A Two-Dimensional Dual Presentation of Bond Market: A Geometric Analysis

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1 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 1 Number 2 Winter 2002 A Two-Dimensional Dual Presentation of Bond Market: A Geometric Analysis Bill Z. Yang * Abstract This paper is developed for pedagogical purposes. It combines the representative-consumer approach and the dual-market framework to develop a two-dimensional dual presentation that links bond market and loanable funds market in price as well as in quantity. It holds that it is incorrect to use the representative-consumer model to explain the behavior of bond price in terms of a given interest rate, and it comments that the existent one-dimensional dual presentations may have prevented one from further exploring the behavior of bond prices. Introduction Traditional bonds pay fixed income and, as a result, their market prices and yields are inversely correlated. This inverse correlation, however, may have been misinterpreted as causation: a rising (falling) interest rate causes bond price to fall (rise). Many textbooks that cover topics of bonds have discussed this issue based on the concept of the present value of future payments, interpreting the market yield to maturity as the discount rate. While mathematically true, this representative-consumer approach incorrectly links bond price and interest rate, and has actually failed to explain the behavior of bond prices. For example, though it has been proved in many textbooks, why the prices of longer term bonds are more volatile than those of shorter term bonds has never been correctly explained, yet. 1 Alternatively, the determination of bond price and interest rate has also been investigated in a dualmarket presentation, in which the equilibrium interest rate is determined in the loanable funds market and the bond price is obtained in the bond market (Hubbard, 2002, pp , and Mishkin, 2000, pp , for example). In their dual presentations, the two markets are connected in quantity only. That is, the quantity of bonds demanded (supplied) equals the quantity of the loanable funds supplied (demanded). However, they have missed another dimension in price that jointly pieces together the two markets. The incompleteness in their one-dimensional dual presentation may have impeded their progress in correctly linking bond price and interest rate, and in further exploring the behavior of bond prices. This paper combines these two approaches the representative-consumer model and the dual-market framework to connect the loanable funds market with the bond market in two dimensions. One dimension is in quantity, as Hubbard and Mishkin did. The other is in price the reservation prices of bond traders along the bond demand and supply curves reveal their desired yields to maturity. Our two-dimensional dual presentation carefully models how the real bond market and the conceptual loanable funds market image each other. It enables us to explain how equilibrium bond price and interest rate are jointly determined, and how they are correlated. Directly motivated by a question from a class of Money and Banking, this research is developed for pedagogical purposes. To be consistent with the teaching practice in classroom as well as with the treatment in current textbooks at upper-level undergraduate financial economics, a geometric method is adopted. This approach has been tested in my own class. According to students, it makes sense to them when studying the determination and the term structure of interest rates and bond prices, though it is different from the treatment given in the textbook for the course. * The author is an assistant professor of economics at Georgia Southern University, Statesboro, GA I am grateful to Amanda King and two referees for their very helpful comments and suggestions on an earlier version of this paper. 1 To my knowledge, the first formal treatment in this regard is Malkiel (1962). It then has become the standard explanation in the literature. For example, for a numerical illustration, see Mishkin (2000, p. 84) and Fabozzi and Modigliani (2003, p.367), and for a formal analysis see Jarrow and Turnbull (2000, p.405), among others.

