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## A theory of the term structure of interest rates, (1985)

Venue: | Econometrika, |

Citations: | 1978 - 3 self |

### Citations

557 |
Generalized Hypergeometric Functions
- Slater
- 1966
(Show Context)
Citation Context ...54) 8=[[(K, l+POjp +2 1 +2(1 -poI)o2'12 (K,+POsI 2 1p 1 2[(1 + 8)2+ KI, M( ) is the confluent hypergeometric function, and F(-) is the gamma function. 17 Proceeding in the same way with Model 2, we obtain the valuation equation: (55) 202rNrr + 2-2y Nyy + po2copypNyp + I2yp2 Npp + [KO- (K +Ak)r]Nr + K2[02 - y]Ny + ypNp + Nt - rN = 0 with N(r, y, p, T, T) = I/p(T). The corresponding valuation formula is: (56) N(r,y,p,t,T) _ e$e[(K +P f)( T-t)]/2 2K /a2 ( + K2 + po-2 rp )( e ( )1 ) + 2e{ x exp ((- ( ) (-)_P+-)P(r, t, T)/p(t), +( + 2 +p po-2)pe (et) lT-)+_2j where [(K2 + pu2op)2 + 2( 12 17 Slater [34] gives properties of the confluent hypergeometric function. J. C. COX, J. E. INGERSOLL, JR., AND S. A. ROSS The term structure of interest rates implied by (54) and (56) can assume a wide variety of shapes, depending on the relative values of the variables and parameters. More complex models incorporating more detailed effects can be built along the same lines. Throughout our paper, we have used specializations of the fundamental valuation equation (6). This equation determines the real value of a contingent claim as a function of real wealth and the state variables. For some empirical purpose... |

331 |
The monetary dynamics of hyperinflation.
- Cagan
- 1956
(Show Context)
Citation Context ...section, one of the explanatory variables is not directly observable. Multifactor generalizations will typically inherit this drawback to an even greater degree. Consequently, it may be very convenient for empirical applications to use some of the endogenously determined prices as instrumental variables to eliminate the variables that cannot be directly observed. In certain instances, it will be possible to do so. Let us choose the spot rate, r, 14 Studies which have expressed expected future spot rates as linear combinations of current and past spot rates include Bierwag and Grove [2], Cagan [4], De Leeuw [8], Duesenberry [11], Malkiel [19], Meiselman [20], Modigliani and Shiller [24], Modigliani and Sutch [25], Van Horne [36], and Wood [38]. Cox, Ingersoll, and Ross [5] examine this issue in a diffusion setting. 400 J. C. COX, J. E. INGERSOLL, JR., AND S. A. ROSS and a vector of long interest rates, 1, as instrumental variables. In general, each of these interest rates will be functions of W (unless the common utility function is isoelastic) and all the state variables. If it is possible to invert this system globally and express the latter as twice differentiable functions of r and... |

31 |
Inflation, Rational Expectations and the Term Structure of Interest Rates,"
- MODIGLIANI, SHILLER
- 1973
(Show Context)
Citation Context ... other investment alternatives. Also, individuals can have specific preferences about the timing of their consumption, and thus have, in that sense, a preferred habitat. Our model thus permits detailed predictions about how changes in a wide range of underlying variables will affect the term structure. 2 We thank Franco Modigliani for mentioning this point. 3 Stiglitz [35] emphasizes the portfolio theory aspects involved with bonds of different maturities, as do Dieffenbach [9], Long [18], and Rubinstein [31], who incorporate the characteristics of other assets as well. Modigliani and Shiller [24] and Sargent [33] have stressed the importance of rational anticipations. INTEREST RATES 387 The plan of our paper is as follows. Section 2 summarizes the equilibrium model developed in Cox, Ingersoll, and Ross [6] and specializes it for studying the term structure. In Section 3, we derive and analyze a model which leads to a single factor description of the term structure. Section 4 shows how this model can be applied to other related securities such as options on bonds. In Section 5, we compare our general equilibrium approach with an alternative approach based purely on arbitrage. In Sectio... |

22 |
Business Cycles and Economic Growth.
- DUESENBERRY
- 1958
(Show Context)
Citation Context ... variables is not directly observable. Multifactor generalizations will typically inherit this drawback to an even greater degree. Consequently, it may be very convenient for empirical applications to use some of the endogenously determined prices as instrumental variables to eliminate the variables that cannot be directly observed. In certain instances, it will be possible to do so. Let us choose the spot rate, r, 14 Studies which have expressed expected future spot rates as linear combinations of current and past spot rates include Bierwag and Grove [2], Cagan [4], De Leeuw [8], Duesenberry [11], Malkiel [19], Meiselman [20], Modigliani and Shiller [24], Modigliani and Sutch [25], Van Horne [36], and Wood [38]. Cox, Ingersoll, and Ross [5] examine this issue in a diffusion setting. 400 J. C. COX, J. E. INGERSOLL, JR., AND S. A. ROSS and a vector of long interest rates, 1, as instrumental variables. In general, each of these interest rates will be functions of W (unless the common utility function is isoelastic) and all the state variables. If it is possible to invert this system globally and express the latter as twice differentiable functions of r and 1, then r and I can be used as ... |

