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**Knowledge emerges from the interaction of individual minds**

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The Futures Market Economist (TFME) is the electronic publication of the ongoing seminar in futures market economics. This seminar is a free voluntary association of individuals interested in the application of economic science towards an understanding of futures and futures options markets.

This seminar is presented biweekly by posting on every other Monday to the following USENET newsgroups:

This seminar is organized and directed by Vern Lyon, Ph.D. His e-mail address is

mailto:vlyon@aros.net

BACK ISSUES: The current and back issues of TFME can be obtained at TFME's home page at the following URL:

http://www.aros.net/~vlyon/

TFME is also, along with many other items of interest to individuals interested in futures markets, available from tradingtactics.com at the following URL:

http://www.tradingtactics.com/

DISCLAIMER: This electronic seminar is for educational purposes only. Any use of information obtained from this seminar is not the responsibility of Vern Lyon, Ph.D. Use it strictly at your own risk.

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Volume 3, Number 12, August 3, 1998

Contents

Yield Curve Prediction

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Yield Curve Prediction

 

In mid-May 1991, I was invited to lunch in the downtown offices of one of the top mysterious Wall Street speculators. ... He asked me if I knew how much of the U.S. public debt was being financed with hot, or speculative, money. I did not. Perhaps $50 billion, he ventured, a sum which he added he himself had materially contributed. Although not naming a number, he said that his holdings, which he was able to finance entirely with borrowed money, yielded him $300,000 in interest income in just one weekend.

                                            Anecdote told by James Grant in his book,
                                                   The Trouble with Prosperity (1996)

I. Introduction

The yield curve, which is the relation between the term to maturity of a bill, note or bond issued by a particular issuer, such as the U.S. Treasury, and the yield to maturity, is one of the more respectable tea leaves used by academics and some Wall Street type economists to make predictions about the future of interest rates, and because of the supposed importance of interest rates on the growth of the "glob"-- otherwise known as the "economy," it is in turn used to predict the course of the business cycle. I myself am more interested in the growth of the money supplies, dollar, yen, deutsche mark, etc., than I am in the growth in GDP, but I share an interest in finding the information contained in the yield curve to be important. However, my interest is less esoteric than what the interest rate "forecasters" see in the yield curve.

Simply stated my interest is to use the information conveyed by the yield curve to attempt to understand financial market players behavior. It is the behavior of active traders and federal reserve interventions that determine prices in financial markets. To go from the information conveyed in the yield curve to a forecast of interest rates cuts out the essential "middleman," the self-interested individual whose interactions with other self-interest individuals is the motor that drives all market phenomena. It is because one person's self-interest is to be a buyer while another person's self-interest is to be a seller that exchange takes place at all.

Macroeconomists recognize the role of self-interest when they refer to the "microfoundations of macroeconomics. However, they take the easy way out when placing their analysis on a microeconomic basis by using a "representative agent." This in effect amounts to considering the economy to consist of just one individual, or an economy with individuals that can not differ too much from one another. This forced homogeneity of individuals retains self interest as the motive for say Robinson Crusoe's behavior, but it precludes individuals from furthering their self-interest by engaging in mutually beneficial exchange because essentially identical individuals have no reason to trade with one another.

It is hard to believe but mainstream academic finance theory has become "macrofied" in that the representative individual has been made welcome in contemporary finance models. That such models leave out as "unimportant " the explanation of such institutions as the New York Stock Exchange, Chicago Mercantile Exchange, Chicago Board of Trade, etc. that are instituted to facilitate exchange seems clearly to be an example of "unhealthy" abstraction. No wonder so many academics vilify "excessive" speculation because the formal implications of their models is that any transaction volume over ZERO is excessive.

