portofolio efficient contruction

1. Portfolio Efficient Construction
Manajemen Investasi dan Portofolio

2. Portfolio Construction
• Where does portfolio construction fit in the portfolio
management process?
• What are the foundations of Markowitz’s Mean-Variance
Approach (Modern Portfolio Theory)? Two-asset to multiple
asset portfolios.
• How do we construct optimal portfolios using Mean Variance
Optimization? Microsoft Excel Solver.
2

3. Portfolio Construction
• How do we incorporate IPS requirements to determine asset
class weights?
• What are the assumptions and limitations of the mean-
variance approach?
• How do we reconcile portfolio construction in practice with
Markowitz’s theory?
3

4. Portfolio Construction within the larger context of
asset allocation
• IPS provides us with the risk tolerance and
return expected by the client
• Capital Market Expectations provide us with
an understanding of what the returns for each
asset class will be
4

5. Portfolio Construction within the larger context of
asset allocation
5
C1: Capital
Market Conditions
I1: Investor’s Assets,
Risk Attitudes
C2: Prediction
Procedure
C3: Expected Ret,
Risks, Correlations
I2: Investor’s Risk
Tolerance Function
I3: Investor’s Risk
Tolerance
M1: Optimizer
M2: Investor’s
Asset Mix
M3: Returns

6. Portfolio Construction within the larger context of
asset allocation
• Optimization, in general, is constructing the
best portfolio for the client based on the client
characteristics and CMEs.
• When all the steps are performed with careful
analysis, the process may be called integrated
asset allocation.
6

7. Mean Variance Optimization
• The Mean-Variance Approach, developed by
Markowitz in the 1950s, still serves as the foundation
for quantitative approaches to strategic asset
allocation.
• Mean Variance Optimization (MVO) identifies the
portfolios that provide the greatest return for a given
level of risk OR that provide the least risk for a given
return.
7

8. Mean Variance Optimization
• TO develop an understanding of MVO, we will
derive the relationship between risk and
return of a portfolio by looking at a series of
three portfolios:
– One risky asset and one risk-free asset
– Two risky assets
– Two risky assets and one risk-free asset
• We will then generalize our findings to
portfolios of a larger number of assets.
8

9. MVO: One risky and one risk-free asset
• For a portfolio of two assets, one risky (r) and one risk-
free (f), the expected portfolio return is defined as:
• Since, by definition, the risk-free asset has zero volatility
(standard deviation), the portfolio standard deviation is:
f
r
r
r
P R
w
R
E
w
R
E *
)
1
(
)
(
*
)
( 


r
r
P w 
 *

9

10. MVO: One risky and one risk-free asset
• With the portfolio return and standard deviation
equations, we can derive the Capital Allocation Line
(CAL):
• Notice that the slope of this line represent the Sharpe
ratio for asset r. It represents the reward-to-risk ratio for
asset r.
p
r
f
r
f
P
R
R
E
R
R
E 

*
]
)
(
[
)
(



10

11. MVO: One risky and one risk-free asset
• With one risky and one risk-free asset, an investor can
select a portfolio along this CAL based on his risk / return
preference.
11

12. MVO: Two risky assets
• With two risky assets (1 and 2), as long as the correlation
between the two assets is less than 1, creating a portfolio
with the two assets will allow the investor to obtain a
greater reward-to-risk ratio than either of the two assets
provide.
12

13. MVO: Two risky assets
• Portfolio expected return and standard deviation can be
calculated as follows:
12
2
1
2
1
2
2
2
2
2
1
2
1 2 




 w
w
w
w
P 


)
(
*
)
(
*
)
( 2
2
1
1 R
E
w
R
E
w
R
E P 

13
1
2 1 w
w 


14. MVO: Two risky assets
• Remember that the correlation coefficient can be calculated as:
Where
and n = number of historical returns used in the calculations.
2
1
2
,
1
12



Cov







n
i
i
i R
R
R
R
n
Cov
1
2
2
1
1
2
,
1 )
)(
(
1
1
14

15. MVO: Two risky assets
• These values (as well as asset returns and standard
deviations) can be easily calculated on a financial
calculator or Excel.
15

16. MVO: Two risky assets
• By altering weights in the two assets, we can construct a minimum-
variance frontier (MVF).
• The turning point on this MVF represents the global minimum
variance (GMV) portfolio. This portfolio has the smallest variance
(risk) of all possible combinations of the two assets.
• The upper half of the graph represents the efficient frontier.
16

17. MVO: Two risky assets
• The weights for the GMV portfolio is determined by the
following equations:
12
2
1
2
2
2
1
12
2
1
2
2
1
2 












w
1
2 1 w
w 

17

18. MVO: Two risky and one risk-free asset
• We know that with one risky asset and the risk-free asset,
the portfolio possibilities lie on the CAL.
• With two risky assets, the portfolio possibilities lie on the
MVF.
• Since the slope of the CAL represents the reward-to-risk
ratio, an investor will always want to choose the CAL with
the greatest slope.
18

19. MVO: Two risky and one risk-free asset
• The optimal risky portfolio is where a CAL is tangent to the
efficient frontier.
• This portfolio provides the best reward-to-risk ratio for the
investor.
• The tangency portfolio risky asset weights can be calculated as:
   
      2
,
1
2
1
2
1
2
2
2
1
2
,
1
2
2
2
1
1
*
)
(
)
(
*
)
(
*
)
(
*
)
(
*
)
(
Cov
r
R
E
r
R
E
r
R
E
r
R
E
Cov
r
R
E
r
R
E
w
f
f
f
f
f
f














