International macroe.. - Free

6.6. NOISE-TRADERS 195and the foreign agent must convert wealth to euros. We also assumethat the price level in each country is Þxed at unity. Individuals thereforeevaluate wealth in national currency units. The portfolio problemis to decide whether to borrow the local currency and to lend uncoveredin the foreign currency or vice-versa in an attempt to exploit deviationsfrom uncovered interest parity, as described in chapter 1.1.The domestic young decide whether to borrow dollars and lend eurosor vice versa. Let λ t be the dollar value of the portfolio position taken.If the home agent borrows dollars and lends euros the individual hastaken a long euro positions which we represent with positive values ofλ t . To take a long euro position, the young trader borrows λ t dollars atthe gross interest rate R t and invests λ t /S t euros at the gross rate R ∗ t .When old, the euro payoff Rt ∗ (λ t /S t ) is converted into (S t+1 /S t )Rt ∗ λ tdollars. If the agent borrows euros and lends dollars, the individualhas taken a long dollar position which we represent with negative λ t .A long position in dollars is achieved by borrowing −λ t /S t euros andinvesting the proceeds in the dollar asset at R t . In the second period,the domestic agent sells −(S t+1 /S t )Rt ∗ λ t dollars in order to repay theeuro debt −Rt ∗ (λ t /S t ). In either case, the net payoff is the numberof dollars at stake multiplied by the deviation from uncovered interestparity, [(S t+1 /S t )R ∗ t − R t ]λ t . We use the approximations (S t+1 /S t ) '(1 + ∆s t+1 )and(R t /Rt ∗ )=(F t /S t ) ' 1+x t to express the net payoffas 12 [∆s t+1 − x t ]Rt ∗ λ t. (6.47)The foreign agent’s portfolio position is denoted by λ ∗t with positivevalues indicating long euro positions. To take a long euro position, theforeign young borrows λ ∗t dollars and invests (λ ∗t /S t ) euros at the grossinterest rate R ∗ t . Next period’s net euro payoff is (Rt ∗ /S t − R t /S t+1 )λ ∗t .A long dollar position is achieved by borrowing −(λ ∗t /S t ) euros andinvesting −λ ∗t dollars. The net euro payoff in the second period is−(R t /S t+1 − R ∗ t /S t )λ ∗t . Using the approximation (F t S t )/(S t S t+1 ) '12 These approximations are necessary in order to avoid dealing with Jenseninequality terms when evaluating the foreign wealth position which render themodel intractable. Jensen’s inequality is E(1/X) > 1/(EX). So we have[(S t+1 /S t )R ∗ t − R t]λ t =[(1 + ∆s t+1 )R ∗ t − (1 + x t)R ∗ t ]λ t, which is (6.47).