Download The Portfolio Balance Model∗

The Portfolio Balance Model∗
U. Michael Bergman
Institute of Economics, University of Copenhagen, Studiestræde 6,
DK1455 Copenhagen K, Denmark
This version: March 16, 2005
1
Introduction
This lecture note covers the portfolio balance model discussed in Sarno and Taylor 4.1.5.
The lecture note is a complement to the textbook, not a substitute. In particular, we will
discuss the portfolio balance model under the maintained assumption that expectations
are static, i.e., the expected rate of depreciation is zero. The portfolio balance model
is described in some detail in other textbooks such as Pilbeam (1998), Hallwood and
MacDonald (2000) and in Branson and Henderson (1985). A discussion of the portfolio
theory of money demand can be found in many textbooks including Mankiw (2000).
The lecture note is organized in the following manner. In the rst section, section 2, we
discuss risk and return in general and show how an investor's portfolio choice is aected
by risk and uncertainty. We also show why it could be optimal to diversify the portfolio,
i.e., to include dierent assets in the portfolio in order to reduce the risk of the portfolio.
In the following section, section 3, we discuss money creation including nonsterilized
and sterilized foreign exchange operations.
In section 4 we set up the portfolio balance model. This section extends and claries
the discussion in section 4.1.5.1 in Sarno and Taylor. We rst derive demand functions for
the assets. There are 3 assets in the model, money, domestic bonds and foreign bonds.
After nding the equilibrium solution to the model we then study the eects of monetary
c 2005 by U. Michael Bergman.
°
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∗
1
policy and interventions on the foreign exchange market. Finally we also discuss the eects
of a change in risk perceptions, one asset becomes relatively more risky. Having discussed
the eects of monetary policy, we turn to scal policy where we contrast the eects of a
moneynanced and a bondnanced scal expansion.
In the last section, section 5, we discuss the adjustment of the model in the long
run paying special attention to adjustments of the trade balance and the current account
corresponding to section 4.1.5.2 in Sarno and Taylor.
2
Risk, return and portfolio choice
A basic assumption underlying the exible price and stickyprice monetary models of
exchange rate determination is that domestic and foreign assets are perfect substitutes
implying that the expected yields on domestic and foreign assets are equalized.
We will now relax this assumption by assuming that international investors regard
domestic and foreign assets as substitutes but not perfect substitutes. Even if domestic
and foreign assets are very similar in most respects, there may be dierences in risk caused
by dierences in liquidity (the ease of which an asset can be sold), tax treatment, default
risk, political risk, ination risk, exchange control risk and exchange rate risks. It may also
be the case that business cycles are not perfectly synchronized such that the rate of return
on domestic and foreign assets are not synchronized implying that investors can hedge
against capital losses by diversifying their portfolios. In other words, we assume that the
risk associated with holding domestic and foreign bonds diers. Remember that an investor
requires a higher expected rate of return on bonds that are more risky to compensate for
the additional risk.
To understand the importance of risk and to motivate why domestic and foreign bonds
are risky assets and why they may have dierent characteristics and thereby dierent risks,
we rst look at the determinants of the risk premium discussed earlier and portfolio choice.
2.1
The risk premium
We have earlier dened UIP as a relationship between the expected change in the nominal
exchange rate and the interest rate dierential. According to the paper by Fama and the
textbook examples, we know that there could exist a time varying risk premium on the
foreign exchange market. For a risk premium to exist, the following three conditions must
be fullled:
1. Domestic and foreign bonds are risky assets but the risk diers such that, for example,
domestic bonds are relatively risky compared to foreign bonds. If they are equally
2
risky and we assume perfect capital mobility, they are perfect substitutes. Therefore,
we have to assume that the risk of holding domestic and foreign bonds diers.
2. Agents are risk averse, i.e., investors accept risk if they expect a higher return on
their investment. If investors are not risk averse, then they would not require a higher
rate of return on relatively risky bonds.
3. There must be a dierence between the riskminimizing portfolio and the actual
portfolio. The riskminimizing portfolio is a theoretical portfolio that would minimize the risks facing investors. Since the amount of domestic and foreign bonds are
given by the issuing authorities it may be the case that investors cannot hold this
portfolio. If they cannot hold the risk minimizing portfolio, then investors require a
risk premium to compensate for the additional risk of the actual portfolio.
If all these three conditions are fullled, there must be a risk premium which compensate
investors for the higher risk exposure.
Note that the risk premium in the UIP relation can be both positive and negative.
If the expected return on domestic bonds is higher compared to the expected return on
foreign bonds, then the risk premium is positive. Why is that? If the expected return
on domestic bonds is higher than the expected return on foreign bonds, then domestic
bonds are more risky than foreign bonds and the risk premium is positive. Similarly, if
the expected return on domestic bonds is lower than the expected return on foreign bonds,
foreign bonds are more risky and the risk premium is negative. From Fama's analysis we
also know that the risk premium is timevarying.
2.2
How can we measure risk and why should investors diversify?
Risk is dened as the variance of capital gains (or losses). If gains and losses are expected
to cancel in the longrun, then the expected value of the gain is zero. A riskless asset has
a very small variance, i.e., there is a very small chance that the capital loss diers from
zero. A risky asset has a high probability of a capital loss and therefore a high variance.
A riskless asset such as money has a zero variance and consequently there is no chance of
a capital loss.
One example of a less risky asset is a bond since the interest payment is known in
advance whereas shares are more risky. The reason for this is that shares pay out a return
in two ways, by dividends (regular payments that the rm makes out of prots) and the
capital gain that investors get is the price of shares increase. If a share can be bought at a
low price and sold (at a later date) at a higher price, there is a capital gain that contributes
to the return (the dividends) while holding the share.
