3.3 A Comparative Static Analysis
3.3.3 Changing Foreign Investment Risks
In the previous chapter we examined the adjustment of a portfolio set up under the new tax condition. In this chapter we study how the investor behaves when the investment risk in the foreign country rises e.g. due to political or economic turmoil.
The fraction invested into the foreign asset is given by (see 3.22) A2min = (1−T)2σ21
(1−T)2σ22+σ12. (3.33)
A change of the return risk of asset 2 in the foreign country influences the optimal fraction according to the first derivative of equation 3.33 with respect to the varianceσ22
∂A2min
∂σ22 =− σ12(1−T)4
[σ12+σ22(1−T)2]2 <0. (3.34)
This derivative is negative what means, that a higher foreign risk leads to smaller investments in the foreign country. What we want to know is how a transactions tax influences this restructuring of the portfolio. Therefore we take the first derivative of equation 3.34 with respect to the tax rate T that
is ∂A2min
∂σ22
∂T =− 4σ14(T −1)3
[σ21+σ22(T −1)2]3 >0. (3.35)
This derivative is positive, since T is smaller than 1. The conclusion is the following:
A higher tax rate raises the first derivative of the foreign fraction with respect to the assumed foreign risk. Since this derivative (equation 3.34) is negative,
its value approximates 0 what means that the restructuring of the portfolio under transactions taxes is lower, since less money will be pulled out of the foreign country, if foreign investment risk goes up.
4 Summary and Conclusions
In this article we examined the effect of transactions taxes on the investor’s portfolio choice. We concentrated on the case of two uncorrelated assets, one available in the home country and the other in a foreign country. Short-selling was not allowed. We found out, that the opportunity set in theµr-σ2r-range is still a parabola with its efficient frontier at the ascending part. We presumed a very risk averse investor reaching to minimize risks. Only in the case of totally correlated assets the ratio between the fractions are independent from the tax rate. Otherwise decreases the investment in the domestic asset with increasing tax rate. Uncorrelated assets assure that diversification without short-selling one asset takes place.
We distinguished two examinations: The portfolio adjustment due to the tax levy or changing tax rates, and the adaptation due to changes in assumed investment risks. For low tax rates the fraction invested in the foreign as-set would increase by adjusting the portfolio due to the tax levy. Hence, as a temporary effect of adjustment the transaction volume on the foreign exchange market would increase and is the opposite effect of what the pro-ponents of the Tobin tax intend. The transactions tax lowers the portfolio risk without necessarily lowering the return. In contrast, a transactions tax has a stabilizing effect when the investment risk abroad increases, since the fraction of the foreign asset would be shifted less to adjust the portfolio.
Further research would be generalizing the approach to correlated assets, introducing a specific utility function of the representative investor to char-acterize his risk aversion and allow riskless lending and borrowing.
Appendix
A1
To delineate the possibilities curve we take the expressions for the two frac-tions (equafrac-tions 3.7 and 3.8)
A1 = µr+T −r¯2(1−T)
¯
r1−r¯2(1−T) +T and
A2 = (¯r1−µr)(1−T)
¯
r1−r¯2(1−T) +T
and plug them into the equation for the portfolio variance (2.4) σr2 =A21σ21+A22σ22+ 2σ12A1A2.
This yields
σ2r = [µr+T −r¯2(1−T)]2σ12 [¯r1 −r¯2(1−T) +T]2
+ [(¯r1−µr)(1−T)]2σ22 [¯r1−r¯2(1−T) +T]2
+ 2σ12
[µr+T −r¯2(1−T)][(¯r1−µr)(1−T)]
[¯r1−¯r2(1−T) +T]2
or
σr2 = 1
[¯r1−r¯2(1−T) +T]2{σ22(1−T)2(¯r21+µ2r−2¯r1µr)
+ σ12[µ2r+T2+ ¯r22(1−T)2+ 2µrT −2µrr¯2(1−T)−2T¯r2(1−T)]
+ 2σ12(1−T)[−µ2r+µr(¯r1−T + (1−T)¯r2) +T¯r1−(1−T)¯r1¯r2]}.
Since we want to display σ2r against µr we can rewrite
σ2r = 1
[¯r1−r¯2(1−T) +T]2{µ2r[σ21+σ22(1−T)2−2σ12(1−T)]
− 2µr[−σ12T +σ12r¯2(1−T) +σ22(1−T)2r¯1−σ12(1−T)(¯r1−T + (1−T)¯r2]
+ σ12[T2+ ¯r22(1−T)2−2T¯r2(1−T) +σ22(1−T)2r¯12]
+ 2σ12(1−T)¯r1[T −(1−T)¯r2]}. (A.1)
We define the term following µ2r in equation A.1
a=σ12+σ22(1−T)2−2σ12(1−T)
and the term following−2µr
b =−σ12T +σ12r¯2(1−T) +σ22(1−T)2¯r1−σ12(1−T)(¯r1−T + (1−T)¯r2
and the last two lines of equation A.1
c=σ12[T2+¯r22(1−T)2−2Tr¯2(1−T)+σ22(1−T)2r¯12]+2σ12(1−T)¯r1[T−(1−T)¯r2.
By defining
A = a
[¯r1−r¯2(1−T) +T]2
B = b a C = c a
we can rewrite
σr2 = A[µ2r−2µrB+C]
= A[µr−B]2+A(C−B2)
= A[µr−B]2+D (A.2)
with
D=A(C−B2).
Equation A.2 is a parabola in the µr-σr2-range.
A2
The condition for real diversification without short-selling is equation 3.19:
A1min ≥0 and A2min ≥0.
Together with equations 3.15 and 3.16 it yields (1−T)[σ22(1−T)−σ12
(1−T)[σ22(1−T)−2σ12] +σ12 ≥ 0 (1−T)[σ12−(1−T)σ12]
(1−T)[σ22(1−T)−2σ12] +σ12 ≥ 0.
We first ignore the denominators and concentrate on the numerators, which are positive for
σ22(1−T)≥σ12 and σ21 ≥(1−T)σ12.
The last conditions divided by σ1σ2 and with ρ12= σσ112σ2 can be rewritten as ρ12≤min{(1−T)σ2
σ1; 1 (1−T)
σ1 σ2}
and is always fulfilled for σ12= 0.
This condition includes that the collective denominator (1−T)[σ22(1−T)− 2σ12] +σ21 is positive, because solving for 2σ12 and dividing by σ1σ2 gives
2ρ12≤(1−T)σ2
σ1 + 1 (1−T)
σ1 σ2.
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