保險及退休基金於國外投資之風險評估：跨國資產組合模型(II)

行政院國家科學委員會專題研究計畫 成果報告

保險及退休基金於國外投資之風險評估：跨國資產組合模

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計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 95-2416-H-004-010-

執 行 期 間 ： 95 年 08 月 01 日至 96 年 10 月 31 日

執 行 單 位 ： 國立政治大學風險管理與保險學系

計 畫 主 持 人 ： 張士傑

計畫參與人員： 博士班研究生-兼任助理：黃雅文

碩士班研究生-兼任助理：曾毓英、胡育寧、吳欣樺

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1 u ] v c  Abstract

In this study, we revisit the investment choice problem in international portfolio management for long-term investors ( i. e. , institutional investors, asset managers, financial planners, and wealthy individuals) where, in particular, the ex change rate risk and the interest rate risk are incorporated. W hile the theoretical literature has made significant development, the case with ex act solution are still relatively few. S tarting with the new perspective in L ioui and P oncet ( 2 0 0 3 ) , they show that the optimal portfolio can be divided into three parts: the international speculative portfolio, the domestic interest rate hedging portfolio and the cross-country interest rate differential hedging portfolio.

S ince the second hedging component presented in L ioui and P oncet ( 2 0 0 3 ) is an indirect solution, we adopt a specific case that all diffusion coefficients appeared in the dynamics of the state variables are constant to clarify the hedging behaviors. T he results show that the optimal strategy follows a four-fund separation theorem and the number of the funds is irrelevant to the number of the assets. F or non-myopic investors, the currency risk -hedging component will not vanish due to the P urchase P ower P arity ( P P P ) deviation and the hedging demand becomes smaller when the investors shorten his time horiz on.

K e y w o rd s: currency rate; interest rate; hedging; separation theorem; P urchase P ower

P a r i t y .

1

Introduction

The trend of globalization and the rising importance of international …nancial markets inspire

an extension of the portfolio theory in considering foreign investments. Popular foreign

investments include stocks, bonds, real estate, mutual funds, and pooled trusts. Foreign

investments are not just diversi…cation components to domestic portfolios; they might help

to mitigate interest rate risk. Campbell, Viceira, and White (2003) argue that domestic

…xed income securities are risky for long-term investors because real interest rates vary over

time and the investments need to be rolled over with uncertain future interest rates. They

illustrate that the interest rate risk can be hedged by holding foreign currency if the domestic

currency tends to depreciate when the domestic real interest rate falls. Hence the major issue

in our analysis has been the optimal investment behaviors for the long-term investors (i.e.,

institutional investors, asset managers, …nancial planners, and wealthy individuals) regarding

the international portfolio selection. International assets bring currency exposure and risk

with them, and so the discussion of speculative, hedging issues and strategic asset allocation

become crucial.

In spite of the evidence on the gains from diversifying internationally, researches have

shown that investor’s portfolios have a disproportionately high share invested in domestic

assets, see French and Porterba (1991). Solnik (1974), Stulz (1981, 1983) and Adler and

Dumas (1983) suggest that the desire to hedge against home in‡ation may increase the

demand for domestic assets relative to foreign assets. For a review on international portfolio

choice, see Uppal (1993). Within this international economy, the changes of real exchange

representative individual trades on available assets to maximize the expected utility of his

…nal wealth. The traditional solution to this problem is derived by using the stochastic

dynamic programming technique pioneered in …nance by Merton (1969, 1971). The investor’s

optimal portfolio strategy is known to contain a speculative element and as many hedge

components as the number of state variables.

Instead of using stochastic control methods, the so-called martingale approach has been

alternatively used by Pliska (1986), Karatzas et al. (1987) and Cox and Huang (1989, 1991)

to study intertemporal consumption and portfolio policies when markets are complete, which

was also the case in the earlier dynamic programming literature. The martingale technology

describes the feasible investment strategy set by an intertemporal budget equation and then

solves the static investment problem in an in…nite dimensional Arrow-Debreu economy. As

mentioned in Vila and Zariphopoulou (1997), the martingale approach is appealing for two

reasons. First, it can be used to solve for the asset demand under very general investment

decisions regarding the stochastic opportunity set. Second, and consequently, it can be

applied in a general setting to solve for the equilibrium investment opportunity set (see

Du¢ e and Huang (1985)).

1.1 Hedging Issues

In addition to the speculative component, two hedging components are obtained in Lioui and

Poncet (2003). The …rst hedging component is associated with domestic interest rate risk

and the second one with the risk brought about by the co-movements of the interest rates

and the market price of risk, which turns out to depend on interest rate di¤erentials across

Poncet, 2003).

PPP is a theory which states that exchange rates between currencies are in equilibrium

when the purchasing power of the two countries are the same. This means that the exchange

rate between two countries should equal the ratio of the two countries’price level of a …xed

basket of goods and services. When a country’s domestic price level is increasing (i.e., a

country experiences in‡ation), that country’s exchange rate must depreciate in order to

return to PPP. The basis for PPP is the law of one price. In the absence of transportation

and other transaction costs, competitive markets will equalize the price of an identical good

in two countries when the prices are expressed in the same currency.