2 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 1 Number 2 Winter A Simple Model of Bond Market A bond is a debt instrument. A traditional bond contract documents a fixed face value and interest payments as well as its maturity. But neither the bond price nor the bond yield is specified in the contract. Instead, both of them are jointly determined in the bond market. The market price of a bond is determined by demand and supply in practice as well as in theory. Let D B (P) and S B (P) be the demand for and supply of the bond, respectively. The equilibrium bond price can be solved from the following condition: D B (P) = S B (P) (1) And the equilibrium bond yield, also known as interest rate, is obtained together with the bond price. In financial economics, interest rate is referred to as yield to maturity, which is formally defined as the solution for variable i from the following equation: n C P = t Pn + (2) t n t= 1 (1+ i) (1+ i) where P = the price of the bond, P n = the face value of the bond, C t = coupon payment per period, and n = the number of periods until maturity. As soon as the very last trade is executed, the market bond yield is instantly revealed by equation (2). It implies that the bond yield moves inversely to the bond price. For simplicity, we briefly denote equation (2) as P = P(i). This model matches the practice in the real-world bond market. The market bond price is an outcome of trading, and the bond yield is no other than an instant by-product of the bond price. A Two-Dimensional Dual Presentation of Bond Market Though bonds are traded in price, not yield (Malkiel, 1962), when a bond issuer sells a bond to a bond buyer, a certain amount of funds is loaned from the bond buyer (= the funds lender) to the bond seller (= the funds borrower). Hence, the quantity of the bond demanded (supplied) equals the quantity of the loanable funds supplied (demanded). On the other hand, along the bond demand (supply) curve each price represents the reservation price of a bond trader: for a buyer it is the willingness to pay, and for a seller it is the willingness to accept. Trader s reservation prices reveal their subjective desired yields to maturity by equation (2). 2 That is, equation (2) maps each price along the bond demand (supply) curve onto an interest rate along the supply (demand) curve in an imaged loanable funds market. Let D F (i) and S F (i) be the demand and supply curves in this conceptual loanable funds market. Then the linkage between the two markets can be formally modeled as D F (i) = S B (P(i)), and S F (i) = D B (P(i)) (3) where P(i) is given in equation (2). And the equilibrium interest rate can be solved from the following equation: D F (i) = S F (i) (4) Equations (1), (2), (3) and (4) present a dual-market framework through two dimensions. Equation (1) determines the equilibrium bond price in the bond market, while equation (4) generates the equilibrium interest rate in the loanable funds market. Equation (2) links the two markets in the dimension of price, and equation (3) pieces them together in the dimension of quantity. That is, equilibrium conditions (1) and (4) mirror image each other through equations (2) and (3). This duality between the two markets warrants that the equilibrium bond price, P *, and interest rate, i *, obtained from (1) and (4), respectively, always satisfy equation (2), P * = P(i * ). It explains how and why the two equilibrium variables are inversely correlated. As 2 A trader s desired yield to maturity reflects the trader s expected opportunity cost of the funds to be traded, which may differ from trader to trader.

3 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 1 Number 2 Winter a corollary, it allows us to solve for the equilibrium bond price and interest rate, either from (1) and (2) in the bond market or from (4) and (2) in the loanable funds market, equivalently. The inverse correlation between the bond price and the interest rate should not be interpreted as causation, however. This is because both variables are market outcomes, and theoretically there is no causal force between two equilibrium variables. Likewise, though the duality allows us to obtain the equilibrium interest rate from (4) first and then to solve for the equilibrium bond price from (2), it by no means implies a causal force from the former to the latter. Recall that equation (4) is the image of equation (1) through equations (2) and (3). So, when equation (4) is employed to solve for the interest rate, the four equations that constitute the dual presentation are all involved. Therefore, if explaining the behavior of bond price in terms of a given interest rate by employing equation (2) alone, one simply confuses correlation with causation. II: P = P(i) P I: Bond market D B a I b I i B a III b III S F 45 0 III: Loanable funds market IV: Funds = Bonds Figure 1. Construction of the Two-Dimensional Dual Presentation F

