RDP 9307: Explaining Forward Discount Bias: Is it Anchoring? Appendix B: The Traders' Asset – Demand Functions
June 1993
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Define Wt as the real wealth a trader manages in period t, xt
as the proportion of Wt held in the foreign nominal asset,
and
as the real returns on the domestic
and foreign assets and zt+1 ≡
−
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
and the excess foreign return zt+1
are given by
where
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
is given by
= (P*St)g(Pt)1−g.
Then,[29]
and hence, from (B3) and (B4),
Equation (B7) implies that
. Since Δpt+1 «
Δst+1 it follows that
. Finally, the variance of exchange
rate changes is closely approximated by
, that is,
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,
should be interpreted as
. A similar comment applies to
. 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 ≤
(Δst+1)/ ≤ 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]