A house holder is currently considering insuring the contents of his house against
theft for one year.
He estimated the contents of his house would cost him £20,000 to replace.
Local crime statistics indicates that there is a probability of 0.03 that his house
will be broken into in the coming year.
In that event his losses will be 10%, 20% or 40% of the contents with
probabilities 0.5, 0.35 and 0.15 respectively.
An insurance policy from Company A costs £150 a year but guarantees to
replace any loss due to the theft.
An insurance policy from Company B will be cheaper at £100 a year but the
householder has to pay the first £x of any loss himself.
An insurance policy from Company C is even more cheaper at £75 a year
but only replaces a fraction of any loss (y%) suffered.
Assume that there can be at most one theft a year.
Draw a decision tree and give your advice to the house holder if x=50, y=40 and
his objective is to maximize the expected monitory value(EMV)?
0.97
150 £
11
3
0.5
0.03
7
12
0.35
13
0.15
14
A
IF INSURED
WHICH 2
COMPANY?
B
0.97
100 £
C
4
0.5
0.03
FINAL 1
DECISION
0.35
8
0.5
9
0.35
0.15
21
22
B
C
NI
23
6
0.5
0.03
A
18
20
0.97
IF NOT INSURED
17
19
5
0.03
16
0.15
0.97
75 £
15
10
0.35
0.15
24
25
26
T - NODE
A
B
C
NI
CALCULATION 0F PROFIT
PROFIT
PROB
T - EMV
11
COST OF INSURING =
(-150)
0.97
(-145.5)
12
2,000 – 150 =
1,850
0.5
925
13
4,000 – 150 =
3,850
0.35
1,347.5
14
8,000 – 150 =
7,850
0.15
1,177.5
15
COST OF INSURING =
(-100)
0.97
(-97)
16
2,000 – 100 – 50 =
1,850
0.5
925
17
4,000 – 100 – 50 =
3,850
0.35
1,347.5
18
8,000 – 100 – 50 =
7,850
0.15
1,177.5
19
COST OF INSURING =
(-75)
0.97
(-72.75)
20
2,000 – [2,000 x (40/100)] – 75 =
1,125
0.5
562.5
21
4,000 – [4,000 x (40/100)] – 75 =
2,325
0.35
813.75
22
8,000 – [8,000 x (40/100)] – 75 =
4,725
0.15
708.75
23
NOT INSURED & NO THEFT =
0
0.97
0
24
20,000 x (10/100) =
(-2,000)
0.5
(-1,000)
25
20,000 x (20/100) =
(-4,000)
0.35
(-1,400)
26
20,000 x (40/100) =
(-8,000)
0.15
(-1,200)
C - NODE
A
B
C
NI
CALCULATION
C - EMV
7
925 + 1,347.5 + 1,177.5 =
3,450
8
925 + 1,347.5 + 1,177.5 =
3,450
9
562.5 + 813.75 + 708.75 =
2,085
10
(-1,000) + (-1,400) + (-1,200) =
3
(3,450 x .03) + (-145.5) =
4
(3,450 x .03) + (-97) =
5
(2,085 x .03) + (-72.75) =
(-10.2)
6
(3,600 x .03) + 0 =
(-108)
(-3,600)
(-42)
6.5
The best choice that the house holder can make is to insure the contents of the
household with the company B, which gives him the highest EMV of 6.5. In such
case if there happens to be a theft at his house he will be safe even though he has
to bear an amount of £50 on his own. This insurance policy comes with a premium
of £100.