Transcript VI-Diversification - University of Cambridge

Risk Management & Real Options
VI. Diversification
Stefan Scholtes
Judge Institute of Management
University of Cambridge
MPhil Course 2004-05
Have a go at diversification
Example courtesy of Sam Savage:
Safe investment
Probability Outcome
40%
-10
60%
50
Expected value= 26
Risky investment Probability Outcome
60%
-10
40%
80
Which of the following portfolios minimizes the probability of loosing
money?
•
100% safe
•
90% safe, 10% risky
•
20% safe, 80% risky
•
None of the above
2 September 2004
© Scholtes 2004
Page 2
What does diversification mean?

Instead of investing all your money in one uncertain payoff, invest it in
several ones
•
Rolling 1 die: 1:6 chance of each number between 1 and 6
•
Rolling 2 dice: (sum of two random numbers)
2 September 2004
̵
2 can only occur as 1+1  1:36 chance
̵
7 occurs in every row and has a 6:36=1:6 chance
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
© Scholtes 2004
Each cell has
1:36 chance
Page 3
Towards the normal curve…
# cells in the
matrix
2
3
4
5
6
7
8
9
10
11
12
• Extremes (low or high) numbers can only occur if BOTH
dice show the same extreme (low or high) numbers
• Numbers in the middle have a higher chance of being realised
2 September 2004
© Scholtes 2004
Page 4
Driver for diversification: The central limit theorem



Central limit theorem: If we aggregate many independent random
effects, the aggregated random variable looks like a normal random
variable.
Mathematically: The sum of n independent random variables is
approximately normal
Mean of this normal = sum of the mean of the individual r.v.’s
Variance of this normal = sum of the variances of the individual r.v.’s
If we take the average of many independent variables with the same
variance, then the shape of the average becomes more and more
normal but also narrower and narrower as we increase the number of
random variables
X 1  ...  X n
Xn
X1
X1
V ( X1) V ( X1)
V(
)  V ( )  ...  V ( )  nV ( )  n

2
n
n
n
n
n
n
2 September 2004
© Scholtes 2004
Page 5
Important Caveat: Statistical Dependence


The statistical independence of the variables you aggregate is crucial for
diversification to work.
Statistical independence of two variables means, for all practical
matters, that there is no common driving source shared by the variables
•

Market uncertainties are often a driver of statistical dependence
•
•

E.g. Volumes of two oil wells can be thought of as independent random
variables, whilst the revenues generated by two oil wells are dependent (with
oil price being the common driver of the latter)
Exchange rate risk, fashion, oil price, etc.
Cannot be diversified away!
Uncertainties that are specific to one company are often called “private
risks”
•
Can be dealt with through diversification
2 September 2004
© Scholtes 2004
Page 6
Correlation
Risky
Safe


80
-10
50
p
0.6-p
0.6
-10
0.4-p
p
0.4
0.4
0.6
For which value of p are the two gambles statistically independent?
What if p=0? If I know that “safe” is higher than expected, then “risky”
MUST be lower than expected
•
•

Payoff
Negative correlation
One variable gives complete knowledge about the other  correlation
coefficient = -1
What if p=0.4?
•
If I know that “safe” is 50 then
̵
̵
2 September 2004
80 is more likely for “risky” (POSITIVE CORRELATION)
But there is still a 1/3 chance that I am wrong (Correlation coefficient = 1 – chance
of being wrong=66%)
© Scholtes 2004
Page 7
Correlation
Risky
Payoff
80
-10
50
0.4
0.2
0.6
-10
0
0.4
0.4
0.4
0.6
Safe

Which of the following portfolios minimizes the probability of loosing
money?
•
100% safe
•
90% safe, 10% risky
•
20% safe, 80% risky
•
None of the above
2 September 2004
© Scholtes 2004
Page 8
Portfolio optimization

Problem: Choose a portfolio of investments under budget constraints
•

Problem: Each portfolio has a shape as its value – some portfolios will
be optimal in some scenarios, others will be optimal in other scenarios
•

Chose which wells to drill if you have an exploration budget of $200 M
BUT: Need to make a decision before scenario is observed
Can’t we go with the portfolio with maximal expectation, assuming that
the size of the portfolio will take care of the risk (diversification
argument)?
•
•
Yes if the portfolio is large and the investments are independent
The latter is unlikely; e.g. wells depend on the movement of the oil price

Need to understand “residual risk” of portfolio

Want to maximise the “value” and minimise the “risk”
2 September 2004
© Scholtes 2004
Page 9
Portfolio optimization





Numbers have natural rank (maximal, minimal, etc), shapes don’t
Boil shapes into numbers
Measures for “value” e.g. expected value
Many possible measures for “risk” , e.g.
•
Variance (Expected squared deviation from expected value)
•
“Semi-variance” (Expected (squared) deviation below the expected value)
•
Probability of loosing money
•
10% value at risk
Two objectives: Maximise value and minimize risk
2 September 2004
© Scholtes 2004
Page 10
Portfolio optimization
Return
Risk-Return Profile
Risk
Which portfolios would you choose?
2 September 2004
© Scholtes 2004
Page 11
Portfolio optimisation - practicality


Fix return expectation and minimise risk, subject to return at least at
expectation and cost of portfolio not exceeding budget
Alternative: fix risk level and maximise return subject to portfolio does
not exceed fixed risk level and cost of portfolio does not exceed budget
2 September 2004
© Scholtes 2004
Page 12
Portfolio optimisation - technicalities
Return
Risk-return profile
Risk


Typically a huge number of possible portfolios e.g. if you have 20 wells to choose
from, the number of possible portfolios is 220 (>1,000,000)
Can use Excel “solver” to solve moderate size problems (e.g. choose from 10-20
wells)
•
see Decision Making with Insight for details on “solver” and other optimization software
2 September 2004
© Scholtes 2004
Page 13
Where are we?
I.
II.
III.
IV.
V.
VI.
Introduction
The forecast is always wrong
I.
The industry valuation standard: Net Present Value
II.
Sensitivity analysis
The system value is a shape
I.
Value profiles and value-at-risk charts
II.
SKILL: Using a shape calculator
III.
CASE: Overbooking at EasyBeds
Developing valuation models
I.
Easybeds revisited
Designing a system means sculpting its value shape
I.
CASE: Designing a Parking Garage I
II.
The flaw of averages: Effects of system constraints
Coping with uncertainty I: Diversification
I.
The central limit theorem
II.
The effect of statistical dependence
III.
Optimising a portfolio
2 September 2004
© Scholtes 2004
Page 14