Our model involves estimating the characteristics of the yield curve and the market prices

of risk only. We consider the economy which consists of two major currencies: a foreign

currency and the domestic one, together with their bond funds and stock portfolios. Then

the parameters describing the current …nancial market, the investment time horizon and

the risk aversion parameter of the investor are fully investigated. Finally, we have obtained

optimal solution in order to clarify the hedging demands under certain market structure.

1.2 Long-Term Issues

Campbell and Viceira (2002) built rigorous theoretical models to show that the optimal

port-folio selections for the long-term investors are not the same as for the short-term investors. If

an investors anticipates that he will learn more by observing …nancial market to update his

preference parameters in response to asset returns, this introduces a new type of

intertem-poral hedging demand into the portfolio selection. In order to fully explore the proposed

martingale method. Most …nancial planning of the investors adopt static portfolio

optimiza-tion models, such as single-period mean variance allocaoptimiza-tion in Markowitz (1959), which are

short-sighted and when rolled forward lead to myopic portfolio rebalancing unless severely

constrained by the portfolio manager’s intuition. The Markowitz’s models are static (i.e.,

single period) and these investment strategies are referred to as short-term investors’ asset

allocation (or tactical asset allocation). The tactical asset allocation is under the assumption

that an investor has a mean-variance criterion in making his …nancial decisions.

Campbell and Viceira (2002) argues that time variation in the opportunity set generate

large di¤erences between optimal portfolios for long-term investors, who concern themselves

expected returns and risks change over time, and short-term investors. Balduzzi and Lynch

(1999) and Barberis (2000) have recently shown that the utility costs of behaving myopically

and ignoring predictability can be substantial. Long-term …nancial planning seems preferable

for the fund managers with a liability benchmark. Merton (1971, 1973) explored the optimal

solution of the dynamic portfolio in a multi-period framework given that the investment

opportunity sets do not vary over time. In our study, the single period short-term theory is

extended to the long-term framework that the opportunity set is time-varying.

Starting with the new perspective in Lioui and Poncet (2003), they show that the optimal

portfolio can be divided into the international speculative portfolio, the domestic interest rate

hedging portfolio and the cross-country interest rate di¤erential hedging portfolio. In this

study, we revisit the portfolio allocation problem where currency rate risk and interest rate

risk are present. In our model, continuous trading is assumed in the international …nancial

domestic currency. The decision variables are the weights of the assets in our opportunities,

i.e., the stock indices, the traded currencies and the bonds in each country that are involved.

We construct the wealth constraint using the martingale methodology to obtain the optimal

international portfolio. The features of this study are summarized in the following

1. We review and investigate the speculative and hedging implication of time-varying risk.

Five sources of uncertainty in the model economy are considered: interest rate risks

represented by the innovations for the domestic and foreign markets, market risks from

the domestic and foreign markets, and the currency rate risk.

2. Lioui and Poncet (2003) obtain an indirect currency risk hedging component to

covari-ances of assets with exogenous variables. The development of our approach adding to

their works in obtaining an explicit strategy with certain market structure to clarify

the hedge e¤ects in …nancial decision allowing for global investors.

3. We show that the optimal international portfolio follow a four-fund separation

theo-rem in maximizing the expected utility. Since the asset prices in the …nancial market

change continuously, the international portfolio must be rebalanced to obtain his

opti-mal solution.

The rest of this paper is organized as follows. Section 2 describes the …nancial market

and the proposed model, starting from the basic framework and followed by the dynamics

of invested opportunity set and the martingale constraints. Section 3 explores its explicit

characteristics regarding the fund wealth on the optimal investment decision incorporating

model with constant parameters. An example with simpli…ed assumptions is fully explored

in Section 5. Conclusions are presented in Section 6.

2

The Market Framework and the Model

2.1 The Market Framework

We consider an economy in which the investor allocate his wealth between a domestic money

market account Bd; a foreign money market account Bf, a domestic discount bond Pd

ma-turing at date Td, a foreign discount bond Pf maturing at Tf; a domestic stock index Sd

and a foreign stock index Sf: These assets comprise a complete market from the domestic

investor’s viewpoint. There are …ve sources of uncertainty across the two economies in terms

of …ve independent Wiener processes Z(t)0= _{Z1(t) Z2(t) Z3(t) Z4(t) Z5(t)} : (here0

denotes transposition). The independence hypothesis on these Brownian motions implies no

loss of generality since we can always shift from uncorrelated to correlated Wiener processes

(and vice versa) via the Cholesky decomposition of the correlation matrix.

First we assume that the currency rate e between the domestic and the foreign market

satis…es de(t) e(t) = e(t)dt + 5 X i=1 ei(t)dZi(t);

where _e(t); ei(t); 1 i 5 are prescribed deterministic functions.

To fully describe the stochastic model for the whole forward-rate curve, the domestic

instantaneous forward interest rate fd is assumed to satisfy

dfd(t; T ) = d(t; T )dt + 5

X

i=1

where _d(t; T ) and di(t; T ); 1 i 5 are prescribed deterministic functions.

