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Un iversa l He dg ing : O pt im iz ing C ur rency R isk

an d R ew ard in Internat ion al E qu ity P ort fo l ios

F ischer B lack

n a w o r l d w h e r e e v e r y o n e c a n h e d g e a g a i n s t

c h a n g e s i n t h e v a l u e o f r e a l e x c h a n g e r a t e s ( t h e

r e l at i v e v a l u e s o f d o m e s t i c a n d f o r e i g n g o o d s ) , a n d

w h e r e n o b a r r i e r s l i m i t i n t e r n a t i o n a l i n v e s t m e n t ,

t h e r e i s a U n i v e r sa l c o n s t a n t t h a t g i v e s t h e o p t i m a l

h e d g e r a t i o - - t h e f r ac t io n o f y o u r f o r e i g n i n v es t -

m e n t s y o u s h o u l d h e d g e . T h e f o r m u l a f o r t h is

o p t i m a l h e d g e r a t i o d e p e n d s o n j u s t t h r e e i n p u t s :

• T h e e x p e c t ed r e tu r n o n t h e w o r l d m a r :

k e t p o r t f o l i o .

• T h e v o l a t il i ty o f t h e w o r l d m a r k e t p o r t -

fol io.

• A V e r a g e e x c h a n g e r a t e v o l a t il i ty .

T h e f o r m u l a i n t u r n y i e l d s th r e e r u l e s :

• H e d g e y o u r f o r e i g n e q u i t ie s .

• H e d g e e q u i t i e s e q u a l l y f o r a l l c o u n t r i e s .

• D o n ' t h e d g e 1 0 0 p e r c e n t o f y o u r f o r-

e i g n e q u i t i e s .

T h i s f o r m u l a a p p l ie s t o e v e r y i n v e s to r w h o

h o l d s f o r e i g n s e c u r i t ie s . I t a p p l i e s e q u a l l y t o a U . S .

i n v e s t o r h o l d i n g J a p a n e s e a s s e t s , a J a p a n e s e i n v e s -

t o r h o l d i n g B r i t i s h a s s e t s , a n d a B r i t i s h i n v e s t o r

h o l d i n g U . S . a s s e t s . T h a t ' s • w h y w e c a l l t h i s

m e t h o d u n i v e r s a l h e d g i n g . , '

W H Y H E D G E A T A L L ?

Y o u m a y c o n s id e r h e d g i n g a z e r o - s u m g a m e .

A f t e r a l l , i f U . S . i n v e s t o r s h e d g e t h e i r J a p a n e s e

i n v e s t m e n t s , a n d J a p a n e s e i n v e s t o r s h e d g e t h e i r

U . S . i n v e s t m e n t s , t h e n w h e n U . S . i n v e s t o r s g a i n

o n t h e i r h e d g e s , J a p a n e s e i n v e s t o r s l os e , a n d v i c e

v e r sa . B u t e v e n t h o u g h o n e s i d e a lw a y s w i n s a n d

t h e o t h e r s i d e a l w a y s l o s es , h e d g i n g redu es risk f o r

b o t h s i d e s.

M o r e o f t e n t h a n n o t , w h e n p e r f o r m a n c e is

m e a s u r e d i n l o c al c u r r e n c y , U . S. i n v e s t o r s g a i n o n

t h e ir h e d g i n g w h e n t h e ir p o r tf o li o s d o b a d l y , a n d

J a p a n e s e i n v e s t o r s g a i n o n t h e i r h e d g i n g w h e n

t h e i r p o r t f o l i o s d o b a d l y . T h e g a i n s f r o m h e d g i n g

a r e s i m i l a r to t h e g a i n s f r o m i n t e r n a t i o n a l d i v e r si -

Reprinted from Financial Analysts Journal (July~August 1989):16 -

22.

f i c a t i o n . B e c a u s e i t r e d u c e s r i s k f o r b o t h s i d e s ,

c u r r e n c y h e d g i n g p r o v i d e s a f r e e l u n c h .

W hy N o t He d g e A l l

I f i n v e s t o r s i n a l l c o u n t r i e s c an r e d u c e r i sk

t h r o u g h c u r r e n c y h e d g i n g , w h y s h o u l d n ' t t h e y

h e d g e 1 0 0 p e r c e n t o f t h e i r f o r e i g n i n v e s t m e n t s ?

W h y h e d g e l es s?

T h e a n s w e r c o n t a i n s o u r m o s t i n t e r e s t i n g

f in d i n g. W h e n t h e y h a v e d i f f e r en t c o n s u m p t i o n

b a s k e t s , i n v e s t o r s i n d i f f e r e n t c o u n t r i e s c a n a l l a d d

t o t h ei r e x p e c t ed r e t u r n s b y t a k in g s o m e c u r r e n c y

r is k i n t h e i r p o r t f o l i o s .

