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Relationship between Treasury bills and Eurodollars: Theoretical and Empirical Analyses

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Abstract

In this paper, we derive an equilibrium relationship between the yields on Eurodollar and Treasury bills based on equivalent martingale results derived by Harrison and Kreps (1979) and Harrison and Pliska (1981, 1983) as well as the corporate debt pricing model developed by Merton (1974). The derived equilibrium relationship incorporates the models used by Booth and Tse (1995) and Shrestha and Welch (2001) as special cases. The equilibrium relationship indicates that the conditional volatility of the yield on Eurodollars explains the variation in the TED spread. We empirically test the equilibrium relationship using a GARCH-M model and the concept of fractional cointegration. We use both the ex ante data implied by the respective futures contracts as well as the ex post spot data with daily, weekly and monthly frequencies. We find empirical support for the Equilibrium relationship.

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Notes

  1. Readers interested in the integration of international financial markets are referred to Lothian (2002).

  2. TED spread is generally defined as the difference between the interest rates implied by the Eurodollar and Treasury bill futures contracts of the same maturity. In this paper, we analyze both the TED spread implied by the futures contracts and the TED spread measured by the difference between the spot Eurodollar and the Treasury bill rates. Since the rate implied by the futures contract can be taken as the expected rate in the future, we will refer the first TED spread (implied by the futures contracts) as the expected or ex ante TED spread and the latter one (computed based on spot rates) as the actual or ex post TED spread.

  3. The mean-reverting process means that the infinite cumulative impulse response is zero. This mean-reverting characteristic, when applied to the relationship between the two interest rates, implies that any deviation from equilibrium at time t will disappear in an infinite number of periods in the future.

  4. For simplicity, we assume that the R fraction of the face value will be paid on maturity even if the default occurs before the maturity date T.

  5. We deal with the issue of cointegration in detail in the next section.

  6. For a detailed discussion of such models, see Duffie and Singleton (2003).

  7. Vassalou and Xing (2004) also uses Merton (1974) model to derive the default probability.

  8. Actually, the short rate does not have to be non-stochastic. We will discuss this issue later (see footnote 10).

  9. It can be shown that \(\frac{{\partial B_d \left( {t,T} \right)}}{{\partial \sigma }} = - Vn\left( { - h_2 } \right)\sqrt {T - t} < 0 \), where n( ) is the standard normal density function.

  10. In the derivation of Eqs. (8) and (9), we assumed the default-free short rate to be non-stochastic. However, this assumption is not necessary. We can allow the short rate to be stochastic in which case e −r(T−t) should be replaced by default-free ZCB price B (t, T) and σ 2(T−t) should be replaced by generalized variance term (see Merton (2000), 395). Alternatively, we can assume that the value of the firm at time T is log-normally distributed with variance σ 2(T−t) in a forward risk neutral world.

  11. See Baillie (1996) for a comprehensive discussion on the long memory process.

  12. The property that the series X t with d ≥ 0.5 have infinite variance would play an important role in the definitions of mean-reverting and stationary fractional cointegrations to be discussed later.

  13. This definition can be generalized by allowing d to be less than 1 but still non-stationary, i.e., 0.5 ≤ d ≤ 1.

  14. Here t represents the time trend. We are including the time trend to capture the observed declining trend in TED spread (please see Fig. 1).

  15. In the variance equation (Eq. (19)) all the parameters are squared in order to avoid the variance being negative in the process of numerical optimization.

  16. Before using the heteroscedastic model given above, the existence of a heteroscedasticity can be tested using the LM test (see Engle (1982)). If the heteroscedasticity is found to be absent, then the cointegrating regression reduces to Eq. (17) with trend term.

  17. The LM statistic is computed by estimating the regression of the squared residual on its own 1 to 5 period lags.

  18. Here we are dealing with non-nested models. For example, GARCH-M(3,1) and GARCH(1,3)-M are non-nested. Therefore, we cannot use likelihood ratio criteria to choose the best-suited model.

  19. This is our conjecture at this time. We intend to follow this in the future.

  20. For all the models, the estimates of the constant term δ 1 (not reported in the paper) were highly significant. Therefore, the misspecification problem for IGARCH with no drift case pointed out by Nelson (1990) does not apply to our case.

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Authors and Affiliations

  1. Department of Finance, School of Business, Rutgers University, Rutgers, USA

    Cheng-few Lee

  2. Department of International Business, Dah-Yeh University, Dah-Yeh, Taiwan

    Cheng-few Lee

  3. Division of Banking and Finance, Nanyang Business School, Nanyang Technological University, 50 Nanyang Avenue, Singapore, Singapore, 639798

    Keshab Shrestha

  4. Department of Finance, Operations and Information Systems, Faculty of Business, Brock University, Brock, Canada

    Robert L. Welch

Authors
  1. Cheng-few Lee
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  2. Keshab Shrestha
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  3. Robert L. Welch
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Correspondence to Keshab Shrestha.

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Lee, Cf., Shrestha, K. & Welch, R.L. Relationship between Treasury bills and Eurodollars: Theoretical and Empirical Analyses. Rev Quant Finan Acc 28, 163–185 (2007). https://doi.org/10.1007/s11156-006-0006-7

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