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QUESTION 1

You invest 500 in a fund this month. For the next following 5 months, you will receive 90, 100, 110, 120, 130, respectively. Suppose the monthly interest rate is 2%, what is the net present value for this investment?

-16.29

-16.61

16.61

16.29

1 points

QUESTION 2

Based on the decision tree on the last page of the slides, what is the expected monetary value if the decision-maker chooses to invest in the new business? Assume that we want to maximize the expected monetary value.

31.5

31

30

29

2 points

QUESTION 3

Based on the decision tree on the last page of the screen, should the decision-maker sells the new business? Assume that we want to maximize the expected monetary value.

True

False

1 points

QUESTION 4

Expected monetary value can take an individual risk attitude into consideration.

True

False

Decision Analysis –
Decision Tree
Dr. Hao-Wei Chen
1
Modeling in Decision Analysis
• Modeling Decisions
• Elements of Decision Problems
• Modeling Uncertainty
• Modeling Preference
2
Basic Elements of Decision Problems
• When facing a difficult decision problem, it is important to first
thought through the problem and identify
•
•
•
•
Values and objectives
Decisions
Uncertain events
Consequences
3
Value and Objectives
• Values – things that matter to us
• Objectives – what we specifically want to achieve
• Most people do not make the effort or take the time to define values and
objectives explicitly, systematically and consciously.
• Focus on the available alternatives and try to find the best one.
• What “best” mean is not carefully defined.
4
A Special Objective – Increase Monetary value
• In modern society, it seems like making money (increase monetary
value) is the objective for most of the decision problems.
• Why is that?
• Proxy objective
• Pricing out
5
A Special Objective – Increase Monetary value
• Advantages to using money as an objective:
• Universally understood
• Measurable
• Simplifies trade-offs because it allows different objectives to be put on the same scale
via pricing out
6
A Special Objective – Increase Monetary value
• Disadvantages:
• Monetary values cannot be assigned to all values and objectives.
• Safety, family, lives, environment
• Can you name some others?
7
Decisions
• A decision exists if there are at least two alternatives
• Immediate decisions: Many situations have decisions that need to be made
right away.
• Identify these decisions immediately
• Some decisions are not urgent. These can wait.
• Identify alternatives
• Timing – When does the decision have to be made?
• Information – You may have to make the decision with less information than you would
like to have.
• Often a trade-off between timeliness and information
8
Sequential Decisions
• Sequential Decisions: Problems often involve multiple decisions to be made
over time, not all at once.
• Identify in what order they need to be made
• Which ones need to be made immediately?
• One decision may trigger another
• A decision may depend on what happened due to a prior decisions
• Called dynamic decision situations
9
Uncertain Event
â–ª Uncertainty: decisions may have to be made without knowing exactly
what will happen in the future or exactly what the ultimate outcome
will be
• Less information that would prefer
â–ª Not all future events are relevant. Determine which ones are; ignore
the rest.
• Relevant future events are those that will impact one or more of your
objectives.
• Information availability should not determine which future decisions you
focus on.
10
Dovetailing uncertain events
â–ª Dovetailing: match relevant future uncertain events with the
decisions for your case.
• This is critical because it helps you to know at each decision exactly what
information is available and what remains unknown.
11
Consequence
â–ª Each objective in the decision context will have a final consequence.
• Multiple objectives mean multiple consequences.
â–ª Determine measures for each consequence so you can determine the
extent to which each objective was met.
• Monetary or non-monetary consequences?
• Can you price out any of the non-monetary consequences?
• What are the trade-offs between various objectives?
12
The Time value of Money
• A stream of cash flows is the most common consequences in personal and
business decisions.
• “A dollar today is worth more than a dollar tomorrow.”
• Trade-offs between current dollars and future dollars
• Why?
• Due to interest rates
• What is the interest rate that you could get for investing your money in the next best
opportunity?
13
Good Deal or not?
• A friend wants you to invest his business
• You pays him $425 now
• One year later, he will pay you $110
• Two years later, he will pay you $121
• Three years later, he will pay you $133.10
• Four years later, he will pay you $146.41
• Total $510.51, which is greater than $425
• Assume that a savings account at 10%, compounded
14
Net Present Value
N PV
=
x0
x1
xn
+
+ ··· +
0
1
(1 + r )
(1 + r )
(1 + r ) n
n
=
i= 0
N PV
=
=
xi
(1 + r ) i
− 425 100.00 121.00 133.10 146.41
+
+
+
+
0
1
2
3
(1.1)
(1.1)
(1.1)
(1.1)
(1.1) 4
− 25
15
Use Excel’s NPV Function
• = -425 + NPV(0.1,B3:B6)
16
Example
• A friend asks you for a loan of $1,000 and offers to pay you back at
the rate of $90 per months for 12 months.
• Using a annual interest rate of 10%, find the net present value
17
Decision Tree
18
Example
• The decision maker wants to decide whether to invest his $50k saving
in a new business to maximize his return on investment for a two
years period.
•
•
•
•
If the business succeeds (prob. 0.3), you will get $100k back.
If the business fails (prob. 0.7), you will get $0 back
If do nothing, you will have $60k in your saving after two years.
Assume all monetary values provided are NPV.
Identify the decision elements for the above question.
19
Construct A Decision Tree
• Decision Three
• From left to right (norm)
• Square : decision nodes
• Circle: chance nodes
20
Example 1 – Decision Tree
success
New Business
Invest?
Y
$100k
0.3
fail
0.7
$0k
N
$60k
21
Decision Tree – Hurricane Example
22
Decision Tree – Uncountable Event
23
Decision Tree – common mistakes
• Out of sequence/ordering
• Probability for each chance node is not sum to 1
• Probability for each chance node is not correct
• Are two events independent?
• If not, what are condition probabilities?
24
Solving A Decision Tree
• We construct a tree from left to right (from a root to several leafs)
• We solve a tree from right to left (solving sub-tree first)
• If the sub-tree is based on a decision node
• Choose the alternative with a higher expected value*
*Assume the objective is to maximize the payoff
• If the sub-tree is based on a chance node
• Calculate the expected value
25
Review: Expected Value
The weighted average of all possible values a random variable can take on.
What?
Sum over all
Possible
outcomes
x P(X = x)
How?
Value (outcome)
of the random
variable
How likely the
value would
occur
xP(X = x)
0
0.3
(0)(0.3) = 0
1
0.5
(1)(0.5) = 0.5
2
0.2
(2)(0.2) = 0.4
Sum over all
possible
values of X
E(X) = 0.9
E(X) = (0)(0.3) + (1)(0.5) + (2)(0.2) = 0.9
26
Calculate the Expected Value
• In a game, there is a 30% chance to lose 50 dollars and a 70% chance
to win 100 dollars. What is the expected payoff of this game?
Let X be the payoff of the game
x
P(x)
-50
0.3
100
0.7
E(X) = (-50)(0.3) + (100)(0.7) = -15 + 70 = 55.
27
Solving a Decision Tree – Example
28
Expected Monetary Values – Issues
• Repeatability is an abstract concept; Many decisions are
one-time decisions
• EMVs is convenient, but it can lead to decisions that may
not seem intuitively appealing
• Example
Game 1
Win $30 with probability 0.5
Lose $1 with probability 0.5
Game 2
Win $2000 with probability 0.5
Lose $1900 with probability
0.5
EMV(Game 1) = 14.50
EMV(Game 2) = 50.00
Which game do you prefer?
29
Quiz
Sale it
$100k
Y
success
New Business
Invest?
Y
N
0.3
$180k
0.5
fail
0.7
$0k
0.5
$30k
N
$31k
30

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