phd appendix slides portfolio solution

8/12/2019 PhD Appendix Slides Portfolio Solution

1/41

International Portoio Choice

Technical Appendix

Spring 2010


2/41

Devereux Sutherland method to solve for international portfolios with

incomplete markets

So far focus on portfolios that replicate the complete markets allocation (at

least up to the rst-order)

Easier to solve : remind that you just have to solve for the ecient allocation

and then back-out the portfolio that replicates it.

What happens when markets are incomplete? Still possible to solve for endoge-

nous portfolios. Purpose of this session.Devereux and Sutherland (forthcoming), Solving for Country Portfolios in Open Economy

Macro Models. Similar method is developed in Tille and van Wincoop (forthcoming) but

less general.


3/41

Idea of DS (2007)

Devereux and Sutherland (2007) develop a method that allows solving for steady

state portfolios in DSGE models, in many cases in closed form.

The main ingredients of the Devereux and Sutherland (2007) approach are:

1. For the rst order dynamics of the non portfolio parts of the system, only

the steady state portfolio matters. This is because the i) portfolio only

enters in the budget constraint and ii) only the zero order component

remains in a rst order approximation of the budget constraint. Why?

2. The steady state portfolio can be derived by looking at the (stochastic)

neighbourhood of the non stochastic steady state and letting the noise go

to zero.


4/41

3. Expected excess returns are zero Et hbrx;t+1i in all time periods in a rstorder approximation of model. This implies that realised excess returnsbrx;t(and thus the portfolio excess returnbrx;t)are zero mean i.i.d randomvariables to a rst order of accuracy.

4. The steady state portfolio can be derived using a rst order solution forthe non portfolio parts and a second order approximation of the portfolio

equation

Since the portfolio excess return is a mean zero i.i.d variable, we can

solve the non portfolio equations conditional on portfolio choice/theportfolio excess return

Through the portfolio excess return, the rst order approximation of

the model is a function of the steady state portfolio


5/41

The second order approximation of the portfolio equation only requires

rst order solutions for the non portfolio parts. This is because thesecond order solution to the portfolio equation includes only second

moments and second order solutions for second moments can be ob-

tained from rst order solutions for realisations.

Substituting terms from the rst order solution of the system into thesecond order approximation of the portfolio equation, we obtain a xed

point problem in the steady state portfolio. We then need to solve for

a steady state portfolio that is consistent with the solution conditional

on the steady state portfolio.

Put dierently, and somewhat less formally, the steady state portfolio depends

on (rst and) second moments of the model economy. These moments depend

on the rst order approximate solution of the model which is itself a function

of portfolio choice.


6/41

Methodological steps

1. The non portfolio parts of the model are approximated up to the rstorder. The portfolio excess return is replaced by the exogenous variable .This system is solved using standard linearisation methods with a vectorof states expanded by :

2. The parts of the model relevant for portfolio choice (the asset Euler equa-tions) are approximated up to the second order.

3. Using the rst order solution of the non portfolio equations conditional

on the portfolio excess return and the second order approximation of theasset Euler equations, a xed point problem is solved for the steady stateportfolio. Devereux and Sutherland (2006) show how this solution can becomputed in closed form under some conditions.


7/41

Advantages

1. The method does not suer from the curse of dimensionality and can

therefore be applied to models with many states and many assets.

2. The practical aspects of the solution method present a relatively small

departure from standard perturbation methods used in macroeconomic

analysis.

3. The method allows computation of portfolios in closed form in cases inwhich we can solve analytically for the rst/second order approximate so-

lution of the model.


8/41

Weaknesses

1. Analytical solutions for rst/second order approximate versions of macro-

economic models are not always possible (the presence of capital accumu-

lation usually prevents obtaining analytical solutions). Closed for solution

for portfolios not always feasible

2. The solution method is based onnth order approximations around the

steady state. Many open economy macro models feature nonstationarity

and therefore make local approximations around the steady state prob-

lematic.

3. The methods are mainly based on rst and second order approximations

which are problematic in models that exhibit strong nonlinearities, such as

are to be expected in the presence of, for example, borrowing constraints.


9/41

Set-up

Two symmetric countries, Home and Foreign. Foreign variables will be denoted

with (*)

Each country produces a good, of quantity YH and YF and price pH, pF;

respectively

Ut=Et

1X=t

t [u (C) + v ()] ;

where C is a bundle of the home and foreign goods, u () is twice continu-

ously dierentiable and v (

) captures parts of preferences not relevant for theportfolio problem

Aggregate price index for Home agents is called P; for foreign agents P. P

used as a numeraire.