However, models do not "live".by their formal properties alone. Stories grow up around them that if spun by the "proper" individuals appear to make the models more explanatory and robust than their formal properties dictate. However, following the prescription of some of these models can be hazardous to the wealth of anybody but academics will be illustrated in this essay as this essay focuses on the so-called "yield curve" and its ability to predict. The most prominent of the theories explaining the yield curve, the expectations theory, in its modern form is modeled in a representative macroeconomic model of the "rational expectations" inspired by the work of macroeconomists such as Nobel laureate Robert Lucas and Thomas Sargent. As we shall see this does not make it correct from the empirical point of view.------------------

Note:

For a book length criticism of representative agent macroeconomics see The Representative Agent in Macroeconomics (1997) by James E. Hartley. For an articulation by an economic theorist, Frank Milne, that glorifies the marriage between modern macroeconomics and modern finance theory see Finance Theory and Asset Pricing (1995).
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This essay will proceed with a section reviewing some of the essentials of yield curve theory. After this a section will discuss what the weight of empirical evidence has to say about the validity of the most widely accepted theory of the yield curve -- the expectations hypothesis . After this will be a concluding section where I will briefly summarize this essay's subject matter in setting the stage for a forthcoming essay in TFME on "Riding the Yield Curve."

II. The Yield Curve

The yield curve, which is the relation between the term to maturity of a bill, note or bond issued by a particular issuer, such as the U.S. Treasury, and the yield to maturity. Discussion of the "yield curve" is a standard topic in academic finance and is covered in standard textbooks in investments. Because if this I will not spend much time discussing "yield curve theory," particularly since, as mentioned above this essay is concerned with using the yield curve for analysis rather than the analysis of the yield curve.

Discussion of the "yield curve" is a standard topic in academic finance and is covered in standard textbooks in investments. Because if this I will not spend much time discussing "yield curve theory," particularly since, as mentioned above this essay is concerned with using the yield curve for analysis rather than the analysis of the yield curve.

The yield curve can have many different forms, or shapes when graphed, but the three main cases of interest are to consider regions of the curve where: (1) the yield curve is increasing with the term to maturity; (2) the yield curve is constant; and (3) the yield curve is decreasing with the term to maturity. In tabular form these three cases are illustrated as follows:

Increasing Yield Curve 

 

Constant (flat) Yield Curve

 

Decreasing Yield Curve 

These three cases do not exhaust the possibilities as a yield curve can have regions of increasing yields followed by a region of decreasing yield, or in general a yield curve can have any shape, but to discuss the theoretical explanations for the yield curve the three shapes, increasing, flat, and decreasing are sufficient.
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Remark:

In an essay in a recent issue of TFME, "Some Thoughts on the Yield Curve," I referred to a paper by University of California at Berkeley economist, Jeffrey Frankel, where he mentioned how in the early 1990's the shapes of the yield curves for the US and Germany differed radically. I took this as evidence that even though for several years capital markets throughout the world have become more and more interdependent that there is still a long way to go until full integration into a truly global capital market takes place. In fact it is this lack of complete integration that makes synthetic banking viable. If the day comes when the politicians no longer have the power to effectively intervene in the market system, which is a day that could come and go, synthetic banking and other forms of interest rate arbitrage will no longer be viable because the interest rate structure would then be determined in the world's capital markets and in effect there would be only one yield curve. Already interest rate arbitraguers in Europe are scrambling to find other ways to make money as the introduction of a common currency in Europe will bring about uniform interest rates in the countries of the common currency area.

Nevertheless there are other reasons that interest rates differ and in discussion of the yield curve the relationship between term to maturity and yield to maturity should, strictly speaking, refer to the fixed income instruments from a single issuer. Therefore, when discussing the interest rate structure in the US, for example, it should be understood that the single issuer is the US Treasury unless otherwise specified.
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There are three generally accepted not inconsistent theories in the literature that have been presented to explain the shape of the yield curve. They are the liquidity or risk premium, the expectations, and the market segmentation or hedging theories. They all are based on what is assumed to be rational investor behavior.

The market segmentation hypothesis assumes that investors choose the maturity of the fixed income investments they make to match their liabilities, which is basically a hedging strategy. For example a life insurance company based on actuarial data knows that a various times in the future it can expect to have to pay claims on life insurance policies. If they expect a lot of claims 20 years from today, then they can match this liability in a low risk way by investing in fixed income securities that mature in 20 years. Because of this consideration they could buy bonds, for example, that might appear to be overvalued by others. The result is that since it is a fact that life insurance companies are major investors in bonds that their "hedging" has an effect on the shape of the yield curve.

