MiF
Art Durnev
International Corporate Finance
Spring 2012
CASE 3: HEDGING CURRENCY RISKS IN
INTERNATIONAL PORTFOLIOS
You are fresh out of the NES MiF program and have joined Troika Dialog’s Emerging Markets group. Your
first assignment is to advise several partners on portfolio allocation strategies for their World Equity Fund.
They have decided that the fund will allocate its capital across Western European, Asian, and North
American countries. Initially, in each of these markets, the fund faces a constraint which makes short-selling
the index prohibitively expensive.
You begin by obtaining monthly market return and exchange rate data over the past 30 years. Combining
this with information contained in forward rates and option prices, you obtain the estimates of annual percent
expected returns, correlations, and risks for the $-valued returns and summarize them in an Excel file
(portfolios_ex.xls).
1. The partners would like for the fund to be fully invested across all countries but would otherwise like to
minimize the fund’s risk. Can you suggest a strategy for doing so? What are the risk, expected return,
Sharpe ratio, and weights of such a minimum-risk all-equity portfolio?
2. Seeing the low returns offered by the minimum-risk portfolio, the partners are interested in learning about
what additional returns they can get if they increase the fund’s risk level. How do the risk-return tradeoffs
(efficiency frontier) offered by the above opportunities look? How do the weights change as you increase
the portfolio risk?
3. Noticing the fund’s strange all-equity stance, you decide to convince the partners to permit the fund to
deposit or borrow under the risk-free rate. a. How much additional expected return can your strategy
offer them when the fund maintains the level of risk calculated in Question 1? How do we allocate the
capital between the risk-free security and equities? What are the weights of the “Tangency” or “Efficient”
portfolio (i.e. weights as a fraction of total equity weight)? b. Can you suggest an alternative way (other
than maximizing the Sharpe Ratio) to find the “Tangency” portfolio? c. How do the risk-return tradeoff
offered by the above opportunities look?
4. How does the efficiency frontier look if you allow to short-sell risky securities? Plot the frontier when you
can invest in a risk-free asset and when you cannot. Compare them to the frontiers in questions 2 and 3.
5. Having convinced the partner’s to relax the all-equity constraint, you decided the next thing to do is to
convince them to relax the countries’ focus. To demonstrate the merits of your plan, you consider the
benefits of diversifying the fund, for the time being, into the RTS, a Russian index. The RTS’ expected
return is estimated to be 24.57% with a standard deviation of 16.35%, and its correlations with the
remaining markets are: 0.37, 0.39. 0.25, 0.25, 0.13, 0.07, 0.27, 0.32, 0.08, 0.17, 0.19, -0.21, -0.24, 0.33,
and 0.17, respectively. a. How much additional return will your plan to obtain on all-equity fund if you
maintain the same risk as in Question 1? b. What would be the risk, return, and weights of portfolio if
investor has the preferences described by function U(R,) =R - 0.5. Assume that short-selling or
investment in a risk-free security is not allowed.
International Corporate Finance
Case3: Portfolios
MiF
Art Durnev
International Corporate Finance
Spring 2012
Next, you set your sights on currency risk. The currency risk, you explain, can be easily hedged away using
money market hedges – simply borrowing in the foreign currency and depositing in dollars to offset a position
in the equity market. To demonstrate this, you set up a spreadsheet similar to the one found in
“portfolios_ex.xls” You obtain decompositions of dollar return covariances into local currency return and
exchange rate covariances. Also, you construct exchange rate expectations assuming uncovered interest
parity holds. The fund has decided that with the uncertainty in financial markets which ensued following the
U.S. presidential elections, the fund will retrench and allocate its capital solely across 4 countries: Korea,
Hong Kong, Singapore, and Taiwan. Again, in each of these markets, the fund faces a constraint which
makes short-selling the index prohibitively expensive.
6. How much return can you obtain on an all-Asian 100% equity portfolio if the partners permit you to take a
position in each currency up to an amount which offsets the equity position (maintaining 18.468% risk )?
Describe this portfolio’s positions.
7. At the presentation of your proposal, one of Troika’s exchange rate forecasters points out that they are
forecasting a 1.5% appreciation of the Taiwanese dollar – not the 1.375% depreciation implied by the
interest differential (that is, the 23.3% dollar-based Taiwanese return estimate actually consists of a