10/41

Set-up

Financial assets

There arenassets and the vector of (gross) asset returns (payos and capital

gains) is given by:r0t=

h r1;t r2;t ::: rn;t

i;

where returns are measured in units of the Home good (numeraire). Returns

are dened to be the sum of the payo of the asset and capital gains expressed

as a percentage of the asset price.


11/41

Set-up

The budget constraint for Home agents is:

Wt=1;t1r1;t+ 2;t1r2;t+ ::: + n;t1rn;t+ Yt Ct

where Y is total disposable income of home agents.

- Wt is the net foreign asset position.

- Assets are dened to be in net zero supply, while the capital stock is ownedby the domestic economy. Claims to capital may be traded indirectly throughan asset that has an identical returns as the capital stock.

- i;t1 are real holdings of wealth (not shares of wealth or shares of assets)

- the budget constraint is written in terms of real home consumption goods:Yt=pH;tYH;t=Pt


12/41

Set-up

Noting thatP

ii;t1=Wt1; we can rewrite the budget constraint as:

Wt=0

t1rx;t+ rn;tWt1+ Yt Ct;

where0t1=

h 1;t1 2;t1 ::: n1;t1

iis the vector of holdings of asset 1 ton 1 and

r0x;t =

h r1;t rn;t r2;t rn;t ::: rn1;t rn;t

i= h rx;1;t rx;2;t ::: rx;n1;t i ;is the vector of excess returns (asset n is used as the numeraire asset)


13/41

Set-up

Dene S to be the nominal exchange rate (Foreign currency units per Home

currency). Then the real exchange rate is given by:

RER=Q =PS

P

The foreign budget constraint is:

1

QtWt =

1

Qt

t1

0

rx;t+ rn;tW

t1

+ Yt C

t ;

where Qt = P

tSt=Pt is the real exchange rate. Note that W

t; ; r are

measured in terms ofhome consumption goods(which is why the real exchange

rate enters) and Y; C are measured in terms of the foreign consumption good.


14/41

Euler equations

Euler equations: For the Home consumer, we have for i < n:

Eth

u0 (Ct+1) ri;t+1i

= Eth

u0 (Ct+1) rn;t+1i

and for the Foreign consumer:

Eth

Q1t+1u0

Ct+1

ri;t+1i

= Eth

Q1t+1u0

Ct+1

rn;t+1i

Again, remember thatri;t+1is written in terms of the home good which implies

that we have to divide the return by the real exchange rate in the portfolio Euler

equations for the foreign agent.

Asset market clearing conditions

t=

t


15/41

Euler equations for H and F and asset market clearing conditions provide

3 (n 1) nonlinear equations in the 3 (n 1) unknowns t; t ; Et hrx;t1iThe other equilibrium conditions are for the moment unspecied. For the exoge-

nous shock processes it is assumed that the innovations are i.i.d. (covariance

matrix of innovations is non time varying). In practice, very dicult com-

putationally to solve this portfolio problem; cannot use conventional loglinear

methods, for two main reasons:

i) the concept of a steady state portfolio is not well dened, as portfolio decisions

are not well dened in the absence of uncertainty,

ii) in a rst order approximation, portfolio choice either suers from indetermi-

nacy or degeneracy. Since only rst order components matter there, an asset

that pays a higher return will be the only asset demanded (usually violating


16/41

market clearing conditions) or all asset pay the same return, implying indeter-

minacy in portfolio choice.

DS (2007) gets around the rst problem by showing that the steady state

portfolio can be understood as the portfolio in the stochastic neighbourhood

of the steady state and letting the uncertainty go to zero.

The second problem is tackled using a second order approximation for the

portfolio equation.


17/41

Steady State

The approximation is based around a point where the vector of non portfolio

variables is X and the vector of portfolio holdings is : In what follows, a

bar is the value of a variable at an approximation point and a hat indicates

log-deviations from the approximation points (unless specied otherwise).

Dene

cWt =

Wt W

C

brx;t =bri;t brn;te = Y


18/41

Note thatcWt is dened in order to avoid dividing by zero. In a symmetricsteady state, we have:W = 0

r1 = r2=:::= rn= 1

rx = 0

Taking a rst order approximation of the budget constraint, we obtain:

cWt= 1cWt1+e0brx;t+bYt bCt;

which implies that only the steady state portfolio enters in the rst order dy-

namics of the system.


19/41

Second order approximation of the portfolio Euler equations

In relative terms:

EthbCt+1

bCt+1

bQt+1=

brx;t+1

i= 0

where = u00 CC=u0 C : These conditions for portfolio holdings and

excess returns need to hold in equilibrium. The steady state portfolio is a solu-

tion to the portfolio equilibrium conditions and this solution is time invariant,

because the moments it depends on are non time varying (by assumption).

This is the portfolio equation that will be key to solve fore!