T o s e e h o w t h is c a n b e , i m a g i n e a n e x t r e m e l y

s im p l e c a se , w h e r e t h e e x c h a n g e r at e b e t w e e n t w o

c o u n t r i e s is n o w 1 :1 b u t w i l l c h a n g e o v e r t h e n e x t

y e a r t o e i t h e r 2 :1 o r 1 :2 w i t h e q u a l p r o b a b i l i t y . C a l l

t h e c o n s u m p t i o n g o o d s in o n e c o u n t r y a p p l e s

a n d t h o s e i n t h e o t h e r o r a n g e s .

I m a g i n e t h a t t h e w o r l d m a r k e t p o r t f o l i o c o n -

t a in s e q u a l a m o u n t s o f a p p l e s a n d o r a n g e s . T o t h e

a p p l e c o n s u m e r , h o l d i n g o r a n g e s i s r i s k y . T o t h e

o r a n g e c o n s u m e r , h o l d i n g a p p l e s i s r i s k y .

T h e a p p l e c o n s u m e r c o u l d c h o o s e t o h o ld o n l y

a p p l e s , a n d t h u s • b e a r n o r i s k a t a l l. L i k e w i s e , t h e

o r a n g e c o r ~ s um e r c o u l d d e c i d e t o h o l d o n l y o r -

a n g e S . B u t; s u r p r i s i n g l y e n o u g h , e a c h w i ll g a i n in

e x p e c t e d r e t u r n b y t r a d i n g a n a p p l e a n d a n o r -

a n g e . A t y e a r ~ e n d , a n o r a n g e w i ll b e w o r t h e i t h e r

t w o a p p l e s o r 0 . 5 a p p l e s . I t s e x p e c t e d v a l u e i s 1 .2 5

a p p l e s . S i m i l a r l y , a n a p p l e w i l l h a v e a n e x p e c t e d

v a l u e o f 1 .2 5 o r a n g e s . S o e a c h c o n s u m e r w i ll g a i n

• r o m t h e s w a p .

Thi s i sn ' t a ma themat i ca l t r i ck . In fac t , : i t ' s

s o m e t i m e s a l l e d S i e g e l ' s p a r a d o x . ''1 I t 's r e a l , a n d

i t m e a n s t h a t i n v e s to r s g e n e r a l ly w a n t t o h e d g e

l e ss t h a n 1 00 p e r c e n t o f th e i r f o r e i g n i n v e s t m e n t s .

T o u n d e r s t a n d S i e g e l' s p a r a d o x , c o n s i d e r h is -

t o r i c a l e x c h a n g e r a t e d a t a f o r d e u t s c h e m a r k s a n d

U . S . d o l la r s . T a b le 1 s h o w s t h e q u a r t e r l y p e r c e n t -

a g e c h a n g e s i n th e e x c h a n g e ra t e s a n d t h e i r a v e r-

a g e s . N o t e t h a t , i n e a c h p e r i o d a n d f o r t h e a v e r -

a g e , t h e g a i n f o r o n e c u r r e n c y e x c e e d s t h e l o ss f o r

t h e o t h e r c u r r e n c y .

F in a n c ia l An a l y s t s J o u r n a l / J a n u a r y - F e b r u a r y 1 9 9 5

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Table 1. Siege l sParadox

Quarter

P e r c e n t a g e

S t a t e -o f - Q u a rt e r C h a n g e s i n

E x c h a n g e R a t e s E x c h a n g e R a t e s

mark dollar mark dollar

dollar mark dollar mark

1Q84 2.75 .362 -5.58 5.90

2Q84 2.60 .384 7.18 -6.69

3Q84 2.79 .358 9.64 -8.79

4Q84 3.06 .326 3.66 -3.52

1Q85 3.17 .315 -1.83 1.84

2Q85 3.11 .321 -2.25 2.30

3Q85 3.04 .328 -13.04 15.01

4Q85 2.64 .377 -7.59 8.21

1Q86 2.44 .408 -4.46 4.67

2Q86 2.33 .427 -6.80 7.29

3Q86 2.17 .459 -7.16 7.73

4Q86 2.02 .494 -5.19 5.46

Q87 .91 .521 -5.11 5.41

2Q87 1.81 .549 0.49 -0.49

3Q87 1.82 .547 1.09 -1.08

4Q87 1.84 .541 - 14.00 16.28

1Q88 1.58 .629 4.29 -4.12

2Q88 1.65 .603 9.83 -8.95

3Q88 1.82 .549 2.27 -2.22

4Q88 1.86 .537 -4.88 5.12

Average - 1.97 2.47

Wh y Universal edg ing?

Why is the optimal hedge ratio identical for

investors everywhere? The answer lies in how

exchange rates reach equilibrium.