15 |
A dynamic general equilibrium model of the asset market and its application to the pricing of the capital structure of the firm, Working Paper No. 497, Sloan School of Management, MIT,
- Merton
- 1970
(Show Context)
Citation Context ...ARBITRAGE METHODS In this section, we briefly compare our methodology to some alternative ways to model bond pricing in continuous time. It is useful to do this now rather than later because the model of Section 3 provides an ideal standard for comparison. Our approach begins with a detailed description of the underlying economy. This allows us to specify the following ingredients of bond pricing: (a) the variables on which the bond price depends, (b) the stochastic properties of the underlying variables which are endogenously determined, and (c) the exact form of the factor risk premiums. In [21], Merton shows that if one begins instead by imposing assumptions directly about (a) and (b), then Ito's formula can be used to state the excess expected return on a bond in the same form as the left-hand side of (4). If the functional form of the right-hand side of (4) were known, then one could obtain a bond pricing equation. For example, if one arbitrarily assumed that bond prices depend only on the spot interest rate r, that the interest rate follows the process given by (17), and that the excess expected return on a bond with maturity date T is Y(r, t, T), then one would obtain (33) j-r2r... |

15 |
The Behavior of Interest Rates.
- ROLL
- 1970
(Show Context)
Citation Context ...lectic analyses of the term structure on explaining and testing the term premiums is desirable, there are two difficulties with this approach. First, we need a better understanding of the determinants of the term premiums. The previous theories are basically only hypotheses which say little more than that forward rates should or need not equal expected spot rates. Second, all of the theories are couched in ex ante terms and they must be linked with ex post realizations to be testable. The attempts to deal with these two elements constitute the fourth strand of work on the term structure. Roll [29, 30], for example, has built and tested a mean-variance model which treated bonds symmetrically with other assets and used a condition of market efficiency to relate ex ante and ex post concepts.3 If rationality requires that ex post realizations not differ systematically from ex ante views, then statistical tests can be made on ex ante propositions by usino ex post data. We consider the problem of determining the term structure as being a problem in general equilibrium theory, and our approach contains elements of all of the previous theories. Anticipations of future events are important, as are ... |

8 | The Term Structure of Interest Rates. - NELSON - 1972 |

7 |
The Term Structure of Interest Rates (Englewood Cliffs:
- Meiselman
- 1962
(Show Context)
Citation Context ...ervable. Multifactor generalizations will typically inherit this drawback to an even greater degree. Consequently, it may be very convenient for empirical applications to use some of the endogenously determined prices as instrumental variables to eliminate the variables that cannot be directly observed. In certain instances, it will be possible to do so. Let us choose the spot rate, r, 14 Studies which have expressed expected future spot rates as linear combinations of current and past spot rates include Bierwag and Grove [2], Cagan [4], De Leeuw [8], Duesenberry [11], Malkiel [19], Meiselman [20], Modigliani and Shiller [24], Modigliani and Sutch [25], Van Horne [36], and Wood [38]. Cox, Ingersoll, and Ross [5] examine this issue in a diffusion setting. 400 J. C. COX, J. E. INGERSOLL, JR., AND S. A. ROSS and a vector of long interest rates, 1, as instrumental variables. In general, each of these interest rates will be functions of W (unless the common utility function is isoelastic) and all the state variables. If it is possible to invert this system globally and express the latter as twice differentiable functions of r and 1, then r and I can be used as instrumental variables in a ma... |

5 |
F.: "A Model of Financial Behavior,"
- LEEUW
- 1965
(Show Context)
Citation Context ...f the explanatory variables is not directly observable. Multifactor generalizations will typically inherit this drawback to an even greater degree. Consequently, it may be very convenient for empirical applications to use some of the endogenously determined prices as instrumental variables to eliminate the variables that cannot be directly observed. In certain instances, it will be possible to do so. Let us choose the spot rate, r, 14 Studies which have expressed expected future spot rates as linear combinations of current and past spot rates include Bierwag and Grove [2], Cagan [4], De Leeuw [8], Duesenberry [11], Malkiel [19], Meiselman [20], Modigliani and Shiller [24], Modigliani and Sutch [25], Van Horne [36], and Wood [38]. Cox, Ingersoll, and Ross [5] examine this issue in a diffusion setting. 400 J. C. COX, J. E. INGERSOLL, JR., AND S. A. ROSS and a vector of long interest rates, 1, as instrumental variables. In general, each of these interest rates will be functions of W (unless the common utility function is isoelastic) and all the state variables. If it is possible to invert this system globally and express the latter as twice differentiable functions of r and 1, then r and... |

