Synthetic money: Addressing the budget‐constraint issue

Abstract

A synthetic currency attempts to mimic a target currency with an optimal portfolio of other currencies, without having a position in the target currency. The original construction methodology of a synthetic currency imposed a budget constraint such that the portfolio weights sum to a non‐zero value, say. However, a synthetic currency is a portfolio of invariant currency indexes, rather than a portfolio of currency positions. In this article, we show that invariant currency indexes are tradable multilateral exchange rates. Consequently, the sum‐to‐x budget constraint unintentionally creates a non‐zero position in the target currency. We address this budget‐constraint issue by replacing the sum‐to‐x budget constraint with a sum‐to‐zero budget constraint, which correctly enforces a zero position in the target currency. Once the budget‐constraint issue is addressed, investors are faced with the fact that synthetic money is unable to mimic significant currency‐specific movements in target currencies.

A Synthetic Chinese Renminbi: Uncovering the Anchor Currencies

“…We estimate a synthetic Chinese renminbi in both the pre-August 2015 and the post-August 2015 periods for three frequencies of data, namely daily, weekly, and monthly. We use a constrained Frankel-Wei regression model subject to a sum-to-zero constraint on the portfolio weights (regression coefficients) (see Kunkler, 2022). Table 1 reports the estimated portfolio weights of the synthetic Chinese renminbi.…”

“…In logarithm terms at time a synthetic Chinese renminbi is given by: subject to a budget constraint such that the portfolio weights sum to zero: where is the synthetic Chinese renminbi that mimics the relative currency rate of the Chinese renminbi (target currency); is an indicator function that equals one when and zero when is the portfolio weight of the relative currency rate; is the relative currency rate; and is the sum of the portfolio weights. It should be noted that any sum-to-budget constraint unintentionally creates a synthetic Chinese renminbi with a non-zero position in the Chinese renminbi (see Kunkler, 2022). The synthetic Chinese renminbi in Eq.…”

“…For example, a synthetic Chinese renminbi is an optimal portfolio of other currencies that mimics the Chinese renminbi, without holding a Chinese renminbi position. To enforce a zero position in the target currency, the optimal portfolio must be subject to a budget constraint such that the portfolio weights sum to zero (Kunkler, 2022).…”

“…The investigation involves estimating a synthetic Chinese renminbi for a group of ten trading partner currencies to test whether the Chinese renminbi is less anchored to the US dollar in the post-August 2015 period, as well as uncovering any other anchor currencies. The synthetic Chinese renminbi is estimated with a constrained Frankel-Wei regression model, where the regression coefficients sum to zero and the common numéraire is an equally-weighted multicurrency numéraire (see Kunkler, 2022).…”