3
An investor who can invest in two assets, money and bonds, will shift the portfolio in
the direction of more bonds and less money if the interest rate rises. A higher return on
bonds compensate for the higher risk, it overcomes the investor's risk aversion. A lower
interest rate will shift the portfolio in the direction of more money and fewer bonds.
Let us now consider two examples of portfolio choice, one with one riskfree and one
risky asset and one example with two risky assets. Assume now that there are two available assets, money which is riskless and pays no return and bonds which yield a return
of i percent. Let α denote that share of the investor's portfolio invested in bonds. In
equilibrium the portfolio consists of α0 percent bonds and 1 − α0 percent money. The rate
of return on the portfolio is
r = α (g + i)
where g is the capital gain. The expected value of the rate of return on the portfolio is
E [r] = E [α (g + i)] = αi
if we assume that E [g] = 0. The rate of return on the portfolio is equal to the percent of
the wealth invested in bonds times the interest rate.
The variance of the rate of return on the portfolio is
E [r − E [r]]2 = σr2 = α2 σg2
implying that the risk of the portfolio (the standard deviation) is equal to the percent of
the wealth invested in bonds times the standard deviation of capital gains (the riskiness of
bonds):
σr = ασg .
How can we determine α0 ? Consider Figure 1. The line AB is the opportunity line
showing the relationship between expected return and riskiness of the portfolio. To obtain
the opportunity line we take the ratio of the expected return on the portfolio and the
standard deviation of the return on the portfolio, i.e.,
E [r] =
i
σ.
σg r
The slope of the opportunity line is i/σg such that a higher interest rate implies a steeper
slope. At point A, α = 0 and the investor earns no return on the portfolio and at point B,
α = 1 and the investor earns a maximum return at the price of maximum risk.
If the investor is risk averse, risk and return are imperfect substitutes, there will be a
set of indierence curves reecting expected utility. As is well known, the investor settles
for point C where the slope of the opportunity line is equal to the slope of the indierence
curve. We have, thus, determined α0 the share of wealth invested in bonds.
4
Figure 1: Portfolio choice.
E [r]
6
B
E [r0 ]
A
C
σr0
σr
-
0
α0
1 α
Consider next the case with two risky assets, domestic bonds and foreign bonds. Let
α be the percentage of the portfolio invested in foreign bonds. The rate of return on the
portfolio is then
r = (1 − α) (g + i) + α (g ∗ + i∗ )
implying that the expected return is (given the assumption that the expected value of the
capital gain or loss is zero for both assets)
E [r] = (1 − α) i + αi∗ .
The variance of the rate of return on the portfolio is
σr2 = E [r − E [r]] = (1 − α)2 σg2 + α2 σg2∗ + 2α (1 − α) σg,g∗
where σg,g∗ is the covariance between capital losses in domestic and foreign bonds. The
riskiness of the portfolio is dependent on the riskiness of the two types of bonds and the
covariance of the capital loss between domestic and foreign bonds.
What would happen if the two risky assets are negatively correlated? In this case a
capital loss on one asset tends to be oset by a capital gain on the other asset. Thus, the
riskiness of the portfolio is reduced. This is important since it helps to explain portfolio
diversication. Empirical evidence also suggest that the covariance between domestic and
5
foreign assets are generally lower than the covariance between dierent domestic assets.
This implies that international diversication is likely to reduce the riskiness of portfolios.
Note that if domestic and foreign bonds are riskless, it follows that the riskiness of our
portfolio is zero and the two assets are perfect substitutes.
If the two assets are independently equally risky, σg = σg∗ , then the riskiness of the
portfolio will be less than the riskiness of domestic bonds if the covariance between capital
losses of the two assets is negative. This further implies that international diversication
makes sense even if interest rates are the same in both countries and the independent
riskiness of the two assets is the same.
Finally, if the wealth of the investor in our example is increased, then the investor will
increase the demand for all assets. If, for example, there is an increase in wealth in the form
of additional bonds, the investor will convert some of the bonds into money. Similarly, if
there is an increase in wealth in the form of additional money, some money will be used to
buy more bonds.
Let us summarize our ndings. We have shown that portfolio diversication is a rational
response to risk for a risk averse investor. A rise in the interest rate leads an investor to
increase bond holdings and reduce money holdings. Thus, a rise in the interest rate leads
to greater risk taking. By diversifying the portfolio, the riskiness of the portfolio can be
reduced.
3
Money creation
The main dierence between the portfolio balance model and the monetary models is that
the source of money creation is important. In the monetary models, it does not matter
how the change in the money stock is created, the eect on the exchange rate is the same
regardless of the source of money creation.
How can the monetary authority create money in the economy? For this purpose it
is useful to consider balance sheets of the central bank, domestic and foreign commercial
banks. The central bank balance sheet is illustrated in Figure 2.
Assume now, for example, that the monetary authority wants to increase the money
supply in the economy (increase the monetary base). To do this, the central bank can
either buy domestic bonds (an open market operation ) or foreign bonds (a nonsterilized
foreign exchange operation ). Consider rst the case where the central bank buys 1 unit
of foreign bonds from the foreign commercial banks. This increases the assets of the
central bank by 1 unit. As a consequence, there is a reduction of assets in the foreign
commercial banks (a reduction of domestic bonds held by the foreign commercial bank)
with 1 unit. If we assume that the foreign commercial bank receives the payment as
6
Figure 2: Central bank balance sheet.
