Expected Values

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What is an expected value?

Expected values are used in decision making when there are many possible outcomes and each outcome has a probability attached to it.

It is an expected value/outcome if the decision will be made.

E.V. = Σ (Each possible outcome x Probability)

Illustration

An ice cream salesman has varying levels of demand for his sales and probabilities for each level of demand occurring. (Table below)

What is the expected value of revenue for the ice cream salesman?

Demand Sales ($) Probability
High 500,000 0.2
Medium 300,000 0.5
Low 200,000 0.3
1

Solution

E.V. = Σ (Each possible outcome x Probability)

=(500,000 x 0.2) + (300,000 x 0.5) + (200,000 x 0.3)
= 310,000

Limitations of E.V.

The expected value assumes that the decision will be taken many times.

Therefore:

  • This is not good for one off decisions, because either the demand will be high, medium or low at any time and will have those corresponding sales - the expected value will show what you will get over the long term.

  • It ignores the investor's attitude to risk, as it is an average of all of the outcomes occurring based on their probability.

  • It ignores the range of all possible outcomes in between our high/medium and low demand

  • It is heavily dependent on the probability %.

Illustration - Joint Probabilities

Overdraft Limit 4,000
Overdraft at Start 2,000
Forecast Cash Flows Period 1 Probability Period 2 Probability
10,000 20% 8,000 40%
6,000 40% 5,000 50%
-5,000 40% -2,000 10%
Closing balance at the end of P1 and P2 using EV -2000 (at start) + (10,000 x 20%) + (6,000 x 40%) + (-5,000 x 40%) = 2,400 -2,000 (at start) + (8,000 x 40%) + (5,000 x 50%) + (-2.000 x 10%) = 5,500
Therefore my EV -,2000 + 2,400 (P1) + 5,500 (P2) = 5,900
What is the chance of a negative cash balance at the end of P2?
Option 1  10,000 (P1) 8,000 (P2) -2,000 (opening OD) 16,000
Option 2  10,000 (P1) 5,000 (P2) -2,000 (opening OD) 13,000
Option 3 10,000 (P1) (2,000) (P2) -2,000 (opening OD) 6,000
Option 4 6,000 (P1) 8,000 (P2) -2,000 (opening OD) 12,000
Option 5 6,000 (P1) 5,000 (P2) -2,000 (opening OD) 9,000
Option 6 6,000 (P1) (2,000) (P2) -2,000 (opening OD) 2,000
Option 7 (5,000) (P1) 8,000 (P2) -2,000 (opening OD) 11,000
Option 8 - NEGATIVE BALANCE (5,000) (P1) (40%) 5,000 (P2)  (50%) -2,000 (opening OD) (2,000)  Chances of this are 40% x 50% = 20%
Option 9 - NEGATIVE BALANCE (5,000) (P1) (40%) (2,000) (P2) (10%) -2,000 (opening OD) (9,000) Chances of this are 40% x 10% = 4%
24% is the chance of having a negative cash balance at the end of P2
What is the probability of going over the OD at the end of P2?  Only in Option 9 - so 4%
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