According to Heath et al. (1992), we simplify HJM model and get the forward rate

fd(t; T ) at time t for the period (T; T + dt) and the short term spot rate process rd(t) at

time t follows the di¤usion process. The domestic spot rate rd(t) is simply given by the

forward-rate for maturity equal to the current date, i.e. rd(t) = fd(t; t): The domestic money

market account Bd(t); starting at Bd(0) = 1; is

Bd(t) = exp

Z t

0

rd( )d :

Upon integration, one …nds that

rd(t) = fd(0; t) + Z t 0 d ( ; t)d + 5 X i=1 Z t 0 di( ; t)Zi( ):

Moreover, for the HJ M model it makes the motion of the spot rate non-Markov. The price

of the domestic discount bond Pd maturing at date Td satis…es

Pd(t; Td) = exp

Z Td

t

fd(t; )d ;

and, with Itô’s lemma, the di¤erential of Pd satis…es

dPd(t; Td) Pd(t; Td) = (rd(t) + hd(t; Td)) dt + 5 X i=1 kdi(t; Td)dZi(t);

where the deterministic function hd(t; Td) is

hd(t; Td) = 1 2 5 X i=1 Z Td t di(t; )d 2 Z _T d t d (t; )d ; and kdi(t; Td) = Z Td t di(t; )d ; 1 i 5:

Following Sorensen (1999), we assume further that the price of the domestic stock index Sd satis…es dSd(t) Sd(t) = ( _d(t) + rd(t)) dt + 5 X i=1 di(t)dZi(t);

where _d(t); di(t); 1 i 5 are deterministic functions.

We adopt the convention that when no confusion arises, all the relations satis…ed by

foreign assets are identical to the corresponding domestic ones, with modi…ed subscript f:

Then we have dff(t; T ) = f(t; T )dt + 5 X i=1 f i(t; T )dZi(t); rf(t) = ff(0; t) + Z t 0 f ( ; t)d + 5 X i=1 Z t 0 f i( ; t)Zi( ); dPf(t; Tf) Pf(t; Tf) = (rf(t) + hf(t; Tf)) dt + 5 X i=1 kf i(t; Tf)dZi(t); where hf(t; Tf) = 1 2 5 X i=1 Z Tf t f i(t; )d 2 Z _T f t f (t; )d ; kf i(t; Tf) = Z Tf t f i(t; )d ; 1 i 5; and dSf(t) Sf(t) = _f(t) + rf(t) dt + 5 X i=1 f i(t)dZi(t):

According to the domestic viewpoint, all prices of foreign assets should be converted by the

real currency rate e: All converted prices are denoted by the symbol _{b. With Itô’s lemma,}

the converted foreign money market cBf := Bf e satis…es

d cBf(t) c Bf(t) = ( _e(t) + rf(t)) dt + 5 X i=1 ei(t)dZi(t):

The converted price of foreign instantaneous stock index cSf := Sf e (see Lioui and Poncet (2003)) dcSf(t) c Sf(t) = _f(t) + rf(t) dt + 5 X i=1 f i(t)dZi(t); where f(t) = e(t) + f(t) + 5 X i=1 ei(t) f i(t); and f i(t) = ei(t) + f i(t); 1 i 5:

The converted foreign discount bond price cPf := Pf e satis…es

dcPf(t; Tf) c Pf(t; Tf) = _f(t; Tf) + rf(t) dt + 5 X i=1 f i(t; Tf)dZi(t); where f(t; Tf) = e(t) + hf(t; Tf) + 5 X i=1 ei(t)kf i(t; Tf); and f i(t; Tf) = ei(t) + kf i(t; Tf); 1 i 5:

2.2 The Martingale Method

The international …nancial market is assumed to be free of frictions and arbitrage

opportu-nities, so there exists a probability measure which is equivalent to the historical probability

measure P with respect to a given numéraire such that the prices expressed in terms of this

We select the numéraire as the riskless asset yielding rd(t) and the corresponding

prob-ability measure Q is the so-called risk neutral probprob-ability. The Radon-Nikodym derivative

dQ=dP is given by dQ dP = (t) = exp Z t 0 ( )0dZ( ) 1 2 Z t 0 ( )0 ( )d ;

and (t), the market prices of risk, is de…ned by means of (t); which is

(t) = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 e1(t) e2(t) e3(t) e4(t) e5(t) kd1(t; Td) kd2(t; Td) kd3(t; Td) kd4(t; Td) kd5(t; Td) f 1(t; Tf) f 2(t; Tf) f 3(t; Tf) f 4(t; Tf) f 5(t; Tf) d1(t) d2(t) d3(t) d4(t) d5(t) f 1(t) f 2(t) f 3(t) f 4(t) f 5(t) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 5 5 ; and (t) = (t) 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 e(t) + rf(t) rd(t) hd(t; Td) f(t; Tf) + rf(t) rd(t) d(t) f(t) + rf(t) rd(t) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; = (t) 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 e(t) hd(t; Td) f(t; Tf) d(t) f(t) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 + (t) 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 0 1 0 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (rf(t) rd(t)) ; = 1(t) + 2(t) (rf(t) rd(t)) ;

where 1(t); 2(t) are 5 1 deterministic functions.

In a complete market, all the risks brought about by the economic factors must be

embedded in the stochastic discount factor (the pricing kernel), so that the market price of

risk sums up all the relevant information available on the market.

3

The Optimization Program

Our problem is the selection of an optimal, self-…nancing portfolio allocation strategy which

maximizes the expected utility. We assume further that the insurer’s horizon T is shorter

than the maturing dates of the domestic and foreign bonds, which ensures that all bonds are

long-lived assets from the insurer’s viewpoint. Here we choose the CRRA utility function

U (W ) such as

U (W ) = 1W ; 0 < < 1;

= ln W; = 0:

The power utility is chosen for two reasons. First, the investors are in general large companies

which de…ne their strategies with respect to the amount of money they are managing, more

or less in a scaling way. This feature is well captured by the use of the power utility function.