Models of international equilibrium generally

assume that the typical investor in any country

consumes a single g ood or basket of goods. 2 The

investor wants to maximize expected return and

minimize risk, measu ring expected return and risk

in terms of his own consumption good.

Given the risk-reducing and return-enhancing

properties of international diversification, an in-

vestor will want to hold an international ly diversi-

fied portfoliO of equities. Given no barriers to

international investment, every investor will hold

a share of a fully diversified portfolio of world

equities. And, in the absence of government par-

ticipation, some investor must lend when another

investor borrows, and some investor must go long

a currency when another goes short.

Whatev er the given levels of market volatility,

exchange rate volatilifies, correlations between ex-

change rates and correlations between exchange

rates and stock, in equilibrium, prices will adjust

until ever yone is willing to hold all stocks and until

someone is willing to take the other side of every

exchange rate contract.

Suppose, for example, that we know the re-

turn on a portfolio in one currency, and we know

the change in the exchange rate between that

currency and another currency. We can thus de-

rive the portfolio return in the other currency. We

can write down an equation relating expected

returns and exchange rate volatilifies from the

points of view of two investors in the two different

currencies.

Suppose that Investor A finds a high correla-

tion between the returns on his stocks in another

country and the corresponding exchange rate

change. He will probably want to hedge in order to

reduce his portfolio risk. But suppose Investor B in

that other country would increase his own portfo-

lio's risk by taking the other side of A's hedge.

Investor A may be so anxious to hedge that he will

be willing to pay B to take the other side. As a

result, the exchange rate contract will be priced so

that the hedge reduces A's expected return but

increases g's.

In equilibrium, both investors will hedge. In-

vestor A will hedge to reduce risk, while Investor

B will hedge to increase expected return. But they

will hedge equally, in proportion to their stock

holdings.

T H E

U N I VE R S A L H E D G I N G F O R M U L A

By extend ing t he above analysis to investors in all

possible pairs of countries, we find that the pro-

portion that each investor wants to hedge depends

on three averages: the average across countries of

the expected excess return on the world market

portfolio; the average across countries of the vola-

tility of the world market portfolio; and the aver-

age across all pairs of countries of exchange rate

volatility. These averages become inputs for the

universal hedging formula:3

~ m - - O ' m2

1 '

/J 'm -- ~O'e2

where

tEJ,m

=

the average across investors of the ex-

pected excess return (return above each

investor's riskless rate) on the world

market portfolio (which contains stocks

from all major countries in proportion

to each country's market value)

o m ~- the average across investors of t he vol-

atility of the world market portfolio

(where variances, rather than standard

deviation, are averaged)

162

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1985-19( , )4

o = t h e a v e r a g e e x c h a n g e r a t e v o l a t i l i t y ( a v -

e r a g e d v a r i a n c e s ) a c r o s s a l l p a i r s o f

c o u n t r i e s

N e i t h e r e x p e c t e d c h a n g e s i n e x c h a n g e r a t e s n o r

c o r r e l a t i o n s b e t w e e n e x c h a n g e r a t e c h a n g e s a n d

s t o c k r e t u r n s o r o t h e r e x c h a n g e r a t e c h a n g e s a f f e c t

o p t i m a l h e d g e r a t i o s . I n e q u i l i b r i u m , t h e e x p e c t e d

c h a n g e s a n d t h e c o r r e l a t i o n s c a n c e l o n e a n o t h e r ,

s o t h e y d o n o t a p p e a r i n th e u n i v e r s a l h e d g i n g

f o r m u l a .

I n t h e s a m e w a y , t h e B l a c k - S c ho l e s o p t i o n

f o r m u l a i n c l u d e s n e i t h e r t h e u n d e r l y i n g s t o c k 's

e x p e c t e d r e t u r n n o r i t s b e t a . I n e q u il i b r i u m , t h e y

c a n c e l o n e a n o t h e r .

T h e C a p i t a l A s s e t P r i c i n g M o d e l i s s i m i l a r .

T h e o p t i m a l p o r t f o l i o f o r a n y o n e i n v e s t o r c o u l d

d e p e n d o n t h e e x p e c t e d r e t u r n s a n d v o l a t i l i t i e s o f

a ll a v a i l a bl e a s se t s . I n e q u i l i b r i u m , h o w e v e r , t h e

o p t i m a l p o r t f o l i o f o r a n y i n v e s t o r i s a m i x o f t h e

m a r k e t p o r t fo l i o w i t h b o r r o w i n g o r le n d i n g. T h e

e x p e c t e d r e t u r n s a n d v o l a t i l i t i e s c a n c e l o n e a n -

o t h e r ( e x c e p t f o r th e m a r k e t a s a w h o l e ) , s o t h e y d o

n o t a f f e c t t h e i n v e s t o r ' s o p t i m a l h o l d i n g s .

