Bayesian evaluation and selection strategies in portfolio decision analysis

E. Vilkkumaa, J. Liesiö, A. Salo

EURO XXV, 8-11 July, Vilnius, Lituhania

The document can be stored and made available to the public on the open internet pages of Aalto University. All other rights are reserved.

Sports Illustrated cover jinx

• Apr 6, 1987: The Cleveland Indians – Predicted as the best team

in the American League– Would have a dismal 61–

101 season, the worst of any team that season

Sports Illustrated cover jinx

• Nov 17, 2003: The Kansas City Chiefs – Appeared on the cover after

starting the season 9-0– Lost the following game and

ultimately the divisional playoff against Indianapolis

Sports Illustrated cover jinx

• Dec 14, 2011: The Denver Broncos– Appeared on the cover after a

six-game win streak– Lost the next three games of the

regular season and ultimately the playoffs

Teams are selected to appear on the cover based on an outlier performance 

0 2 4 6 8 10 12 14 16 180

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True value v

Est

ima

te vE

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True value v

Ba

yes

est

ima

te vB

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Prior mean

Post-decision disappointment in portfolio selection

• Selecting a portfolio of projects is an important activity in most organizations

• Selection is typically based on uncertain value estimates vE

• The more overestimated the project, the more probably it will be selected

• True performance revealed → post-decision disappointment

= Selected project = Unselected project Size proportional to cost

Bayesian analysis in portfolio selection

• Idea: instead of vE, use the Bayes estimate vB=E[V|vE] as a basis for selection

• Given the distributions for V and VE|V, Bayes’ rule states

• E.g., V~N(μ,σ2), VE=v+ε, ε~N(0,τ2) → V|vE~N(vB,ρ2), where

f(V|VE) f(V)·f(VE|V) →

dvvvvfvVv )|(]|[E EEB

.,22

42

22

2

22

2

EB vv

Bayesian analysis in portfolio selection

• Portfolio selected based on vB

– Maximizes the expected value of the portfolio given the estimates– Eliminates post-decision disappointment

• Using f(V|VE), we can– Compute the expected value of additional information– Compute the probability of project i being included in the optimal

portfolio

Example

• 10 projects (A,...,J) with costs from 1 to 12 M$• Budget 25M$

• Projects’ true values Vi ~ N(10,32)

• A,...,D conventional projects– Estimation error εi ~ N(0,12)

– Moreover, B can only be selected if A is selected

• E,...,J novel, radical projects– More difficult to estimate: εi ~ N(0, 2.82)

Example cont’d

True value = 52Estimated value = 62

True value = 55Estimated value = 58

= Selected project = Unselected project Size proportional to cost

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True value v

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E

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True value vB

aye

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vB

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Prior meanPrior mean

Value of additional information

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.05

0

0.05

0.1

0.15

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0.25

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A

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CG

H E

Probability of being in the optimal portfolio

EV

I (M

EU

R)

F

• Knowing f(V|vE), we can compute – Expected value (EVI) of

additional information VE

– Probability that project i is included in the optimal portfolio

Probability of being in the optimal portfolio close to 0 or 1

EV

I fo

r si

ngle

pro

ject

re-

eval

uatio

n

= Selected project = Unselected project Size proportional to cost

Value of additional information

1 2 3 491

92

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97

Number of evaluation rounds

Por

tfolio

val

ue (

% o

f the

opt

imum

)

k=100

k=30

Best 30

Random 30

1 2 3 464

66

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70

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78

Number of evaluation rounds

Sha

re o

f cor

rect

cho

ices

(%

)

k=100

k=30

Best 30

Random 30

• Selection of 20 out of 100 projects

• Re-evaluation strategies1. All 100 projects

2. 30 projects with the highest EVI

3. ’Short list’ approach (Best 30)

4. 30 randomly selected projects

Conclusion

• Estimation uncertainties should be explicitly accounted for because of– Suboptimal portfolio value– Post-decision disappointment

• Bayesian analysis helps to– Increase the expected value of the selected portfolio– Alleviate post-decision disappointment– Obtain project-specific performance measures– Identify those projects of which it pays off to obtain additional

information