Lectures 2 & 3: Portfolio Balance
• Motivation
– How can we allow for effects of risk?
• Currency risk & equity risk (Lecture 2).
• Country risk (Lecture 3).
• Key parameters:
– Risk-aversion, ρ
– Variance of returns, V
– Covariances among returns, Cov.
Each investor i at time t allocates shares
of his or her portfolio to a menu of assets,
as a function of expected return & risk:
xi,t = βi (Et rt+1 , risk) .
Sum across investors i to get the aggregate demand
for assets, which must equal supply in the market.
Invert the function to determine what Etrt+1 must be,
for supplies xt to be willingly held.
The general portfolio balance case:
Tobin (1958, 1969)
 lots of assets (M, Bonds, Equities),
with different attributes &
 lots of investors with different preferences.
But we will focus more on one-period bonds, and
assume uniform preferences among relevant investors.
Lecture 2 assumption (most relevant for rich countries):
exchange risk is the important risk.
We will also consider risk in equity markets.
Lecture 3 assumption (most relevant for developing countries):
default risk is important.
Portfolio Diversification
Starting point: Most investors care not just about expected
returns, but also about risk. => rp ≠ 0 => UIP fails.
Motivating questions for Portfolio Balance Model:
« What determines the risk premium?
How large is it?
« How should you manage a portfolio, e.g., a Sovereign Wealth Fund?
« How can we bring more information to bear
on the structure of investors’ asset demands?
« How do we think about effects of:
• Current account deficits,
• Budget deficits, and
• (sterilized) forex intervention, which had no effects in monetary models?
Open-economy portfolio balance model
Demand for foreign bonds by investor i:
x i, t = Ai + Bi Et (r ft+1 – r dt+1) ;
where x is the share of the portfolio allocated to foreign assets, vs. domestic.
« For now, assume foreign assets all denominated in $ (and/or €, ¥, etc.),
and domestic assets all denominated in dirham (domestic currency);
Then portfolio share xi ≡ S Fi / Wi ,
Di ≡ domestic assets held,
where Wi ≡ Di + S Fi ≡ total wealth held;
Fi ≡ foreign assets held,
« Assume, for now, no default risk.
and S ≡ exchange rate.
Then expected real return differential
= exchange risk premium rpt ≡ i $t – i d t + Et ∆s t+1 .
So
x i, t = Ai + Bi rpt .
Sum asset demands across all investors in the marketplace:
Total demand for foreign assets
≡
xt ≡ Σ [ x i, t ]
= Σ [Ai + Bi rpt ]
For now assume investors to have identical parameters Ai=A and Bi=B:
xt = A + B rpt
« where aggregate portfolio share xt ≡ St Ft / Wt ,
«
«
«
W ≡ D + SF ≡ total wealth held,
F ≡ total foreign ($) assets held, &
D ≡ total domestic assets held.
Financial market equilibrium: assets held = assets supplied….
How do asset supplies get into the market?
« Domestic debt is issued by the government:
t
Dt 
 ( BudgetDeficit )dt

« In “small-country case,” only local residents’ holdings are relevant.
Then aggregate supply of foreign assets given by:
t
Ft   (CurrentAccount )dt

Note 1: forex intervention, even if sterilized, would subtract from D & add to F.
« Note 2: in the general case, the behavior of all investors matters,
though x foreigners > x local residents .
<= Home bias. See appendix.
(The extreme small-country case is: xforeigners = 1.)
Now assume investors diversify optimally
Tobin: “Don’t put all your eggs in one basket.”