1 |
A General Theory of Asset Valuation Under Diffusion Processes,"
- GERMAN
- 1977
(Show Context)
Citation Context ...s the process given by (17), and that the excess expected return on a bond with maturity date T is Y(r, t, T), then one would obtain (33) j-r2rPrr+ K(O- r)Pr + Pt-rP = Y(r, t, T). If there is some underlying equilibrium which will support the assumptions (a) and (b), then there must be some function Y for which bond prices are given by (33). However, as Merton notes, this derivation in itself provides no way to determine Y or to relate it to the underlying real variables. An arbitrage approach to bond pricing was developed in a series of papers by Brennan and Schwartz [3], Dothan [10], Garman [14], Richard [28], and Vasicek [37]. Arguments similar to those employed in the proof of Theorem 2 of Cox, Ingersoll, and Ross [6] are used to show that if there are no arbitrage opportunities, Y must have the form (34) Y(r, t, T) = aql(r, t)P,(r. t, T), where fr is a function depending only on calendar time and not on the maturity date of the bond. This places definite restrictions on the form of the excess expected return; not all functions Y will satisfy both (33) and (34). 398 J. C. COX, J. E. INGERSOLL, JR., AND S. A. ROSS There are some potential problems, however, in going one step further... |

1 |
Value and Capital, 2nd edition. London:
- HICKS
- 1946
(Show Context)
Citation Context ...e expected values of future spot rates or holdingperiod returns. In its simplest form, the expectations hypothesis postulates that bonds are priced so that the implied forward rates are equal to the expected spot rates. Generally, this approach is characterized by the following propositions: (a) the return on holding a long-term bond to maturity is equal to the expected return on repeated investment in a series of the short-term bonds, or (b) the expected rate of return over the next holding period is the same for bonds of all maturities. The liquidity preference hypothesis, advanced by Hicks [16], concurs with the importance of expected future spot rates, but places more weight on the effects of the risk preferences of market participants. It asserts that risk aversion will cause forward rates to be systematically greater than expected spot rates, usually ' This paper is an extended version of the second half of an earlier working paper with the same title. We are grateful for the helpful comments and suggestions of many of our colleagues, both at our own institutions and others. This research was partially supported by the Dean Witter Foundation, the Center for Research in Security P... |

1 |
The Term Structure of Interest Rates: Excpectations and Behavior Patterns.
- MALKIEL
- 1966
(Show Context)
Citation Context ...not directly observable. Multifactor generalizations will typically inherit this drawback to an even greater degree. Consequently, it may be very convenient for empirical applications to use some of the endogenously determined prices as instrumental variables to eliminate the variables that cannot be directly observed. In certain instances, it will be possible to do so. Let us choose the spot rate, r, 14 Studies which have expressed expected future spot rates as linear combinations of current and past spot rates include Bierwag and Grove [2], Cagan [4], De Leeuw [8], Duesenberry [11], Malkiel [19], Meiselman [20], Modigliani and Shiller [24], Modigliani and Sutch [25], Van Horne [36], and Wood [38]. Cox, Ingersoll, and Ross [5] examine this issue in a diffusion setting. 400 J. C. COX, J. E. INGERSOLL, JR., AND S. A. ROSS and a vector of long interest rates, 1, as instrumental variables. In general, each of these interest rates will be functions of W (unless the common utility function is isoelastic) and all the state variables. If it is possible to invert this system globally and express the latter as twice differentiable functions of r and 1, then r and I can be used as instrumental v... |

1 |
Bessel Functions of Integer Order," Handbook of Mathematical Functions,
- OLIVER
- 1965
(Show Context)
Citation Context ...e-cor where w = 2K/o-2 and -=-2KG/ 02. The steady state mean and variance are 0 and G20/2K, respectively. Consider now the problem of valuing a default-free discount bond promising to pay one unit at time T'0 The prices of these bonds for all T will completely determine the term structure. Under our assumptions, the factor risk premium in (12) is (21) (at f-I +( 1 Q1 c1n-,1Y -Ay 9 Processes similar to (17) have been extensively studied by Feller. The Laplace transform of (18) is given in Feller [12]. See Johnson and Kotz [17] for a description of the noncentral chi-square distribution. Oliver [27] contains properties of the modified Bessel function. 10 A number of contractual provisions are sufficient to preclude default risk and make the value of a bond independent of the wealth of its seller. For example, the terms of the bond could specify that the seller must repurchase the bond at the price schedule given by (23) whenever his wealth falls to a designated level. INTEREST RATES By using (15) and (21), we can write the fundamental equation for the price of a discount bond, P, most conveniently as (22)' o72rPrr+ K( - r)Pr+ P,-ArPrPrP = O, with the boundary condition P(r, T, T) = 1. Th... |