20/41

Note also from the linearised rst order conditions for consumption:

EthbCt+1+bri;t+1i = Et hbCt+1+brn;t+1i

) Eth

rx;t+1i

= 0

Implies that realised excess returns,brx;tare mean zero i.i.d variables. Portfolioexcess returnsebrx;t are therefore also mean zero i.i.d random variables andreplaced by the exogenous variable t in the budget constraint:

cWt=

1

cWt1+

bYt

bCt+ t;

where t=e0brx;t:


21/41

First order approximation of the model

A1

" st+1Et [ct+1]

# = A2

" stct

#+ A3xt+ Bt

xt = N xt1+ "t;

where s is a vector of predetermined variables, c is a vector of jump variables,

x is a vector of exogenous forcing processes and " is a vector of i.i.d shocks(innovations); B is a column vector with one in the row corresponding to the

evolution of net wealth and zero in all other rows. The solution to this system

can be written as:

st+1 = F1xt+ F2st+ F3t

ct = P1xt+ P2st+ P3t;


22/41

Excess returns

Using the appropriate rows in the previous system, rewrite the vector of excess

returns

brx;t+1= R1t+1+ R2"t+1;where R1,R2 are matrices. Note thatbrx;t+1 cannot depend on the value ofstate variables,xtorstas such, but only on i.i.d innovations, as we know from

above that it is a zero mean i.i.d variable to a rst order.

Using t=e0brx;tt+1=e0R1t+1+e0R2"t+1


23/41

Towards the portfolio equation

Step 1: write the excess returns as a function of the innovations

t+1=e0R2

1

e0R1

"t+1= H"t+1

where H =e0R21e0R1

brx;t+1 = R1t+1+ R2"t+1= R1 e0R21 e0R1+ R2! "t+1

= (R1H + R2) "t+1= R"t+1

where R = R1H + R2. Note that portfolios are in H


24/41


Step 2: write the relative marginal utilities of C as a function of the innovations

Using the solution of the model, write:

bCt+1 bCt+1 bQt+1= =D1t+1+ D2"t+1+ D3 " xtst+1 #

Substitute for in the previous equation:

bCt+1 bCt+1 bQt+1= = D"t+1+ D3 " xtst+1 # ;where D =D1H + D2


25/41


Postmultiplying bybrx;t+1 and taking expectations at time t:Et

hbCt+1

bC

t+1

bQt+1=

brx;t+1

i = 0

Et "D"t+1+ D3 " xtst+1 #! (R"t+1)# = 0Dropping the terms involving D3 as xt; st+1 are uncorrelated with "t+1 by

assumption and since D"t+1 is a scalar, we get:

Et hR"t+1"0

t+1D0

i= RD0 = 0

with he variance-covarince matrix of innovations: =Eth

"0t+1"t+1i


26/41

The portfolio equation

We need to solve fore such that:R(

e)D0(

e) = 0

This is a system of equations unknowne = the Portfolio equation.Substituting for R;D from above, we get:

(R1H(e) + R2) (D1H(e) + D2)0 = 0

R2D0

2+ R2H0D1+ R1HD

0

2+ R1HH0D1 = 0

Looks like a quadratic equation ine but not...


27/41

By substituting H =e0R2

1e0R1 and multiplying by1 e0R1

2

0 =

1 e0R12R2D02+ 1 e0R1R2R02eD1+

1 e0R1R01eR2D02+ R01eR2R02e0D1Note thate0R1,1 e0R1 and D1 are scalars. So this simplies into:

0 =D01R2R0

2e R2D02R01e + R2D02This gives the following closed form solution for portfolios:

e= hR2D02R01D1R2R02i1R2D02The nal part is to show that this solution is consistent with using the non-

stochastic steady state of the non portfolio variables. The authors do this by


28/41

showing that the approximation error for this solution goes to zero as the in-

novations go to zero, i.e. the solution can be seen as a bifurcation point in the

set of non stochastic equilibria.

In order to obtain the full solution of the model, we now need to combine the

expression for the portfolio with the rst order solution of the model conditional

on the portfolio.


29/41

Remark: Locally complete markets:

If portfolio is such that ratio of marginal utilities are equalized to RER up-to

the 1st order, thenbCt+1bCt+1bQt+1=must not be aected by innovations:D2+ D1H = 0

Solving fore; we obtain:e0 = D2 [R1D2 D1R2]1

=

hD0

2R0

1 D1R0

2i1D0

2

Note that in this case (as in the previous lectures) the portfolio is independant

on the covariance-variance matrix of shocks.


30/41

Issues and remarks

Wealth Dynamics I

Note that the fact that rst and higher order moments of portfolios are irrele-

vant for the evolution of wealth/net foreign assets (of course) does not meanthat wealth/net foreign assets are constant over time. It does mean, however,

that it is (endogenous) returns on assets rather than rebalancing that matter

for the rst order evolution of wealth. Also note that nothing here says that

portfolios are constant over time, but rather that these changes in portfolios

are not relevant for the rst order dynamics of the system.


