Problem 1.
Markov chains (steady state):
XYZ insurance company charges its customers according to their accident history. If you
have not had accidents the last two years will be charged for the new policy $ 1,730,000
(state 0); if you have had an accident in each of the last two years you will be charged $
2,280,000 (State 1); If you had accidents the first of the last two years you will be
charged $ 1,650,000 (state 2) and if you had an accident the second of the last two
years will be charged $ 1,670,000 (State 3). The historical behavior of each state is
given by the following cases of accident, taken in four different events.
Table 1. Historical accident data
According to Table 1 by applying the Markovian processes, ie finding the transition
matrix and solving the respective equations of p * q, where p is the transition matrix and
q the vector [W X Y Z]. Answer:
a. What is the transition matrix resulting from proportionality according to the
accident history?
b. What is the average premium paid by a customer in Payoff, according to historical
accident rate?
Transition probability
matrix
STATE E0 E1 E2 E3 TOTAL
E0 0,3 0,2 0,4 0,1 1
E1 0,1 0,3 0,5 0,1 1
E2 0,2 0,1 0,1 0,6 1
E3 0,3 0 0,2 0,5 1
What is the average premium paid by a customer in Payoff, according to
historical accident rate?
STATE E0 E1 E2 E3
E0 0,3 0,2 0,4 0,1
E1 0,1 0,3 0,5 0,1
E2 0,2 0,1 0,1 0,6
E3 0,3 0 0,2 0,5
STATE E0 E1 E2 E3
E0 0,22 0,16 0,28 0,34
E1 0,19 0,16 0,26 0,39
E2 0,27 0,08 0,26 0,39
E3 0,28 0,08 0,24 0,4
E0 E1 E2 E3
0,24 0,12 0,26 0,38
1580000 2150000 1630000 1550000
379200 258000 423800 589000
1650000
Accidents in the year
STATE E0 E1 E2 E3 TOTAL
E0 2961 1974 3948 987 9870
E1 1100 3300 5500 1100 11000
E2 2500 1250 1250 7500 12500
E3 4380 0 2920 7300 14600
a) With the last frequencies, we calculate the probabilities of
trasnsition, keeping in the same order in table (E0,E1,E2, E3) is:
0,3 0,2 0,4 0,1
0,1 0,3 0,5 0,1
P=
0,2 0,1 0,1 0,6
0,3 0 0,2 0,5
As we are taking two months as reference, we have to get
0,3 0,2 0,4 0,1
P
2 0,1 0,3 0,5 0,1
=
0,2 0,1 0,1 0,6
0,3 0 0,2 0,5
P2 0,3 0,2 0,4 0,1
0,1 0,3 0,5 0,1
0,2 0,1 0,1 0,6
0,3 0 0,2 0,5
2
=
0,22 0,16 0,28 0,34
P
0,19 0,16 0,26 0,39
0,27 0,08 0,26 0,39
0,28 0,08 0,24 0,4
a. What is the transition matrix resulting from proportionality according to the
accident history?
R/= This is the transition matrix resulting from proportionality according to the
accident history to the w, x,y,z.
b. What is the average premium paid by a customer in Payoff, according to
historical accident rate?
E0 E1 E2 E3
0,24 0,12 0,26 0,38
1.580.0 2.150. 1.630. 1.550.
00 000 000 000
379. 258. 423. 589.
200 000 800 000
Problem 2. Markov chains (Initial state multiplication):
In Colombia there are 5 main mobile operators such as Tigo, Comcel, Movistar, ETB
and Uff, which we will call states. The following chart summarizes the odds that each
client has to stay in their current operator or make a change of company.
Table 2. Probabilities of change and permanence in the company of
Telephony (Transition Matrix)
The current percentages of each operator in the current market are for Tigo 0.25 for
Comcel 0.2, for Movistar 0.3, for ETB 0.1 and 0.15 for Uff (initial state).
MOVISTA
STATE TIGO COMCEL R ETB UFF
TIGO 0,18 0,28 0,19 0,18 0,17
COMCEL 0,21 0,23 0,17 0,25 0,14
MOVISTA
R 0,19 0,16 0,23 0,26 0,16
ETB 0,18 0,19 0,23 0,21 0,19
UFF 0,22 0,23 0,19 0,17 0,19
Find the probability that each user stays with the mobile company for
the next period.