4 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 1 Number 2 Winter A Geometric Analysis of Dual Presentation In this section, we continue to provide a geometric version of the two-dimensional dual presentation. It helps compare our two-dimensional dual presentation with the two existent approaches that analyze bond price, as shown in the next section. It is also a useful pedagogical tool to explain the determination of, and the linkage between, interest rate and bond price. The geometric two-dimensional dual presentation is given in Figure 1 on page 13. Corresponding to the four equations in the previous section, it consists of four quadrants: Quadrant I describes the bond market, quadrant II shows the curve, P = P(i), equation (2), quadrant III depicts the loanable funds market, and quadrant IV indicates that bonds equal funds in quantity. Quadrants I and III are one-to-one mapped to each other as follows. Step 1, starting from a point on a demand curve in quadrant I (or on a supply curve in III), draw a vertical (horizontal) segment until it reaches the 45-degree line in quadrant IV and then make a right-angle turn to quadrant III (or I); Step 2, from the same point in quadrant I (or III), draw a horizontal (vertical) segment until it reaches the curve, P = P(i) in quadrant II, and then make a right-angle turn to quadrant III (or I); Step 3, the crossing point in quadrant III (or I) of the lines drawn in steps (1) and (2) determines the image of the starting point in quadrant I (or III). In Figure 1, for example, we only use two points on the demand curve in the bond market (a I and b I ) and their counterparts on the supply curve in the loanable funds market (a III and b III ) that image each other. 3 The construction of this 4-quadrant diagram characterizes the two-dimensional dual presentation as follows. First, the demand and supply curves in the bond market and the loanable funds market one-to-one image each other through a two-dimensional non-linear mirror in quantity and price. The 45-degree line in quadrant IV indicates how they are linked in quantity, while the curve, P = P(i), in quadrant II displays how they are connected in price. Second, the equilibrium bond price, P *, obtained from quadrant I and the interest rate, i *, solved from quadrant III are always on the curve P = P(i), by construction. It implies that the equilibrium solutions obtained from quadrants I and II, or from quadrants III and II, are the same. Third, a shift of the bond demand (supply) curve corresponds to a shift of the funds supply (demand) toward the same direction. Hence, a rising (falling) bond price accompanies falling (rising) interest rate. Why Is The Two-Dimensional Dual Presentation Necessary? The geometric two-dimensional dual presentation established in the last section immediately shows that each of the two approaches in the current bond theory to analyzing the behavior of bond prices only involves a part of the four-quadrant model, as illustrated in Figure 1. The representative-consumer approach has focused on quadrant II only, using equation (2) with the interest rate as an expository variable for bond price. In contrast, the existent dual analysis has just missed quadrant II. In this section, we discuss why lack of quadrant II in a dual presentation may prevent one from correctly linking bond price and interest rate, and comment why it is incorrect to use equation (2) alone to explain the behavior of bond prices. Lack of quadrant II may lead to using it separately Dual representations of the bond market have been introduced, for example, in Hubbard (2002) and Mishkin (2000), among others. However, they have linked the bond market and the loanable fund framework through a dimension in quantity only. 4 That is, their dual presentations have missed quadrant II, compared with the two-dimensional dual presentation developed in this paper. The benefit from adopting a dual presentation is two fold. First, it helps account for the linkage between bond price and interest rate. Second, it helps understand the behaviors of both variables integrally. 3 For the sake of clarity, we deliberately only draw one demand curve in quadrant I and its image, a supply curve, in quadrant III. Also, we draw linear demand and supply curves for simplicity, though the equation (2) obviously implies the non-linearity if we map those curves in the two markets at three or more points. 4 Though both bond price and interest rate are involved, their linkage has not been explicitly addressed and modeled.