Assets
Net domestic currency bonds
Net foreign currency bonds
foreign money
gold
Liabilities
monetary base = currency +
required reserves held at the central bank
net worth
a deposit in the domestic commercial banks, this implies that the assets of the foreign
commercial bank increase (deposits held at domestic commercial banks) by 1 unit. In
domestic commercial banks, both assets and liabilities increase by 1 unit (an increase in
deposits held by foreign commercial banks and an increase in reserves). The total eect of
this open market operation is a 1 unit increase in the monetary base. These transactions
are illustrated in Figure 3.
Figure 3: A nonsterilized foreign exchange operation.
Central bank
Domestic commercial banks
Assets
Liabilities
Assets
Liabilities
Foreign bonds +1 monetary base +1 Reserves +1 deposits of foreign banks +1
Foreign commercial banks
Assets
Liabilities
Foreign bonds -1
deposits at domestic banks +1
An alternative strategy is to buy domestic bonds from the domestic commercial banks,
i.e., an open market operation. The total eect would in this case also be a 1 unit increase
in the monetary base, but there may be other eects also if domestic and foreign bonds
are not perfect substitutes. If this is the case, domestic commercial banks may want to
adjust their portfolios, i.e., their holdings of both domestic and foreign bonds which in
turn may aect the interest rate and the exchange rate. This could also happen in the
sterilized foreign exchange operation above. The main question is, however, whether the
eects on the domestic interest rate and the exchange rate are identical in these two cases.
According to the monetary models, both the FPMM and the SPMM models, the eects
must be identical since it is implicitly assumed that domestic and foreign bonds are perfect
substitutes.
7
A third alternative is to combine a foreign exchange operation and an open market operation, a sterilized foreign exchange operation. In particular, we may consider the dierence
between an expansionary foreign exchange operation expanding the money supply and a
restrictive open market operation that fully sterilizes the increase in the money supply.
In other words, if the monetary authority wishes to keep the money supply at its original
level (the level before the foreign exchange operation), they can sell or buy domestic bonds
from the public so that the money supply held by the public returns to its initial level.
The net eect is that the public holds less (more) foreign bonds and more (less) domestic
bonds. This combined policy is called a sterilized foreign exchange operation.
Consider again the foreign exchange operation above. The resulting eect was an
increase in the money supply with 1 unit. Assume now that the monetary authority wants
to reduce the money supply by 1 unit so that the initial money supply is restored. One
way to do this would be to sell domestic bonds to the public. If the monetary authority
sells domestic bonds to the public, the deposits in commercial banks will decrease by 1
unit, the reserves held by commercial banks will also decrease by 1 unit since the public is
using its deposits to pay for the bonds. At the central bank, the holdings of domestic bonds
decrease by 1 unit and the monetary base is also decreased by one unit. These transactions
are illustrated in Figure 4. In the rst stage, the central bank buys foreign bonds from
foreign commercial banks and in the second stage the central bank sells domestic bonds to
the domestic households or the domestic commercial banks. The net eect, from the central
bank's perspective, is a change in currency positions from domestic bonds to foreign bonds
leaving the monetary base unchanged. If the central bank buys foreign bonds from the
public and sells domestic bonds to the public, there is a change in the currency positions
in private portfolios from foreign bonds to domestic bonds.
Figure 4: A sterilized
Central bank
Liabilities
Assets
Foreign bonds +1 monetary base +1
Domestic bonds -1 monetary base -1
foreign exchange operation.
Domestic commercial banks
Assets
Liabilities
Reserves +1 deposits of foreign banks +1
Reserves -1
deposits of households -1
Foreign commercial banks
Liabilities
Assets
Foreign bonds -1
deposits at domestic banks +1
In the SPMM and FPMM models discussed earlier it does not matter how the monetary authority creates money. The reason is that domestic and foreign bonds are perfect
8
substitutes. As a consequence, a nonsterilized foreign exchange operation and an open
market operation must have exactly the same eects if the change in the money supply
in both cases is equal. However, within the portfolio balance model discussed below, the
eects are not identical. This suggest that a sterilized foreign exchange operation can aect
the exchange rate without changing the money supply. This will, in fact, be shown below
when discussing the portfolio balance model in detail.
4
The portfolio balance model
Let us now assume that expectations are static, i.e., we assume that the expected rate of
depreciation is zero. We also focus on the shortrun adjustment. This implies that we
assume that both domestic prices and output are xed. The model economy we study is a
small open economy such that the rest of the world can be taken as given. There will be
no reaction from the rest of the world.
There are 3 assets in the model, money M , domestic bonds, B (denominated in domestic
currency), and foreign bonds, F (denominated in foreign currency). We assume that there
is a xed net supply of domestic bonds which is the sum of bond holdings of households
and bond holdings of the monetary authority
B = Bp + Ba
where Bp is bonds held by the public and Ba bonds held by the monetary authority.
Similarly, foreign bonds are held by the public and the monetary authority
F = Fp + Fa
but the holdings of foreign assets can increase or decrease over time via the current account
surplus or decit. The current account decit, thus, reects the accumulation of foreign
assets and is dened as the partial derivative of foreign bond holdings of the public with
respect to time:
∂F
CA =
= Ḟ = T + i∗ (F p + F a)
∂t
where T is the trade balance and i∗ (F p + F a) is the interest rate receipt from net holdings
of foreign assets. The trade balance is assumed to be a function of the real exchange rate
and domestic income
∂T
∂T
T = T (S/P, Y ) where
> 0,
< 0.
∂S
∂Y
The monetary base is dened as the sum of domestic and foreign bond holdings of the
monetary authority
M = Ba + SF a.
9
Note that foreign bonds are denominated in the foreign currency implying that we have
to multiply with the exchange rate to convert the value of foreign bonds into the domestic
currency.
Total nancial wealth is given by the following identity
W = M + Bp + SF p = Ba + SF a + Bp + SF p = B̄ + SF .