21.8% expected local return and a 1.5% anticipated appreciation). Assume that the fund maintains the
level of risk in Question 6. With this new information, how do your recommendations change? Why?
It is now one year later. Your fund has been an enormous success since your arrival. As a result, you find
yourself with a large team of market and currency researchers and forecasters. After many weeks of work,
they produce for you updated estimates of risks and correlations which remain similar to those in the
Template. However, interest rates and your researcher’s expectations of market and currency returns have
changed substantially. The following new returns and interest rates are anticipated (note that UIP is not
expected to hold):
Country
Singapore
Korea
Taiwan
Hong Kong
Risk-free rate
Market Exchange Rate Interest Rate
15.5
2.5
4.0
18.5
5.5
3.0
23.5
6.5
1.0
21.5
6.0
0.0
5.5
8. You have been informed that the returns of your fund will now be compared to the returns of other funds
(i.e. other groups) during each month of the subsequent year. Your employers would like the average
monthly ranking of your fund to be as high as possible. Your only constraints are that 1) capital is
allocated across the 4 markets and risk-free security; 2) no markets are sold short; 3) leverage is no
higher than 2 (i.e. the weights on the 4 markets add up to no more than 2); 3) no currency is sold short in
greater magnitude than the long position in the country’s equity market. a. Can you suggest a set of
portfolio weights that will accomplish this? b. Assuming that you are managing pension money, would it
be a good investment strategy?
International Corporate Finance
Case3: Portfolios
MiF
Art Durnev
International Corporate Finance
Spring 2012
NOTE: As the case output, please, send me your Excel spreadsheet with all necessary formulas, graphs, and
explanations.
International Corporate Finance
Case3: Portfolios
MiF
Art Durnev
International Corporate Finance
Spring 2012
Appendix A
Optimal Portfolios
A Three-Country Example Using Excel
Refer to portfolios_ex.xls
Step 1:
Correlation coefficients and standard deviations appear in cells B5:E8. Create the
Variance/Covariance Matrix V using the following formula: Cov(A,B)=Corr(A,B)*SD(A)*SD(B)).
Place it in cells B18:D20.
Step 2:
Enter returns from Question 1. Then set up the vector of returns in excess of US risk-free rate, RRf: Enter this formula as a column vector in cells E11:E13.
Step 3:
Set up a vector of initial weights W for each country by entering 0.0 in cells B23:B25. Enter the
total risky portfolio weight in B26 (“=SUM(B23:B25)”).
Step 4:
Set up a row vector of weights WT by entering the following formulas in cells C22 through E22:
C22: “=23”
D22: “=B24”
E22: “=B25”.
Step 5:
Create the expected return cell by entering the following formula into B29:
“=B14+MMULT(C22:E22,E11:E13)”
This calculates Rf + WTR …the expected return given certain weights in each of the 3 risky markets
and the reminder in the risk-free security.
Step 6:
Create the portfolio standard deviation cell by entering the following formula into cell B30:
International Corporate Finance
Case3: Portfolios
MiF
Art Durnev
International Corporate Finance
Spring 2012
“=MMULT(MMULT(C22:E22,B18:D20),B23:B25))^0.5”
This calculates (WTVW)1/2. The expression “MMULT(C22:E22,B18:D20)” multiplies the row vector
of weights (WT) in cells C22:E22 times the Variance-Covariance matrix in B18:D20. The rest
multiplies this times the column vector of weights (W) in B23:B25 and takes it square root.
Step 7:
Run solver:
-to solve for the minimum risk portfolio, enter B30 as the “Target Cell”. Set “Equal To:” to “Min”.
Enter cells B23:B25 in the “By Changing Cells:” box. This says solver will attempt to minimize the
risk level (B30) by altering the portfolio weights (B23:B25). To avoid investment in a risk-free
security add B26=1 constraint. No short-sale constraint is B23:B25>=0.
-to solve for the maximum return for a given level of risk, enter the desired level of risk in cell C30.
Then enter B29 as the Target Cell. Set Equal To: to Max. Then add the constraint B30=C30.
-to construct the efficient frontier, add an additional constraint that the risk is equal to some level in
cell, B30=C30. By changing the number in C30 you will be getting a new risk-return combination.
- to construct the optimal portfolio, that is the portfolio that maximizes the
Sharpe Ratio = (R – Rf)/ enter =(B29-B14)/B30 in cell B31 and set the target cell =B31 in the
Step 8:
To allow borrowing at a risk-free rate, delete the B26=1 constraint.
Step 9:
To allow short-sales, delete the B23:B25>=0 constraint.
International Corporate Finance
Case3: Portfolios
MiF
Art Durnev
International Corporate Finance
Spring 2012
Appendix B
Unhedged $ Portfolio Return
Expected $ return for an unhedged portfolio with N $-denominated risky securities:
R p  (1  1   2  ... N )r f  1 R1   2 R2  ... N RN 
r f  1 R1  r f    2 R2  r f   ... N RN  r f

