1. JOHN VON NEUMANN INSTITUTE
VIETNAM NATIONAL UNIVERSITY HOCHIMINH CITY
BLACK-LITTERMAN PORTFOLIO
OPTIMIZATION
Hoang Hai Nguyen
nguyen.hoang@jvn.edu.vn
HCM City, July- 2012
1
2. Abstract: In practice, mean-variance optimization results in non-intuitive and extreme
portfolio allocations, which are highly sensitive to variations in the inputs. Generally,
efficient frontiers based on historical data lead to highly concentrated portfolios. The
Black-Litterman approach overcomes, or at least mitigates, these problems to a large
extent. The highlight of this approach is that it enables us to incorporate investment
views (which are subjective in nature). These aspects make the Black-Litterman model a
strong quantitative tool that provides an ideal framework for strategic/tactical asset
allocation. In this project, we will apply the Black-Litterman model for the context of
VietNam equity markets. To represent the VietNam equity markets, we select top K = 10
most market capitalization stocks of the Ho Chi Minh city stock exchange with historical
data of at least 1 year for tactical asset allocation. As extension part, we empirically
compare the performance of the two approaches. The study found that the BlackLitterman efficient portfolios achieve a significantly better return-to-risk performance
than the mean-variance optimal approach/strategy.
2
3. 1. Introduction
Since publication in 1990, the Black-Litterman asset allocation model has gained wide
application in many financial institutions. As developed in the original paper, the BlackLitterman model provides the flexibility of combining the market equilibrium with
additional market views of the investor. The Black-Litterman approach may be contrasted
with the standard mean-variance optimization in which the user inputs a complete set of
expected returns 1 and the portfolio optimizer generates the optimal portfolio weights.
Because there is a complex mapping between expected returns and the portfolio weights,
and because there is no natural starting point for the expected return assumptions, users
of the standard portfolio optimizers often find their specification of expected returns
produces output portfolio weights which do not seem to make sense. In the BlackLitterman model the user inputs any number of views, which are statements about the
expected returns of arbitrary portfolios, and the model combines the views with
equilibrium, producing both the set of expected returns of assets as well as the optimal
portfolio weights.
Although Black and Litterman concluded in their 1992 article [Black and Litterman,
1992]:
“. . . our approach allows us to generate optimal portfolios that start at a set of neutral
weights and then tilt in the direction of the investor’s views.”
they did not discuss the precise nature of that phenomenon. As we demonstrate here, the
optimal portfolio for an unconstrained investor is proportional to the market equilibrium
portfolio plus a weighted sum of portfolios reflecting the investor’s views. Now the
economic intuition becomes very clear. The investor starts by holding the scaled market
equilibrium portfolio, reflecting her uncertainty on the equilibrium, then invests in
portfolios representing her views. The Black-Litterman model computes the weight to put
on the portfolio representing each view according to the strength of the view, the
covariance between the view and the equilibrium, and the covariances among the views.
We show the conditions for the weight on a view portfolio to be positive, negative, or
zero. We also show that the weight on a view increases when the investor becomes more
bullish on the view, and the magnitude of the weight increases when the investor
becomes less uncertain about the view.
The rest of the article is organized as follows. In Section 2, we review the basics of the
Black-Litterman asset allocation model. In Section 3, we present our empirical findings
of the study and data description. Then we present our main results in Section 4
3
4. 2. The Black-Litterman model
The Black and Litterman (1990, 1991, 1992) asset allocation model is a sophisticated
asset allocation and portfolio construction method that overcomes the drawbacks of
traditional mean-variance optimization. The Black-Litterman model uses a Bayesian
approach to combine the subjective views of investors about the expected return of assets.
The practical implementation of the Black-Litterman model was discussed in detail in the
context of global asset allocation (Bevan and Winkelmann, 1998), sector allocation
(Wolfgang, 2001) and portfolio optimization (He and Litterman, 1999). In order to
incorporate the subjective views of investors, the Black-Litterman model combines the
CAPM (Sharpe, 1964), reverse optimization (Sharpe, 1974), mixed estimation (Theil,
1971, 1978), the universal hedge ratio/Black‟s global CAPM (Black and Litterman
1990, 1991, 1992; Litterman, 2003), and mean-variance optimization (Markowitz,
1952). The Black-Litterman model creates stable and intuitively appealing mean-variance
efficient portfolios based on investors‟ subjective views and also eliminates the input
sensitivity of the mean-variance optimization. The most important input in mean-variance
optimization is the vector of expected returns. The model starts with the CAPM
equilibrium market portfolio returns starting point for estimation of asset returns, unlike
previous similar models started with the uninformative uniform prior distributions. The
CAPM equilibrium market portfolio returns are more intuitively connected to market and
reverse optimization of the same will generate a stable distribution of return estimations.