Next,we specify the demand for the three assets in the model. First, money demand is
a function of the interest rate, expected change in the exchange rate, output and nancial
wealth,
³
h i
´
M = m i, E Ṡ , Y, W
where
∂m
∂m
∂m
∂m
h i < 0,
< 0,
> 0,
> 0.
∂i
∂Y
∂W
∂ E Ṡ
The demand for money is inversely related to the interest rate and the expected change in
the exchange rate and positively related to domestic income and wealth.
The demand for domestic bonds is also a function of the same variables as the demand
for money, i.e.,
´
³
h i
Bp = b i, E Ṡ , Y, W
where
∂b
∂b
∂b
∂b
h i < 0,
> 0,
< 0,
> 0.
∂i
∂Y
∂W
∂ E Ṡ
Thus, the demand for domestic bonds is inversely related to the expected change in the
exchange rate and domestic income and positively related to the interest rate and nancial
wealth.
Finally, the demand for foreign bonds is given by
³
h i
SF p = f i, E Ṡ , Y, W
where
´
∂f
∂f
∂f
∂f
h i > 0,
< 0,
< 0,
> 0.
∂i
∂Y
∂W
∂ E Ṡ
The demand for foreign bonds is inversely related to the domestic interest rate and domestic
income and positively related to the expected change in the exchange rate and nancial
wealth.
Having specied the model, we can now start our analysis. The rst step is to take the
total dierential of the wealth identity with respect to nancial wealth W :
dW −
∂b
∂f
∂m
dW −
dW −
dW = 0
∂W
∂W
∂W
10 which implies that
∂m
∂b
∂f
+
+
= 1.
∂W
∂W
∂W
This relation is implied since an increase in wealth can be held as either money, domestic
bonds or foreign bonds. The change in the demand for the three assets must sum to one.
This relation is known as the balance sheet constraint and is an identity.
Taking the total dierential of the wealth identity with respect to the interest rate and
the expected change in the exchange rate, we nd
<0
z}|{
>0
z}|{
<0
z}|{
∂m ∂b
∂f
+
+
=0
∂i
∂i
∂i
and
<0
z }| {
∂m
h i+
∂ E Ṡ
<0
z }| {
∂b
h i+
∂ E Ṡ
>0
z }| {
∂f
h i = 0.
∂ E Ṡ
Why do these conditions hold? If, for example, the interest rate rises, then the investor
adjusts its portfolio. Given the signs of the partial derivatives, the investor increases domestic bond holdings and decreases money and foreign bond holdings. A similar argument
applies to the second condition which states how the portfolio adjusts when the expected
exchange rate changes.
4.1
Derivation of asset demand functions
We will now derive asset market equilibrium schedules in the exchange rateinterest rate
plane. Our aim is to nd relationships between exchange rates and interest rates where
the three asset markets are in equilibrium.
Take the total dierential of the
wealth equation with respect to i, W and S under the
h i
maintained assumption that dE Ṡ = 0. Then we obtain
dW = F p dS
∂m
∂m
di +
dW
∂i
∂W
∂b
∂b
0 = di +
dW
∂i
∂W
∂f
∂f
di +
dW .
F p dS =
∂i
∂W
0=
11 (1)
(2)
(3)
(4)
The money market schedule (all combinations of the interest rate and the exchange rate
where the money market is in equilibrium) can be found if we insert equation (1) into (2)
such that
∂m
∂m
di
=−
F p dS
∂i
∂W
implying that the slope of this schedule is
∂m
dS
= − ∂m∂i > 0
di
Fp
∂W
∂m
since ∂W
> 0 and ∂m
< 0.
∂i
The domestic bond schedule (all combinations of the interest rate and the exchange
rate where the market for domestic bonds is in equilibrium) can be found by inserting
equation (1) into (3) such that
∂b
∂b
di = −
F p dS
∂i
∂W
implying that the slope of this schedule is
∂b
dS
= − ∂b∂i < 0
di
Fp
∂W
∂b
since ∂b
> 0 and ∂W
> 0.
∂i
Finally, the foreign bonds schedule (all combinations of the interest rate and the exchange rate where the market for foreign bonds is in equilibrium) is found if inserting
equation (1) into (4) implying that
F p dS =
∂f
∂f
di +
F p dS
∂i
∂W
such that the slope of this schedule is
dS
=³
di
1−
∂f
∂i ´
∂f
Fp
∂W
<0
∂f
since ∂f
< 0 and ∂W
> 0.
∂i
Let us now plot all these three schedules in the exchange rateinterest rate plane as
in Figure 5.1 The portfolio balance model is in equilibrium when all three markets are in
equilibrium, i.e., where the three schedules intersect. The ME schedule is upward sloping and describes equilibrium in the domestic money market. The explanation is that a
depreciation of the exchange rate (an increase in S ) leads to an increase in the domestic
1 This
is the same graph as Figure 4.8 in Sarno and Taylor.
12 investor's wealth (foreign assets are worth more after the depreciation). The increase in
wealth leads to an increase in the demand for money. But since the money supply is xed,
the increase in the money demand can only be oset by an increase in the interest rate.
The BE schedule is downward sloping since a depreciation that raises the demand for
domestic bonds increases the price of bonds leading to a fall in the interest rate which will
reduce the demand for domestic bonds. A depreciation must then be oset by a fall in the
interest rate.
Finally, the FE schedule depicting equilibrium on the market for foreign bonds is also
downward sloping. The reason for this is that a depreciation of the exchange rate leads
to an increased demand for domestic bonds and therefore investors are inclined to sell
money and foreign bonds to buy domestic bonds. Alternatively, a rise in the interest rate
makes domestic bonds more attractive and the exchange rate must depreciate in order to
maintain equilibrium on the market for foreign bonds, i.e., to increase the value of foreign
bonds measured in the domestic currency.
