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Heterogeneous Agents and the Indifference Pricing of Property Index Linked Swaps

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Abstract

Swap spreads predicted by the traditional risk-neutral valuation models are much lower than the quoted market spreads for property index linked swaps (Patel and Pereira, Journal of Real Estate Finance and Economics, 36:5–21, 2008). This paper attempts to develop a utility indifference-based model for evaluating the reservation spreads of swap receivers and payers based on the principle of expected wealth utility equivalence rather than the traditional risk-neutral argument. Under the proposed model framework, this paper addresses the determination of the swap spreads. When the incompleteness of real estate markets and heterogeneity of representative agents are taken into consideration, it is shown that the agents’ risk preferences and heterogeneous beliefs about expected future property returns are the remarkable determinants for the swap spreads. Our model also identifies market power and the settlement rules in the event of counterparty default as important factors in determining the swap spreads. Our model provides a possible interpretation for the difference between the spreads predicted by the traditional models and the actual market spreads.

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Notes

  1. Despite such a growth, development of the real estate swap market has still been limited by the non-tradability of commercial real estate indices in the cash or spot market, noise and lag issues in these indices, and poor correlation between these indices and most real estate portfolios (Geltner and Fisher 2007).

  2. Buttimer et al. (1997), Bjork and Clapham (2002), and Patel and Pereira (2008) provide good attempts to price property index linked swaps. They develop risk-neutral valuation models to predict the fair market value for this category of property derivatives. The risk-neutral valuation framework is dependent on the notion that there could exist underlying assets that are continuously traded without frictions.

  3. The risk-neutral argument implies that the expected returns on all financial assets are the risk-free interest rate under the risk-neutral probability measure. As a result, all the investors are indifferent toward risk and are risk preference free in that they require no compensation for their investment risk.

  4. In a complete market setting, standard risk-neutral models can uniquely determine a fair market value of derivative securities, based on risk-neutral valuation argument.

  5. The assumption of market completeness means that the spanning condition holds in financial markets. That is, the payoff of any financial product in this market can be accurately replicated in any state by creating a portfolio consisting of traded financial assets, such that the risk associated with the product can be completely hedged due to building the perfectly replicating portfolio. In contrast, however, the payoff of a real estate investment project is usually not completely spanned by setting up the traded asset portfolio due to market illiquidity, high transaction costs, and substantial unhedgeable idiosyncratic risk associated with this project.

  6. There are possibly a number of causes leading to market incompleteness, such as transaction costs, trading constraints, stochastic volatility, illiquidity and unobservability of underlying assets. Among these causes, these latter two can be thought of as the typical features in real estate markets as discussed above.

  7. See Staum (2007) for a more detailed discussion on the pricing issue of derivative securities under market incompleteness.

  8. The risk-neutral valuation framework can also be extended to evaluate derivatives whose underlying assets are likely to be neither continuously traded nor continuously observable, while this usually requires estimating the market price of risk based on the capital asset pricing model (CAPM) (Hull 2003, Chapter 28).

  9. For a recent survey on heterogeneous beliefs about expectation, see Hommes (2006).

  10. Another alternative pricing approach is the CAPM-type equilibrium model, which is also likely to be developed for evaluating property index linked swaps. Several earlier paradigms, such as Rubinstein (1976), Merton (1973), Constantinides (1978) and Breeden and Litzenberger (1978), develop this category of models for pricing derivatives securities. However, the CAPM-type models are typically subject to several limitations. First, in these models, future uncertain cash flows generated from a derivative are discounted using a discount rate adjusted to the market risk premium, which is determined by the covariance between its return and the return on a market portfolio. This suggests that a market portfolio (or aggregate consumption) needs to be defined and considered in the models, so that the risk premium of the given derivative can be priced. However, the market portfolio is unobservable, and therefore a proxy for it has to be chosen. As a result, the pricing results of the models could be sensitive to the selection of the proxy in place of the market portfolio (Jagannathan and Wang 1996). Second, the CAPM-type models usually assume that all asset returns in an economy are normally distributed, while the normality assumption is typically improper for derivative securities (Altug and Labadie, Chapter 4, 2008). Third, investors are assumed to have homogeneous expectations on future asset returns, while heterogeneous expectations could be important in explaining swap transactions. Our model takes into account heterogeneous expectations. Fourth, these models are the equilibrium models of asset pricing, whereas our model has flexibility in allowing for market disequilibrium. Finally, there is still a lack of empirical evidence in favor of the CAPM (e.g., Jagannathan and Wang 1996).

  11. For simplicity, in the subsequent analysis we focus only on fixed rate payments the owner will receive, while it is straightforward to extend our model to allow for floating rate payments.

  12. We can also explain the risk aversion coefficient as the precautionary saving motive here as discussed in Miao and Wang (2007).

  13. Without loss of generality, we assume that the real estate returns derived from the owner’s own property are perfectly correlated with the real estate index linked returns.

  14. Our model is built on the standard representative-agent based framework (see, e.g., Lucas 1978), which implies that the aggregate behavior of all agents of the same kind in an economy might effectively be described by the behavior of a representative agent. Similar to Basak and Croitoru (2000, 2006), this model extends this approach to allow for two market agents, who have heterogeneous beliefs and risk preferences. These heterogeneities result in swap transactions. We do not further extend this model by considering more market participants with multiple utility functions and even search cost, in that this, to a large extent, will complicate our analysis without additional insights. Actually, Duffie et al. (2005) have provided a detailed discussion about the effects of market search and bargaining on various asset prices in over-the-counter markets by developing a search-and-bargaining based model. Their model considers two categories of agents, investors and market makers, and these agents are further classified according to multiple market attributes. Their findings show that bid-ask spreads in an economy will become smaller, and bid and asking prices will gradually approach the Walrasian equilibrium prices, as multiple market makers become more accessible or investors find other investors more quickly for a direct trade. However, fast intermediation by a monopolistic market maker does not make the bid and ask prices approach the equilibrium prices due to her full bargaining power (see also Duffie et al. 2007).

