Saturday, 12 December 2015

A Portfolio Balance Model of Exchange Rates

In my last post, I explained how I like to think about the relationship between exchange rates and international balances.  The key point was that current flows do not matter much in themselves.  What matters is the build up of balances, in particular where those involve entities having to take positions in their non-functional currency. 

I like models, so I'm using this post to set out a little model of this process.  This is based on the models of Godley and Lavoie.  The most notable departure is that I am setting the expected exchange rate equal to the actual outcome.  I'm doing this because the results of this type of model can depend heavily on how exchange rate expectations are formed.  To fully understand this, we need to see how the models behave when we eliminate any systematic expectation error.

Unfortunately, open economy models with floating exchange rates generally require many more equations than closed economy models.  To keep this manageable enough to contain in a blog post, I have made the model as simple as I can whilst retaining enough to show the key dynamics.  The main point here is that I have assumed a "small" economy, so that I can take what goes on in the rest of the world as exogenous.   As usual, I have relegated the equation listing to the end. 

There are three sectors: a public sector, a domestic private sector and the rest of the world.  There are two financial assets - domestic government bonds and foreign government bonds.  Both are held by both domestic and foreign investors.  This is shown in the balance sheet matrix below.  As I have consolidated foreign investors and foreign issuers into one, foreign holdings of foreign bonds do not appear.  Foreign bonds are recorded at their foreign currency value and divided by the exchange rate so the matrix is all in the domestic currency.

Private Sector
Rest of World
Domestic bonds
Foreign bonds
Fd / e

- Fd / e

The net wealth of each sector is determined by historical flows to date (subject to valutaion at the prevailing exchange rate).  The exchange rate must then adjust to ensure that investors wish to hold domestic and foreign bonds in the proportions in which they are in issue.

There are two portfolio decisions to be made here.  First, the domestic private sector has to decide how to allocate its financial wealth between domestic and foreign bonds.  This is assumed to be a function of the domestic interest rate (r) and the expected return on foreign bonds (rrf).  This latter return needs to take into account expected exchange rate movements.

Fd / e = f1 ( V, r, rrf )  

The amount of domestic bonds that foreign investors wish to hold is also assumed to depend on rates.  Here the rates are the domestic rate adjusted for expected exchange rate movements and the foreign rate.  The function also depends on total overseas financial wealth (Vf), converted to its domestic equivalent value.  As we are assuming a small economy, Vf is treated as exogenous.

Bf = f2 ( Vf / e, rrd , rf )

The other behavioural equations required in the model are those describing expenditure - total private expenditure, exports and imports - and those describing how domestic prices adjust.  I have used similar expenditure functions to those that appear in G&L with prices determined by a Phillips curve type relationship based on adaptive expectations.  (It would not be too difficult to adapt this to include some more micro-founded behavioural assumptions and a forward looking Phillips curve.)  I have also assumed there is no intermediate production (imports are not used in production of exports).

The accounting structure of the model is best captured by the flow of funds matrix.  (This includes a production account so that all rows sum to zero, as well as all columns.)

Private Sector
Rest of World
Factor income

- Y
- T

Domestic interest
r . Bd
- r . B
r . Bw

Foreign interest
rf. Fd / e

- rf . Fd / e

Domestic consumption
- d . p

d . p
Government spending

- g . p

g . p

- x . p
x . p
- m . pf / e

m .  pf / e

Change in domestic bonds
- Δ Bd
- Δ Bw

Change in foreign bonds
- Δ Fd / e

Δ Fd / e


There are lots of things we can explore with this model, including many of those typically explored within Mundell-Fleming style models.  As an example, I have looked at what happens when there is a sudden unexpected change in the preferences of domestic investors towards foreign assets.  The charts below show some of the results.



The immediate impact is a sharp drop in the exchange rate.  Domestic investors purchase more foreign bonds (reflected in the jump in gross overseas investment), but in the short run the current account flows are insufficient to finance this.  The counterpart must therefore be purchase by foreign investors of the domestic bonds that domestic investors are selling.  The exchange rate falls until foreign investors expect sufficient future currency gains to make them want the additional domestic bonds.  The initial impact on net foreign investment is principally due to the upwards revaluation of overseas investment due to the exchange rate movement.

The drop in the exchange rate gives a boost to net exports, which together with rising import prices gives a brief increase in domestic inflation.  However, having initially fallen, the exchange rate now rises (which delivers the higher returns expected by foreign investors).  The combined effect erodes the price advantage in foreign trade and the initial increase in the trade balance is soon reversed.  However, by then the net foreign investment position has improved enough that net interest income from abroad outweighs the negative trade balance.

