Loss aversion and the term structure of interest rates

Abstract

This article studies how the loss averse behaviour affects the term structure of real interest rates. Since the pro-cyclical conditional expected marginal rate of substitution, implied from the US consumption data, is consistent with the proposition of loss aversion, we incorporate the loss averse behaviour of prospect theory into the consumption-based asset pricing model. Motivated by the similarity between habit formation and the prospect theory utility, habit formation is exploited to determine endogenously the reference point of this behavioural finance utility. The highly curved characteristic of the term structure of real interest rates can thus be captured by the additional consideration of loss aversion. This model also fits the downward sloping volatility of the real yield curve in the data of US Treasury Inflation-Protection Securities (TIPS). Moreover, depending on the effective risk attitude of the representative agent with the loss averse behaviour of prospect theory, our model is capable of generating a normal or an inverted yield curve.

Notes

¹ Other alternatives include: (1) nonexpected utilities in Weil (1989) and Epstein and Zin (1990); (2) habit formation in Abel (1999), Constantinides (1990) and Campbell and Cochrane (1999); (3) some types of market incompleteness, such as Rietz (1988), the asymmetric underlying process in Hung (1994), the transaction cost in Aiyagari and Gertler (1991) and Heaton and Lucas (1996), heterogeneous agents in Mankiw (1986), Mankiw and Zeldes (1991), Weil (1992), Lucas (1994), Constantinides and Duffie (1996), etc.

² Tversky and Kahneman (1992) further extended the original prospect theory to the cumulative prospect theory to solve a variety of experimental evidence inconsistent with standard expected utility theory. Afterwards many studies tried to analyse the characteristic of this theory, including Schmidt (2003), Law and Peel (2009), Schmidt and Zank (2008), Cain et al. (2008), etc.

³ For example, one of the pioneer articles to apply the loss aversion of prospect theory to solving the equity premium puzzle is Benartzi and Thaler (1995), in which they consider a wealth-based loss averse utility for investors. In addition, in a more recent research, Lien (2001) study the effect of the loss aversion of prospect theory on the optimal futures hedge ratio, in which the investor is assumed to maximize the expected utility on his period-end wealth.

⁴ Routledge and Zin (2003) report a similar exercise to that of Melino and Yang (2003).

⁵The other solution of λ₁ is −1.014, which contradicts the assumption that the marginal utility with respect to the consumption must be positive.

⁶These 60 000 sets of random samples are employed to describe the possible states of nature of the world in our model. They are sampled once and used to calculate all the results under different values of the parameters. Using common random samples helps to isolate the effects of applying different values of the parameters from the effects of different realizations of the simulated samples on the interest rate term structure.

⁷The numerical results in Section ‘The effect of β in the exponential utility case’ will show that if the values of the parameters are adjusted to let the interest rates back to the normal magnitude, the term premium will become much smaller.

⁸This result seems contradictory to previous studies, in which habit formation is thought to be a valid way to increase the degree of risk aversion and therefore enlarge the degree of the risk or term premium. This could be attributed to the fact that our model employs the exponential utility, whereas the previous models employ the power utility. In the power utility, when the consumption level c_{t + i} is close to the reference point v_{t + i} , the degree of risk aversion is enlarged effectively. As for the exponential utility, except the case in which the value of c_{t + i}  − v_{t + i} is negative and very small, the degree of risk aversion will not be enlarged effectively. In our model, however, since v_{t + i} keeps a close trace behind c_{t + i} , the value of c_{t + i}  − v_{t + i} is not small enough to enlarge the degree of risk aversion effectively. Therefore, the exponential utility with the habit reference point cannot generate a high enough term premium.

⁹In fact, the probability of c_t  ≥ v_t is about 90% in our data set.

¹⁰ The reason why we do not use the results in Wachter (2006), the final publication of Wachter (2004), is that the longest maturity of the risk-less interest rates in Wachter (2006) is only 5 years, that is too short for us to compare the whole spectrum of the term structure.

¹¹These data sets are within a page titled ‘The US Real Term Structure of Interest Rates with Implicit Inflation Premium’ on his website.

¹²The first issuance of the US TIPS is from 1997, so our studying period begins from that year.

¹³We take the arithmetic average of every three monthly real yield curves on the website of McCulloch to derive the quarterly real yield curves for comparisons.