The Black-Litterman model converts these CAPM equilibrium market portfolio returns to
implied return vector as a function of risk-free return, market capitalization, and
covariance with other assets. Implied returns are also known as CAPM returns, market
returns, consensus returns, and reverse optimized returns. Equilibrium returns are the set
of returns that clear the market if all investors have identical views.
The following is the Black-Litterman formula (Equation 1) along with detailed
description of each of its components. In this project, K represents the number of views
and N represents the number of assets in the model.
[ ] = ( ∑)
+ ′Ω
( ∑) ∏ + ′Ω
(1)
where,
E[R] is the new (posterior) combined return vector (N × 1 column vector);
τ, a scalar;
Σ, the covariance matrix of excess returns (N × N matrix);
P, a matrix that identifies the assets involved in the views (K × N matrix or 1 × N
4
5. row vector in the special case of 1 view);
Ω, a diagonal covariance matrix of error t erms from the expressed views
representing the uncertainty in each view (K × K matrix);
∏, the implied equilibrium return vector (N × 1 column vector);
Q, the View Vector (K x 1 column vector)
The Black-Litterman model uses the equilibrium returns as a starting point and the
equilibrium returns of the assets are derived using a reverse optimization method using
Equation 2
∏ =
∑
(2)
where,
∏, is the implied equilibrium excess return vector;
, a risk aversion coefficient;
∑, the covariance matrix, and
, is the market capitalization weight of the assets.
The risk aversion coefficient characterizes the expected risk-return tradeoff and it acts as
a scaling factor for the reverse optimization. The risk aversion coefficient can be
calculated using equation 3
=
(3)
The implied equilibrium return vector is nothing but the market capitalization-weighted
portfolio. In the absence of views, investors should hold the market portfolio. However,
Black-Litterman model allows investors to incorporate their subjective views on the
expected return of some of the assets in a portfolio, which may differ from the implied
equilibrium returns. The subjective views of investors can be expressed in either absolute
or relative terms.
where, Q, the view vector, which is k × 1 dimension; k, the number of views, either
absolute or relative. The uncertainty of views results in a random, unknown,
independently, normally distributes error term vector ( ) with mean 0 and covariance
matrix Ω. Thus a view has the form Q+
5
6. Q+ =
:
:
:
+ :
(4)
Investor views on the market and their confidence level on the views form the basis for
arriving at new combined expected return vector. With respect to investor views, we
need to consider the following aspects while developing the Black-Litterman model:
1. Each view should be unique and uncorrelated with the other.
2. While constructing the views, we need to ensure that the sum of views is either
0 or 1, which ensures that all the views are fully invested.
The investor view matrix (P) was constructed differently by various authors. He and
Litterman (1999) and Izorek (2005) used a market capitalization weighted scheme.
However, market capitalization weighted scheme is applicable only in relative views.The
expected return on the views is organized as a column vector (Q) expressed as Kx1
vector.
Omega, the covariance matrix of views, is a symmetric matrix with non-diagonal
elements as 0s. For calculating it, we have assumed that the variance of the views will be
proportional to the variance of the asset returns, just as the variance of the prior
distribution is. This method has been used by He and Litterman (1999) and Meucci
(2006). Using these expected return, risk aversion coefficient (λ) and covariance matrix
(∑), new asset weights can be allocated using equation 5.
= ( ∑)
* E[R]
(5)
Before we attempt to detail the empirical examination of the Black-Litterman model, it
might be useful to give an intuitive description of the major steps, which are presented in
Figure 1
6
7. Figure 1: Major steps behind the Black-Litterman model.
3. Empirical findings of the study and Data description
Data description
The current study is based on various stocks constructed and maintained by the Ho chi
minh city stock exchange (HSE), VietNam. We select top K = 10 most market
capitalization of Ho chi minh city stock exchange with historical and data of at least 1
year and use daily closing prices from January 1st, 2011 to January 31st, 2012.