Second, pension funds are regulated in such a way that they can not reach negative values.

This is true also in the power utility case, thanks to the in…nite marginal utility at zero.

The wealth W (t) of the investors at each time t is

W (t) = Bd(t)Bd(t) + _Bc_f(t) cBf(t)

where ( i(t) : i 2 fBd; cBf; Pd; cPf; Sd; cSfg) stand for the numbers of units of each asset.

Applying Itô’s lemma under the consideration of self-…nancing strategy and noting that the

domestic money market account is a riskless asset from the insurer’s viewpoint, we have (c.f.

Merton (1971)) dW (t) W (t) = ( )dt + (t) 0 _{(t)dZ(t); (1)} where (t)0 = Bcf_(t) Pd_(t) Pcf_(t) Sd_(t) Scf_(t) ;

is the portfolio weight vector of the risky assets and ( ) denotes an irrelevant function, a

notation which will be frequently used in the sequel.

De…ne the optimal growth portfolio (t) as (also see Merton (1992) and Long (1990))

(t) = Bd(t) (t) 1; then (t) = exp Z t 0 ( )0dZ( ) + Z t 0 rd( ) + 1 2 ( ) 0 _{( ) d :}

The investor’s international portfolio selection problem is written as

max E [U (W (T ))] ; 0 < < 1

with the martingale constraint

E W (T )

(T ) = W (0):

Here E [ ] is the expectation operator under the historical probability measure P: Following

Lioui and Poncet (2003) and according to Cox and Huang (1989, 1991), the …rst order

condition of the optimization problem is

W (T ) = 11 _{(T )} 1 1 _;

where the Lagrange multiplier is characterized by

W (0) = 11_{E (T )}1 _:

The optimal wealth V (t) at time t is equal to

V (t) = 11 _(t)E t h (T )1 i (2) = 11 _(t) 1 1 _P d(t; T ) 1Et (t; T ) 1 ; where (t; T ) = Pd(T; T ) (t) Pd(t; T ) (T ) = (t) Pd(t; T ) (T ) ; (3)

and Et[ ] is the expectation operator under the probability measure P and conditional with

respect to zt; the …ltration at time t: De…ning Et (t; T )

1

as J ( ; t; T ) and invoking

Itô’s lemma, we have formally

dJ ( ; t; T )

J ( ; t; T ) = ( )dt + J( ; t; T )

0_dZ(t);

where J( ; t; T ) is the 5 1 di¤usion vector of the process dJ ( ; t; T )=J ( ; t; T ):

Applying Itô’s lemma to (2), we have

dV (t) V (t) = ( )dt + 1 1 (t) 0 1 Pd(t; T ) 0₊ J( ; t; T )0 dZ(t); (4) where Pd(t; T )0 = k_d1(t; T_d) k_d2(t; T_d) k_d3(t; T_d) k_d4(t; T_d) k_d5(t; T_d) : (5)

Identifying the di¤usion terms of (1) and (4), we obtain the expression of optimal allocation

strategy (t) of risky assets as

(t) = (t) 1 1

Lastly, turning to the benchmark case of the logarithmic utility, (t) 1(₁1 (t)) in equation

(6) readily reveals the investor’s myopic behavior, i.e., the speculative component. While

(t) 1₍

1 Pd(t; T ) + J( ; t; T )) are the hedge terms in the optimal solution. Since

prices in the …nancial market change continuously, the optimal portfolio must be rebalanced

continuously in order to maintain the proposed weights.

4

Constant Parameter Models

In this section, we adopt the foregoing model and the methodology to a speci…c case, in

which all di¤usion coe¢ cients appeared in the dynamics of the state variables are constants

instead of deterministic functions. The following proposition is the summary of the optimal

asset allocation strategy in this constant case, and note that all coe¢ cients without argument

notation are all constants.

Proposition 1 (An International Investment Model - a four-fund theorem) Given

the dynamics of the investment opportunity set follow the di¤ usion process in equation (13),

(14), (17), (18) and (20), the domestic CRRA investor’s optimal allocation strategy (t) of

risky assets is divided into three parts: the international myopic portfolio 1, the

domes-tic interest rate hedging portfolio 2 and the cross-country interest rate di¤ erential hedging

portfolio 3. It constitutes a four-fund theorem in optimal investment strategy. In four-fund

theorem, the international portfolio invests in the following four funds to maximize the

ex-pected utility: the international myopic portfolio wM with level ₁a ; the domestic interest

rate hedge portfolio wY with level ₁b ; the cross country interest rate di¤ erential hedge

The optimal allocation strategy (t) of risky assets is given by (t) = 1+ 2+ 3; (7) = 1 1 (t) 1 _(t) 1 (t) 1 kd1(t; T ) kd2(t; T ) kd3(t; T ) kd4(t; T ) kd5(t; T ) 0 + 1 (t; T ) (t) 1 l1 l2 l3 l4 l5 0 ; = a 1 wM b 1 wY + c 1 wE: where (t) = 1(t) + 2(t) (rf(t) rd(t)) (t; T ) =n( (T ) (t)) (rf(t) rd(t)) + e(T ) e(t) (T ) + (t) o and (T ) = Z t 0 2( )0 2( )d ; e(t) = Z t 0 1( )0 2( )d ; (t) = Z t 0 ( )q ( )d :