I npu t s f o r the Fo rmu la

H i s t o r i c a l d a t a a n d j u d g m e n t a r e u s e d t o c r e -

a t e i n p u t s f o r t h e f o r m u l a . T a b l e s 2 t h r o u g h 8 g i v e

s o m e h i s t o r i c a l d a t a t h a t m a y b e h e l p f u l .

T a b l e 2 l is ts w e i g h t s t h a t c a n b e a p p l i e d t o

d i f f e r e n t c o u n t r i e s i n e s t i m a t i n g t h e t h r e e a v e r -

a g e s . J a p a n , t h e U n i t e d S t a t e s a n d t h e U n i t e d

K i n g d o m c a r ry t he m o s t w e i g h t .

T a b l e s 3 t o 5 c o n t a i n s t a t i s t i c s f o r 1 9 8 6 - 8 8 a n d

T a b l e s 6 t o 8 c o n t a i n s t a t i s t i c s f o r 1 9 8 1 - 8 5 . T h e s e

s u b p e r i o d s g i v e a n i n d i c a t io n o f h o w s t a ti s ti c s

c h a n g e f r o m o n e s a m p l e p e r i o d t o a n o th e r .

W h e n a v e r a g i n g e x c h a n g e r a t e v o l a ti l it i e s

o v e r p a i r s o f c o u n t r ie s , w e i n c l u d e t h e v o l a t i li t y o f

a c o u n t r y ' s e x c h a n g e r a t e w i t h i t s e l f . T h o s e v o l a -

t i l i t i e s a r e a l w a y s z e r o ; t h e y r u n d i a g o n a l l y

t h r o u g h T a b l e s 3 a n d 6 . T h i s m e a n s t h a t t h e

a v e r a g e e x c h a n g e r a t e v o l a t i l i t i e s s h o w n i n T a b l e s

5 a n d 8 a r e l o w e r t h a n t h e a v e r a g e s o f t h e p o s i t i v e

n u m b e r s i n T a b l e s 3 a n d 6 .

T h e e x c e s s r e t u r n s i n T a b l e s 4 a n d 7 a r e

a v e r a g e s f or t he w o r l d m a r k e t r e t u r n i n e ac h

c o u n t r y ' s c u r r e n c y , m i n u s t h a t c o u n t r y ' s r i s k le s s

i n t e r e s t r a te . T h e a v e r a g e e x c e s s r e t u r n s d i f f e r

Table 2. Capitalizations and Capitalization Weights

Domestic Companies Listed on the Major

Stock Exchange as of December 31, 1987

Companies in the FT-Actuaries World

Ind ice s as of Decemb er 31, 1987-t

Capitalization Capitalization

(U.S. $ billions) W eight (%) (U.S. $ billions) Weight (%)

Japa n 2700 40 2100 41

Unite d States 2100 31 1800 34

Unite d Kin gdom 680 10 560 11

Cana da 220 3.2 110 2.1

Ge rm any 220 3.2 160 3.1

France 160 2.3 100 2.0

Austr alia 140 2.0 64 1.2

Sw itzerland 130 1.9 58 1.1

Italy 120 1.8 85 1.6

Neth erlands 87 1.3 66 1.3

Sw eden 70 1.0 17 0.32

Ho ng Kong 54 0.79 38 0.72

Belgium 42 0.61 29 0.56

Den ma rk 20 0.30 11 0.20

Singa pore 18 0.26 6.2 0.12

N ew Zealan d 16 0.23 7.4 0.14,

No rw ay 12 0.17 2.2 0.042

Au stria 7.9 0.12 3.9 0.074

To tal 6800 100 5300 100

* From Activities and

Stat is t ics:

1987 Report by Federation Internationale de s Bourses de Valeurs (page 16).

t The FT-Actuaries World Indices

M

are jointly compiled by The Financial Times Limited, Goldm an, Sac hs & Co., and County

NatW est/Wood M ackenzie in conjunction with the Institute of Actuaries and the Faculty of Actuaries. This table excludes Finland,

Ireland, Malaysia, Mexico, South Africa and Spain.

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Table 3. Exchan ge Rate Volati l ities, 1986 -1988

C a n - G e r - A u s - S w i t z e r - N e t h e r - S w e - H o n g B e l- D e n - S i n g a - N e w Z e a - N o r - A u s -

J a p a n U . S . U . K . a d a m a n y F r a n ce t ra li a l a n d I ta ly l a n d s d e n K o n g g i u m m a r k p o r e l a n d w a y t r i a