Allocation of Portfolio between Bonds & Equity in the Pula Fund
“MANAGEMENT OF COMMODITY REVENUES – BOTSWANA’S CASE”
by Linah Mohohlo, Governor, Bank of Botswana, 1999-2016
% allocations to bonds (“fixed income”) vs. equity
very safe
½&½
very risky
Efficient Frontier: Allocation of Portfolio
between Bonds & Equity
“MANAGEMENT OF COMMODITY REVENUES – BOTSWANA’S CASE”
by Linah Mohohlo, Governor, Bank of Botswana, 1999-2016
very risky
½&½
very safe
Optimally Diversified Portfolios
xt
=
A
+
= Minimum-variance
B rpt
+
Speculative
portfolio
portfolio
Problem: Choose xt to maximize Et [ U(Wt+1 ) ]
Certain assumptions => same problem as Mean-Variance optimization:
maximize Φ [E(W+1), V(W+1)], Φ1>0, Φ2<0.
End-of-period wealth W+1
 Wx(1  r$1 )  W (1  x)(1  rd1 )
 W [(1  r1 )  ( x)(r1  r1 )]
d
[
$
EW1  W [(1  Er1 )  ( x) E(r1  r1 )]
d
$
d
d
V (W1 )  W 2 V ( rd1 ) x2V ( r$1rd1 )2 xCov( rd1,r$1rd1 ) 
Optimal diversification
d
dE ()
dV ()
 1
 2
= 0.
dx
dx
dx
First-order condition:
1WE (r$1  rd1 )  2W 2 2[V (r$1  rd1 ) x  Cov(rd1 , r$1  rd1 )]  0
Define
2
rp  E(r1  r1 ), RRA ≡   
2W , &
1
$
d
Then
This matches
V  V( r$+1 – rd+1).
rp  [Vx  Cov(rd1 , r$1  rd1 )] .
1
1
rp  B x  B A
So for the optimal-diversification case B-1 = ρ V
and
}
A  V 1Cov(rd1 , r$1  rd1 ) .
A is the minimum-variance portfolio
(in x = A + [ρV] -1 rp):
It’s what an investor holds if risk-aversion ρ = ∞.
For example, if goods prices are non-stochastic
and s+1 is the only source of uncertainty,
then V = Var (s+1)
and A = α, the share of foreign goods in consumption basket.
E.g., if all consumption is domestic (A=α = 0), domestic bonds are safe;
very risk-averse investors do not venture abroad (because Cov(rd, r$-rd) = 0).
Also, depending how rp is defined, rp may differ from i - i* - Es by a convexity term = (α – ½) V .
(if s+1 is distributed normally, as in the resolution of the Siegal paradox mentioned in an appendix to the forward bias lecture.)
Equities: Whatever is risk-aversion ρ , the optimal portfolio allocates
a substantial share abroad, because the min-variance portfolio does.
Who
holds
what
portfolio?
x ≥ 1.0
●
x=.75
●
Moderately risk-averse
Very risk-tolerant
x=.3 ●
The most risk-averse
●
x=0
A foolishly underdiversified American
Appendix: Home bias in portfolio holdings
•
In practice, investors in each country hold relatively more
of their own country’s stocks & bonds than the optimaldiversification model seems to say they should.
•
•
Statistics show that home bias, though high, is declining slowly.
Implications for the portfolio balance model?
•
The “small-country” model assumes extreme home bias:
•
•
Most finance models go to the opposite extreme:
•
•
Foreigners hold none of the domestic country’s assets.
all investors have the same portfolio preferences.
The realistic case, e.g., the 2-country model,
assumes foreigners have a relatively greater preference
for their own assets than do domestic residents.
Macroeconomic Policy Analysis II, Professor Jeffrey Frankel,
In practice, most equities are held by domestic residents
but this “home bias” has slowly declined.
Home bias in equity holdings has slowly declined.
The 2-country portfolio-balance model
Foreign residents are in the market for domestic vs. foreign
assets, alongside home residents, with weights wH vs. wF.
Now aggregate:
rp  B 1 x  B 1  wi Ai .
i
A difference in consumption preferences, H < F , for home vs. foreign
residents => some preference for local assets, AH < AF (home bias).
If the domestic country runs a CA surplus
=> Its share of world wealth, wH, rises over time,
and foreigners’ share falls.
=> Domestic preference, AH , receives increasing weight
in total global demand.
=> Global demand for domestic assets rises.
=> Required expected return falls.