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Financial linkages

Debt and equity finance

The total net consolidated foreign asset position (investment position) of each area consists of a net debt position (fixed income instruments) and the net equity position (shares in capital)

\[ \mathit{netiip}_t = \mathit{netf}_t + \mathit{netb}_t \]

There are several fundamental differences between how debt positions versus equity positions are modeled in GEES.

  • International equity has explicit gross positions (households holding equity placed in another area, while their own area's capital shares being held by nonresidents), international debt is only modeled through the implied net positions.

  • Equity is held and handled directly by households while debt is held and handled by a simple financial system (an agent on on behalf of households as principals).

  • Behaviorally, both equity and debt give rise to no-arbitrage conditions in their respective markets. Optimal equity pricing leads to household capital pricing equations. Optimal debt pricing leads to interest parity in forex markets.

  • Their respective non-arbitrage conditions are insufficient to determine the actual volumes of equity or debt held and traded (typical of DSGE models). Equity is assumed to be held and shared in fixed real proportions (i.e. an area's households claims on another area's real capital is fixed). The volume of debt is, on the other hand, determined as the result of stock-flow relationships between the current and financial accounts of the BOP, together with a global net zero supply (clearing) condition.

  • Absent any physical investment, changes in the nominal value of equity are purely the result of capital price gains or losses (valuation effects); this means that changes in the value of equity is not associated with any transactions (no equity being actually traded). Valuation does not enter the BOP, only the IIP accounts. The only equity-related transactions on the BOP is the physical investment (additions to physical capital). Increases or reductions in debt, on the other hand, consist of both transactions (new debt created, old debt paid down) and valuation (through forex, depending on the currency of denomination).

The behavioral foundations of international equity is part of the description of households, its accounting implications for the IIP are described in the IIP/BOP section.

The behavioral and accounting foundations of international debt is described here.

International debt and the financial system

Each area has a simple financial system connecting three parties: local households, the local government, and the rest of the world. At the end of each period, the financial system has zero net worth, balancing exactly its positions with the three parties:

\[ \mathit{bh}_t + \mathit{bg}_t + \mathit{netf}_t = 0 \]

where

  • \(\mathit{bh}_t\) is the net claims of the financial system on local households,

  • \(\mathit{bg}_t\) is the net claims of the financial system on the local government,

  • \(\mathit{netf}_t \equiv \mathit{netf}_t^\mathrm{lcy} + \mathit{netf}_t^\mathrm{fcy}\) is the net claims of the financial system on the financial systems of all other areas, consisting of local currency and foreign currency denominations, in general.

All debt positions are assumed of one-period duration. Then, the ex-post profits of the local financial systems at the beginning of each period (after settling all obligations) are given by

\[ \Pi_{b,t} \equiv \left( \mathit{rh}_{t-1} - \mathit{zh}_t \right) \; \mathit{bh}_{t-1} + \mathit{r}_{t-1} \; \mathit{bg}_{t-1} + \mathit{r}_{t-1} \; \mathit{netf}_{t-1}^\mathrm{lcy} + \tfrac{\mathit{ex}_{t}}{\mathit{ex}_{t-1}} \; \mathit{r}_{t-1}^\mathrm{fcy} \; \mathit{netf}_{t-1}^\mathrm{fcy} \]

where

  • \(r_t\) and \(r_t^\mathrm{fcy}\) are risk-free gross rates of interest traded between the financial sectors of the individual areas, or between the financial sector and the local government;

  • \(\mathit{rh}_t\) is the gross rate of interest charged on the positions between the financial sector and local households;

  • \(\mathit{zh}_t\) is the cost of maintaining the positions with households, including intermediation costs and credit risk expectations; this cost function is detailed below.

The financial system acts as an agent of the households, maximizing its expected profits. There being no intertemporal connections between the positions at two consecutive times, the optimization problem is quasi static

\[ \max\nolimits_{\{ \mathit{bh}_t, \, \mathit{bg}_t, \, \mathit{netf}_t^\mathrm{lcy} \}} \mathrm E_t \left[ \beta \; \mathit{vh}_{t+1} \; \Pi_{b,t+1} \right] \]

Optimal behavior and no-arbitrage conditions

The first-order optimality conditions effectively imply no-arbitrage conditions.