19

20. MVO: All risky assets (market) and one risk-free
asset
• We can generalize our previous results by considering all
risky assets and one risk-free asset. The tangency
(optimal risky) portfolio is the market portfolio. All
investors will hold a combination of the risk-free asset
and this market portfolio.
• In this context, the CAL is referred to as the Capital
Market Line (CML).
20

21. Investor Risk Tolerance and CML
• To attain a higher expected return than is
available at the market portfolio (in exchange
for accepting higher risk), an investor can
borrow at the risk free-rate.
• Other minimum variance portfolios (on the
efficient frontier) are not considered.
21

22. Portfolio Possibilities Combining the Risk-Free Asset and Risky
Portfolios on the Efficient Frontier
22
p

)
E(R p
RFR
M

23. Assumptions / Limitations of Markowitz
Portfolio Theory
 Investors take a single-period perspective in
determining their asset allocation.
◦ Drawback: Investors seldom have a single-period
perspective. In a multiple-period horizon, even Treasury
bills exhibit variability in returns
◦ Possible Solutions:
 Include the “risk-free asset” as a risky asset class.
 If investors have a liquidity need, construct an efficient frontier and
asset allocation on the funds remaining after the liquidity need is
satisfied.
23

24. Assumptions / Limitations of Markowitz
Portfolio Theory
• Investors base decisions solely on expected return
and risk. These expectations are derived from
historical returns.
– Drawback: Optimal asset allocations are highly sensitive to
small changes in the inputs, especially expected returns.
Portfolios may not be well diversified.
– Potential solutions:
• Conduct sensitivity tests to understand the effect on asset
allocation to changes in expected returns.
24

25. Assumptions / Limitations of Markowitz
Portfolio Theory
 Investors can borrow and lend at the risk-free rate.
◦ Drawback: Borrowing rates are always higher than lending
rates. Certain investors are restricted from purchasing
securities on margin.
◦ Potential solutions:
 Differential borrowing and lending rates can be easily incorporated
into MVO analysis. However, leverage may be practically irrelevant
for many investors (liquidity, regulatory restrictions).
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26. Practical Application of MVO
• MVO can be used to determine optimal
portfolio weights with a certain subset of all
investable assets.
• An efficient frontier can be constructed with
inputs (expected return, standard deviation
and correlations) for the selected assets.
26

27. Practical Application of MVO
• MVO can be either unconstrained, in which
case we do not place any constraints on the
asset weights, or it can be constrained.
27

28. Practical Application of MVO
• Unconstrained Optimization
– The simplest optimization places no constraints on
asset-class weights except that they add up to 1.
– With unconstrained optimization, the asset
weights of any minimum variance portfolio is a
linear combination of any other two minimum
variance portfolios.
28

29. Practical Application of MVO
• Constrained Optimization
– The more useful optimization for strategic asset
allocation is constrained optimization.
– The main constraint is usually a restriction on
short sales.
29

30. Practical Application of MVO
• Constrained Optimization
– We can determine asset weights using the corner
portfolio theorem. This theorem states that the
asset weights of any minimum variance portfolio
is a linear combination of any two adjacent corner
portfolios.
– Corner portfolios define a segment of the efficient
frontier.
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31. Practical Application of MVO
• Excel Solver is a powerful tool that can be
used to determine optimal portfolio weights
for a set of assets.
• To use the tool, we need expected returns and
standard deviations for our assets as well as a
set of constraints that are appropriate for the
portfolio.
31

32. Readings for this week
◦ Chapters Seven and Eight in book
draft
◦ Online to the various sites that
define statistics like mean,
variance, standard deviation,
covariance and correlation

33. Harry Markowitz
As a graduate student in Economics
at University of Chicago in the 1950s,
Harry wanted to know how to optimally
construct a portfolio of stocks
In order to make any headway,
Harry had to decide how to describe
(define) a stock. So, what to do?

34. A Stock is a Probability
Distribution of Returns
according to Harry
Returns
mean

35.  
n
x
x
i
n
i



1

MEAN
MEAN

36. Correlation Coefficient
1,2  1
,2
1
2
-1 <= 1,2 <= 1

37. Standard Deviation

The square root of variance

38. Mean-Variance (Harry Markowitz, 1955)
• Each asset defined as:
– Probability distribution of returns
– Mean and Variance of the distribution known
– Covariances of returns between any two assets
are known
– Assume no riskless asset (all variances > 0)
• Portfolio is
– A collection of assets with a mean and a variance
that can be calculated
– Also an asset (no difference between portfolio and

39. Diagram with 2 Assets
Me
an
Standard Deviation =
√(Variance)
Asset 1 (μ1, σ1)
Asset 2 (μ2, σ2)

40. of assets 1 and 2
Me
an
Asset 1 (μ1, σ1)
Asset 2 (μ2, σ2)
Portfolio (μP, σP)
σ
Where should the portfolio be in
the diagram?
Asset 1 (μ1, σ1)
Asset 2 (μ2, σ2)
Portfolio (μP, σP)
σ

41. Investors will Choose some portfolio among those on the
efficient frontier
• Those who wish less risk choose portfolios
that are further to the left on the efficient
frontier. These portfolios are those with lower
mean and lower standard deviation
• Investors desiring more risk move to the right
along the efficient frontier in search of higher
mean, higher standard deviation portfolios

42. Portfolio Choice
Me
an
σ
σ
Less risk
More risk

43. Readings
• RB 7
• RB 8 (pgs. 229-239)
• RM 3 (5, 6.1.1 – 6.1.4)
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