International financial assets
Table of contents
G-Cubed models track each region’s net international financial assets and it does so by currency of denomination. It is important to track what currency the assets are denominated in because exchange rate movements lead to asset-valuation effects.
The total USD value of the international financial assets held by a region are represented by ASSU(region) variable. That total is comprised of international financial assets denominated in each region’s currency.
The value of the international financial assets held in a specific currency is represented by ASSE(currency,region). ASSE(currency,region) is interpreted as the value of international financial assets held by the given region and denominated in the given currency. It is measured in Local Currency Units (LCU) for the specified currency rather than in US dollars.
ASSU is related to ASSE by the equation:
ASSU = sum( currency, exp(REXN(currency))*ASSE ) ;
Note that the real exchange rate, REXN, is used to convert from LCUs to US Dollars.
International financial asset accumulation
The G-Cubed model includes rules for how these stocks of international financial assets (one stock per currency), evolve for each region based on the region’s current account balance. These rules are expressed by the following three equations:
ABUY = sum( abroad, ainv*CURR(abroad) );
lead(ASSE) = ASSE*(1-labgrow(owner)) + ( ashr*ABUY + aeye*(CURR(owner)-ABUY) ) / exp(REXN(currency)) ;
ASSU = sum( currency, exp(REXN(currency))*ASSE ) ;
ABUY(region) is defined as the region’s new purchases of international financial assets, denominated in currencies other than the region’s own currency, but all valued in US dollars.
Current account balances, measured in US dollars, CURR(region), are given by:
CURR(region) = TBAU(region) + PBAL(region) + IRAS(region)
Simplify by ignoring exogenous PBAL(region) for now. In most G-Cubed models, PBAL(region) is equal to zero.
TBAU(region) is the trade balance (exports minus imports) for the region, in US dollars and it is related to the trade balance in the region’s currency TBAL(region) by:
TBAU = TBAL * exp(REXN) ;
Each region earns interest, IRAS(region), on its international financial asset holdings. IRAS(region) is again valued in US dollars.
IRAS = sum( currency, INTR(currency)*exp(REXN(currency))*ASSE ) ;
Note that the assets denominated in a given currency ASSE(currency,owner)m earn interest for the owner region at the real interest rate prevailing in the region with the specified currency.
Increments to ASSE(currency,owner)
To understand the evolution of ASSE(currency,owner), we need to undestand the increments for each currency, in US dollar values in the relevant equation:
lead(ASSE) = ASSE*(1-labgrow(owner)) + ( ashr*ABUY + aeye*(CURR(owner)-ABUY) ) / exp(REXN(currency)) ;
As an example, using ASSE(AUS,JPN), Japanese holdings of Australian denominated financial assets:
ASSE(AUS,JPN) = self.ASSE(AUS,JPN)*(1-labgrow(JPN)) + (ashr(AUS,JPN)*ABUY(JPN) + aeye(AUS,JPN)*(CURR(JPN)-ABUY(JPN)))/exp(REXN(AUS))
aeye is an identity matrix so it is zero for all currency/owner pairs except where they are equal (in those cases it is one).
ashr is an upper-triangular matrix. Each column documents how that region’s purchases of foreign current denominated financial assets is split across those foreign currencies. All columns except the first, which is all zeros, sum to one. Being upper-triangular, the ashr matrix implies that each region only invests in international financial assets that are denominated in its currency or the currencies of regions that come before the owner region in the G-Cubed model’s list of regions, as specified in the SYM files.
The design of the ashr and aeye matrices ensure that the sum of the increments, in US dollar values, across all currencies for a given region, must be equal to the current account balance for that region, CURR(region). Each region invests a part of their current account balance in assets denominated in their own currency, CURR(region)-ABUY(region) and the remainder, ABUY(region), is denominated in currencies of other regions.
The split of ABUY(region) across foreign currency denominations is determined by the order of the regions and the nominal GDP of those regions in the parameter calibration year.
The order of the regions matters because each region only invests in its assets denominated in its own currency or the currency of regions that come before it in the model’s list of regions.
The relative size of the regions matters because the proportion of foreign currency denominated investment that is invested in each foreign currency is the nominal GDP of the region with that currency, divided by the nominal GDP of the other regions with currencies that investments are denominated in.
For example, if Australia is third in the region list, after the USA and Japan, then Australia splits ABUY(AA) between the USA and Japan and the fraction invested in the USA is the nominal GDP of the USA divided by the sum of nominal GDP for the USA and Japan.
All nominal GDP values are converted to USD before comparison. All nominal GDP values based on the parameter calibration year.
Market-clearing conditions to determine the equation for ABUY
To determine the split of the current account balance between those invested in assets that are denominated in the region’s own currency and assets denominated in other currencies, the model needs to determine ABUY(region).
The purchases by a given region, of international financial assets denominated in the currency of that same region, are equal to CURR(region) - ABUY(region). The vector of such purchases, with an element for each region, is given by CURR - ABUY. The vector purchases of international financial assets denominated in each currency by other regions is sum(abroad, ashr * ABUY). We require that the total purchases of international financial assets with a given denomination, summing across all regions, are equal to zero. This is because the sales of those assets must, by definition, equal the purchases of those assets. This restriction can be written as:
CURR - ABUY + sum(abroad, ashr * ABUY) = 0
Solving for ABUY, we have:
ABUY = inv(I-ashr) * CURR
where inv() is the matrix inverse function. The G-Cubed model includes the parameter ainv = inv(I-ashr).
This equation for ABUY ensures that all international financial asset markets clear.