List of 10 stocks are selected such as:
No.
Stocks
Code
1
Baoviet Holdings
PetroVietnam Fertilizer and
Chemicals Company
Vietnam export import Bank
FPT Corporation
Hoang Anh Gia Lai JSC
Masan Group Corporation
Saigon Securities Inc
Sai Gon Thuong Tin Bank
Vingroup
Vinamilk Corp
BVH
Market
capitalization
(billion VND)
46,272
DPM
12,160
4.61%
EIB
FPT
HAG
MSN
SSI
STB
VIC
VNM
15,523
10,610
13,645
64,409
7,549
14,040
35,595
47,095
5.89%
4.02%
5.17%
24.39%
1.79%
5.31%
13.48%
17.84%
2
3
4
5
6
7
8
9
10
Proportion
17.51%
7
8. Empirical findings of the study
As VietNam is an emerging economy that could withstand the after-effects of global
financial meltdown, several foreign institutional investors are keen on parking their
investments in the country. Each of them has different long-term and short-term views on
different sectors of the VietNam equity market. This has motivated to empirically
examine the tactical asset allocation across different sectors of VietNam equity market
through Black-Litterman approach.
The study has considered the monthly closing price of ten stocks of HSE from January
1st, 2011 to January 31st, 2012. The daily closing price of stocks has been taken to
compute the continuous compounded return of daily these stocks by taking the natural
logarithmic of price difference. This is represented as follows:
= ln( ) − ln(
)
where,
is the return at time t
, price at time t, and
, price at time t-1
A risk-return profile of 10 stocks over a one years, from 1st, 2011 to January 31st, 2012,
is presented in the Table 1 and Figure 2.
Table 1 and Figure 2 indicate the risk-return profiles of ten stocks of HSE.
Table 1.Historical risk-return profile of different sectors
(1st, 2011 to January 31st, 2012)
No.
1
2
3
4
5
6
7
8
9
10
Stocks Risk (%)
BVH
DPM
EIB
FPT
HAG
MSN
SSI
STB
VIC
VNM
52.61%
35.56%
18.52%
29.26%
41.33%
48.04%
40.62%
25.00%
42.67%
27.32%
Return (%)
10%
12%
15%
15%
16%
15%
18%
20%
15%
25%
8
9. 30%
VNM
25%
STB
Return
20%
HAG
FPT
EIB
15%
SSI
VIC
10%
MSN
DPM
BVH
5%
0%
0%
10%
20%
30%
40%
50%
60%
Risk
Figure 1. Scatter plot of risk-return profile of different sector
(1st, 2011 to January 31st, 2012)
Traditional mean variance optimization often leads to highly concentrated, undiversified
asset allocations. When developing an opportunity set, one should select non-overlapping
mutually exclusive asset classes that reflect the investors‟ investable universe. In this
project, we have presented two types of graphs – efficient frontier graphs and efficient
frontier asset allocation area graphs. Efficient frontier displays returns on the vertical axis
and the risk (standard deviation) of returns on the horizontal axis. Efficient frontier is the
locus of points, which represents the different combination of risk and return on an
efficient asset allocation, where an efficient asset allocation is one that maximizes return
per unit of risk. This is presented in Figure 3.
35%
Assets
Implied_EF
30%
Return
25%
20%
15%
10%
5%
0%
10%
20%
30%
40%
50%
60%
Risk
Figure 2: Efficient frontier, historical return versus risk.
9
10. Efficient frontier asset allocation area graphs complement the efficient frontier graphs.
They display the asset allocations of the efficient frontier across the entire risk spectrum.
Efficient frontier area graphs display risk on the horizontal axis. The efficient frontier
area graph displays all the asset allocation on the efficient frontier. This is helpful to
visualize the efficient frontier graphs and the efficient frontier asset allocation area graphs
together because one can simultaneously see the asset allocations associated with the
respective risk-return point on the efficient frontier, and vice versa.
