NOTATION Domestic () interest rate: R; foreign: R $ Spot exchange - - PDF document

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NOTATION Domestic (€) interest rate: R; foreign: R$ Spot exchange rate $S per €1 Forward rate $F per €1 Basic forex parity relationships

• 1. Without forward cover, given E(S)=Se,

€1 €(1+R) $Se(1+R) }Assume cert-

• r:

} ainty, or risk €1  $S  $S(1+R$) }neutrality For no net speculative capital flows: (1) Se S =1+R$ 1+R or (1a) eS S  R$  R Uncovered interest parity, UIP. [uncertainty+risk aversion  modify UIP.] What, e.g., if R$ < R + eS S ?

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• 2. Using the forward market:

€1  €(1+R)  $F(1+R) with certainty. Given the equally certain: €1  $S  $S(1+R$) it's clear that arbitraging implies that: (2) F S = 1+R$ 1+R or (2a) FS S = R$R 1+R  R$R i.e. (covered) interest parity, CIP: fwd premium/discount = interest differential.

• Q. under CIP, why should hedgers use fwd

cover, given that spot transactions can  same result?

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F as a predictor of future S Under certainty, where UIP should hold exactly, Se = S1+R$ 1+R = F, and F (assumed always to satisfy CIP) is the market's expectation of the spot rate. If UIP doesn't hold, an opportunity for profit

• arises. Assume everything relates to '1 period'.

e.g. Se S1+R$ 1+R , then buy €1 for $S borrowed $. After one period, you have accumulated €(1+R), i.e. $Se(1+R), more than you need to repay the debt of $S(1+ R$). Given Se, S (and F) rise. Risky world of risk neutralitysimilar result: UIP gives the rational Exp. of the future spot rate: if you held any other, you'd believe riskless profits

• available. The rational expectation is that such
• pportunites have been exploited. Then F is the

rational expectation.

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Risk aversion: UIP contains a risk-premium term: e.g. from the $ perspective, the expected appreciation eS S of the € needs to above the interest differential R$  R for risk-aversion to be overcome and for funds to start to flow into €. Then, F becomes a biased estimate of the future spot rate, even if expectations are rational. Note that bias can also arise because of expectational errors (i.e. non-rational expectations which, unlike RE, will be systematically falsified)  i.e. UIP & hence F may indicate the current market expectation (e.g. under risk neutrality) but if expectations are not rational, then F will not be a rational expectation, i.e will be biased ex post.

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• 3. Purchasing power parity PPP

[P=the goods price-level & Ǐ: E(inflation rate)] Under certainty/risk neutrality, to exclude goods-market arbitrage: PS=P$ : i.e. absolute PPP, unlikely with transactions costs & trade barriers, but we may have relative PPP: At t=0 buy one unit of IR goods priced at P, expected to be worth €P(1+Ǐ) at t=1, or (i) $SeP(1+Ǐ). Alternatively €P=$SP at t=0, so buy $SP/P$ US

• goods. Expected value for t=1:

(ii) $(SP/P$)P$(1+Ǐ$). To exclude arbitrage, (i)=(ii): $Se(1+Ǐ)=$S(1+Ǐ$),

• r Se/S=(1+Ǐ$)/(1+Ǐ) which in approx. form is:

(3a) eS S  Ǐ$ Ǐ

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Finally using (1a) + (3a): R$  R eS S  Ǐ$ Ǐ  (4a) R$  Ǐ$ = R  Ǐ 'Real interest parity' (I. Fisher): under UIP and PPP, (expected) real returns should be equated internationally.  relevant to Q. of international diversification. Notice that the parity relationships may be put in a slightly different form involving logs: e.g.take logs of (1) Se S =1+R$ 1+R and ln(Se)ln(S) =ses= ln(1+R$)ln(1+R)R$R. If S & F were € per $1 (rather than $ per €1) then, in all the parity relationships, swap R & R$.

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Siegel’s paradox One reason for writing the relationships in log form is to avoid Siegel’s paradox. If E($S)=$F, then you might think that E(€1/S)=€1/F: not so, because the expectation of a function of a random variable is not generally equal to the function of the expectation. In fact when the function is convex, as here, then according to Jensen’s inequality, E(1/S)1/E(S). So if E(S)=F, then E(1/S)1/F: i.e. E(S)=F and E(1/S)=1/F can’t both hold. However, if we use E(log S)= log F, then E(log 1/S)=E(logS)= E(logS)= logF=log1/F. See Baillie and McMahon (1989), p. 166; also Harrison, Michael and Patrick Waldron (2011), Mathematics for Economics and Finance, London: Routledge, p. 356ff. and p. 362ff.

