RDP 9307: Explaining Forward Discount Bias: Is it Anchoring? Appendix B: The Traders' Asset – Demand Functions

Define Wt as the real wealth a trader manages in period t, xt as the proportion of Wt held in the foreign nominal asset, Inline Equation and Inline Equation as the real returns on the domestic and foreign assets and zt+1Inline EquationInline Equation as the real excess return on the foreign asset. Following Frankel and Engle (1984), real wealth in period t+1, Wt+1 is

By assumption, traders maximise a function of the expected value and variance of end-of-period real wealth, F(EtWt+1, VtWt+1). Differentiating F(•,•) with respect to xt and introducing the identity of traders with the subscript k gives[28]

where γ ≡ −2WtF2/F1 is the coefficient of relative risk aversion evaluated at Wt, common to both types of traders.

The real domestic return Inline Equation and the excess foreign return zt+1 are given by

where Inline Equation is the price index, measured in domestic currency, for the traders' consumption basket. Each trader spends the fraction g of her consumption expenditure on foreign goods, so Inline Equation is given by Inline Equation = (P*St)g(Pt)1−g. Then,[29]

and hence, from (B3) and (B4),

Equation (B7) implies that Inline Equation. Since Δpt+1 « Δst+1 it follows that Inline Equation. Finally, the variance of exchange rate changes is closely approximated by Inline Equation, that is, Inline Equation where k = r or a.[30] Substituting these approximations into the asset demand functions (B2) leads directly to equation (9) in the text.[31]

Footnotes

Since the anchored traders' behaviour is determined by their own subjective expectations rather than by true mathematical expectations, Inline Equation should be interpreted as Inline Equation. A similar comment applies to Inline Equation. As we shall see however, this subtlety makes no important difference to the analysis. [28]

The parameter values derived in Section 5 imply Δpt+1 ≈ (i − i*)t « Δst+1 « 1, which justifies the first-order Taylor series approximations used to derive (B5) – (B7). [29]

The rational traders' estimate of the variance of exchange rate changes is larger than the anchored traders' estimate (because the former understand that nominal money shocks contribute to this variance). But the effect is small. For almost all the results we report 1 ≤ Inline Equation (Δst+1)/Inline Equation ≤ 1.1. Hence, this approximation is a good one. [30]

We note here a refinement which slightly modifies the analysis but makes no important difference to the results. Krugman (1981) points out that (B5) – (B7) are not quite correct. The variance of exchange rate changes is so large that second-order Taylor expansions should be used for all the St+1/St terms. This refinement implies (Frankel (1983)) that the first term in equation (9), g, should be replaced by g – (g – 1/2)/γ. Implementing this refinement leads to a minimal change to the model. To be precise, the special case of the model introduced in Section 4.4 now applies when the proportion of foreign assets managed by traders is g – (g – 1/2)/γ. Assuming any other proportion of foreign assets leads to the more general form of the model described in Section 4.7. [31]