L24 & L25: PORTFOLIO BALANCE
• Motivation
– How can we allow for effects of risk?
• Currency risk (Lecture 24).
• Country risk (Lecture 25).
• Key parameters:
– Risk-aversion, ρ
– Variance of returns, V
– Covariances among returns, Cov.
Each investor 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 24 assumption (most relevant for rich countries):
exchange risk is the important risk.
Lecture 25 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 this lecture, assume foreign assets all denominated in $ (and/or €, ¥, etc.),
and domestic assets all denominated in dirham (domestic currency);
Then portfolio share xi ≡ SFi / 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 general,
x foreigners > x local residents
(Home bias).
In extreme “small-country case,” xforeigners = 1
=> only local residents’ holdings are relevant.
t
Then aggregate supply of foreignassets given by: Ft 
 (CurrentAccount )dt

Note: forex intervention, even if sterilized, would subtract from D & add to F.
To repeat,
Now invert:
xt = A + B rpt .
rp t = B-1 x t - B-1 A .
We see that asset supplies are a determinant of the risk premium.
Special case :
| B-1 | = 0
=>
• perfect
substitutability
( |B|=∞ ),
•
A large x forces up the expected return
that portfolio holders must be paid.
no risk premium
(rpt = 0).
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
very safe
½&½
very risky
Efficient Frontier: Allocation of Portolio
between Bonds (“Fixed Income”) & Equity
“MANAGEMENT OF COMMODITY REVENUES – BOTSWANA’S CASE”
by Linah Mohohlo, Governor, Bank of Botswana
very risky
½&½
very safe
Optimally Diversified Portfolios
xt
=
=
A
Minimum-variance
portfolio
+
B rpt
+
Speculative
portfolio
Problem: Choose xt to maximize Et [ U(Wt+1 ) ]
Under certain assumptions,
=> same problem as to maximize Φ [E(W+1), V(W+1)], Φ1>0, Φ2<0.
End-of-period wealth W+1
$
d
 Wx (1  r1 )  W (1  x)(1  r1 )
 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 V ( rd1 ) x2V ( r$1rd1 )2 xCov( rd1,r$1rd1 ) 
2
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
$
V  V( r$+1 – rd+1).
d
Then
This matches
rp  [Vx  Cov(r1 , r1  r1 )] .
d
1
d
1
rp  B x  B A
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 .
(See resolution of Siegal paradox, in latter part of Addendum 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
But in practice: Home bias in equity holdings.
Most equities are held by domestic residents.
Home bias in equity holdings has slowly declined
though more so among Advanced Markets than EMs
Appendix: Beyond the small-country 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.
Home Bias in International Allocation of Equity Portfolios
In practice,
Americans hold most
of their portfolio
in US securities,
Japanese hold theirs
in Japanese securities,
Etc.
Further evidence on home bias in US
Equity shares are from 1997 comprehensive survey of US residents’ holdings of foreign securities.
Bias column ≡ 1 minus (foreign equity share / world market share).
If US investors held foreign securities in proportions equal to those in the world equity market benchmark, bias would = 0.
From: G.Baekert & R.Hodrick, Intl. Fin.Management, 2004, Table 14.16, Panel A
Country
Share in U.S.
Equity Portfolio
US
89.90
UK
1.82
Japan
1.14
France
0.71
Canada
0.59
Germany
0.54
Italy
0.35
Netherlands 0.89
Switzerland 0.52
Sweden
0.32
Bias
0.79
0.88
0.75
0.75
0.85
0.76
0.55
0.79
0.72
Source: Based on Table 1, in Ahearne, Griever & Warnock (2002).
Country
Share in U.S.
Equity Portfolio
Spain
0.21
Australia
0.26
Hong Kong 0.23
Mexico
0.29
Brazil
0.26
India
0.05
China
0.02
Taiwan
0.04
Russia
0.07
South Africa 0.08
Bias
0.83
0.79
0.87
0.56
0.76
0.91
0.98
0.97
0.87
0.92
International diversification has risen
Proportion of foreign bonds + equities in total equity +
bond portfolios of residents in the reported countries.
From a 2002 UBS Asset Management study.
Source: Baekert & Hodrick, op.cit., Table 14.16, Panel B
1991
US
4%
Japan
12%
The Netherlands 12%
UK
23%
Switzerland
11%
Australia
14%
Sweden
4%
2000
11%
27%
62%
26%
21%
19%
25%
…though since the 2008 GFC, we have seen some signs of a return to home bias.