The liquidity or risk premium theory differs from the segmentation hypothesis in concentration on "speculative" rather than "hedging" behavior of fixed income investors.

While it is true that short term interest rates are more volatile than long term interest rates, the capital gains (losses) associated with equivalent interest rate changes are greater for long term fixed income securities. Because of the possibility that an investor might need to sell a long term security before its maturity, and hence possibly suffer a capital lose, it is argued that speculators buying bonds, for example, require a compensation for this risk in terms of a higher yield. Because of this it is argued that regardless of other considerations that yields will be greater on longer term securities than they would otherwise be if not for the added risk of holding long term securities.

Considerations of risk probably do have effects on the shape of the yield curve, but since to analyze these effects in detail requires a extended discussion of risk premium theory, which, as regards the conventionally accepted risk premium theory, I find totally inadequate, I will leave further discussion of effect of risk on the yield curve for another time. However, it is important to note that it is not possible to empirically distinguish between risk and expectation effects on economics variable, such as yields to maturity, which fogs up the implications of any empirical analysis.

The expectations hypothesis, which is sometimes referred to as the "unbiasedness" hypothesis because if it is true then the implicit forecasts of interest rates that are impounded in the yield curve are unbiased in the sense that they are optimal forecasts given available information.

Specifically what this means is that if the yield curve slopes up or is increasing then the optimal forecast is that interest rates, both short-term and long-term, are expected to rise, if it slopes down, interest rates, both short-term and long-term, are expected to decline, and if it is flat then interest rates are expected to remain at their current levels.

The reasoning behind the expectations hypothesis is an arbitrage argument. This can be seen by considering a simple case of the relationship between the yield on a one year security, r(1), the expected yield on a one year security to be issued one year from today, r(e,1), and the yield on a two year security, r(2). According to the expectations hypothesis arbitrage free interest rates will occur when the long-term interest rate is equal to the average of the 1 year rate and the expected 1 year rate a year from now, or

[r(1) + r(e,1)]/2 = r(2) (*)

For example, if r(1) = 0.03 and r(e,1) = 0.05), then when profitable arbitrage has be exhausted r(2) must equal 0.04, the average of 0.03 and 0.05.
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Remark:

Since not only the principle, but also the interest income can be reinvested at the beginning of the second year the notion of the "geometric" mean, rather than the arithmetic mean is thought to be the appropriate mean to use. Analogous to (*) above is the following where the geometric average is used:

sqrt[R(1 + r(1) * R( 1 + r(e.1)] = 1 + r(2),

which for r(1) = 0.03 and r(e,1) = 0.05 gives

1 + r(2) = 1.039951922, or

r(2) = 0.039951922

which is close enough to 0.04 to, at least for this simple 2 period example, to use the simpler notion of the arithmetic average for illustrative purposes.
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In the language of arbitrage theory r(1), r(e,1) and r(2) that satisfy (*) above are arbitrage free prices or expected prices (interest or expected interest rates). If this condition does not hold then it is possible for an arbitraguer to make a profit. To see this consider the two cases that contradict (*) holding:

First assume that

[r(1) + r(e, 2)]/2 < r(2)

In this case a profit seeking arbitraguer could buy the long-term security yielding r(2) and finance this purchase by borrowing at r(1) for 1 year and then refinancing for the second year at the expected rate of r(e,1). If their expectation on interest rates is fulfilled they will make a profit.

Alternatively assume that

[r(1) + r(e,1)]/2 r(2).

In this case a profit seeking arbitraguer would buy a short-term security yielding r(1) and finance this purchase by taking out a two year loan at rate of interest r(2). Then after the first year they would reinvest the proceeds in another short-term security that they expect to yield r)e,1). Once again if their expectations are fulfilled they will profit.

Since this arbitrage activity will always present an expected profit when (*) doesn't hold as an equality, arbitrage activity will be such as to, for example, in the first case to drive up short-term rates and drive down long term rates, conversely for the second case , until (*) holds.

This is an example of risk arbitrage. It is risk arbitrage because of the uncertainty of the short-term interest rate at the beginning of year two. Because of this the expectations theory is often combined with the above mentioned liquidity or risk preference theory, particularly since expectations can not be directly observed as I mentioned above.

