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Review of Quantitative Finance and Accounting

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01.05.2013 | Original Research

The quality of public information and the term structure of interest rates

verfasst von: Frederik Lundtofte

Erschienen in: Review of Quantitative Finance and Accounting | Ausgabe 4/2013

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Abstract

This paper analyzes the term structure of interest rates in an exchange-only Lucas (Econometrica 46:1429–1445, 1978) economy where consumers learn about a stochastic growth rate through observations of the endowment process and an external public signal. We allow for deluded consumers, who exaggerate the degree of covariation between the external public signal and the growth rate. With such consumers, there can be a premium for noisy external public information in long-term bonds and the social value of more precise public information can be negative. Moreover, our model can create excessive yield volatility and deviations from the expectations hypothesis.

Vorheriger Artikel Outperformance Certificates: analysis, pricing, interpretation, and performance

Nächster Artikel Nominal interest rates and stationarity

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Note that the instantaneous correlation matrix associated with the vector process (\(Z_t^{D},\,Z_t^{\mu},\,Z_t^{s}\)) is valid (i.e., positive semidefinite) if and only if \(\rho_{\mu D}^2 \le 1\), ϕ² ≤ 1, and \(\rho_{\mu D}^2 +\phi^2 \le 1\). We assume that consumers only assign values to ϕ such that these conditions are fulfilled.

This is similar to the way Scheinkman and Xiong (2003) and Dumas et al. (2009) model overconfidence.

It is not a coincidence that, e.g., Xiong and Yan (2009) restrict their attention to the log utility case.

For general aggregation results, see Rubinstein (1974).

One set of parameters that yields an estimation error that is increasing with increasing precision of external public information is the following: ϕ = −0.2, κ = 0.2, ρ_μD = 0.8, σ_μ = 0.015, σ_D = 0.13, σ_s = 0.10.

In order to see this, note that \({\rm d}r_{t}=\gamma {\rm d}\widehat{m} _{t}\) and \(\overline{r}-r_{t}=\gamma (\overline{m}-m_{t})\).

However, while the interest rate in our model displays linear mean reversion, many studies find that real-world interest rates exhibit nonlinear mean reversion and some other studies argue that stationarity might be volatility induced (see Jones 2003, and the references therein).

Note that the following two parameter sets yield a variance of the interest rate that is increasing in the precision of public information: (i) ϕ = 0, γ = 3, κ = 0.2, σ_D = 0.13, σ_s = 0.10, σ_μ = 0.015, ρ_μD = −0.8; and (ii) ϕ = −0.2, γ = 3, κ = 0.2, σ_D = 0.13, σ_s = 0.10, σ_μ = 0.015, ρ_μD = −0.8.

Since the yield is given by \(R=-\ln P/(T-t)\), the instantaneous yield volatility is proportional to the instantaneous volatility of bond returns, i.e., Vol _t[dR] = Vol _t[dP/P]/(T − t). From Eq. (28), \(Vol_{t}[{\rm d}P/P]=\gamma \left( {\frac{1-e^{-\kappa \tau _{M}}}{\kappa}}\right) \left( \left( \frac{\widehat{v}_{t}+\rho_{\mu D}\sigma_{\mu}\sigma_{D}}{\sigma_{D}}\right)^{2}+\left(\frac {\widehat{v}_{t}+\phi \sigma_{\mu}\sigma_{s}}{\sigma_{s}}\right)^{2}\right)^{1/2}.\)

The following are examples of sets of input values that generate an upward-sloping term structure: (i) \(t=0, \, \phi =0, \, \gamma =3, \, \beta =0.05 , \, m_{0}=0.025, \, v_{0}=0.01^{2}, \, \overline{\mu}=0.030, \, \kappa =0.2, \, \sigma_{D}=0.13, \, \sigma_{\mu}=0.015, \, \sigma_{s}=0.10, \, \rho_{\mu D}=-0.5\). (ii) \(t=0, \, \phi =0.2, \, \gamma =3, \, \beta =0.05, \, m_{0}=0.025, \, v_{0}=0.01^{2}, \, \overline{\mu}=0.030, \, \kappa =0.2, \, \sigma_{D}=0.13, \, \sigma_{\mu}=0.015, \, \sigma_{s}=0.10, \, \rho_{\mu D}=-0.5\).

We find the limits by applying l’Hôpital’s rule.

The following are examples of parameter sets that give rise to a long-run yield that is higher than the long-run mean of the short interest rate: (i) γ = 3, σ_D = 0.13, σ_s = 0.10, σ_μ = 0.015, ρ_μD = −0.5, κ = 0.2 and ϕ = 0. (ii) γ = 3, σ_D = 0.13, σ_s = 0.10, σ_μ = 0.015, ρ_μD = −0.5, κ = 0.2 and ϕ = 0.2.

See also Svensson (2006) and Morris et al. (2006).

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Titel: The quality of public information and the term structure of interest rates
verfasst von: Frederik Lundtofte
Publikationsdatum: 01.05.2013
Verlag: Springer US
Erschienen in: Review of Quantitative Finance and Accounting / Ausgabe 4/2013
Print ISSN: 0924-865X
Elektronische ISSN: 1573-7179
DOI: https://doi.org/10.1007/s11156-012-0295-y

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