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OVERSHOOTING

Assume: foreign & domestic bonds perfect substitutes, +UIP & short run: sticky goods prices & long run: %P=%M and PPP. A monetary contraction at t=0 (a) Short run: P constant, M

P , R

(b) Long run: P, so (i) M

P & R  initial values &

(ii) by PPP, S (c) Assume rat. exps: Se in SR & LR. R in SR only, during which S overshoots its LR (PPP) value. t=0 t R(t) R R0 t=0 t S S1 S0 S(t)

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While RR0, depreciation of S after t=0 offsets the additional interest, UIP holding at each t. From UIP, R + eS

S =R$

Se is the LR equilibrium value, by rational expectations, and R$ is given. At each step in the SR, R is above its LR value. R rises initially, capital inflows occur, S rises above Se, and the expected depreciation of S

• ffsets the higher R.

As P falls, R falls along a path to its original value, and at each stage is accompanied by a lower S, so the expected depreciation is lower. This may be seen alternatively using a version of UIP in log format: s  se=RR$.

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The efficient set with exchange-rate risk 2 risky securities, European & US Real returns: Investors IRL US Securities: € (1) RǏ (3) R+xǏ$ $ (2) R$xǏ (4) R$Ǐ$ [x  eS S ]

•  asset, E(real return) differs  investor
• unless PPP,  Ǐ$=Ǐ+x,  (1)=(3), (2)=(4)
• O/W suppose Ǐ$=Ǐ+x+e, e random with

E(e)=0 & payoff is (1) RǏ (3) RǏe (2) R$Ǐ$+e (4) R$Ǐ$ Then a given asset has: *Same E(Real return)  investors *More risk for foreign investor.

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Intuition (& calculus) suggests that if the US asset becomes riskier from an IRL perspective because of increased exchange-rate risk, the IRL investor’s optimal portfolio will involve less of the US asset. An equivalent conclusion applies to the IRL asset/US investor.

 100% in IRL IRL investor US investor E(Real return) 100% in US

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Possible explanations for home bias: financial & informational frictions

• departures from PPP;
• institutional barriers, capital controls, etc;
• transactions costs & taxes [but see Tesar & Werner:

most investors trade foreign shares more often than domestic: odd, if foreign=excess costs] Uppal: these are not enough. Other possibilities:

• benefits of diversification less than expected?
• inefficiency of small stock-markets?
• investors diversify via stock in multinationals? (but

these are more correlated with other domestic firms, than with foreign)

• unfamiliarity of foreign assets? (a 'transaction

cost'?)

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Do investors buy the foreign market portfolio? Kang & Stultz - foreign inv. in Japanese securities:

• mainly in large firms: 7% of top size quintile, 1¼%

bottom, N=1452 1975-1991;

• foreign owned averages 3.8% (equally weighted),

6.4% (value weighted);

• i.e. not the Japanese market PF;
• foreign PF has =5.4%, v. 4.8% for J. market PF.;
• foreign PF has no extra return.

Product market frictions Obstfeld & Rogoff: start by considering the 'traded '

• v. 'non traded' classification of goods. (T, NT)

Significance: equity claims on either type of firm can be traded, but earnings must be redeemed in traded goods.

• so hold a globally diversified PF, but stock in "NT"

industries is all held domestically.

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• i.e. home bias: domestic investors hold all the "NT"

stock, & international diversification only applies to the rest of their PFs. Doesn't explain observed biases. For O&R: goods occupy a place in the continuum: Traded v. Non-traded (v. dichotomy). Trade-costs explicity modelled,  model that predicts high & realistic level of home bias. Baxter & Jermann: to hedge human capital risk need short position in domestic traded assets, i.e. % of foreign equity should be higher. Heathcote & Perri: returns to domestic equity covary negatively with labour income, so domestic equity is a good hedge v. labour income risk: 'the international diversification puzzle is not as bad as you think'. Lane & Milesi-Ferretti (2008) [Our] most striking result is that bilateral equity investment is strongly correlated with the underlying pattern of trade in

• goods. Informational linkages, such as a common

language, are also important.