Jap an 0 11 9 12 7 7 14 7 8 7 7 11 9 8 10 17 9 8

Un ite d Stat es 11 0 11 5 11 11 11 12 10 11 8 4 11 11 6 15 10 11

Unite d Kingdo m 9 10 0 11 8 8 14 9 8 8 7 11 9 8 10 16 9 9

Ca na da 12 5 11 0 12 11 12 13 11 11 9 6 12 11 8 15 10 12

Ge rma ny 7 11 8 12 0 3 15 4 3 2 5 11 6 4 10 17 8 5

Fran ce 7 11 8 11 2 0 14 5 3 3 5 11 6 4 10 17 7 5

Au stral ia 14 11 14 12 14 14 0 15 14 14 12 11 14 14 12 14 14 14

Switz er land 7 12 9 13 4 5 15 0 5 5 7 12 8 6 11 18 9 7

Italy 8 10 8 11 3 3 14 5 0 3 5 11 6 4 10 17 7 5

Ne the r lan ds 7 11 8 11 2 3 14 5 3 0 5 11 6 4 10 17 7 5

Sw ede n 7 8 7 9 5 5 12 7 5 5 0 8 6 4 8 16 6 5

Ho ng Ko ng 11 4 11 6 11 11 11 12 10 11 8 0 11 11 5 14 10 11

Bel gium 9 11 9 12 6 6 14 8 6 6 6 11 0 6 10 17 8 6

De nm ark 8 11 8 11 4 4 14 6 4 4 4 11 6 0 10 17 7 5

Sin gap ore 10 6 10 8 10 10 12 11 10 10 8 5 10 10 0 15 10 10

Ne w Zea land 17 15 16 15 17 17 14 18 17 17 15 14 17 17 15 0 16 17

Nor wa y 9 10 9 10 7 7 13 9 7 7 5 10 8 7 10 16 0 7

Au stri a 8 11 9 12 5 5 15 7 5 5 5 11 6 5 10 17 8 0

S o u r c e : F T - A c t u a r i e s W o r l d I n d i c e s d a t a b a s e .

b e t w e e n c o u n t r i e s b e c a u s e o f d i f f e r e n c e s i n e x -

c h a n g e r a t e m o v e m e n t s .

T h e e x c e s s r e t u r n s a r e not n a t i o n a l m a r k e t

r e t u r n s . F o r e x a m p l e , t h e J a p a n e s e m a r k e t d i d

b e t t e r t h a n t h e U . S . m a r k e t i n 1 9 8 7 , b u t t h e w o r l d

m a r k e t p o r t f o l i o d i d b e t t e r r e l a t i v e to i n t e r e s t r a t e s

i n t h e U n i t e d S t a t e s t h a n i n J a p a n .

B e c a u s e e x c h a n g e r a t e v o l a t i l i t y c o n t r i b u t e s t o

a v e r a g e s t o c k m a r k e t v o l a t i l i t y , O'm2 s h o u l d b e

g r e a t e r t h a n ½o-e . E x c h a n g e r a t e v o l a t i l i t y a l s o

c o n t r i b u t e s t o t h e a v e r a g e r e t u r n o n t h e w o r l d

m a r k e t , s o ~ m s h o u l d b e g r e a t e r t h a n ~cre , t o o .

Table 4.

Wo r l d M ad m t E xcess R e tu rn s an d R e tu m

Volalf i it ies in D i fferent Currencies, 1 986 -

1988

E x c e s s R e t u r n R e t u r n V o l a t i li t y

Cu rre nc y 1986 1987 1988 1986 1987 1988

J a p a n 8 - 12 21 14 26 15

Un i ted S ta tes 29 12 14 13 25 11

U n i t e d K i n g d o m 2 3 - 1 4 1 6 1 4 2 6 1 5

Ca na da 26 4 5 14 24 11

G e r m a n y 8 - 5 3 0 1 5 2 7 1 4

F r a n c e 1 1 - 7 2 7 1 4 2 6 1 4

A u s t r a l i a 2 3 - 2 - 6 1 9 2 5 1 4

S w i t z e r l a n d 8 - 8 3 6 1 5 2 7 1 5

I ta ly 2 -6 23 15 27 14

N e t h e r l a n d s 8 - 7 3 0 1 5 2 7 1 4

S w e d e n 1 6 - 6 1 9 1 3 2 5 1 3

H o n g K o n g 3 0 1 3 1 7 1 3 2 5 1 1

B e l g i u m 7 - 8 2 8 1 5 2 7 1 4

D e n m a r k 8 - 1 0 2 6 1 5 2 7 1 4

S ing apo re 36 6 16 12 25 12

N e w Z e a l a n d 1 5 - 2 2 1 3 2 0 2 9 1 4

N o r w a y 1 9 - 1 1 1 5 1 4 2 6 1 2

A u s t r i a 7 - 6 3 0 1 5 2 7 1 4

S o u r c e : F T - A c t u a r i e s W o r l d I n d i c e s

TM

d a t a b a s e .

