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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 Ã¢â‚¬â€œ
Decision Tree
Dr. Hao-Wei Chen
1
Modeling in Decision Analysis
Ã¢â‚¬Â¢ Modeling Decisions
Ã¢â‚¬Â¢ Elements of Decision Problems
Ã¢â‚¬Â¢ Modeling Uncertainty
Ã¢â‚¬Â¢ Modeling Preference
2
Basic Elements of Decision Problems
Ã¢â‚¬Â¢ When facing a difficult decision problem, it is important to first
thought through the problem and identify
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Values and objectives
Decisions
Uncertain events
Consequences
3
Value and Objectives
Ã¢â‚¬Â¢ Values Ã¢â‚¬â€œ things that matter to us
Ã¢â‚¬Â¢ Objectives – what we specifically want to achieve
Ã¢â‚¬Â¢ Most people do not make the effort or take the time to define values and
objectives explicitly, systematically and consciously.
Ã¢â‚¬Â¢ Focus on the available alternatives and try to find the best one.
Ã¢â‚¬Â¢ What Ã¢â‚¬Å“bestÃ¢â‚¬Â mean is not carefully defined.
4
A Special Objective Ã¢â‚¬â€œ Increase Monetary value
Ã¢â‚¬Â¢ In modern society, it seems like making money (increase monetary
value) is the objective for most of the decision problems.
Ã¢â‚¬Â¢ Why is that?
Ã¢â‚¬Â¢ Proxy objective
Ã¢â‚¬Â¢ Pricing out
5
A Special Objective Ã¢â‚¬â€œ Increase Monetary value
Ã¢â‚¬Â¢ Advantages to using money as an objective:
Ã¢â‚¬Â¢ Universally understood
Ã¢â‚¬Â¢ Measurable
Ã¢â‚¬Â¢ Simplifies trade-offs because it allows different objectives to be put on the same scale
via pricing out
6
A Special Objective Ã¢â‚¬â€œ Increase Monetary value
Ã¢â‚¬Â¢ Monetary values cannot be assigned to all values and objectives.
Ã¢â‚¬Â¢ Safety, family, lives, environment
Ã¢â‚¬Â¢ Can you name some others?
7
Decisions
Ã¢â‚¬Â¢ A decision exists if there are at least two alternatives
Ã¢â‚¬Â¢ Immediate decisions: Many situations have decisions that need to be made
right away.
Ã¢â‚¬Â¢ Identify these decisions immediately
Ã¢â‚¬Â¢ Some decisions are not urgent. These can wait.
Ã¢â‚¬Â¢ Identify alternatives
Ã¢â‚¬Â¢ Timing Ã¢â‚¬â€œ When does the decision have to be made?
Ã¢â‚¬Â¢ Information Ã¢â‚¬â€œ You may have to make the decision with less information than you would
like to have.
Ã¢â‚¬Â¢ Often a trade-off between timeliness and information
8
Sequential Decisions
Ã¢â‚¬Â¢ Sequential Decisions: Problems often involve multiple decisions to be made
over time, not all at once.
Ã¢â‚¬Â¢ Identify in what order they need to be made
Ã¢â‚¬Â¢ Which ones need to be made immediately?
Ã¢â‚¬Â¢ One decision may trigger another
Ã¢â‚¬Â¢ A decision may depend on what happened due to a prior decisions
Ã¢â‚¬Â¢ Called dynamic decision situations
9
Uncertain Event
Ã¢â€“Âª Uncertainty: decisions may have to be made without knowing exactly
what will happen in the future or exactly what the ultimate outcome
will be
Ã¢â‚¬Â¢ Less information that would prefer
Ã¢â€“Âª Not all future events are relevant. Determine which ones are; ignore
the rest.
Ã¢â‚¬Â¢ Relevant future events are those that will impact one or more of your
objectives.
Ã¢â‚¬Â¢ Information availability should not determine which future decisions you
focus on.
10
Dovetailing uncertain events
Ã¢â€“Âª Dovetailing: match relevant future uncertain events with the
Ã¢â‚¬Â¢ This is critical because it helps you to know at each decision exactly what
information is available and what remains unknown.
11
Consequence
Ã¢â€“Âª Each objective in the decision context will have a final consequence.
Ã¢â‚¬Â¢ Multiple objectives mean multiple consequences.