PROBABILITY 0
MOVISTA
TIGO COMCEL R ETB UFF
0,25 0,2 0,3 0,1 0,15
MOVISTA
STATE TIGO COMCEL R ETB UFF
TIGO 0,1971 0,2185 0,1992 0,2185 0,1667
COMCEL 0,1942 0,2186 0,2022 0,2158 0,1692
MOVISTA
R 0,1935 0,213 0,2064 0,2158 0,1713
ETB 0,1956 0,2145 0,2038 0,2161 0,17
UFF 0,1964 0,2209 0,1998 0,2145 0,1684
PROBABILITY 1
TIGO COMCEL MOVISTA ETB UFF
R
0,195 0,2175 0,202 0,2195 0,166
COMPANIES
STATE TIGO COMCEL MOVISTAR ETB UFF
E0 TIGO 0,18 0,28 0,19 0,18 0,17
E1 COMCEL 0,21 0,23 0,17 0,25 0,14
E2 MOVISTAR 0,19 0,16 0,23 0,26 0,16
E3 ETB 0,18 0,19 0,23 0,21 0,19
E4 UFF 0,22 0,23 0,19 0,17 0,19
C. Find the probability that each user stays with the mobile company for the next
period.
PROBABILITY 0
TIGO COMCEL MOVISTAR ETB UFF
0,25 0,2 0,3 0,1 0,15
PROBABILITY 1
P0 * T = P1 TIGO COMCEL MOVISTAR ETB UFF
P1 = NEXT
PERIOD 0,195 0,2175 0,202 0,2195 0,166
Problem 3. Markov chains (Initial state multiplication):
In Colombia there are 6 main mobile operators such as Avantel, Tigo, Comcel, Movistar,
ETB and Uff, which we will call states. The following chart summarizes the odds that
each client has to stay in their current operator or make a change of company.
Table 3. Probabilities of change and permanence in the company of
Telephony (Transition Matrix)
The current percentages of each operator in the current market are for Avantel 0.1, Tigo
0.15 for Comcel 0.15, for Movistar 0.35, for ETB 0.1 and 0.15 for Uff (initial state).
AVANT TIG COMC MOVIST
STATE EL O EL AR ETB UFF
AVANT
EL 0,19 0,18 0,17 0,15 0,19 0,12
TIGO 0,17 0,15 0,16 0,16 0,18 0,18
COMCE
L 0,16 0,19 0,17 0,17 0,16 0,15
MOVIST
AR 0,18 0,18 0,19 0,18 0,15 0,12
ETB 0,15 0,16 0,19 0,15 0,18 0,17
UFF 0,15 0,16 0,17 0,19 0,18 0,15
PROBABILITY 0
AVANT TIG COMC MOVIST
EL O EL AR ETB UFF
0,1 0,15 0,15 0,35 0,1 0,15
AVANT TIG COMC MOVIST
STATE EL O EL AR ETB UFF
AVANT 0,17 0,17 0,14
EL 0,1674 01 0,175 0,1645 4 9
0,16 0,17 0,14
TIGO 0,1662 99 0,1753 0,1667 37 82
COMCE 0,16 0,17 0,14
L 0,167 98 0,1747 0,1664 31 9
MOVIST 0,17 0,17 0,14
AR 0,1681 11 0,1748 0,1658 26 76
0,17 0,17 0,14
ETB 0,1656 01 0,175 0,1667 32 94
0,17 0,17 0,14
UFF 0,1666 03 0,1758 0,1667 24 82
PROBABILITY 1
AVANT TIG COMC MOVIST
EL O EL AR ETB UFF
0,17 0,16 0,14
0,169 2 0,1775 0,171 75 3
COMPANIES
COMCE
STATE AVANTEL TIGO L MOVISTAR ETB UFF
E0 AVANTEL 0,19 0,18 0,17 0,15 0,19 0,12
E1 TIGO 0,17 0,15 0,16 0,16 0,18 0,18
E2 COMCEL 0,16 0,19 0,17 0,17 0,16 0,15
E3 MOVISTAR 0,18 0,18 0,19 0,18 0,15 0,12
E4 ETB 0,15 0,16 0,19 0,15 0,18 0,17
E5 UFF 0,15 0,16 0,17 0,19 0,18 0,15
PROBABILITY 0
COMCE
AVANTEL TIGO L MOVISTAR ETB UFF
0,1 0,15 0,15 0,35 0,1 0,15
PROBABILITY 1
COMCE
P0 * T = P1 AVANTEL TIGO L MOVISTAR ETB UFF
P1 = NEXT
PERIOD 0,169 0,172 0,1775 0,171 0,1675 0,143
Problem 4. Markov chains (Initial
state multiplication):
Suppose that 4 types of soft drinks are obtained in the market: Colombian, Pepsi Cola,
Fanta and Coca Cola when a person has bought Colombian there is a probability that
they will continue to consume 40%, 20% of which will buy Pepsi Cola, 10% that Fanta
buys and 30% that Coca Cola consumes; when the buyer currently consumes Pepsi
Cola there is a probability that he will continue to buy 30%, 20% buy Colombiana, 20%
that Fanta consumes and 30% Coca Cola; if Fanta is currently consumed, the likelihood
of it continuing to be consumed is 20%, 40% buy Colombian, 20% consume Pepsi Cola
and 20% go to Coca Cola. If you currently consume Coca Cola the probability that it will
continue to consume is 50%, 20% buy Colombian, 20% that consumes Pepsi Cola and
10% that is passed to Fanta.