A n E xa m p l e

T a b l e s 5 a n d 8 s u g g e s t o n e w a y t o c r e a t e

i n p u t s f o r t h e f o r m u l a . T h e a v e r a g e e x c e s s r e t u r n

o n t h e w o r l d m a r k e t w a s 3 p e r c e n t i n t h e e a r l i e r

p e r i o d a n d 1 1 p e r c e n t i n t h e l a t e r p e r i o d . W e m a y

t h u s e s t i m a t e a f u t u r e e x c e s s r e t u r n o f 8 p e r c e n t .

T h e v o l a t i l i t y o f t h e w o r l d m a r k e t w a s h i g h e r

i n t h e l a t e r p e r i o d , b u t t h a t i n c l u d e d t h e c r a s h , s o

w e m a y w a n t t o u s e t h e 1 5 p e r c e n t v o l a t i l i t y f r o m

t h e e a r l i e r p e r i o d . T h e a v e r a g e e x c h a n g e r a t e v o l -

a t i l i ty o f 1 0 p e r c e n t i n t h e e a r l i e r p e r i o d m a y a l s o

b e a b e t t e r e s t i m a t e o f t h e f u t u r e t h a n t h e m o r e

r e c e n t 8 p e r c e n t .

T h i s r e a s o n i n g l e a d s t o t h e f o l l o w i n g p o s s i b l e

v a l u e s f o r t h e i n p u t s :

~ m = 8 % ,

orm = 1 5 % ,

cr = 10%.

G i v e n t h e s e i n p u t s , t h e f o r m u l a t e l l s u s t h a t 7 7 p e r

c e n t o f h o l d i n g s s h o u l d b e h e d g e d :

0 . 0 8 - 0 . 1 5 2

- 0 . 7 7 .

1

0 . 0 8

-

~ 0 . 1 0 ) 2

T a b le 5 . W o r l d A v e r a g e V a lu e s , 1 9 8 6 - 1 9 8 8

E x c h a n g e

E x c e s s R e t u r n R a t e

R e t u r n V o l a t i l i t y V o l a t i l i t y

1986 17 14 9

1987 -3 26 8

1988 1 8 13 8

1986-88 11 18 8

164 Financia l Analysts

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1 9 8 5 - 1 9 9 4

Table 6. Exchang e Rate VolaBities, 1981-1985

United United

Japan St at es Ki ngd om Canada Germany Fra nce Austra lia Switzerland Italy Netherlands

Japan 0 12 13 11 10 10 12 11 9 10

United States 11 0 12 4 12 13 11 13 10 12

United Kingdom 12 13 0 12 10 11 14 12 11 10

Canada 11 4 11 0 11 12 10 12 10 11

Germany 10 12 10 12 0 5 13 7 5 2

France 10 13 11 12 4 0 12 8 5 5

Australia 12 10 13 10 12 12 0 13 11 12

Switzerland 11 14 12 13 7 8 14 0 8 7

Italy 9 10 11 10 5 5 12 8 0 5

Netherlands 10 12 10 11 2 5 12 7 5 0

Source: FT-Actuaries World Indices" data base.

To compare the results of using different in-

puts, we can use the historical averages from both

the earlier and later periods:

/-~m = 3% or 11%,

O"m 15% or 18%,

~e = 10% or 8%.

With the historical averages from the earlier period

as inputs, the fraction hedged comes to 30 per

cent:

0.03 - 0.152

0.30.

0.03 - ~(0.1012

Using averages from the later period gives a frac-

tion hedged of 73 per cent:

0.11

0.18 2

-0.73.

0.11 - 1~(0.08)2

Generally, s traight historical averages va ry too

much to serve as useful inputs for the formula.

Estimates of long-run average values are better.

Table 7.

Wor l d M a ne t E xce s s Re tu r ns a nd R e t um

Volat il it ies in Di f ferent Cu rrencies, 198 1-

1985

Currency Excess Return Return Volatility

Japan 3 17

United States - 1 13

United Kingdom 10 16

Canada 2 13

Germany 8 15

France 7 16

Australia 7 18

Switzerland 9 16

Italy 4 15

Netherlands 8 15

Optimization

The universal hedging formula assumes that

you put into the formula your opinions about wha t

investors around the world expect for the future. If

your own views on stock markets and on exchange

rates are the same as those you attribute to inves-

tors generally, then you can use the formula as it

is.

If your views differ from those of the consen-

sus, you may want to incorporate them using

optimization methods. Starting with expected re-

turns and covariances for the stock markets and

exchange rates, you would find the mix that max-

imizes the expected portfolio return for a given

level of volatility.

The optimization approach is fully consistent

with the universal hedging approach. When you

put the expectations of investors aroun d the world

into the optimization approach, you will find that

the optimal currency hedge for any foreign inw~st-

ment will be given by the universal hedging for-

mula.