1 = 1 1 (t) 1 _{(t) =} a 1 wM; wM = (t) 1 _(t) 10₅ (t) 1 _(t); a = 10₅ (t) 1 (t); 2 = 1 (t) 1 kd1(t; T ) kd2(t; T ) kd3(t; T ) kd4(t; T ) kd5(t; T ) 0 = b 1 wY; wY = (t) 1 _{kd1(t; T ) kd2(t; T ) kd3(t; T ) kd4(t; T ) kd5(t; T )} 0 10 5 (t) 1 kd1(t; T ) kd2(t; T ) kd3(t; T ) kd4(t; T ) kd5(t; T ) 0; b = 10₅ (t) 1 _{kd1(t; T ) kd2(t; T ) kd3(t; T ) kd4(t; T ) kd5(t; T )} 0 ; 3 = 1 (t; T ) (t) 1 l1 l2 l3 l4 l5 0 = c 1 wE; wE = (t; T ) (t) 1 _{l1 l2 l3 l4 l5} 0 10 5 (t; T ) (t) 1 l1 l2 l3 l4 l5 0; c = 10₅ (t; T ) (t) 1 _{l1 l2 l3 l4 l5} 0 :

given a, b, and c are real constants. (see the Appendix for the de…nitions of the notations

and the related lemmas).

The explicit expression of the optimal allocation strategy (7) is a revision to the

Propo-sition 2 appeared in Lioui and Poncet (2003). Here we reproduce the last paragraph in

page 2227 of their paper: "Sheer inspection of (A.14) shows that E_tphb(t; ) =( 1)i will be

random at time t only because (t) is stochastic. As shown in (A.12), the latter is random

because of the interest rate di¤erential (rf(t) rd(t)): In a Gaussian framework, any

an a¢ ne function of the instantaneous di¤erential. It follows that b J ( ; t; ) E_tp h b(t; ) 1 i = eA( ;t; )+B( ;t; )(rf(t) rd(t))_;

where A( ) and B( ) are deterministic functions." But in their previous paper, Lioui and

Poncet (2001), also under (nearly) identical assumptions, they claim that the expectation

is in the form of exp A( ; t; ) + B( ; t; )(rf(t) rd(t))2 ; i.e. a quadratic function of the

instantaneous di¤erential; see (13) of this paper. The latter observation is correct (however,

the formula (13) of Lioui and Poncet (2001) requires revisions). In fact, by a simple example

and the standard device of stochastic analysis, one can easily refute the argument appeared

in Lioui and Poncet (2003) as follows. Take r(t) = Z(t); where Z(t) is the standard

one-dimensional Wiener process. According to (10), (11) and (12) in Appendix, we have

Et exp k Z T t r( )2d = exp (t)r(t)2 '(t) ; where (t) = r k 2tanh p 2k(T t); '(t) = 1 2ln h coshp2k(T t)i:

In the appendix of Lioui and Poncet (2001), the authors put considerable e¤ort on the

evaluation of the conditional expectation but, some revisions are required on the last term

in (A.5) of their paper by taking the term 1b(t; D)=( 2 s( D t) 2) out of the integral.

5

Illustrative Example

With the explicit expression for the hedging demands, we now analyze the analytical results

by the parameters. We begin with several assumptions and then state formally the result in

the proposition. The market assumptions are as follows

(t) = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 e1 0 0 0 0 0 d2(Td t) 0 0 0 0 0 f 3(Tf t) 0 0 0 0 0 d4 0 0 0 0 0 _{f 5} 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 5 5 (t) = (t) 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 e+ rf(t) rd(t) hd f + rf(t) rd(t) d f + rf(t) rd(t) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 e= e1 hd= d2(Td t) f= f 3(Tf t) d= d4 f= f 5 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1= e1 0 1= f 3(Tf t) 0 1= _{f 5} 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (rf(t) rd(t)) ; = 1(t) + 2(t) (rf(t) rd(t)) ; d (rf( ) rd( )) = q ( )dt + 5 X i=1 lidZi( ); q ( ) = q; (d (rf( ) rd( )))2 = 5 X i=1 l2_id ;

(t) = Z t o 2( )> 2( )d = ( 1 2 e1 + 1₂ f 5 )t + 1 f 3 ( 1 Tf t 1 Tf ); (t) = Z t 0 ( )q ( )d = ( 1₂ e1 + 1₂ f 5 )q t 2 2 + q f 3 (ln( Tf Tf t ) t Tf ); e(t) =Z t 0 1( )> 2( )d = ( ₂e e1 + ₂f f 5 )t + f f 3 ( 1 Tf t 1 Tf ); J( ; t; T )> = 1 n ( (T ) (t)) (rf(t) rd(t)) + e(T ) e(t) (T ) + (t) o l1 l2 l3 l4 l5 ; = 1 8 > > > > > > < > > > > > > : [( 12 e1 + 12 f 5 )(T t) + 1 f 3( 1 Tf T 1 Tf t] (rf(t) rd(t)) +( e 2 e1 + f 2 f 5 )(T t) + f f 3( 1 Tf T 1 Tf t) ( 12 e1 + 12 f 5 )q (T2₂ t2)+ q f 3(ln( Tf T Tf t) + T t Tf )) 9 > > > > > > = > > > > > > ; l1 l2 l3 l4 l5 : Pd(t; T )>= 0 _d2(T t) 0 0 0 :