The way that the yield curve is used to make prediction about future interest rates can also be illustrated using the above two period example.

To see this solve for r(e,1) in (*) above. This will give

r(e,1) = 2 * r(2) - r(1).

This equation in turn can be expressed as

r(e,1) = r(2) + r(2) - r(1) (**)

Nominal interest rates and hence expectations of them, i.e., r(e,1), to be meaningful economically must be positive, or r(e,1) > 0. Given this restriction there are three cases to consider corresponding to whether the yield curve slopes up, is flat or slopes down.

Case 1: Yield curve slopes up [r(2) > r(1)]

In this case it is clear from (**) that

r(e,1) > r(2) > r(1) ,

or interest rates are expected to rise [r(e,1) > r(1)].

Case 2: Yield curve flat [r(2) = r(1)]

In this case from (**)

r(e,1) = r(2) = r(1),

or interest rates are expected to remain unchanged.

Case 3: Yield curve slopes down [r(2) < r(1)]

In this case (**) implies

0 < r(e,1) < r(2) < r(1),

or interest rates are expected to decline [r(e,1) < r(1)].

While expectations can not be directly observed, the existence of futures markets in the securities involved gives a way to indirectly infer what, for example, r(e,1) is. For example let us assume that it is June 20, 1998 and there is a futures contract for 1 year US treasury notes that expires on June 20, 1999 which if it is not offset must be satisfied by the acceptance or delivery of a 1 year US treasury security maturing in June, 2000. The price of the Jun 99 futures can then be used to infer the expected yield on 1 year notes available in the spot market in June of 1999. In other words, since the prices of fixed income securities and their yields have a fixed inverse relationship, the futures price of Jun 99 can be used as an estimate of the price of 1 year notes a year from June 1998, or equivalently the expected yield, r(e,1). In short the futures market makes "visible" what market participants on average expect short-term interest rates to be in 1 year.

There are, however, some complications involved in using futures prices as "forecasts" of futures prices:

First of all there are considerations of various carrying charges. One of the great debates in the futures market literature is that between Keynes with his theory of "Normal Backwardation," which implies that futures prices are biased estimates of future spot prices, and Holbrook Working who argued that with the proper accounting of carrying charges that futures prices are unbiased estimates of future spot prices.

Secondly, in my extended discussion of synthetic banking in past issues of TFME I have referred several times to the literature that concludes in the majority of cases that, at least for foreign currency futures prices, that they are biased estimates of future spot prices.

Therefore, in the context of interest rate futures there is the possibility that futures prices, while indeed estimates of futures spot price, could be biased estimates. I will discuss more on this issue in the forthcoming essay on "Riding the Yield Curve."

III. Empirical Testing the Expectations Hypothesis

Frankel in his article "The Power of the Yield Curve to Predict Interest Rates (or Lack Thereof)," in his book Financial Markets and Monetary Policy (1995), empirical studies attempting to verify the expectations or unbiasedness hypothesis have not been kind. In other words, the r(e,1) in the two period case I have discussed above have in the majority of of the tests found to be biased estimates of the future spot interest rate. This has been particularly true for the case of long-term interest rates. In fact the bias has been such that the expectations theory seems to have gotten it backwards, i.e., long-term interest rates have tended to move in the opposite direction from the predictions of the theory.

Frankel, as an epigraph to his paper quotes from and important paper summarizing many of these empirical studies by Robert Shiller, James Campbell, and Kermit Schoenholtz, "Forward Rates and Future Policy: Interpreting the Term Structure of Interest Rates" in Brookings Papers in Economic Activity (1983), the following:

The simple expectations theory . . . has been rejected many times in careful econometric studies. But the theory seems to reappear perennially . . . as if nothing had happened to it. It is uncanny how resistant superficially appealing theories in economics are to contrary evidence. We are reminded of . . . Tom and Jerry cartoons . . . . The villain, Tom the cat, may be buried under a ton of boulders, blasted through a brick wall (leaving a cat-shaped hole) or flattened by a steamroller. Yet seconds later he is up again plotting his evil deed.