A N o t e o n t he C u r r e n c y H e d g e

The formula assumes that investors hedge real

(inflation-adjusted) exchange rate changes, not

changes due to inflation differentials between

countries. To the extent that currency changes are

the result of changes in inflation, the formula is

only an approximation.

In other words, currency hedging only ap-

proximates real exchange rate hedging. But most

Table 8. W or ld Average Va lues , 1981-1985

Excess Return Exchange Rate

Return Volatility Volatfli W

3 15 10

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1 9 8 5 - 1 9 9 4

c h a n g e s i n c u r r e n c y v a l u e s , a t l e a s t i n c o u n t r i e s

w i t h m o d e r a t e i n f l a t i o n r a t e s , a r e d u e t o c h a n g e s

i n r e a l e x c h a n g e r a t e s . T h u s c u r r e n c y h e d g i n g w i l l

n o r m a l l y b e a g o o d a p p r o x i m a t i o n t o r e a l e x -

c h a n g e r a t e h e d g i n g .

I n c o n s t r u c t i n g a h e d g i n g b a s k e t , i t m a y b e

d e s i r a b l e t o s u b s t i t u t e h i g h l y l i q u i d c u r r e n c i e s f o r

l e s s l iq u i d o n e s . T h i s c a n b e s t b e d o n e b y b u i l d i n g

a c u r r e n c y h e d g e b a s k e t t h a t c l o s el y tr a ck s t h e

b a s k e t b a s e d o n t h e u n i v e r s a l h e d g i n g f o r m u l a .

W h e n t h e r e i s t r a c k i n g e r r o r , t h e f r a c t i o n h e d g e d

s h o u l d b e r e d u c e d .

I n p ra c ti c e, t h e n , h e d g i n g m a y b e d o n e u s i n g

a b a s k e t o f a f e w o f th e m o s t l i q u id c u r r e n c i e s a n d

u s i n g a fr a c ti o n s o m e w h a t s m a l l e r th a n t h e o n e t h e

f o r m u l a s u g g e s t s .

T h e f o r m u l a a l s o a s s u m e s t h a t t h e r e a l e x -

c h a n g e r a t e b e t w e e n t w o c o u n t r i e s i s d e f i n e d a s

t h e r e l a ti v e v a l u e o f d o m e s t i c a n d f o r e i g n g o o d s .

D o m e s t i c g o o d s a re t h o s e c o n s u m e d a t h o m e , n o t

t h o s e p r o d u c e d a t h o m e . I m p o r t s t h u s c o u n t a s

d o m e s t i c g o o d s . F o r e i g n g o o d s a r e t h o s e g o o d s

c o n s u m e d a b r o a d , n o t t h o s e p r o d u c e d a b r o a d .

C u r r e n c y c h a n g e s s h o u l d b e e x a m i n e d t o s e e

i f t h e y t r ac k r e a l e x c h a n g e r a t e c h a n g e s s o d e f i n e d .

W h e n t h e c u r r e n c y r a t e c h a n g e s b e t w e e n t w o

c o u n t r i e s d i f f e r f r o m

real

e x c h a n g e r a t e c h a n g e s ,

t h e h e d g i n g d o n e i n t h a t c u r r e n c y ca n b e m o d i f i e d

o r o m i t te d .

I f e v e r y o n e i n t h e w o r l d e v e n t u a ll y c o n s u m e s

t h e s a m e m i x o f g o o d s a n d s e r v ic e s , a n d p r i c e s o f

g o o d s a n d s e r v i c e s a r e t h e s a m e e v e r y w h e r e ,

h e d g i n g w i l l n o l o n g e r h e l p .

A P P L Y IN G T H E F O R MU L A T O O T H E R T Y P E S

O F P O R T F O U O S

H o w c a n y o u u s e t h e f o rm u l a i f y o u d o n ' t h a v e a

f u l l y d i v e r s i f i e d i n t e r n a t i o n a l p o r t f o l i o , o r i f f o r -

e i g n e q u i t i e s a r e o n l y a s m a l l p a r t o f y o u r p o r t f o -

l io ? T h e a n s w e r d e p e n d s o n w h y y o u h a v e a s m al l

a m o u n t i n f o r e i g n e q u i t i e s . Y o u m a y b e

( a) w a r y o f f o r e i g n e x c h a n g e r is k ;

( b ) w a r y o f f o r e i g n e q u i t y r i s k , e v e n i f i t i s

o p t i m a l l y h e d g e d ; o r

( c) w a r y o f f o r e i g n e x c h a n g e r i s k a n d f o r e i g n

e q u i t y r i sk , i n e q u a l m e a s u r e .

I n c a s e ( a ) , y o u s h o u l d h e d g e m o r e t h a n t h e

f o r m u l a s u g g e s t s . I n c a s e ( b ) , y o u s h o u l d h e d g e

l e s s t h a n t h e f o r m u l a s u g g e s t s . I n c a s e ( c ) , i t

p r o b a b l y m a k e s s e n s e t o a p p l y t h e f o r m u l a a s

g i v e n t o t h e f o r e i g n e q u i t i e s y o u h o l d .