(t) = (t) 1 1 1 (t) 1 Pd(t; T ) + J( ; t; T ) (8) = 1 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 e+rf(t) rd(t) 2 e1 hd d2(Td t)2 f+rf(t) rd(t) f 3(Tf t)2 d 2 d4 f+rf(t) rd(t) 2 f 5 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 d2(T t) d2(Td t) 0 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 + 1 8 > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : [( 12 e1 + 1 2 f 5 )(T t) + 1 f 3( 1 Tf T 1 Tf t] (rf(t) rd(t)) +( e 2 e1 + f 2 f 5 )(T t) + f f 3( 1 Tf T 1 Tf t) ( 12 e1 + 12 f 5 )q (T2₂ t2) + q f 3(ln( Tf T Tf t) + T t Tf )) 9 > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > ; 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 l1 e1 l2 d2(Td t) l3 f 3(Tf t) l4 d4 l5 f 5 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 :

As there is no reason to assume that any of these hypothetical cases will occur, it is likely that

empirical tests using them will in general underestimate the size of the currency risk premia.

In the general case where investors are not myopic, however, the market price of currency

risk will not vanish. This is because the expected rates of return on all assets embedded

in J ( ; t; T ) will, in particular, be in‡uenced by J( ; t; T ), i.e. by currency-related risk.

The latter, which is tantamount to PPP deviation risk, will be hedged at equilibrium, and

hence priced. Since deviations from PPP imply that the national real spot rates will di¤er,

currency risk is related to the risk involved in the random ‡uctuations of real interest rate

spreads across countries which is discussed in Lioui and Poncet (2003).

utility, the hedging demand becomes smaller when the investor shortens his time horizon.

Hence, equilibrium rates of return are consistent with the market evidence.

6

Concluding Remarks

The bene…ts of international diversi…cation have been known for many decades, but it is only

recently that investors have started allocating a signi…cant portion of their portfolio holdings

in foreign assets. To manage the risk of international portfolios, investors need to know the

speculative and hedging demands in the cross-country variation in global return uncertainty.

This study investigates the international asset allocation for global investors, which

in-corporates the hedge demands in controlling the stochastic variation due to PPP deviation.

The development of our approach adding to the previous works of Lioui and Poncet (2003)

is that we compare the obtained optimal strategies with certain market structure in order to

clarify the hedge e¤ects in …nancial decision allowing for global investors. Finally,

hypothet-ical mutual funds are constructed in our work to ful…ll the proposed demands. The optimal

investment strategies are a leveraged growth optimal portfolio, but with contingent leverages

as time goes by.

Following the four-fund theorem stated in Rudof and Ziemba (2004), the optimal

portfo-lio consists of into four components: the international myopic portfoportfo-lio, the domestic interest

rate hedge portfolio, the cross country interest rate di¤erential hedge portfolio and the

do-mestic riskless asset. With respect to the most common approach used in the literature, the

market structure and the certain utility employed to describe the investor’s attitude toward

fund separation methodology.

7

Appendix

7.1 Evaluation of a Certain Conditional Expectation

Theorem 2 (Feynman-Kac Formula, c.f. Lamberton et al (1991), Theorem 5.1.7)

Let u be a well-behaved function de…ned on [0; T ] Rn: If u satis…es

@u @t + Atu ru = 0; 8(t; x) 2 [0; T ] R n and u(T; x) = f (x); then u(t; x) = E f (X_Tt;x) exp Z T t r( ; Xt;x)d ;

where the At is the in…nitesimal operator of the n dimensional di¤ usion process dXt =

b(t; Xt)dt+ (t; Xt)dWt. The conditional expectation is taken with respect to t; where Xt= x:

We consider a conditional expectation u(t; x) as the following

u(t; x)) = E exp k Z T

t

Z( )2d (9)

which is conditioned at t and Xt = Z (t) = x; where Z(t) is a standard one-dimensional

Wiener process and k is a constant. Note that, the conditional expectation (9) is akin to

(30) modulo a deterministic factor and the evaluation of the more general (30) may bene…t

PDE satis…ed by u; which is @u @t = 1 2 @2u @x2 kx 2_u

and subject to the boundary condition

u(T; x) = 1:

Assume that u satis…es the form

u(t; x) = exp (t)x2 '(t) ; (10)

then the boundary condition becomes

(T ) = 0; '(T ) = 0:

After the separation of variables, we have

x2 0(t) + 2 (t)2 k = 0

and

(t) = '0(t):

Solution of the above two ODEs yields

(t) = r k 2tanh p 2k(T t) (11) and '(t) =1 2ln h coshp2k(T t) i : (12)

7.2 Evaluation of Constant Parameter Models

The following list is the summary of the underlying dynamics in this constant case, and note

that all coe¢ cients without argument notation are all constants.