Figure 5: Equilibrium in the portfolio balance model.
S 6
S̄
Md > Ms
Bd < B s
Fd < Fs
Md > Ms
Bd > B s
Fd < Fs
ME
Md < Ms
Bd > B s
Fd < Fs
Md > Ms
Bd < B s
Fd > Fs
FE
Md < Ms
BE Bd > Bs
Fd > Fs
Md < Ms
Bd < Bs
F d > Fs
-
ī
i
We note here that the BE schedule is steeper than the FE schedule. The reason for this
is that if this would not be the case, then the portfolio balance model is unstable. In order
to have a stable model, we assume that changes in the interest rate aects the demand for
13 domestic bonds more than it aects the demand for foreign bonds | ∂f
|<| ∂b
| such that
∂i
∂i
the BE schedule is steeper than the FE schedule.
To check that this assumption implies stability we can consider a point where the
exchange rate is equal to S but where the interest rate is above its equilibrium value. If
the exchange rate is xed and the interest rate is above i, then the demand for money
must be lower than at equilibrium, there is excess supply of money. This implies that all
points above the ME schedule there is excess demand of money and all points below the
ME schedule correspond to excess supply.
Similarly, if the interest rate is above its equilibrium level indicating lower prices on
domestic bonds and therefore greater demand for domestic bonds, there is excess demand
for bonds above the BE schedule and excess supply of domestic bonds below the BE
schedule.
At the same combination of interest rate and exchange rate, the demand for foreign
bonds must decrease since a higher interest rate increases the demand for domestic bonds,
a fall in the demand for money and the demand for foreign bonds. Therefore, above the
FE schedule there is excess supply of foreign bonds. Alternatively, assume that the interest
rate is at its equilibrium value i. If the exchange rate is below its equilibrium value S < S
(the exchange rate has appreciated) there is excess demand for foreign bonds which will
drive up the exchange rate S , i.e., depreciate the currency so that equilibrium on the foreign
exchange market is restored.
The money market schedule will shift up to the left if there is excess supply of money,
the domestic bond schedule will shift up to the right if there is excess supply of domestic
bonds and the foreign bond schedule will shift down to the left if there is excess supply of
foreign bonds.
4.2
Monetary policy
In this section we will consider the shortrun eects of monetary policy on the domestic
interest rate and the exchange rate. In particular, we will study three dierent operations
on the foreign exchange market and the domestic money market, i.e., the three market
operations discussed in section 3.
1. An open market operation where the monetary authority expands the money supply.
2. A nonsterilized foreign exchange operation where the monetary authority (the central bank) intervenes on the foreign exchange market by exchanging domestic money
for foreign bonds. This open market operation is called a nonsterilized intervention.
14 3. A sterilized foreign exchange market intervention where the monetary authority exchanges domestic bonds for foreign bonds such that the domestic money supply is
unchanged.
4.2.1
Open market operation
Assume on the monetary authority increases the private sectors holdings of money, i.e.,
they increase the money supply. To create more money in the economy, the monetary
authority purchases domestic bonds from the public and sell money. This implies that M
increases, B declines whereas W is unchanged. In other words
dM = −dBp = dBa.
What are the eects of this expansionary monetary policy? Assume initially that the
economy is in full equilibrium in Figure 6. The interest rate is equal to i and the exchange
rate is S .
The monetary authority sells money and buys domestic bonds. This will drive up
the demand for domestic bonds and therefore lead to an increase in bond prices implying
that the interest rate must fall. The BE schedule must therefore shift down to the left to
BE'. The excess supply of money in the portfolios that the public hold will increase the
demand for both domestic and foreign bonds which results in a fall in the interest rate and
a depreciation of the exchange rate.
The ME schedule will shift up to the left to ME'. The FE schedule is unchanged because
the open market operation involves only a swap of money for domestic bonds. In short, the
open market operation leads to an increase in the demand for domestic bonds and excess
supply of money.
The total eect from this open market operation is that the interest rate falls from i to
0
0
i and the exchange rate depreciates from S to S , see Figure 6.
4.2.2
Nonsterilized foreign exchange operation
Assume now that the monetary authority buys foreign bonds from the private sector and
sells money. In this case
dM = −S dF p = S dF a.
This market operation is called a nonsterilized foreign exchange operation since the money
supply is increased.
As above we initially assume that the model is in full equilibrium. Also as above, there
is excess supply of money leading to a shift in the ME schedule up to the left to ME', see
Figure 7. For a given interest rate, the money supply has increased and there is excess
15 Figure 6: The eects of an open market operation.
S 6
ME'
ME
S̄ 0
S̄
FE
BE'
ī0
BE
-
ī
i
supply of money and excess demand for foreign bonds. The FE schedule shifts up to the
right such that the exchange rate depreciates and the interest rate falls. The shortage of
foreign bonds in the portfolios requires the exchange rate to depreciate which in turn tends
to increase the domestic currency value of investor's remaining holdings of foreign bonds.
The fall in the interest rate is required to encourage investors to hold money. The BE
schedule is unchanged since the monetary authority swaps money for foreign bonds.
The main eects compared to the case when the monetary authority buys domestic
bonds are the same, in the shortrun the interest rate must fall and the exchange rate
must depreciate.
