Page 119 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes 0
1 0.7 0.3
or [s s ] =
A B 0.2 1 0.8 0
0.3 0.3 0
or [s s ] =
A B 0.2 0.2 0

Note that I – P is a singular matrix and hence, effectively, there is only one equation, given by
0.3s – 0.2s = 0.
A B
In order to find s and s , we need another equation. This equation is provided by the fact that the
A B
sum of market shares is unity i.e. s + s = 1. Thus, solving 0.3s – 0.2s = 0 and s + s = 1,
A B A B A B
simultaneously, we get the equilibrium values of the market shares s and s . In the above
A B
example, these values are s = 0.4 i.e. 40% and s = 0.6, i.e. 60%.
A B
Example
The price of an equity share of a company may increase, decrease or remain constant on any
given day. It is assumed that the change in price on any day affects the change on the following
day as described by the following transition matrix:
Change Tomorrow
Increase Decrease Unchange

Increase 0.5 0.2 0.3
Change Today Decrease 0.7 0.1 0.2
Unchange 0.4 0.5 0.1

(i) If the price of the share increased today, what are the chances that it will increase, decrease
or remain unchanged tomorrow?
(ii) If the price of share decreased today, what are the chances that it will increase tomorrow?

(iii) If the price of the share remained unchanged today, what are the chances that it will
increase, decrease or remain unchanged day after tomorrow?
Solution:

(i) Given that the price of the share has increased today, the probability of its going up
(today) is 1 and probability of each of events, decreasing or remaining unchanged is equal
to zero.

Thus, the initial state vector is R = [1 0 0]. Now the tomorrow’s state vector
0
0.5 0.2 0.3
R = 1 0 0 0.7 0.1 0.2 0.5 0.2 0.3
1
0.4 0.5 0.1

Hence, the chances that the price will rise, fall or remain unchanged tomorrow are 50%,
20%, 30% respectively (given that it has increased today).
(ii) The initial state vector is R = [0 1 0] and the chances of price increase tomorrow are 70%.
0
(iii) Here R = [0 0 1]
0

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