be about 0.005% and the total
debt will increase by a factor of 1000.
July 19, 2001.
Note added December 3, 2001.
Stephen Hawking said in the Foreword to his
best-selling book The Brief History of Time that he had been hesitant
to include equations because every equation was liable to cut sales of the
book into half. When I privately circulated this paper in July, I found that
every equation cut the interest of my friends in my thesis into half.
Frustrated, I decided to put off publication of my paper awaiting further
evidence.
It came on September 11 with the subsequent
declaration of an open-ended war, to which the price of oil and the rate of
interest responded by falling. Mr. Greenspan's compulsive cutting of
interest-rates for the tenth time this year despite the reappearance of
irrational exuberance in the stock market, strikes me as measure of utter
desperation. Whoever has heard of a central banker deliberately cutting
interest rates in the face of a shooting war and calls for unconditional surrender
of the enemy? Either Mr. Greenspan does not know what he is doing, or he is
being swept off his feet by the dogs of deflation. What we are witnessing is
not a war between righteous and terrorist forces. It is something bigger. It
is a tug-of-war between inflation and deflation.
Dr. Antal
E. Fekete
|
|