Frankel finds this and what he considers to be confusion on the issue of "great interest, because the theory's failure would seem to offer traders and investors a valuable opportunity to make money in financial markets. Indeed, confusion among traders and investors about what ought to hold in theory, what is observed in practice, and the implication of the discrepancy, is probably what keeps traders and investors from fully exploiting the opportunity and thereby allows it to persist [p. 131]."

The confusion that Frankel alludes to is illustrated by considering the "costs" and "benefits" of following the prescription dictated by the expectations hypothesis. First, however, note that I have illustrated the yield curve as a predictor in the above two period example as a prediction about r(2), a short-term rate. However, the theory applies to all rates in that they all move up and down together, although not necessarily by the same magnitude. Because of this I will analyze things from the perspective of bond investors who have the most to worry about with respect to capital losses.

Suppose bond traders are faced with an upward sloping yield curve. According to the yield curve this is a prediction that interest rates will rise. Therefore bond traders who accept the yield curve as an unbiased predictor will sell bonds or even establish short positions. To the extent that they dominate the bond market the result will be that their activity will tend to bring about the yield curve's prediction in that their selling will put up rates. Therefore they will be happy and even perhaps be believers in higher education.

On the other hand suppose bond traders find the yield curve fanciful and when they see that long-term rates are greater than short-term rates they see a way to make a profit by borrowing at the short-term rate to buy the bonds the higher yield, and hence to profit from the interest rate differential. Furthermore, they might find this a particularly attractive trade because when US treasuries are involved there is no required margin, i.e., they can finance the whole deal as did the bond trader that Grant discusses in the epigraph to this essay.

Well if there are plenty of banks willing to finance this type of a deal then if these bond traders dominate the market relative to the "believers" in the yield curve, then their activity will itself tend to bring down rates and in the process falsify the yield curves prediction.

In this battle between the believers and the non-believers the empirical evidence tends to support the non-believers. Frankel summarizes some of the empirical studies and concludes that the results, particularly for yield curve prediction of long-term rates are not accurate. With respect to long-term rates he writes:

The usual finding here is a particularly devastating rejection of the [expectations hypothesis]. Long-term rates often move in the opposite direction from that which is forecast by the term structure, falling when the yield curve's slope is steep, and rising when it is flat [p. 136].

IV. Conclusion

There are many fixed income investment strategies that are what can be grouped under what can be called "Riding the Yield Curve" strategies. This strategies all depend on their profitability on the long-term interest rate being greater than the short-term interest rate. For example in my discussion of synthetic banking I have followed a strategy that consists of taking a long position in U.S. Treasury bond futures and a short position in Japanese yen futures. The bond futures side of this synthetic banking strategy is a "riding the yield curve" strategy. It provides an edge because when long-term interest rates are greater than short-term interest rates the further out futures contract will be priced at a discount to less further out contracts, and all of these will be priced at a discount to the underlying cheapest to deliver spot bond. The "ride" consists of the price of the futures contract, assuming nothing else changes, increasing until at maturity it is equal to the underlying cash bond.

According to the expectation hypothesis riding the yield curve will result in zero profit on average. However, the above conclusion of noted economist Jeffrey Frankel says that riding the yield curve will result in positive profits on average. Who or what are you going to believe? The big time professional bond market trader discussed by Grant in the epigraph to this essay clearly doesn't believe the professors, even if he is aware of their work.

Analyzing the subject of "Riding the Yield Curve" will be the subject of the next issue of TFME.

Before closing there is one more remark I wish to make. In the discussion of the yield curve above it was noted that two different predictors of interest rates were considered: the yield curve itself and if there is an appropriate futures market, the interest rate implied by the price of a futures contract.

One can then ask the question as to which predictor is the best? Well, arbitrage between the cash and futures markets should bring about equality of these two predictors. This does not mean that because they make the same prediction that they are correct. Instead based on the emprical evidence they will probably both be wrong. This does not matter to the arbitrageur. The only thing that matters is that in the predictable here and now a price differential exists. Exploiting price differentials has great appeal to someone like myself who is too dumb to be able to predict the unknowable future. Particulaly since, as synthetic banking illustrates, the power of government can result in arbitrage opportunities that persist.

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Logon, learn, enjoy. Knowledge is too important to be left to the professors, or any other special interest group.