31/41

Issues and remarks

Wealth Dynamics 2

With incomplete markets, the wealth distribution is often nonstationary. When

we are approximating around a steady state, the approximation is usually onlyrelatively accurate in the neighbourhood of it. In a model with nonstationarity

we should thus expect poor accuracy at longer horizons. Moreover, the very

concept of a steady state is not very meaningful in a model with nonstationarity,

as variables do not return to their initial value, after a shock.


32/41


33/41

A simple example

We present the simple model in Devereux and Sutherland (2007) in order illus-

trate the solution method. The model is characterised by:

- two countries, single good, symmetric power utility over consumption,

Ut=Et

1Xr=

tC1t1

- output and monetary shocks (in logs), innovations uncorrelated

yt = tyt1+ "y;t

yt = ty

t1+ "y;t


34/41

= 2666642y 0 0 0

0 2

y 0 00 0 2m 00 0 0 2m

377775

- quantity theory of the price level:

Mt=PtYt ; Mt =PtYt

- Law of one price holds and real exchange rate is constant:

P =S P

- two nominal bonds are the only assets traded

rB;t =RB;tPt1

Pt; rB;t =R

B;t

Pt1Pt

;


35/41

where RB;t; RB;t are the nominal returns of the bonds

- home budget constraint:

Wt=B;t1rB;t+ B;t1r

B;t+ Yt Ct

- rst order conditions for consumption and bond holdings:

Ct = Eth

Ct+1rB;t+1i

= Eth

Ct+1rB;t+1i

Ct = Eth

Ct+1rB;t+1i

= Eth

Ct+1rB;t+1i

- the resource constraint;

Ct+ C

t =Yt+ Y

t


36/41

Linearisation

In the steady state

rB =r

B =RB =RB = 1

- Euler equations:

bct = Et hbct+1+brB;t+1ibct = Et hbct+1+brB;t+1ibct = Et hbct+1+brB;t+1ibct = Et hbct+1+brB;t+1i

which imply:

Eth

brB;t+1

i= Et

h

brB;t+1

i


37/41

- Resource constraint, budget constraint and money demands:

bct+bct =byt+bytcWt= 1

cWt1+ 1

Y brx;t +byt bct

cmt=bpt+byt;cmt =bpt +byt- real returns

brB;t = [log RB;t

bpt+

bpt1

brB;t = [log RB;t bpt +bpt1


38/41

FromEt hbrB;t+1i= Et hbrB;t+1i:[log RB;t Et1 [bpt] +bpt1 = [log RB;t Et1 [bpt ] +bpt1

[log RB;t [log R

B;t = Et1 [bpt] bpt1 Et1 [bpt ] bpt1

brx;t =

brB;t

brB;t

= [log RB;t bpt+bpt1 [log RB;t bpt +bpt1= Et1 [bpt] bpt (Et1 [bpt ] bpt )

Thus we have:

s0

t = hcWt1 Et1 hbPti Et1 hbPt i ic0

t =hbCt bCt Et hbrB;t+1i bPt bPt brB;tbrx;t i

x0t =hbytbytcmtcmt i


39/41

Solving for portfolios

*First: get the excess returns as a function of innovations...

brx;t = "y;t "m;t "y;t "m;tR1 = 0; R2= h

1 1 1 1

iIn this specic case, excess returns are not aected by the portfolio...


40/41

*Then get the ratio of marginal utilities as a function of the innovations and

exccess returns... with the same notations as before, the log-linearization gives:

cWt= 12 (1y)1ybyt 12 (1y)1ybyt +cWt1+ btbct= 1 cWt1+ (1 ) t+ 2(1+y)2(1y)byt+ 12 (1y)1ybyt

ct = 1

cWt1 (1 ) t+ (1y)2(1y)byt+ 12 2(1+y)1y byt

bct bct = 2 1 cWt1+ 2 (1 ) t+ 11ybyt 11ybytD1 = 2 (1 )

D2 =

11y

11y

0 0


41/41

The portfolio equation gives the solution for the portfolio

The solution for steady state portfolios is then:

eB = eB = R2D02R01 D1R2R021 R2D02=

2y

2 2y+ 2m 1 yHome consumers short home currency bonds (and go long in foreign currency

bonds). Home price level here is countercyclical so Home currency bonds have

high returns when output is high at Home. Bad hedge for output risk, Homeagents prefer foreign bonds. Note also that monetary volatility reduces the

holdings of bonds by reducing their hedging properties.

phd appendix slides portfolio solution

Documents