4.2.3
Sterilized foreign exchange operation
Let us now combine the two market operations discussed above. Assume therefore that
the monetary authority buys foreign bonds from the public in the rst stage and then in
the second stage they sell domestic bonds to the public such that the money supply is
unchanged. This implies that
dM = −S dF p
16 Figure 7: The eects of a nonsterilized foreign exchange operation.
S 6
ME'
ME
S̄ 0
S̄
FE'
FE
BE
ī0
-
ī
i
and
−dM = dBp
so that −S dF p = dBp. Thus, the sterilized foreign exchange operation only aects the
currency positions in private portfolios which change from foreign bonds to domestic bonds.
What are the net eects of these operations? As above we assume that the model is in
full equilibrium as depicted in Figure 8, the interest rate is i and the exchange rate is S .
Since the money supply is unchanged, the ME schedule is unaected.
There is excess demand for foreign bonds (the monetary authority buys foreign bonds)
which will lead to a shift up to the right from FE to FE'. The excess supply of domestic
bonds causes a shift in the BE schedule up to the right to BE'. The net eect is a higher
domestic interest rate and a depreciated currency. The reason why the currency depreciates
is that there is excess demand for foreign bonds requiring an exchange rate depreciation
to restore equilibrium. The excess supply of domestic bonds leads to a fall in bond prices
and thus a rise in the interest rate. Note that if the two assets are perfect substitutes as in
the FPMM and the SPMM models, a swap of domestic for foreign bonds is an exchange
of identical assets that cannot have any eect whatsoever on interest rates and exchange
rates.
17 Figure 8: The eects of a sterilized foreign exchange operation.
S 6
ME
S̄ 0
S̄
FE'
BE'
BE
ī
4.2.4
ī0
FE
-
i
A comparison of the effects of open market operations
We have now analyzed three dierent market operations but we have said nothing about
the relative eects. The open market operation and the nonsterilized foreign exchange
operation lead to similar eects, a lower interest rate and a depreciated currency. From a
practical point of view it would be interesting to contrast these eects, i.e., to answer the
question whether the implied eects are identical in size.
In Figure 9, we contrast the eects of the three dierent market operations. As can
be seen in the graph, an open market operation (I) leads to a strong eect on the interest
rate and a weak eect on the exchange rate whereas we obtain opposite predictions for a
nonsterilized foreign exchange operation (II). The reason for this dierence is that an open
market operation creates a greater shortage of domestic bonds by creating a greater excess
demand for domestic bonds which can only be oset by a large fall in the interest rate.
The stronger eect on the exchange rate from a nonsterilized foreign exchange operation
stems from the fact that this operation creates a greater excess demand for foreign bonds
which can only be satised by a stronger depreciation of the exchange rate.
A sterilized foreign exchange operation (III) also leads to a depreciated currency but
the interest rate rises. The reason for this is that this operation creates an excess supply
18 Figure 9: Comparison of eects of an open market operation (I), a nonsterilized foreign
exchange operation (II) and a sterilized foreign exchange operation (III).
S 6
ME'
II
ME
I
III
S̄
FE'
BE'
BE
FE
BE
-
ī
i
of domestic bonds leading to lower bond prices and therefore a higher interest rate.
4.3
A change in risk perceptions
There are also other reasons for changes in interest rates and exchange rates in this model.
Assume for example that, for some reason, foreign bonds become more risky compared to
domestic bonds. As a result of this change in risk, there will be a decreased demand for
foreign bonds and an increased demand for domestic bonds.
In terms of our model, this implies that both the BE and the FE schedules shift, the
ME schedule is unaected since the supply of money is unchanged, see Figure 10. The
fall in the demand for foreign bonds induces a shift of the FE schedule down to the left
and the increased demand for domestic bonds also leads to a shift down to left of the
BE schedule. In the new equilibrium, the interest rate is lower and the exchange rate
has appreciated. The decreased demand for foreign bonds induces an appreciation of the
exchange rate whereas the increased demand for domestic bonds leads to higher bond prices
and therefore lower interest rate.
19 Figure 10: A change in risk perceptions, foreign bonds become more risky.
S 6
ME
S̄
S̄ 0
FE
BE
FE'
BE'
ī0
4.4
-
ī
i
Fiscal policy
We will now continue our study of changes in various asset supplies on the equilibrium
interest rate and exchange rate by considering the eects of scal policies.
There are two ways the government can nance an increase in government expenditures.
One way is to borrow from the central bank (which at the moment is forbidden in the
EU). In this case both M and W increase by the amount of the decit (or the change in
government expenditures). The alternative is to borrow from the public by selling bonds.
In this case B and W increase by the amount of the decit.
Let us now consider the rst case, the government borrows from the central bank. This
is called moneynancing meaning that the central bank prints money. As was mentioned
before, this implies that M and W increase by the amount of the government decit. The
rise in wealth increases the demand for both B and F as wealth holders try to rebalance
their portfolios. Thus, there is an excess supply of money and excess demand for both
domestic and foreign bonds.
First, there is a upward shift in the ME schedule since there is an excess supply of
money (the money supply is increased), see Figure 11. Since wealth also increases, there
is excess demand for both domestic and foreign bonds. At the initial exchange rate there
20 is excess demand for domestic bonds which drives up the price of bonds and thereby lower
the interest rate. The BE schedule therefore shifts down to the left to BE'. At the same
time, there is excess demand for foreign bonds implying that a higher exchange rate is
needed to eliminate the excess demand (for a given interest rate). The FE schedule shifts
up to the right to FE'.
Figure 11: A moneynanced government budget decit.
ME'
S 6
S̄ 0
ME
S̄
FE'
FE
BE
BE'
0
ī
-
ī
i
There is a new full equilibrium where the exchange rate is higher (the exchange rate
depreciates) and a lower interest rate, see Figure 11. A budget decit nanced by printing
money lowers the interest rate and leads to a depreciated currency.
