94 Mathematics for Finance
5.2 Two Securities
We begin a detailed discussion of the relationship between risk and expected
return in the simple situation of a portfolio with just two risky securities.
Example 5.2
Suppose that the prices of two stocks behave as follows:
Scenario Probability ReturnK 1 ReturnK 2
ω 1 0. 5 10% −5%
ω 2 0. 5 −5% 10%If we split our money equally between these two stocks, then we shall earn 5%
in each scenario (losing 5% on one stock, but gaining 10% on the other). Even
though an investment in either stock separately involves risk, we have reduced
the overall risk to nil by splitting the investment between the two stocks. This is
a simple example of diversification, which is particularly effective here because
the returns are negatively correlated.
In addition to the description of a portfolio in terms of the number of shares
of each security held (developed in Section 4.1), we shall introduce another very
convenient notation to describe the allocation of funds between the securities.
Example 5.3
Suppose that the prices of two kinds of stock areS 1 (0) = 30 andS 2 (0) = 40
dollars. We prepare a portfolio worthV(0) = 1,000 dollars by purchasing
x 1 =20sharesofstocknumber1andx 2 = 10 shares of stock number 2. The
allocation of funds between the two securities is
w 1 =30 × 20
1 , 000
= 60%,w 2 =10 × 40
1 , 000
= 40%.
The numbersw 1 andw 2 are called theweights. If the stock prices change to
S 1 (1) = 35 andS 2 (1) = 39 dollars, then the portfolio will be worthV(1) =
20 ×35 + 10×39 = 1,090 dollars. Observe that this amount is no longer split
between the two securities as 60% to 40%, but as follows:
20 × 35
1 , 090∼= 64 .22%,^10 ×^39
1 , 090
∼= 35 .78%,
even though the actual number of shares of each stock in the portfolio remains
unchanged.