I f t h e b a r r i e r s t o f o r e i g n i n v e s t m e n t a r e s m a l l ,

y o u s h o u l d g a in b y i n v e st in g m o r e a b r o a d a n d b y

c o n t i n u i n g t o h e d g e t h e o p t i m a l f r a c t io n o f y o u r

f o r e i g n e q u i t i e s .

Fore ign Bond s

W h a t i f y o u r p o r t f o li o c o n t a i n s f o r e i g n b o n d s

a s w e l l a s f o r e i g n s t o c k s ?

T h e a p p r o a c h t h a t l e d t o t h e u n i v e r s a l h e d g -

i n g f o r m u l a f o r s t o c k s s u g g e s t s 1 00 p e r c e n t h e d g -

i n g f o r fo r e i g n b o n d s . A p o r t f o l io o f f o r e ig n b o n d s

t h a t i s h e d g e d w i t h s h o r t - t e r m f o r w a r d c o n t r a c t s

s t i l l h a s f o r e i g n i n t e r e s t r a t e r i s k , a s w e l l a s t h e

e x p e c t e d r e t u r n t h a t g o e s w i t h t h a t r i sk .

A n y f o r e ig n b o n d s y o u h o l d u n h e d g e d c a n b e

c o u n t e d a s p a r t o f y o u r t o ta l e x p o s u r e t o f o r ei g n

c u r r e n c y r i s k . T h e l e s s y o u h e d g e y o u r f o r e i g n

b o n d s , t h e m o r e y o u w i ll w a n t t o h e d g e y o u r

f o r e i g n s t o c k s .

A t t i m e s , y o u m a y w a n t t o h o l d u n h e d g e d

f o r e i g n b o n d s b e c a u s e y o u b e l i e v e t h a t t h e e x -

c h a n g e r a t e w i l l m o v e i n y o u r f a v o r i n t h e n e a r

f u t u r e . I n t h e l o n g ru n , t h o u g h , y o u w i ll w a n t t o

h e d g e y o u r fo r e ig n b o n d s e v e n m o r e t h a n y o u r

f o r e i g n e q u i t i e s .

C O N C L U S I O N

T h e f o r m u l a ' s r e s u l t s m a y b e t h o u g h t o f a s a b a s e

c a s e . W h e n y o u h a v e s p e c i a l v i e w s o n t h e p r o s -

p e c t s f o r a c e rt a in c u r r e n c y , o r w h e n a c u r r e n c y ' s

f o r w a r d m a r k e t i s i l l i q u i d , y o u c a n a d j u s t t h e

h e d g i n g p o s i t i o n s t h a t t h e f o r m u l a s u g g e s t s .

W h e n y o u d e v i a t e f r o m t h e f o r m u l a b e c a u s e

y o u t h i n k a p a r t i c u l a r c u r r e n c y i s o v e r p r i c e d o r

u n d e r p r i c e d , y o u c a n p l a n t o b r i n g y o u r p o s i t io n

b a c k t o n o r m a l a s t h e c u r r e n c y r e t u r n s t o n o r m a l .

Y o u m a y e v e n w a n t t o u s e o p t i o n s , s o t h a t y o u r

e f f e c ti v e h e d g e c h a n g e s a u t o m a t i c a l l y a s t h e c u r -

r e n c y p r i c e c h a n g e s .

F o o r N O T E S

1. J.J. Siegel, Risk , Inter est Rates, and the F orward Ex-

change, Quarterly Journal of Economics (M ay 1972).

2. See, for example, B.H. Solnik, An Equilibrium Mo del ofth e

International Capital Market, Journal of Economic Theory

1 6 6 F in a n c ia l A n a l ys t s Jo u ma l / Ja n u a r y - F e b r u a r y 1 9 9 5

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1985-1994

(August 1974); F.L.A. Grauer , R.H. Litzenberger, and R.E.

Stehle, "Shar ing Rule=~ and Equilibrium in an In terna tional

Capital Market Under Uncertainty,'"

ournal of Financial Eco-

nomics (June 1976); P. Sercu, "A Generaliza tion of the Inter-

national Asset Pricing Model,"

Revu e de l Associat ion Francaise

de Finance,

(June 1980); and R. Stulz, "A Model of Interna -

tional Asset Pricing,"

Journal of Financial Economics

(Decem-

ber 1981).

3. Thederivation of the formula is desqribed in detail in F.

Black, "Equilibrium Exchange Rate Hedging," National Bu-

reau of Economic Research Working Paper No. 2947 (April

1989).

F inanc ia l Ana lys t s Journa l / January-February 1995

6 7