The alternative for the government to nance a budget decit is to borrow from the
public. The government sells bonds to the private sector implying that both B and W
increase by the same amount as the government decit. The increase in total wealth raises
the demand for foreign bonds. Therefore, a bondnanced government decit leads to
excess supply of domestic bonds, excess demand for money and excess demand for foreign
bonds.
Since there is excess demand for foreign bonds, the FE schedule will shift up to the
right to FE'. The increase in the supply of domestic bonds creating an excess supply of
21 domestic bonds leads to lower bond prices and therefore a higher interest rate. The BE
schedule therefore shifts up to the right to BE'. Finally there is excess demand for money
leading to a shift in the ME schedule down to the right to ME'. These movements are
depicted in Figure 12. The total eect is a depreciated currency and a higher interest rate.
Figure 12: A bondnanced government budget decit.
S 6
ME
ME'
S̄ 0
S̄
FE'
BE
ī
0
ī
FE
BE'
-
i
Is this the only possible outcome of a bondnanced increase in government expenditures? Consider the case when the eects from the interest rate on money demand and
the demand for domestic bonds are large. In other words, there is a larger shift in the ME
and the BE schedules. This case is shown in Figure 13. There is a large downward shift in
the ME schedule to ME' and a large shift up to the right in the BE schedule. There is a
new longrun equilibrium where the exchange rate appreciates and where there is a large
increase in the interest rate. Thus, the eect on the exchange rate is ambiguous!
How can this ambiguous eect be explained? If domestic and foreign bonds are close
substitutes, then any rise in the interest rate will produce a substitution from foreign bonds
to domestic bonds. In this case, the substitution eect will dominate over the wealth eect
and the price of foreign exchange will decline, i.e., the currency will appreciate. If the
wealth eect dominates over the substitution eect, the currency will depreciate.
Let us now summarize our analysis of the portfolio model and the eects of monetary
22 Figure 13: A bondnanced government budget decit.
S 6
ME
ME'
S̄
S̄ 0
BE
ī
FE
FE'
BE'
ī0
-
i
and scal policy on the exchange rate and the interest rate. Table 1 shows the direction
of the changes in the interest rate and the exchange rate. Expansionary monetary policy
always leads to a depreciated currency. A nonsterilized expansionary monetary policy
lowers the interest rate whereas a sterilized monetary policy leads to a higher interest rate.
Bondnanced expansionary scal policy leads to a higher interest rate but the eect on
the exchange rate is ambiguous. If the budget decit is nanced by printing money (the
government borrows from the central bank) the exchange rate depreciates and the interest
rate declines.
4.5
The risk premium, imperfect and perfect substitutability
One basic assumption underlying the portfolio balance model is that domestic and foreign
bonds are imperfect substitutes such that the domestic interest rate tends to diverge from
the foreign interest rate. Another important assumption is that the change in expected
future
changes in the exchange rates are zero, i.e., we have assumed static expectations
h i
dE Ṡ = 0.
23 Table 1: Eects of economic policy on the interest rate and the exchange rate.
Expansionary monetary policy
Domestic bonds (∆M = −∆Bp)
Foreign bonds (∆M = −∆Bp)
Sterilized intervention (∆M = −S∆F p = ∆Bp)
Expansionary scal policy
Bond nanced (∆B = ∆W )
Moneynanced (∆M = ∆W )
∆i
∆S
−
−
+
+
+
+
+
−
+/−
+
Remember that we have dened UIP as
h i
i − i∗ = E Ṡ + ρ
where ρ is the risk premium. Take the rst dierence of this relation such that
h i
di − di∗ = dE Ṡ + dρ.
h i
If dE Ṡ = 0 and di∗ = 0, then
di = dρ .
That is, all changes in the domestic interest rate reect changes in the risk premium. For
example, a rise in the domestic interest rate can be interpreted as a rise in the risk premium
on domestic bonds.
Compare now with the case when domestic and foreign bonds are perfect substitutes
as in the SPMM and the FPMM models. In this case the risk premium is zero and the
domestic interest rate is equal to the foreign interest rate. Any changes in the domestic
interest rate must be caused by a change in the foreign interest rate. In terms of our
portfolio balance model, the BE and the FE schedules must coincide and be vertical at the
foreign interest rate. This is illustrated in Figure 14. As can be easily seen, there is no
dierence between the eects of an open market operation where the monetary authority
buys domestic bonds from the public and a foreign exchange operation where the monetary
authority buys foreign bonds from the public. The ME schedule will shift up to the left in
both cases whereas the BE/FE schedule is unchanged (there is no change in the interest
rate) such that the exchange rate depreciates.
A sterilized foreign exchange operation leaves the ME schedule unchanged and the
BE/FE is also unaected implying no eect on neither the exchange rate nor the interest
24 Figure 14: Perfect substitutability of domestic and foreign bonds and expansionary monetary policy.
S 6
BE/FE
S̄ 0
ME'
ME
S̄
ī = ī∗
-
i
rate. Expansionary scal policy, either nanced by moneyprinting or borrowing from the
public has no eect on the interest rate. The exchange rate depreciates in the former case
and appreciates in the latter case.
5
Adjustment of the interest rate and the exchange rate and
the current account
In this section we consider the adjustment of the exchange rate and the current account.
The model we have discussed so far only explains what happens in the shortrun, how the
shortrun equilibrium is aected by economic policy and changes in risk perceptions. Our
next step is to consider the implied eects on the current account and the feedback from
the current account on the exchange rate.
Remember that the current account was dened as
CA =
∂F
= Ḟ = T + i∗ (F p + F a) = T + i∗ F
∂t
where i∗ F is net income from foreign investment. The shortrun asset market equilibrium
25 may imply either a current account surplus or a decit by a corresponding surplus or decit
in the capital account (the net income from foreign investment) if the trade balance is zero.