5 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 1 Number 2 Winter However, if quadrant II is not explicitly incorporated into a dual presentation, the first aspect cannot be analyzed. Also, the second point may not be fully explored. Hubbard (2002, pp ) and Mishkin (2000, pp ) both have demonstrated in dual presentation that the same reasons may change the demand for (supply of) bonds and the supply of (demand for) loanable funds in the same direction. As a result, the bond price and the interest rate move in opposite directions. This is correct. However, lack of quadrant II in their dual presentations has impeded their progress in linking the two variables correctly within the dual presentation. In fact, they both end up discussing how a given yield to maturity determines the bond price by using equation (2) alone in a separate chapter (Hubbard, 2002, pp , and Mishkin, 2000, pp ). So, what is the problem to use equation (2) alone to analyze the behavior of bond prices? It is incorrect to use quadrant II alone to explain bond price Equation (2), P = P(i), has been almost universally employed to evaluate bonds, interpreting the market interest rate as the discount rate. Based on the concept of the present value of future income flows, many textbooks have proved mathematically that (i) bond price moves inversely in response to a change in the bond yield; and (ii) longer term bond prices are more volatile that shorter term bond prices. While mathematically true, point (i) in fact gives a right answer the discount rate inversely affects the present value, to a wrong question how does the bond yield affect the bond price? And the proof for point (ii) is based on an unrealistic assumption as well as an incorrect theoretic reasoning. When evaluating the price of a bond by using equation (2), P = P(i), a larger discount rate i indeed leads to a smaller value in P. It is correct mathematically. However, one would make a basic mistake in economics, if the yield of a bond were interpreted as the discount rate to evaluate the bond itself. It is like measuring the opportunity cost of a choice in terms of the choice per se! To calculate the present value of future payments, a discount rate is needed. In practice, the average interest rate of treasury bonds is usually employed as the discount rate to evaluate stocks, compensations in legal settlements, etc. The economic rationale behind this practice is the concept of opportunity cost of the funds, implicitly assuming one would otherwise invest the funds in treasury bonds the best alternative forgone. If one evaluates the price of a bond with its own yield to maturity as the discount rate, then any price can be justified as the right one by equation (2), because this is how the yield to maturity is defined. That is, to evaluate a bond with its existing yield using equation (2), one would find that the value of the bond so calculated exactly equals the bond price currently traded in the bond market! As a result, the bond price would never change! This is obviously incorrect. 5 If the interest rate is used as an explanatory variable for the bond price in equation (2), a natural question is where this market interest rate comes from? The representative-consumer approach does not have an answer for this question, because the interest rate is taken as given. But the two-dimensional dual presentation can easily answer it. Institutionally, bond yield is a market variable obtained jointly with the market bond price by the definition of yield to maturity. This point is captured in quadrants I and II. Theoretically, in the dual-market framework the same factors that cause the bond price to fall also kick the interest rate up. This inverse correlation between the two equilibrium variables is quantitatively characterized in equation (2). When attempting to explain how interest rate determines bond price, one simply misinterprets the correlation as causation another basic mistake in economics. 6 Some textbooks employ a loanable funds framework to obtain the market interest rate, and then use equation (2) to explain the bond price. This treatment essentially employs quadrants III and II and assumes a causal force from interest rate to bond price. This is also incorrect. These authors have ignored that the 5 However, if the discount rate in equation (2) is interpreted as the opportunity cost of the funds, the equation can be used to explain many interesting observations. For example, bond prices move inversely with stock indexes more often than not in history. In bull markets, the opportunity cost of holding bonds becomes higher. So does the discount rate. As the discount rate increases, the bond prices decrease, because more funds move to the stock market from the bond market. Symmetrically, the same notion can also explain why stocks perform better when bond yields are lower, because the interest rate is used as the discount rate to evaluate stocks. We all know that the opportunity cost of a choice is defined as the best alternative forgone, but not the choice itself. 6 While current textbooks continue to misinterpret this inverse correlation as causation, our former students current business columnists have corrected this mistake by action. One can easily find a report in the WSJ, for example, every day on Credit Markets like this: At 4. p.m.[friday], the benchmark 10-year note was up 12/32 point at 102 8/32. Its yield fell to 4.092% from 4.139% Thursday, as yields move inversely to prices. See the Wall Street Journal, Monday, October 28, 2002, C13, by Rebecca Christie and Deborah Lagomarsino.

