- Portfolio Management 95
The weights are defined byw 1 =x 1 S 1 (0)
V(0),w 2 =x 2 S 2 (0)
V(0),
wherex 1 andx 2 are share numbers of stock 1 and 2 in the portfolio. This
means thatwkis the percentage of the initial value of the portfolio invested in
security numberk. Observe that the weights always add up to 100%,
w 1 +w 2 =x^1 S^1 (0) +x^2 S^2 (0)
V(0)=V(0)
V(0)
=1. (5.1)
If short selling is allowed, then one of the weights may be negative and the
other one greater than 100%.
Example 5.4
Suppose that a portfolio worthV(0) = 1,000 dollars is constructed by taking
a long position in stock number 1 and a short position in stock number 2 in
Example 5.3 with weightsw 1 = 120% andw 2 =−20%. The portfolio will
consist of
x 1 =w 1V(0)
S 1 (0)
= 120%×
1 , 000
30
=40,
x 2 =w 2 V(0)
S 2 (0)=−20%×^1 ,^000
40
=− 5
shares of type 1 and 2. If the stock prices change as in Example 5.3, then this
portfolio will be worth
V(1) =x 1 S 1 (1) +x 2 S 2 (1) =V(0)(
w 1S 1 (1)
S 1 (0)
+w 2S 2 (1)
S 2 (0)
)
=1, 000
(
120%×
35
30
−20%×
39
40
)
=1, 205
dollars, benefiting from both the rise of the price of stock 1 and the fall of
stock 2. However, a small investor may have to face some restrictions on short
selling. For example, it may be necessary to pay a security deposit equal to
50% of the sum raised by shorting stock number 2. The deposit, which would
amount to 50%×200 = 100 dollars, can be borrowed at the risk-free rate and
the interest paid on this loan will need to be subtracted from the final value
V(1) of the portfolio.