Ã¢â€“Âª Determine measures for each consequence so you can determine the
extent to which each objective was met.
Ã¢â‚¬Â¢ Monetary or non-monetary consequences?
Ã¢â‚¬Â¢ Can you price out any of the non-monetary consequences?
Ã¢â‚¬Â¢ What are the trade-offs between various objectives?
12
The Time value of Money
Ã¢â‚¬Â¢ A stream of cash flows is the most common consequences in personal and
Ã¢â‚¬Â¢ Ã¢â‚¬Å“A dollar today is worth more than a dollar tomorrow.Ã¢â‚¬Â
Ã¢â‚¬Â¢ Trade-offs between current dollars and future dollars
Ã¢â‚¬Â¢ Why?
Ã¢â‚¬Â¢ Due to interest rates
Ã¢â‚¬Â¢ What is the interest rate that you could get for investing your money in the next best
opportunity?
13
Good Deal or not?
Ã¢â‚¬Â¢ A friend wants you to invest his business
Ã¢â‚¬Â¢ You pays him \$425 now
Ã¢â‚¬Â¢ One year later, he will pay you \$110
Ã¢â‚¬Â¢ Two years later, he will pay you \$121
Ã¢â‚¬Â¢ Three years later, he will pay you \$133.10
Ã¢â‚¬Â¢ Four years later, he will pay you \$146.41
Ã¢â‚¬Â¢ Total \$510.51, which is greater than \$425
Ã¢â‚¬Â¢ Assume that a savings account at 10%, compounded
14
Net Present Value
N PV
=
x0
x1
xn
+
+ Ã‚Â·Ã‚Â·Ã‚Â· +
0
1
(1 + r )
(1 + r )
(1 + r ) n
n
=
i= 0
N PV
=
=
xi
(1 + r ) i
Ã¢Ë†â€™ 425 100.00 121.00 133.10 146.41
+
+
+
+
0
1
2
3
(1.1)
(1.1)
(1.1)
(1.1)
(1.1) 4
Ã¢Ë†â€™ 25
15
Use ExcelÃ¢â‚¬â„¢s NPV Function
Ã¢â‚¬Â¢ = -425 + NPV(0.1,B3:B6)
16
Example
Ã¢â‚¬Â¢ 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.
Ã¢â‚¬Â¢ Using a annual interest rate of 10%, find the net present value
17
Decision Tree
18
Example
Ã¢â‚¬Â¢ 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.
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
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
Ã¢â‚¬Â¢ Decision Three
Ã¢â‚¬Â¢ From left to right (norm)
Ã¢â‚¬Â¢ Square : decision nodes
Ã¢â‚¬Â¢ Circle: chance nodes
20
Example 1 Ã¢â‚¬â€œ Decision Tree
success
Invest?
Y
\$100k
0.3
fail
0.7
\$0k
N
\$60k
21
Decision Tree Ã¢â‚¬â€œ Hurricane Example
22
Decision Tree Ã¢â‚¬â€œ Uncountable Event
23
Decision Tree Ã¢â‚¬â€œ common mistakes
Ã¢â‚¬Â¢ Out of sequence/ordering
Ã¢â‚¬Â¢ Probability for each chance node is not sum to 1
Ã¢â‚¬Â¢ Probability for each chance node is not correct
Ã¢â‚¬Â¢ Are two events independent?
Ã¢â‚¬Â¢ If not, what are condition probabilities?
24
Solving A Decision Tree
Ã¢â‚¬Â¢ We construct a tree from left to right (from a root to several leafs)
Ã¢â‚¬Â¢ We solve a tree from right to left (solving sub-tree first)
Ã¢â‚¬Â¢ If the sub-tree is based on a decision node
Ã¢â‚¬Â¢ Choose the alternative with a higher expected value*
*Assume the objective is to maximize the payoff
Ã¢â‚¬Â¢ If the sub-tree is based on a chance node
Ã¢â‚¬Â¢ 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
Ã¢â‚¬Â¢ 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
Ã¢â‚¬Â¢ Repeatability is an abstract concept; Many decisions are
one-time decisions
Ã¢â‚¬Â¢ EMVs is convenient, but it can lead to decisions that may
not seem intuitively appealing
Ã¢â‚¬Â¢ 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
Invest?
Y
N
0.3
\$180k
0.5
fail
0.7
\$0k
0.5
\$30k
N
\$31k
30