,
where Rp = $ portfolio return, wi = weight on security i, rf = risk-free rate and Ri = $expected return
for security i.
In matrix notations,
R p  rf   T R e  rf  1 2 ... N 
 R1  rf 


 R2  rf  .
...



 RN  rf 
The variance-covariance matrix is
V p   T V ,
where
 var( R1 )
cov( R R )
2 1
V 
.

cov( R N R1 )
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cov( R1 R2 ) ... cov( R1 R N ) 
var( R2 )
... cov( R2 R N ) 
.
.
. . . . 

cov( R N R1 ) ... var ( R N )

Case3: Portfolios
MiF
Art Durnev
International Corporate Finance
Spring 2012
Appendix C
Hedged $ Portfolio Return
Expected return of the hedged portfolio with N risky securities:
R p  r f  1 R1  %S1  r f    2 R2  %S 2  r f   ... N RN  %S N  r f  
1c r f  r1  %S1    2c r f  r2  %S 2   ...1c r f  r2  %S 2 
,
where Ri = local-currency return on a security in country i Si = expected return on currency i, ri = interest
rate in country i and ic = weight on currency i. Note that hedging currency i involves borrowing currency at
rate ri, exchanging it to $, and depositing $ at rf.
In matrix notations,
R p  rf   T R e  rf  1 2 ... N 1c 2 c ... N 
 R1  %S1  rf 


 R2  %S 2  rf 
...



 RN  %S N  rf 

 .
r

r

%

S
f
1
1


r  r  %S 
2
2
 f

...



rf  rN  %S N 
For practical purposes, it is more convenient to write the portfolio return as
R p  r f   T R  r f   1  2 ... N  1c  2 c ... N 
 R1  %S1  r f 


 R2  %S 2  r f 
...



 R N  %S N  r f 

 .
%

S

r

r
1
1
f


%S  r  r 
2
2
f


...



%S N  rN  r f 
In this case, the optimization program will choose the currency weights as –w1c, -w2c, …, -wNc.
International Corporate Finance
Case3: Portfolios
MiF
Art Durnev
International Corporate Finance
Spring 2012
The variance-covariance matrix is:
V p   T VRR   T VSS    T VSR   T VRS  .
where VRR, VSS, VSR, VRS are variance-covariance matrixes of returns, exchange rates and returns with
exchange rates, respectively. They are:
V RR
cov( R1 R2 ) ... cov( R1 R N ) | 0 0...0 
 var( R1 )
cov( R R ) var( R )
... cov( R2 R N ) | 0 0...0 
2 1
2

.
.
.
. . . 


cov( R N R1 ) cov( R N R1 ) ... var ( R N )
| 0 0...0


,
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 


0
... 0
| 0 0...0 
0
.
.
.
|
.


0
... 0
| 0 0...0 
0
To shorten notations we drop %notation. So Si stands for %Si.
VSS
cov( S1 S 2 ) ... cov( S1 S N ) | var( S1 )
cov( S1 S 2 ) ... cov( S1 S N ) 
 var( S1 )
cov( S S ) var ( S )
... cov( S 2 S N ) | cov( S 2 S1 ) var ( S 2 )
... cov( S 2 S N ) 
2 1
2

.

.
.
| .
.
.


cov( S N S1 ) cov( S N S 2 ) ... var ( S N ) | cov( S N S1 ) cov( S N S 2 ) ... var ( S N ) 
                     |                      


cov( S1 S 2 ) ... cov( S1 S N ) | var( S1 )
cov( S1 S 2 ) ... cov( S1 S N ) 
 var( S1 )
cov( S S ) var ( S )
... cov( S 2 S N ) | cov( S 2 S1 ) var ( S 2 )
... cov( S 2 S N ) 
2 1
2


.

.
.
|.
.
.


cov( S N S1 ) cov( S N S 2 ) ... var ( S N ) | cov( S N S1 ) cov( S N S 2 ) ... var ( S N ) 


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Case3: Portfolios
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Art Durnev
International Corporate Finance
Spring 2012
VSR
VRS
0 
cov( S1 R1 ) cov( S1 R2 ) ... cov( S1 R N ) | 0 0 ...
cov( S R ) cov( S R ) ... cov( S R ) | 0 0 ...
0 
2 1
2 2
2 N

.

.
.
.|.
.


cov( S N R1 ) cov( S N R2 ) ... cov( S N R N ) | 0 0 ... 0 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- | - - - - - - - - - -  ,


0 
cov( S1 R1 ) cov( S1 R2 ) ... cov( S1 R N ) | 0 0 ...
cov( S R ) cov( S R ) ... cov( S R ) | 0 0 ...
0
2 1
2 2
2 N


.

.
.
. .
.


cov( S N R1 ) cov( S N R2 ) ... cov( S N R N ) | 0 0 ... 0 
cov( R1 S1 ) cov( R1 S 2 ) ... cov( R1 S N ) | cov( R1 S1 ) cov( R1 S 2 ) ... cov( R1 S N )

cov( R S ) cov( R S ) ... cov( R S ) | cov( R S ) cov( R S ) ... cov( R S )

2 1
2 2
2 N
2 1
2 2
2 N


.

.
.
.
.
.


cov( R N S1 ) cov( R N S 2 ) ... cov( R N S N ) | cov( R N S1 ) cov( R N S 2 ) ... cov( R N S N ) 
                       |                        


0
0
| 0
0
0
0

0

0
0
| 0
0
0


.
.
.|
.
.
.

0

0
0
|0
0
0


International Corporate Finance
Case3: Portfolios
MiF
Art Durnev
International Corporate Finance
Spring 2012
International Corporate Finance
Case3: Portfolios