Download Utility Change

The Dark Side of International Diversification:
Implications for Imperfect Risk Sharing
In Joon Kim (Yonsei Univ.)
So Jung Kim (Yonsei Univ.)
Sun-Joong Yoon (Dongguk Univ.)
2012. 6. 15
0
Motivation
 International diversification

Unless a new asset is perfectly correlated with existing assets, its
inclusion in portfolio expands asset universe and makes the portfolio
more efficient.

Such a belief has lead most counties to gradually open their economies
to foreign investors.

Currently, the beneficial effect of international diversification becomes
one of the common beliefs in asset investment strategies.
1
Literature Reviews
 Initial Empirical Studies I

International diversification effects are measured as the comovement of
international markets and cross-market correlation is the most straightforward
method.

Longin and Solnik (1995): correlations between international stock returns
vary over time. That is, international diversification benefits change over time.

Longin and Solnik (2001) and Ang and Chen (2002) show that correlation
increases in bear markets and in period of high volatility.
2
Literature Reviews
 Initial Empirical Studies II

Forbes and Rigobon (2002) indicate that the correlation is conditional on
market volatility and upwardly biased.
 The dynamic conditional correlation (DCC) model of Engle (2002) and Tse
and Tsui (2002)
 The dynamic equicorrelation (DECO) model of Engle and Kelly (2009).
3
Literature Reviews
 Recent Empirical Studies I

As globalization advances, the correlation between capital markets becomes
higher and as a result decreases the size of international diversification
benefits.
 Beine et al. (2010) shows that the introduction of the euro increases
comovement across the entire return distribution and reduce the benefits
of portfolio diversification within the euro area.
 In particular, the increase in correlation intensified during the global crisis,
resulting in “market contagion,” where a shock to a particular economy
spreads sequentially to other economies and triggers a revelation of
systemic risk.
4
Literature Reviews
 Recent Empirical Studies II

Similarly, various studies investigate patterns and trends in correlations for
developed markets and emerging markets.
 Christoffersen et al. (2011) find that correlations between DMs have
exceeded correlations between EMs.
 Christoffersen et al. (2011) and Beine and Candelon (2011) show that the
correlations of emerging markets has increased, that is, diversification
benefit of emerging markets has decreased over time.
5
Motivation
 The benefits of international diversification thus decrease from
ongoing market integration.

Nevertheless, no study has yet shown the welfare losses (negative
effects) of domestic investors from market integration.

This is because even if, at worst, the correlation increases by one, the
diversification effect converges to zero and not to negative values.
 Therefore, this paper becomes the first theoretical paper that
examines the negative side effect of market integration.
6
Equilibrium
 A mean-variance analysis of Sharpe (1964).

However, investors know the distribution of future payoffs rather than
the rate of returns on assets.

The asset prices are endogenously determined in equilibrium (Mossin,
1966; Lucas, 1978).
 Preference
U i  fi ( yi ,1 , yi ,2 )  yi ,1   i yi ,2

n 1
 Portfolio return:
yi ,1    j xi , j  xi ,n ,
j 1
n 1 n 1
 Portfolio variance:

yi ,2   jk xi , j xi ,k
j 1 k 1
xi , j denotes the j-th asset holdings of i-th investor and  jk denotes
the covariance between j-th asset and k-th asset.
7
Equilibrium
 Budget constraints
n 1
 p x
j 1

where
j
i, j
 xi , j   q  xi ,n  xi ,n   0
p j is the price of j-th asset and
q
is the price of n-th asset
 Market clearing condition
x
m
i 1
i, j
 xi , j   0
8
Three Economies
 Domestic economy (non-integrated economy)

Two investors coupled with two financial assets
 One risky asset and one risk-free asset:
x1,1  0, x2,1  0,
x1,2  0, x2,2  0.
 We know the expected payoff returns of two assets as well as payoff
variances.
9
Three Economies
 Integrated economy I

Domestic economy achieved equilibrium before integration and all domestic
*
*
*
*
investors hold mean-variance efficient portfolios ( x1,1 , x1,2 , x2,1 , x2,2 )

The foreign investors hold the entire foreign asset , x3,3 .

Asset holdings before integration
1 1
1 2
 *

*
*
x

x
,
x

x
,
x

0,
1,1
1,1
2,1
1,1
3,1


1 1 1  2
1 1 1  2


 *


1 2 
2
2
 1 x1,1  x1,1 ,
 x1,2 

 1 
1


1

1


1

1
2 
1
2


  Equilibrium





1

2
*
2
*
2
x  x    
 1 x1,1 
x1,1  , x 3,2  0, 
2,2
2,2
1


1 1 1  2

 1  1  1  2

 *

*
*
 x1,3  0, x 2,3  0, x3,3  x3,3 .

10
Three Economies
 Integrated economy II

Domestic economy did not achieve equilibrium yet before integration and the
domestic investors do not hold mean-variance efficient portfolios.

The foreign investors hold the entire foreign asset , x3,3

Asset holdings before integration
 x1,1  0, x2,1  0, x3,1  0, 


 x1,2  0, x2,2  0, x3,2  0,   Equilibrium
 x  0, x  0, x  0. 
2,3
3,3
 1,3

11
About Asset Universe
 Why is the foreign asset risky?

The foreign asset always accompanies exposure to exchange rate risk.