de(t) e(t) = edt + 5 X i=1 eidZi(t); (13) dfd(t; T ) = d(t; T )dt + 5 X i=1 didZi(t); (14) rd(t) = fd(0; t) + Z t 0 d ( ; t)d + 5 X i=1 diZi(t); (15) Bd(t) = exp Z t 0 rd( )d ; dPd(t; Td) Pd(t; Td) = (hd(t; Td) + rd(t)) dt + 5 X i=1 kdi(t; Td)dZi(t); where hd(t; Td) = 1 2(Td t) 2 5 X i=1 2 di Z Td t d (t; )d ; kdi(t; Td) = di(Td t); 1 i 5; (16) dSd(t) Sd(t) = ( _d+ rd(t)) dt + 5 X i=1 didZi(t); (17) dff(t; T ) = f(t; T )dt + 5 X i=1 f idZi(t); (18) rf(t) = ff(0; t) + Z t 0 f ( ; t)d + 5 X i=1 f iZi(t); (19) Bd(t) = exp Z t 0 rd( )d ; dPf(t; Tf) Pf(t; Tf) = (rf(t) + hf(t; Tf)) dt + 5 X i=1 kf i(t; Tf)dZi(t);

where hf(t; Tf) = 1 2(Tf t) 2 5 X i=1 2 f i Z Tf t f (t; )d ; kf i(t; Tf) = f i(Tf t); 1 i 5; dSf(t) Sf(t) = _f + rf(t) dt + 5 X i=1 f idZi(t); (20) d cBf(t) c Bf(t) = ( _e+ rf(t)) dt + 5 X i=1 eidZi(t); dcSf(t) c Sf(t) = _f + rf(t) dt + 5 X i=1 f idZi(t); where f = e+ f+ 5 X i=1 ei f i; f i = ei+ f i; 1 i 5; dcPf(t; Tf) c Pf(t; Tf) = _f(t; Tf) + rf(t) dt + 5 X i=1 f i(t; Tf)dZi(t); where f(t; Tf) = e+ hf(t; Tf) + 5 X i=1 eikf i(t; Tf); f i(t; Tf) = ei+ kf i(t; Tf); 1 i 5; and (t) = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 e1 e2 e3 e4 e5 kd1(t; Td) kd2(t; Td) kd3(t; Td) kd4(t; Td) kd5(t; Td) f 1(t; Tf) f 2(t; Tf) f 3(t; Tf) f 4(t; Tf) f 5(t; Tf) d1 d2 d3 d4 d5 f 1 f 2 f 3 f 4 f 5 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ;

and (t) = (t) 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 e hd(t; Td) f(t; Tf) d f 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 + (t) 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 0 1 0 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (rf(t) rd(t)) = 1(t) + 2(t) (rf(t) rd(t)) : (21)

From the de…nition of Pd(t; T ); we have

Pd(t; T ) = exp Z T t fd(t; )d = exp ( Z T t fd(0; )d Z T t Z t 0 d (u; )dud 5 X i=1 di(T t) Zi(t) ) and since fd(t; T ) = fd(0; T ) + Z t 0 d (u; T )du + rd(t) fd(0; t) Z t 0 d (u; t)du

and (15), the expression of rd(t)

rd(t) = fd(0; t) + Z t 0 d ( ; t)d + 5 X i=1 diZi(t); we thus obtain Pd(t; T ) = exp Z T t (fd(0; ) fd(0; t)) d Z T t Z t 0 d (u; )dud exp (T t)rd(t) + (T t) Z t 0 d (u; t)du : (22) We have also Z T t rd( )d = Z T t fd(0; ) + Z 0 d (u; )du + 5 X i=1 diZi( ) ! d :

From Z T t Zi( )d = Z T t (T )dZi( ) + (T t)Zi(t); it follows that Z T t rd( )d = Z T t fd(0; ) + Z 0 d (u; )du d + 5 X i=1 di Z T t (T )dZi( ) + (T t)Zi(t) = Z T t fd(0; ) + Z 0 d (u; )du d + 5 X i=1 di Z T t (T )dZi( ) +(T t)rd(t) (T t) fd(0; t) (T t) Z t 0 d (u; t)du: (23)

Thus, by substituting (22) and (23) into (3), the de…nition of , we have

(t; T ) = exp Z T t ( )0dZ( ) Z T t rd( ) + 1 2 ( ) 0 _{( ) d P} d(t; T ) 1 = exp Z T t ( )0dZ( ) Z T t 1 2 ( ) 0 _{( )d} exp ( ₅ X i=1 Z T t di(T )dZi( ) + 2(T t)fd(0; t) ) :

Upon inspection, only the …rst term in the last equality would generate stochastic components

after taking conditional expectations. We proceed to carry out the calculation.

Applying the decomposition of (t) in (21) to the integral

exp Z T t ( )0dZ( ) + Z T t 1 2 ( ) 0 _{( )d}

we have exp Z T t ( )0dZ( ) + Z T t 1 2 ( ) 0 _{( )d (24)} = exp Z T t 1( )0dZ( ) + Z T t (rf( ) rd( )) 2( )0dZ( ) exp 1 2 Z T t 1( )0 1( )d + Z T t 1( )0 2( ) (rf( ) rd( )) d exp 1 2 Z T t 2( )0 2( ) (rf( ) rd( ))2d :

We neglect the integrals 1₂R_tT 1( )0 1( )d ,

RT

t 1( )0dZ( )on the right-hand side of (24)

because of the deterministic contributions after taking the conditional expectations. There

are three stochastic integrals left, namely

1 2 Z T t 2( )0 2( ) (rf( ) rd( ))2d ; Z T t 1( )0 2( ) (rf( ) rd( )) d and Z T t (rf( ) rd( )) 2( )0dZ( ).

