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	Government	Rest of World
Domestic bonds	Bd	-B	Bw
Foreign bonds	Fd / e		- Fd / e
Total	V	-B	NFI

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	Government	Rest of World	Production
Factor income	Y			- Y
Taxes	- T	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
Exports			- x . p	x . p
Imports	- m . pf / e		m .  pf / e
Change in domestic bonds	- Δ Bd	Δ B	- Δ Bw
Change in foreign bonds	- Δ Fd / e		Δ Fd / e
Total	0	0	0	0

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 . p_f / 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

Variables

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

B	Domestic bonds
Bd	Domestic bonds held by domestic investors
Bw	Domestic bonds held by overseas investors
D	Domestic expenditure by the domestic private sector
e	Exchange rate (units of foreign currency per unit of domestic currency)
Fd	Overseas bonds held by domestic investors (in foreign currency terms)
g	Real government expenditure
m	Real imports
p	Price of domestic output
pf	Price of foreign output
r	Interest rate on domestic bonds
rf	Interest rate on foreign bonds
rrd	Expected return to foreign investors on domestic bonds
rrf	Expected return to domestic investors on foreign bonds
T	Taxes
V	Net financial wealth of domestic private sector
Vf	Financial wealth of rest of world (in foreign currency terms)
X	Real exports
x0	Base level of real exports
y	Real output
YD	Disposable income of domestic private sector
π	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.

International Balances and the Exchange Rate

I've read a few things recently on the complex relationships between interest rates, exchange rates and trade balances.

I like to approach this by starting with the stock variables.  We can imagine a simple global balance sheet matrix like that set out below.  This looks at the position of Country X, which uses Currency X (and we'll assume for simplicity that only Country X uses Currency X).  The assets in other currencies are converted into Currency X at prevailing spot rates, so that we can do some adding up.

So at a particular point in time, investors of Country X hold total assets of 100, which they need to allocate.  Being based in Country X, there will be some preference for investing in currency X assets.  However, those investors will probably also wish to hold some level of other assets to diversify their portfolio.  The balance will depend to some extent on potential returns in different currencies, with the return being a function of nominal interest rates in the respective currencies and potential exchange rate movements.

	Currency X Assets	Other Currency Assets	Total
Country X Investors	80	20	100
Rest of World Investors	50	850	900
Total	130	870	1,000

Likewise, investors in the rest of the world will also want to diversify their investments, so they will want to include some Currency X assets in their portfolio.  Again they will take potential returns into account.  There will be many other factors, of course, effecting what assets investors choose to hold.  But, all other things being equal, the better the prospects for returns in a currency, the more of it investors will want to hold.

Just as Country X investors have a preference for investing in their own currency, so will investors in the rest of the world.  So, the greater the share of total investments held by Country X investors, the greater the overall demand for Currency X assets.

This is what determines exchange rates in the very short term.  At any point in time, the amount of assets in each currency is given, as is the assets of each investor class.  Taking interest rates as given, the exchange rate must find a level where investors want to hold the assets that are available in each currency.

Generally, if there are more Currency X assets than investors wish to hold, then Currency X needs to become cheaper and it will depreciate until it finds the right level.  However, since the right level depends on potential future movements of the currency, which in turn depends on what the exchange rate will be in the future, this can be hard to pin down.  I will come back to this.

So far we have said nothing about trade flows.  In fact, current account items do not matter in themselves.  There is no need for the current account to balance in either the short or the long term.  The relevance of trade flows and investment income is that, over time, they change the balance of assets.

So, if country X is running a persistent current account deficit, then there will be corresponding changes in our matrix.  This could be reflected in a growth of Currency X assets, as Country X finances its spending with borrowing[1], or it might be reflected in a reduction in the total assets of Country X investors (or some combination of the two).

The impact depends on what is going on with investors and assets in the rest of the world.  In combination, these changes may result in the ratios of values within the matrix remaining relatively stable.  However, they could lead to significant changes.  For example, Country X's persistent deficits might lead to Currency X assets becoming a greater and greater proportion of the whole.  In that event we might expect to see some consequences for interest rates and/or exchange rates.

We can see what is happening with interest rates, but we are left with the question of what drives expectations of exchange rate movements.  If today's exchange rate falls, we might hazard a guess that it is more likely to return gains in the future, but it all depends on where the exchange rate needs to be going into the distant future.  We seem to need some kind of target exchange rate to which the actual exchange rate will move over time.

In some models, this target exchange rate role is filled by a measure of purchasing power parity.  The framework I'm using here suggests a slightly different approach.  What it requires is that the exchange rate moves towards a level that allows the values in the matrix to remain in balance.  In a growth scenario, this would mean all the values ultimately growing at the same rate.  Any exchange rate that fails to meet this criterion will lead to an unsustainable position of ever diverging values.

We can write down models that reflect this.  By making certain assumptions about behaviours, we can trace out exchange rate paths that tend towards these steady state positions, while fulfilling relative return requirements along the way.  But it should be noted that there are many factors coming into play in these models, not least how fiscal and monetary policy are expected to develop into the future.  However clearly the equilibrium exchange rate might be defined under perfect foresight, the reality of exchange rate determination is probably closer to blind guesswork.

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[1] I have assumed here that Country X borrowers only borrow in Currency X, but in reality they might borrow in different currencies.  The same issues apply, but this explanation is much simpler if we ignore that.  On the whole, it is much less common for an entity to borrow (other than on a matched basis) outside its functional currency than it is to invest.