6 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 1 Number 2 Winter conceptual loanable funds framework is an integral part of the dual presentation. It is the duality of the model that ensures the equivalence in equilibrium solutions, no matter whether they are solved from the bond market by quadrants I and II, or from the loanable funds market by quadrants III and II. However the equilibrium is solved, conceptually, the bond price and the interest rate are both the outcomes of the dualmarket framework. Hence, there is no causation between the two. 7 Finally, we point out that the existent proofs in the literature for why longer term bond prices are more volatile than shorter term bond prices are incorrect. All these proofs basically show that the curve, P = P(i), in quadrant II is steeper for longer term bonds than for shorter-term bonds. Assuming that yields of bonds with different terms to maturity change equally, say, from 5% to 6%, it concludes that the longer the term to maturity, the larger the changes in price. So, what is wrong with this proof? First, it is based on an unrealistic assumption that the yields of all bonds change equally. Observations as well as the expectations theory of interest rate show that shorter term interest rates fluctuate more in percentage point than longer term rate. If the change in short rate is significantly larger than the change in long rate, then even a steeper P = P(i) may not necessarily guarantee more volatile longer term bond prices! Second, this proof implicitly assumes that interest rate is an explanatory variable for bond price. This is an incorrect starting point, as we discussed earlier. Therefore, this well-received phenomenon has in fact never been explained, yet, though so many textbooks have adopted this so-called proof for years. Applying the two-dimensional dual presentation, we can provide an alternative explanation for why longer term bond prices are more volatile, if they indeed are. Graphically, for a longer term bond, the curve P = P(i) is steeper. It then leads to steeper demand and supply curves in quadrant I. A change in economic environment and policy may shift the demand and supply curves in both loanable funds and bond markets toward the same direction horizontally 8. And steeper demand and supply curves in the bond market results in a larger change in equilibrium bond price, because steeper demand and supply curves shift more in vertical distance than flatter ones, provided that they shift almost equally in horizontal distance. Conclusions This note has developed a dual presentation that images bond market and loanable funds market through a two-dimensional mirror. The two notions behind the duality are that the quantity of the bond demanded (supplied) equals the quantity of the loanable funds supplied (demanded), and that the reservation price along the demand and supply curves in the bond market reveals the desired yield to maturity along the supply and demand curves in the loanable funds market. The duality in this model warrants that the equilibrium solutions obtained from either the bond market or the loanable funds market are equivalent, and the market bond price and interest rate are inversely correlated. However, this correlation should not be interpreted as causation, since both variables are the market outcomes without a causal force between the two. The two-dimensional dual presentation combines two existent models that analyze bond price and interest rate in the bond literature the representative-consumer model and the dual-market framework. These two existing models turn out to be two exclusive subsets of the two-dimensional dual presentation. The representative-consumer approach has misinterpreted the inverse correlation as causation between the bond price and the interest rate, and hence has incorrectly linked the two variables. On the other hand, the existent one-dimensional dual presentation has missed the dimension in price when piecing together the bond market and the loanable funds market, which may have impeded the progress in correctly exploring how the two variables are linked in the literature such as why the prices of longer term bonds are more volatile than those of shorter term bonds. 7 Another possible mistake in obtaining the interest rate first in a loanable funds framework and then explaining the bond price is to interpret the interest rate as an aggregate variable in macroeconomics. This treatment is not appropriate when discussing micro issues such as the term structure of interest rates and bond prices. 8 Notice the direction of the axes in quadrant III and read it accordingly as if the interest rate is labeled in the vertical axis and the loanable funds in the horizontal axis.

7 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 1 Number 2 Winter References Fabozzi, Frank J., and Franco Modiglinani, Capital Markets: Institutions and Instruments. 3 rd Edition, Prentice Hall Hubbard, R. Glenn, Money, the Financial System and the Economy. 4 th Edition, Addison-Wesley Jarrow, Robert, and Stuart Turnbull, Derivative Securities. 2 nd Publishing, Thomas Learning Edition, South-Western College Malkiel, Burton, Expectations, bond prices, and the term structure of interest rates, Quarterly Journal of Economics 76: Mishkin, Frederic S., The Economics of Money, Banking and Financial Markets. 6 th Addison-Wesley-Longman Edition,