- Risky Assets 53
Remark 3.3
If the stock pays a dividend of div(n) at timenand this is reflected in the price
S(n), then the following version of the logarithmic return should be used:
k(n)=lnS(n)+div(n)
S(n−1)Consecutive one-step logarithmic returns can be combined in an additive
manner to find the return during the overall time period.
Exercise 3.8
For the data in Example 3.2 find the random variablesk(1),k(2) and
k(0,2). Comparek(0,2) withk(1) +k(2).Proposition 3.2
If no dividends are paid, then
k(n, m)=k(n+1)+k(n+2)+···+k(m).Proof
On the one hand,
S(m)=S(n)ek(n,m)
by the definition of the logarithmic return. On the other hand, using one-step
logarithmic returns repeatedly, we obtain,
S(m)=S(n)ek(n+1)ek(n+2)···ek(m)=S(n)ek(n+1)+k(n+2)+···+k(m).The result follows by comparing these two expressions.
3.1.2 Expected Return
Suppose that the probability distribution of the returnKover a certain time
period is known. Then we can compute the mathematical expectationE(K),
called theexpected return.
Example 3.5
We estimate the probabilities of recession, stagnation and boom to be 1/4,
1 /2, 1/4, respectively. If the predicted annual returns on some stock in these