The price of a foreign security confronted by domestic investors is the product
of the local price and the exchange rate.

Because the exchange rate varies continuously, even if the local asset is
riskless, it becomes a risky asset for domestic investors.
 Why one risky asset in domestic economy?

In this paper, the number of a risky asset is one in both markets, respectively.

As shown by the two-fund separation theory of Cass and Stiglitz (1970) and
Ross (1978), investment universes can be represented by one risky asset (the
market portfolio) and one riskless asset.

In this sense, the risky asset can be interpreted as an index fund.
12
Welfare Change Calculations
 Welfare changes

The utility changes of domestic investors before and after integration
 Comparison

Utility change 1: Domestic economy  Integrated economy I
 Changes in Utility I = U(integrated economy I) – U(domestic economy)

Utility change 2: Domestic economy  Integrated economy II
 Changes in Utility II = U(integrated economy II) – U(domestic economy)
13
Equilibrium in Domestic Economy
 <Asset holdings in equilibrium>

Proposition 1: Given one risky asset and one riskless asset in a domestic
economy, the equilibrium portfolio holdings of two investors whose
preferences are characterized by the mean-variance utility function are
as follows:
x1,1 
1 1
x1,1 ,
1 1 1  2
x1,2 
1 2
2 2
1 x1,1 
 12 x1,12 ,
2
1 1 1  2
1  1  1  2 
x2,1 
1 2
x1,1 ,
1 1 1  2
x2,2  x2,2 
1 2
2 2
1 x1,1 
12 x1,12
2
1 1 1  2
1  1  1  2 
14
Equilibrium in Integrated Economy I
 <Asset holdings in equilibrium>

Proposition 2: Given that domestic investors hold efficient portfolios before
integration, the addition of a new foreign asset changes their portfolio
holdings in equilibrium as follows:
xi ,1 
1 i
x1,1 ,
1 1 1  2 1  3



1 i
2
xi ,2  xi*,2   3 
 32 x3,3   1,3 x1,1  
x3,3  x i*,3 

1 1 1  2 1  3

 1  1  1  2  1  3



1 i
2
2
* 
  1 

x


x
x

x

1 1,1
1,3 3,3  
1,1
i ,1  ,
1 1 1  2 1  3

 1  1  1  2  1  3

xi ,3 
1 i
x3,3
1 1 1  2 1  3
15
Equilibrium in Integrated Economy II
 <Asset holdings in equilibrium>

Proposition 3: Given that domestic investors hold inefficient portfolios
before integration, the addition of a new foreign asset changes their
portfolios as follows:
xi ,1 
1 i
x1,1 ,
1 1 1  2 1  3



1 i
2
2
xi ,2  xi ,2   3 
 3 x3,3  1,3 x1,1   1   1   1  x3,3  xi,3 
1


1


1

1
2
3
2
3

 1




1 i
2
2
  1 
1 x1,1  1,3 x3,3   1   1   1  x1,1  xi,1  ,
1


1


1

1
2
3
2
3

 1

xi ,3 
1 i
x3,3
1 1 1  2 1  3
16
Price Changes
 Proposition 4:

The equilibrium price ratios between risky assets and the riskless asset
before and after market integration are as follows:
p1non-integrated
2
2




x1,1
1
1
non-integrated
q
1 1 1  2
p1integrated
2
2




1
1 x1,1   1,3 x3,3
integrated
q
1 1 1  2 1  3

p3integrated
2
2




3
3 x3,3   1,3 x1,1
qintegrated
1 1 1  2 1  3



 Affecting factors:
 returns, volatility, size, and correlation of the new asset
 the risk aversion of the foreign investor
17
Price Changes
 Price ratios between the domestic risky asset and the riskless asset
Graph A. gamma=1,

Graph B. gamma=0.1,
The parameters used are as follows:
Graph C. gamma=10
 1   2  1,   3, 1  1 and
x1,1  1 .
18
Utility - Reference
 Original utilities

Here, the welfare changes are measured as the utility changes,
characterized by mean-variance utility.

The maximum utilities of domestic investors before integrating markets
U1  1 x1,1 
U 2  x2,2 
2  2 1 1
1  1  1  2 
1 2
1  1  1  2 
2
2 2

1 x1,1 ,
2
 12 x1,12 .
19
Utility Change I
 Utility changes in Integrated economy I (Efficient portfolio holders)

Proposition 5: Given that a domestic economy achieves equilibrium and
domestic investors therefore hold efficient portfolios, the addition of a
foreign asset always enhances the welfare of all domestic investors:
U1  U1integrated  U1non-integrated
2


1 
1
1
1
 

 3 x3,3   0,
  1 x1,1 

1   1 1 1  2 1  3 1 1 1  2 
1 1 1  2 1  3

U 2  U 2integrated  U 2non-integrated
2


1 
1
1
1
 

 3 x3,3   0.
  1 x1,1 

 2   1 1 1  2 1  3 1 1 1  2 
1 1 1  2 1  3

20
Utility Change II
 Utility changes in Integrated economy II (Inefficient portfolio holders)

Proposition 6: When a domestic economy does not achieve equilibrium and
thus domestic investors hold inefficient portfolios, the addition of a foreign
asset can damage the welfare of some domestic investors:

1 1
2  2 1 1  2 2
2
 x
U1  


 1   1   1  2 1  1  1  2  1  3 1   1  2  1 1,1
1
2
3
1
2




2 1
2
  1,3 x1,1 x3,3 ,

 x 

2
2

1 1 1  2 1  3 
1  1  1  2  1  3 
 1  1  1  2  1  3 

1 1
2 2
3 3,3

 2 2
1 2
1 2
 x
U 2  

 1   1   1  2 1   1  2  1 1,1
1
2
3
1
2


1 2
2 2
2 2


x

 1,3 x1,1 x3,3 .
3 3,3
2
2
1  1  1  2  1  3 
1  1  1  2  1  3 
21
Utility Change - Examples
 Assumptions:
x1,1  1 , 1  1 ,  1   2   3
 Example 1:

The case of perfectly positive correlation (
  1, x3,3  1,  3  1)
4
5
 7 1

2
U1      32 x3,3
  3 x3,3      0
9
36
 36 9

2
 5 1
 7
2
U 2       32 x3,3
  3 x3,3     0
9
 36 9
 36
 Example 2

  1, x3,3  1,  3  1)
The case of perfectly negative correlation (
4
 7 1
 3
2
U1      32 x3,3
  3 x3,3     0
9
 36 9
 4
2
9
 5 1

2
U 2       32 x3,3
  3 x3,3      0
9
36
 36 9

22
Sensitivity Test
 Utility Changes of Inefficient Portfolio Holders

Gamma1=gamma2=gamma3=1
23
Sensitivity Test – Integrated Economy II
 The utility changes of Investor 1 in the integrated economy II

The investors holding the more risky asset than efficient portfolio
U1
4
   1 3 x1,1 x3,3  0

9
U1
2
   3 x3,3  2  1 x1,1 
  3 x3,3  9
24
Sensitivity Test – Investor 1
 For correlation

The utility of investor 1 decreases in correlation.

This means that the addition of a highly correlated asset decreases demand
on the existing risky asset and decreases its price.

Hence the welfare of investor 1 who holds excessive risky asset will
decrease.
25
Sensitivity Test
 For volatility * quantity of the foreign asset

The sensitivity of utility changes with respect to vol*quantity is
complicated.

If the correlation between two risky assets is negative, first derivative is
always positive, which guarantees the utility enhancement of investor 1
by market integration.

For a positive correlation, the inclusion of the foreign asset  3 x3,3  2  1 x1,1
reduces the domestic risky asset demand and its price. In this range the
first derivation of the utility changes of investor 1 can be negative and
thus the utility change decreases in  3 x3,3.

However, as  3 x3,3 increases in the range  3 x3,3  2  1 x1,1 , the first
derivation is positive and thus the utility change increases in and, more
specifically, convexly.
26
Sensitivity Test – Integrated Economy II
 The utility changes of Investor 2 in the integrated economy II

The investors holding more risk-free asset than efficient portfolio
U 2 2
  1 3 x1,1 x3,3  0

9
U 2
2
   3 x3,3   1 x1,1 
  3 x3,3  9
27
Sensitivity Test – Investor 2
 For correlation

The utility of investor 2 increases in correlation because first derivation
is always positive, regardless of size and volatility.

This means that the addition of a highly correlated risky asset increases
the riskless asset price due to its scarcity and thus the wealth of investor
2 increases.
28
Sensitivity Test – Investor 2
 For volatility * quantity of the foreign asset

If the correlation between two risky assets is positive, the first
derivation becomes positive, which also guarantees the utility
enhancement of investor 2 through market integration.

For a negative correlation, market integration can decrease the utility
change of investor 2 according to the value of  3 x3,3.
 For example, the utility changes of investor 2 can be negative in the
range  3 x3,3    1 x1,1
 Otherwise, they increase in  3 x3,3 .
29
Implications for Open Economy System
 Policy implications for less-developed countries

Usually, less-developed counties open their economy to attract foreign capital
in the development process.

This policy is supported by the conventional belief that the market
integration expands the investment universe and thus is always beneficial to
all domestic investors.

However, only in a circumstance that domestic economy achieves
equilibrium and thus investors hold efficient portfolio, their welfares
increase through market integration.

To guarantee the beneficial welfare enhancement of all domestic investors,
therefore, the policy makers should firstly seek the equilibrium of domestic
economy, and secondly integrate the markets.
30
Conclusion
 Summary

We investigate the welfare changes of domestic investors by market
integration.

Only the efficient portfolio holdings of domestic investors guarantee the
welfare enhancement of all domestic investors.

To minimize the welfare losses of domestic investors, the policy makers of
less-developed economies should first pursue equilibrium so that investors
hold efficient portfolios and, second, decide to open their economy.
 Limitations

The assumption of mean-variance preference.

The extension of utility function to CRRA or Epstein and Zin (1989) may
enrich our implications for international diversification
31