Holding the trade balance unchanged, a current account surplus aects the asset markets
since the supply of foreign bonds is increased, if CA > 0 then Ḟ > 0 if T = 0. As the
accumulation of foreign assets continues, net income from foreign assets i∗ F also increase
and tend to widen the surplus such that F increases further. A current account decit
does the opposite.
The trade eect stemming from the real exchange rate can overcome the foreign investment income eect. If there is a large amount of income from foreign investments, then
the country can have a permanent trade decit and therefore a lower real price of foreign
exchange, i.e., the country can be less competitive.
The adjustment process can be illustrated in the following way. Monetary policy, for
example an open market operation leading to an increase in the money supply, has an
immediate eect on the exchange rate and the interest rate. The exchange rate depreciates
and the interest rate falls. Since the current account is the sum of the trade balance and
net foreign investment income, a current account surplus is reected by an accumulation
of foreign bonds which feeds back into the asset markets leading to an appreciation of the
currency. We will therefore get an additional eect on the exchange rate in the longrun.
This together with adjustments of the price level will aect the trade balance. In the long
run, the current account is balanced such that a positive net foreign investment income
i∗ F requires a decit in the trade balance.
Asset markets


M 

 →
B 

F  →
6
 S
 i
i∗ F







Current Account
−→
T + i∗ F
Accumulation
of foreign assets
= ∆F
We will now consider an open market operation where the government is nancing a
budget decit by printing money following the discussion in Sarno and Taylor on page
119120. We know from above that there is a shortterm exchange rate depreciation and a
lower interest rate, see Figure 11. Assume that the economy is in full equilibrium initially,
the three asset markets are in equilibrium, net foreign investment income is zero and the
trade balance is zero such that the current account is zero. We also normalize the exchange
rate and the price level such that they are both equal to unity.
Figure 15 depicts this initial equilibrium in the time space. At time t0 , S0 = P0 = 1
(point A) and the trade balance is zero (point F). At time t0 the government is nancing
26 a budget decit by printing money. We know from our analysis above that the exchange
rate depreciates immediately, it increases to S1 (point C). At the same time the interest
rate falls as is shown in Figure 11. As a result of the depreciated currency, there is an
improvement in the trade balance, there is an increase in exports and a fall in imports.
The trade balance improves to point G in Figure 15. As a consequence, there is a current
account surplus implying that there is an accumulation of foreign assets. There is now
an excess supply of foreign bonds and the households try to rebalance their portfolios by
selling foreign bonds and buying domestic bonds. In terms of our analysis in Figure 11
above, there is a shift in the FE schedule to the left. As a result the exchange rate starts
to appreciate. In Figure 15 this is illustrated as a movement of the exchange rate from S1
at point C towards point E. The appreciated currency leads to a decline in competitiveness
such that the trade balance deteriorates, there is a movement in Figure 15 from point G
to point H.
Figure 15: The dynamics of an open market purchase of domestic bonds.
S ,P
6
S1
P1
S2
P0 , S0 = 1
C
K
E
D
A
t0
t1
F
0
G?
B
- Time
-
H
I
?
Trade balance
The increase in the money supply will eventually lead to an increase in the price level to
the new longrun level P1 , the price level increase from point A to point E and B such that
27 the longrun level is equal to P1 . Remember that the exchange rate immediately jumps to
point C and then starts to appreciate, i.e., to move down towards point E. At point E, the
exchange rate is equal to the price level such that the real exchange rate is unity and equal
to the initial real exchange rate. This also implies that the trade balance is zero as is also
shown in Figure 15 at point I. The current account, however, is in surplus implying that
households are receiving interest income from abroad i∗ F . The accumulation of foreign
bonds still implies an excess supply of foreign bonds leading to shifts in the FE schedule
down to the left such that the exchange rate is still appreciating. For the current account
to be zero in this situation, the trade balance must go into a decit (the positive net foreign
investment income must be balanced by a trade decit if the current account should be
zero). As is shown in Figure 15, there will be a trade balance decit in the longrun (point
H). In order to get a trade balance decit, the competitiveness must deteriorate, i.e., the
real price of foreign currency must fall. At the longrun price level P1 , the exchange rate
must appreciate to S2 such that the ratio of S/P is lower than initially, a value less than
unity. This implies that the path of the exchange rate crosses the pricelevel path leading
to a lower real exchange rate. The longrun eect of a moneynanced budget decit is
therefore a depreciated currency (the longrun exchange rate is S2 ), a higher price level
(P1 ), a lower real interest rate, a trade balance decit and a current account equal to zero.
During the adjustment towards the longrun equilibrium, the exchange rate overshoots
its new longrun level (the dierence between S1 and S2 ) similar to what is predicted by
the Dornbusch model. The dierence, however, is that this overshooting eect does not
only rely on an assumption of stickyprices. If prices adjust immediately, there will still be
an overshooting eect if the shortterm exchange rate S1 exceeds the longrun price level
P1 . Exchange rate overshooting, therefore, does not rely on an assumption of stickyprices.
28 References
Branson, W. H. and D. W. Henderson, (1985), The Specication and Inuence of Asset Markets, in, Jones, R. W. and P. B. Kenen, (ed.), Handbook of International
Economics, Volume, NorthHolland, Amsterdam.
Hallwood, C. P. and R. MacDonald, (2000), International Money and Finance, Blackwell,
Oxford, Third Edition.
Mankiw, N. G., (2000), Macroeconomics, Worth Publishers, New York, Fourth Edition.
Pilbeam, K., (1998), International Finance, MacMillan, Hampshire, Second Edition.
29