Note that, from (15) and (19) the dynamics of rf(t) rd(t) is

rf(t) rd(t) = q(t) + 5 X i=1 liZi(t); (25) where q(t) = ff(0; t) + Z t 0 f ( ; t)d fd(0; t) Z t 0 d ( ; t)d ; li = f i di; i = 1 to 5: Applying (25), we have d (rf( ) rd( )) = q ( )dt + 5 X i=1 lidZi( ); (26) (d (rf( ) rd( )))2 = 5 X i=1 l2_id ;

where q ( ) = dq( )=d :

De…ne (t) such that

(t) = Z t

o

2( )0 2( )d : (27)

Integration by parts and the application of Itô’s lemma with (rf( ) rd( ))2render the

integral 1 2 Z T t 2( )0 2( ) (rf( ) rd( ))2d ; into 1 2 Z T t 2( )0 2( ) (rf( ) rd( ))2d ; (28) = 1 2(rf(T ) rd(T )) 2 (T ) 1 2(rf(t) rd(t)) 2 (t) Z T t ( ) (rf( ) rd( )) d (rf( ) rd( )) 1 2 Z T t ( ) (d (rf( ) rd( )))2:

After substituting (26) into (28), it is clear that the only term we need to specify is Z T t ( ) (rf( ) rd( )) d (rf( ) rd( )) ; and Z T t ( ) (rf( ) rd( )) d (rf( ) rd( )) = Z T t ( ) (rf( ) rd( )) q ( )d + X i Z T t qi ( )dZi( ) = (rf(T ) rd(T )) (T ) (rf(t) rd(t)) (t) Z T t ( )q ( )d X i Z T t li ( )dZi( ) + X i Z T t li ( )dZi( );

through repeated integration by parts, where

(t) = Z t

0

Thus, we may summarize our results in the following lemmas

Lemma 3 With the assumptions of our …nancial model, there exist two deterministic

func-tions (T ) in equation (27) and (T ) in equation (29) such that

1 2 Z T t 2( )0 2( ) (rf( ) rd( ))2d (30) = 1 2(rf(T ) rd(T )) 2 (T ) 1 2(rf(t) rd(t)) 2 (t) (rf(T ) rd(T )) (T ) + (rf(t) rd(t)) (t) + X i Z T t ( )dZi( ) + ( ):

Lemma 4 With the assumptions of our …nancial model, there exist two deterministic

func-tions e (T ) such that the integral Z T

t

1( )0 2( ) (rf( ) rd( )) d

may be treated in a similar fashion. The …nal result is Z T t 1( )0 2( ) (rf( ) rd( )) d (31) = (rf(T ) rd(T )) e (T ) (rf(t) rd(t)) e (t) + X i Z T t ( )dZi( ) + ( ); where e(t) =Z t 0 1( )0 2( )d : (32)

Lemma 5 With the assumptions of our …nancial model, substituting the expression (25)

into the stochastic integral Z T

t

we have Z T t (rf( ) rd( )) 2( )0dZ( ) = Z T t q( ) + 5 X i=1 liZi( ) ! 2( )0dZ( ) = X i Z T t ( )dZi( ) + X i;j Z T t li 2j( )Zi( )dZj( ); (33)

where 2j( ); 1 j 5 denotes the j th component of the 5 1 function 2( ):

Collecting all the results of (30),(31) and (33) obtained above, we compute J ( ; t; T ) as

J ( ; t; T ) = Et (t; T ) 1 = A( ; t; T ) exp 2 (1 )(rf(t) rd(t)) 2 ( (T ) (t)) exp 1 (rf(t) rd(t)) e (T ) e(t) (T ) + (t) ;

where A( ; t; T ) is a deterministic function. Here we utilize the independence property of

(rf(T ) rd(T )) (rf(t) rd(t)) with respect to the conditional expectation operator Et[ ]

because of the expression (25), and the fact such that the expressionR_tT li 2j( )Zi( )dZj( )

is independent with respect to Et[ ] is also used. Applying Itô’s lemma, we have

dJ ( ; t; T ) J ( ; t; T ) = 1 n ( (T ) (t)) (rf(t) rd(t)) + e(T ) e(t) (T ) + (t) o d (rf(t) rd(t)) + ( ) dt = 1 n ( (T ) (t)) (rf(t) rd(t)) + e(T ) e(t) (T ) + (t) o 5 X i=1 lidZi(t) + ( ) dt:

Proposition 6 The instantaneous conditional ( ₁) moment of the Arrow-Debreu prices of

the reference country bond of maturity T is given by

J ( ; t; T ) = Et (t; T ) 1 = A( ; t; T ) exp 2 (1 )(rf(t) rd(t)) 2_{( (T ) (t))} exp 1 (rf(t) rd(t)) e (T ) e(t) (T ) + (t) ;

where A( ; t; T ) is a deterministic function. The di¤ usion vector J( ; t; T )> of the process

of dJ ( ;t;T )_{J ( ;t;T )} is given by J( ; t; T )0 (34) = 1 n ( (T ) (t)) (rf(t) rd(t)) + e(T ) e(t) (T ) + (t) o l1 l2 l3 l4 l5 :

Substituting the expressions of (t); (t) and e (T ) in (27), (29) and (32), respectively

and (5), (16), (21) and (34) into (6), we obtain the expression of optimal allocation strategy

(t) of risky assets.

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