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01.08.2014 | Research Article

Portfolio management with stochastic interest rates and inflation ambiguity

verfasst von: Claus Munk, Alexey Rubtsov

Erschienen in: Annals of Finance | Ausgabe 3/2014

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Abstract

We solve, in closed form, a stock-bond-cash portfolio problem of a risk- and ambiguity-averse investor when interest rates and the inflation rate are stochastic. The expected inflation rate is unobservable, but the investor can learn about it from observing realized inflation and stock and bond prices. The investor is ambiguous about the inflation model and prefers a portfolio strategy which is robust to model misspecification. Ambiguity about the inflation dynamics is shown to affect the optimal portfolio fundamentally different than ambiguity about the price dynamics of traded assets, for example the optimal portfolio weights can be increasing in the degree of ambiguity aversion. In a numerical example, the optimal portfolio is significantly affected by the learning about expected inflation and somewhat affected by ambiguity aversion. The welfare loss from ignoring learning or ambiguity can be considerable.

Vorheriger Artikel Hidden persistent disasters and asset prices

Nächster Artikel The determinants of a cross market arbitrage opportunity: theory and evidence for the European bond market

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For simplicity of notation we suppress the dependence of \(\sigma _P\) on \(\bar{T}-t\).

As Scheinkman and Xiong (2003), Dumas et al. (2009), and Branger et al. (2013), we assume that learning was long enough so that the variance of the estimation error has reached a steady state.

To simplify the notation, we write \({\varvec{\Pi }}\) instead of \(\{(\varPi _s^S, \varPi _s^P)\}_{s \in [t,T]}\) and \(e\) instead of \(\{e_s\}_{s \in [t,T]}\). The expectation operator with respect to the probability measure \(\mathbb P ^e\) is defined as \(E_{t,\mathbf{y}}^\mathbb{P ^e}[\cdot ] \triangleq E^\mathbb{P ^e}[\cdot |X_t=x,Z_t=z,r_t=r,\hat{\beta }_t=\hat{\beta }]\).

For a critique of this approach, see Pathak (2002).

The hedge against the inflation risk is also zero if \(\theta =1-\gamma \), but since empirical studies support \(\gamma >1\) and \(\theta \) has to be positive, this is unlikely to be the case.

It can be shown that the HJB equation for the case with observable \(\beta \)—so that the dynamics in (5) is used instead of (9)—is similar to the HJB equation (19), but some coefficients to second-order derivatives involving \(\beta \) are different.

Note that the ambiguity aversion parameter depends on the precise model set-up and source of ambiguity and therefore has to be estimated on a case-by-case basis.

Since Brennan and Xia (2002) estimate the parameters for real interest rates, we have adjusted their parameters to be applicable in our model.

Recall that we assume a correlation of 0.3 between realized and expected inflation, \(\rho _{Z\beta }\). If, in the case with an observable expected inflation rate, we use \(\rho _{Z\beta }=-0.3\), the term \(\pi _{\beta }^S\) will in fact be slightly negative, so that the hedge against expected inflation and thus the entire portfolio weight of the stock will be decreasing with the horizon—in contrast with typical investment advice. Using \(\rho _{Z\beta }=-0.3\) in our general model with unobserved expected inflation, the portfolio weight of the stock would still be increasing with the horizon.

In settings with multiple risky assets, the effect of ambiguity about a traded asset price on the optimal portfolio is not clear-cut. Flor and Larsen (2013) report that the speculative component of the bond (stock) demand increases (decreases) in \(\theta \) if the investor is uncertain about the stock price process only.

It is easy to check that the assumptions of the theorem are satisfied because entries in all matrices are constant.

This is the same equation as Eq. (25) but without the supremum.

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Titel: Portfolio management with stochastic interest rates and inflation ambiguity
verfasst von: Claus Munk
Alexey Rubtsov
Publikationsdatum: 01.08.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Annals of Finance / Ausgabe 3/2014
Print ISSN: 1614-2446
Elektronische ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-013-0238-1

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