samlet årgang - Økonomisk Institut - Københavns Universitet

438
i i v12 1 + v212 + v313 (14)
1–t 1–t v32 3 + v113 + v223 r e 1 = + 1 re 3 – , 1
i i v222 + v112 + v323 (15)
1–t 1–t v3 2 3 + v113 + v223 r e 2 = + 2 re 3 – , 2
The variables 1 and 2 in (14) and (15) may be either smaller or greater than unity,
so a personal tax on the return to equity will not necessarily drive up the required
returns on unquoted shares, as already pointed out by Apel and Södersten (1999). To
provide some intuition for this result, consider an investor with a well-diversified portfolio
where unquoted shares carry a relatively small weight. In this case where v 1 → 0
and v 2 → 0, it follows from (14) and (15) that
i i 13 r e 1 = + 1 re 3 – , 1 1–t 1–t 2 3
i i 23 r e 2 = + 2 re 3 – , 2 1–t 1–t 2 3
NATIONALØKONOMISK TIDSSKRIFT 2005. NR. 3
(16)
(17)
The variance of the return on the »market portfolio« tends towards 2 3 when v1 → 0
and v2 → 0. Our variables 1 and 2 then become roughly equal to the »beta« known
from the conventional Capital Asset Pricing Model, as indicated in (16) and (17). For
unquoted shares with a relatively high covariance with the market portfolio (j > 1),
the required risk premium will be high, because the holding of such shares adds to the
aggregate risk on the investor’s portfolio. For such relatively risky shares the benefit
from the income insurance offered by a symmetric shareholder income tax (with full
loss offset) outweighs the fact that the tax reduces the average net rate of return relative
to the net return on safe assets. Accordingly, it follows from (16) and (17) that the
tax will reduce the required return on unquoted shares when j > 1. 2 Conversely, for
unquoted shares with a relatively low j ( < 1), the tax will drive up the required return.
A personal tax on the full return to shares will thus distort the pattern of risk-taking
and the cost of capital within the sector of small companies whose shares are not
traded on the international stock market.
2. The mean-variance model of Apel and Södersten (1999, p. 85) also implies that j3 / 2
3 > 1 (j = 1, 2) is a
necessary and sufficient condition for dr e j / dt < 0, when unquoted shares have a negligible weight in the investor’s
portfolio.