  1. An interest parity between local currency and foreign currency interest rates effectively describes the currency risk component in the relationship between two risk-free rates:
\[ r_t = \mathrm E_t \left[ \tfrac{\mathit{ex}_{t+1}}{\mathit{ex}_t} r_t^\mathrm{fcy} \right] \]
  1. The local household interest rate is marked up over the risk-free rate, with the sprad determined by the intermediation cost and credit risk:
\[ \mathit{rh}_t = r_t + zh_t \]

International debt clearing and underlying assumptions

Net global debt position is zero at all times, a natural constraint in a fully closed macroeconomic system:

\[ \sum\nolimits_{a} \; \mathit{netf}_t^a = 0 \]

Furthermore, in the baseline version of the model, we assume that all international debt positions are always denominated in the so-called reference currency, specifically in the US dollars.

An immediate implication of this assumption is that the US economy is the only area that is not exposed to the forex valuation effects from its debt positions.

Finally, the cost function for household positions, \(\mathit{zh}_t\) is driven by the net debt position of the country as a whole as a fraction of the value of production capital:

\[ \mathit{zh}_t = \mathit{zh}_t^\mathrm{aut} + \theta_1 \; \tfrac{1}{\theta_2} \; \left[ \exp -\theta_2 \; \frac{\mathit{netf}_t}{\mathit{pk}_t \; k_t} \right] + \varepsilon_{\mathit{zh}, t} \]

GEES: International finance linkages module

Declare quantities

!parameters(:finance :steady)

    !for ?H=<areas> !do
        !for ?K=<areas> !do
            "Corporate equity portfolio share !! $\phi_\mathrm{?K}$" ?H_phi_?K
        !end
    !end

!parameters(:finance :dynamic)

    !for ?H=<areas> !do
        "Forward capital adjustment cost parameter !! $\xi_k$" ?H_xi_k
    !end

Define equations

!equations

    !for ?K=<areas> !do
        "Average arbitrage-free condition"
        !for ?H=<areas> !do
            + ?H_phi_?K * ( ...
                - ?H_beta * ?H_vh{+1} * ?H_rh * ?K_pk * ?H_e / ?K_e ...
                - ?H_vh * ?K_xi_k * ?K_pk * [ log(?K_k) - log(?K_k{+1}/&?K_roc_k) ] ...
                + ?H_vh * ?K_pu * ?K_u * ?H_e / ?K_e ...
                + gg_zk * ?H_zk * ?H_beta * ?H_vh{+1} * (1-?K_delta) * ?K_pk{+1} * ?H_e{+1} / ?K_e{+1} ...
            ) ...
        !end
        ;
    !end


    !for ?H=<areas> !do
        "BOP FA corporate equity transactions to GDP ratio"
        ?H_bpfeq_to_ngdp * ?H_ngdp = ...
        !for ?K=<areas> !do
            + ?H_phi_?K * ?H_e / ?K_e * ( ...
                + (1 - ?K_delta) * ?K_pk * ?K_k{-1} ...
                - ?K_pk * ?K_k ...
            ) ...
            - ?K_phi_?H * ( ...
                + (1 - ?H_delta) * ?H_pk * ?H_k{-1} ...
                - ?H_pk * ?H_k ...
            ) ...
        !end
        ;

        "BOP CA corporate equity primary income to GDP ratio"
        ?H_bpceq_to_ngdp * ?H_ngdp = ...
        !for ?K=<areas> !do
            + ?H_phi_?K * ?H_e / ?K_e * ( ...
                + ?K_pu * ?K_uk ...
            ) ...
            - ?K_phi_?H * ( ...
                + ?H_pu * ?H_uk ...
            ) ...
        !end
        ;
    !end


    !for ?H=<areas> !do
        "Value of corporate equity portfolio"
        ?H_kk = !for ?K=<areas> !do + ?H_phi_?K * ?H_e / ?K_e * (?K_pk * ?K_k) !end;
    !end