At present, each Colombian brand, Pepsi Cola, Fanta and Coca Cola have the
following percentages in market share respectively (30%, 25%, 15% and 30%)
during week 3.
c. Find the transition matrix.
d. Find the probability that each user stays with the mark or change to another for
period 4 (problem 4) and period 5 (problem 5).
PEPSI
STATE COLOMBIANA COLA FANTA COCA-COLA
COLOMBIANA 0,40 0,2 0,1 0,3
PEPSI COLA 0,2 0,3 0,2 0,3
FANTA 0,4 0,2 0,2 0,2
COCA-COLA 0,2 0,2 0,1 0,5
PEPSI
COLOMBIANA COLA FANTA COCA-COLA
0,30 0,25 0,15 0,3
PEPSI
STATE COLOMBIANA COLA FANTA COCA-COLA
COLOMBIANA 0,30 0,22 0,13 0,35
PEPSI COLA 0,28 0,23 0,15 0,34
FANTA 0,32 0,22 0,14 0,32
COCA-COLA 0,26 0,22 0,13 0,39
PEPSI
COLOMBIANA COLA FANTA COCA-COLA
0,29 0,225 0,14 0,345
PEPSI
STATE COLOMBIANA COLA FANTA COCA-COLA
COLOMBIANA 0,29 0,222 0,135 0,357
PEPSI COLA 0,286 0,223 0,138 0,353
FANTA 0,292 0,222 0,136 0,35
COCA-COLA 0,278 0,222 0,135 0,365
PEPSI
COLOMBIANA COLA FANTA COCA-COLA
0,29 0,2225 0,1365 0,355
COMPANIES
PEPSI COCA-
STATE COLOMBIANA COLA FANTA COLA
E0 COLOMBIANA 0,4 0,2 0,1 0,3
E1 PEPSI COLA 0,2 0,3 0,2 0,3
E2 FANTA 0,4 0,2 0,2 0,2
E3 COCA-COLA 0,2 0,2 0,1 0,5
PROBABILITY 0
PEPSI COCA-
COLOMBIANA COLA FANTA COLA
0,30 0,25 0,15 0,3
PROBABILITY 1
PEPSI COCA-
P0 * T = P1 COLOMBIANA COLA FANTA COLA
P1 = PERIOD 1 0,29 0,225 0,14 0,345
PROBABILITY 2
PEPSI COCA-
P1 * T = P2 COLOMBIANA COLA FANTA COLA
P2 = PERIOD 2 0,29 0,2225 0,1365 0,355
PROBABILITY 3
PEPSI COCA-
P2 * T = P3 COLOMBIANA COLA FANTA COLA
P23= PERIOD 3 0,2845 0,22225 0,1359 0,35735
PROBABILITY 4
PEPSI COCA-
P3 * T = P4 COLOMBIANA COLA FANTA COLA
P4 = PERIOD 4 0,28408 0,222225 0,135815 0,35788
Problem 5. Markov chains (Initial state multiplication):
Suppose you get 6 types of Jeans brands in the Colombian market: Brand 1, Brand 2,
Brand 3, Brand 4, Brand 5 and Brand 6. The following table shows the odds that you
continue to use the same brand or change it.
Table 4. Probabilities of change and permanence in the brand
At present, brand, have the following percentages in market share respectively (19%,
18%, 17%, 15%, 19% y 12%) during week 4.
a. Find the transition matrix.
b. Find the probability that each user stays with the mark or change to another for
period and period 5 (problem 5).
CONCLUSIONS
We learned to apply the strategies of how to evaluate a situation, when we don´t
know what is going to happen to a financial of a company. Ivan Gomez
We could understand the probabilities of occurring when a decision in taken. Ivan
Gomez
A strategy was used to could know what will it happen in the future, with the
clients or user of a service provided by a supplier. Ivan Gomez