  15. For the purpose of this study, we pay no attention to the reasons for generating heterogeneous beliefs on expectation, and only focus on the significant effect of heterogeneous beliefs on real estate swap deals. See Xiong and Yan (2010) for a survey on the possible interpretations for heterogeneous expectations.

  16. The 2002 Agreement is an amendment to the 1991 Agreement and is viewed as a new market standard agreement for swap transactions.

  17. For a more detailed discussion on this category of US court cases, see, e.g., Cuillerier and Zylberberg (2009).

  18. Appendix A provides a more specific expression for (33).

  19. It is worth noting that our model is developed according to the principle of wealth utility indifference, while Buttimer et al. (1997) and Bjork and Clapham (2002) are built on the risk-neutral valuation framework. There two categories of pricing models are different in model and parameter specifications. Therefore, it is difficult to carry out an accurate comparison between these models. Despite such, to highlight their significant differences through an approximate comparison, we set all the basic input parameter values to approximate as closely as possible those in Buttimer et al. (1997) and Bjork and Clapham (2002).

  20. In this numerical analysis, our reservation swap spreads are shown to be much higher than about 0.125 basis points predicted in Buttimer et al. (1997). Patel and Pereira (2008) show that over the period from 11/05 to 03/06, the average spread for several LIBOR-IPD swaps is about 300 basis points and the lowest spread is around 100 basis points. Most of our estimated reservation swap spreads are also greater than 100 basis points. These results imply that, compared with the risk-neutral valuation models, our utility indifference-based model is likely to better approximate the actual market spreads for property index linked swaps.

  21. See Bliss and Panigirtzoglou (2004) for a detailed discussion on the estimated values of risk aversion coefficient.

  22. Some empirical results suggest that the estimated coefficients of absolute risk aversion could also be very small (e.g., Lien 2002; Bliss and Panigirtzoglou 2004).

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Authors and Affiliations

  1. School of Insurance, Southwestern University of Finance and Economics, 55 Guanghuacun Street, Chengdu, 610074, China

    Ming Pu

  2. Department of Real Estate Studies, Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seoul, 143-701, Korea

    Gang-Zhi Fan

  3. Department of Real Estate, National University of Singapore, 4 Architecture Drive, Singapore, 117566, Singapore

    Seow Eng Ong

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  1. Ming Pu
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Correspondence to Gang-Zhi Fan.

Appendix

Appendix

Equation (33) can be further written as

$$ \begin{array}{*{20}{c}} {U({R_{{fix}}}) = E\left[ { - \frac{1}{{{\gamma_1}}}\exp \left( { - {\gamma_1}({W_0} + \sum\limits_{{i = 1}}^K {{e^{{ - ir}}}{R_{{fix}}}} + \sum\limits_{{i = K + 1}}^N {{e^{{ - ir}}}{R_i}} )} \right)} \right]} \hfill \\{ = \sum\limits_{{j = 1}}^N {E\left[ { - \frac{1}{{{\gamma_1}}}\exp \left( { - {\gamma_1}({W_0} + \frac{{1 - {e^{{ - rj}}}}}{{{e^r} - 1}}{R_{{fix}}} + \sum\limits_{{i = j + 1}}^N {{e^{{ - ir}}}{R_i}} )} \right)} \right]\Pr (K = j)} } \hfill \\{ = \sum\limits_{{j = 1}}^N {E\left[ { - \frac{1}{{{\gamma_1}}}\exp \left( { - {\gamma_1}({W_0} + \frac{{1 - {e^{{ - rj}}}}}{{{e^r} - 1}}{R_{{fix}}} + k(j + 1){R_j} + \sum\limits_{{i = j + 1}}^N {k(i){X_i}} )} \right)} \right]\Pr (K = j)} } \hfill \\{ = \sum\limits_{{j = 1}}^N {\{ - \frac{1}{{{\gamma_1}}}\exp [ - {\gamma_1}({W_0} + \frac{{1 - {e^{{ - rj}}}}}{{{e^r} - 1}}{R_{{fix}}} + k(j + 1)[{R_0} + j\mu ] - \frac{1}{2}{\gamma_1}{k^2}(j + 1)j{\sigma^2}} } \hfill \\{ + \sum\limits_{{i = j + 1}}^N {k(i)\mu - \frac{1}{2}{\gamma_1}{\sigma^2}\sum\limits_{{i = j + 1}}^N {{k^2}(i)} } )]\} \Pr (K = j).} \hfill \\\end{array} $$

The research funding from the Southwestern University of Finance and Economics “Project 211” grants is acknowledged. We thank Jun Liu, Yu Wu, Yan Dong, Lin Huang, and seminar participants at Southwestern University of Finance and Economics and at the 2009 Global Chinese Real Estate Congress Meeting for their helpful discussions and comments. Any errors are our own.

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Pu, M., Fan, GZ. & Ong, S.E. Heterogeneous Agents and the Indifference Pricing of Property Index Linked Swaps. J Real Estate Finan Econ 44, 543–569 (2012). https://doi.org/10.1007/s11146-010-9298-4

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