- Risky Assets 59
later chapters.
To begin with, we shall work out the dynamics of expected stock prices
E(S(n)).Forn=1
E(S(1)) =pS(0)(1 +u)+(1−p)S(0)(1 +d)=S(0)(1 +E(K(1))),where
E(K(1)) =pu+(1−p)d
is the expected one-step return. This extends to anynas follows.
Proposition 3.4
The expected stock prices forn=0, 1 , 2 ,...are given by
E(S(n)) =S(0)(1 +E(K(1)))n.Proof
Since the one-step returnsK(1),K(2),...are independent, so are the random
variables 1 +K(1),1+K(2),.... It follows that
E(S(n)) =E(S(0)(1 +K(1))(1 +K(2))···(1 +K(n)))
=S(0)E(1 +K(1))E(1 +K(2))···E(1 +K(n))
=S(0)(1 +E(K(1)))(1 +E(K(2)))···(1 +E(K(n))).Because theK(n) are identically distributed, they all have the same expecta-
tion,
E(K(1)) =E(K(2)) =···=E(K(n)),
which proves the formula forE(S(n)).
If the amountS(0) were to be invested risk-free at time 0, it would grow to
S(0)(1 +r)nafternsteps. Clearly, to compareE(S(n)) andS(0)(1 +r)nwe
only need to compareE(K(1)) andr.
An investment in stock always involves an element of risk, simply because
the priceS(n) is unknown in advance. A typical risk-averse investor will re-
quire thatE(K(1))>r, arguing that he or she should be rewarded with a
higher expected return as a compensation for risk. The reverse situation when
E(K(1))<rmay nevertheless be attractive to some investors if the risky re-
turn is high with small non-zero probability and low with large probability.
(A typical example is a lottery, where the expected return is negative.) An
investor of this kind can be called a risk-seeker. We shall return to this topic