In the long-term steady-state position, lower domestic inflation is matched by currency appreciation, so the real exchange rate is constant.  The higher current account balance is offset by the impact of currency appreciation on the capital losses to domestic investors on foreign bondholdings.  

There are many interesting dynamics that are brought out by this model.  I might look at some more in subsequent posts.

Equation Listing

Real output is the sum of domestic private expenditure, government expenditure and exports.

(1)          y =  d + g + x

The real value of domestic private expenditure is total nominal private expenditure less nominal imports, divided by the price of domestic goods.

(2)          d = ( C - m . pf / e ) / p

Total private expenditure is based on disposable income and the stock of bonds held:

(3)          C = α1 . YD  + α2 . ( Bd-1 + Fd-1 / e )

where disposable income is based on current flows, but excludes capital gains and losses (part of the reason for doing this is to avoid blips in disposable income due to revaluations in foreign bonds when the exchange rate jumps.)

(4)          YD = y . p - T + r-1 . BD-1 + rf . Fd-1 / e

Taxes are levied as a proportion (τ) of production income and nominal interest income.

(5)          T = τ .  ( y . p + r-1 . Bd-1 + rf . Fd-1 / e )

The ratio of real imports to real domestic private expenditure is based on the terms of trade.

(6)          m = d . µ . ( e . p / pf )σ1

Real exports are based on the terms of trade.

(7)          x = x0 . ( e . p / pf )σ2

The definition of disposable income implies that private wealth evolves according to the following accounting identity:

(8)          V = YD - C + Bd-1 + Fd-1 / e

The domestic private sector allocates a portion of its wealth to foreign bonds based on relative expected returns.

(9)          Fd = e . λ1 . V. ( rrf / r )κ1

Foreign holdings of domestic bonds is based on the relative rates of return and the stock of the rest of the world's savings (converted into the domestic currency equivalent).   

(10)        Bw = λ2 . Vf / e . ( rrd / rf )κ2

The stock of domestic bonds evolves in line with the government budget constraint.

(11)        B = B-1 . ( 1 +r-1 ) + g . p - T

The balance of wealth is made up of domestic bonds

(12)        Bd = V - Fd / e

The balance of domestic bonds are held by overseas investors.  

(13)        Bw = B - Bd

(We already have an equation for Bw.  For the purposes of solving the model, this equation (10) is re-arranged to define the exchange rate)

The expected rate of return (expressed in foreign currency terms) on domestic bonds depends on the nominal domestic interest rate and expected exchange rate movements.

(14)        rrd = ( 1 + rd ) . E[ e+1 ] / e - 1

(15)        rrf = ( 1 + rf ) . e / E[ e+1 ] - 1

Price setting here is based on inflation expectations and a measure of the output gap.

(16)        p = p-1 . E[π] . ( y / y* )ψ

Inflation expectations are adaptive.

(17)        E[π] = E[π]-1 + ε . ( π-1 - E[π]-1 )

Where inflation is taken as the change in the price of domestic output (expressed as 1 plus the change).

(18)        π = p / p-1


The variables are listed below.  Uppercase variables denote nominal values.

Domestic bonds
Domestic bonds held by domestic investors
Domestic bonds held by overseas investors
Domestic expenditure by the domestic private sector
Exchange rate (units of foreign currency per unit of domestic currency)
Overseas bonds held by domestic investors (in foreign currency terms)
Real government expenditure
Real imports
Price of domestic output
Price of foreign output
Interest rate on domestic bonds
Interest rate on foreign bonds
Expected return to foreign investors on domestic bonds
Expected return to domestic investors on foreign bonds
Net financial wealth of domestic private sector
Financial wealth of rest of world (in foreign currency terms)
Real exports
Base level of real exports
Real output
Disposable income of domestic private sector
Inflation of domestic output prices

Policy variables are the level of government expenditure, the tax rate (τ) and the domestic interest rate.  Foreign variables (Vf, x0 and rf) are taken as exogenous in accordance with the "small" economy assumption.


  1. "I might look at some more in subsequent posts."

    Please do! I find these models a great tool for thinking about these complex dynamics.

    1. Thank you. I think these models are invaluable for thinking about the dynamics. With many SFC models, I have a good sense of what they will do before I run the simulation, but ones like this are rich enough that they often surprise me and I learn something new by working through why the results are what they are.