To avoid the limitation of efficient frontiers based on historical data leads to highly
concentrated portfolios in the mean variance approach of Markowitz‟ s theory, the BlackLitterman model (1992) proposed a better solution. This was further researched and
emphasized by Von Neumann, Morgenstern and James Tobin. A rich literature on this
was well documented by Sharpe (1964, 1974), respectively. The pivotal point of BlackLitterman model is implied returns. Implied returns (otherwise known as equilibrium
returns) are the set of sectoral indices returns that clear the market if all investors have
identical views. This means the market follows the strong form efficiency of the efficient
market hypothesis or leads to a perfect competitive market. To compute the equilibrium
returns of the sectoral indices, we need an input parameter, that is, risk aversion
coefficient. The risk aversion coefficient characterizes the risk-return trade off. Risk
aversion coefficient is the ratio of risk-return and variance of the benchmark portfolio.
The mathematical representation of risk aversion coefficient (denoted by λ) is as follows:
=
−
where,
is the return on benchmark;
, the risk free rate, and
, is the variance of the benchmark.
This project considered HSE as the benchmark index to compute the risk aversion
coefficient. We have considered the risk free rate to be 8%. By computing the ratio of
risk premium and variance of HSE, we have calculated the risk aversion coefficient (λ) at
4.2%. The risk aversion coefficient characterizes the risk return trade off. From the daily
return series of stocks, we have generated the covariance matrix. This is represented in
Table 2
10
12. No.
1
2
3
4
5
6
7
8
9
10
Stocks
BVH
DPM
EIB
FPT
HAG
MSN
SSI
STB
VIC
VNM
Risk (%)
52.61%
35.56%
18.52%
29.26%
41.33%
48.04%
40.62%
25.00%
42.67%
27.32%
Total implied
return* (%)
44.84%
24.51%
5.25%
12.84%
27.04%
42.38%
22.22%
3.13%
21.51%
12.97%
Table 4.Implied return (∏ = λ∑
) of stocks - risk profile
(January 1st, 2011 to January 31st, 2012).
*Total implied return = implied excess return + risk free rate
After generating the implied return and risk of the stocks, we have generated the
optimized portfolio efficient frontier. Here, it is understood that implied returns are
considered as the E[R] of the respective stocks.
These implied returns are the starting point for the Black-Litterman model. However, it
has been observed that most investors stop thinking beyond this point while selecting the
optimal portfolio. If investors or market participants do not agree with implied returns,
the Black-Litterman model provides an effective framework for combining the implied
returns with the investor’s unique views or perception regarding the markets, which result
in well diversified portfolios reflecting their views.
To implement the Black-Litterman approach, an asset manager has to express his or her
views in terms of probability distribution. Black-Litterman assumes that the investor has
two kinds of views absolute and relative. For now, we assume that the investor has k
different views on linear combinations of E[R] of the n assets. This is explained in details
as an equation (Equation 1) in the methodology section.
In this project, we have considered the combination of one absolute and one relative view
on list of our stocks. These views are expressed as follows:
Absolute view
View 1
VNM will generate an absolute return of 10%.
Relative view
12
13. Views 2
MSN outperform HAG by 8%.
These two views are expressed as follows:
µVNM = 0.1
strong view:
= 0.0019
µMSN - µHAG= 0.08 weaker view:
Thus P =
0 0
0 0
0
0
0
0 0 0 0
0 −1 1 0 0
0
0
= 0.0065
1
0.1
0.0019
,q=
and Ω =
0
0.08
0
0
0.0065
Applying formula (1) to compute E[R], we get
E[R]
BVH
DPM
EIB
43.56% 23.97% 5.25%
FPT
HAG
MSN
SSI
STB
12.55% 27.77% 39.43% 22.01% 3.04%
VIC
VNM
21.60% 11.46%
Set up the quadratic problems for portfolion optimization:
min
¸
μ x≥R
Ax = 1
x≥0
where,
x: weight vector of portfolio
H: covariance matrix of our stocks
μ: new combined return vector
R: expected return contraint of portfolio
A: unity vector
13
15. Figure 3. Efficient Frontier and the Composition of Efficient Portfolios
using the Black-Litterman approach
Extension part: comparision of two approaches
Figure 4 plots the efficient frontier generated by implied return and Black Litterman
return. It can be concluded that Black-Litterman model provides the optimal portfolio
with maximum return and minimum risk in comparison to implied return based and mean
variance based portfolio optimization.
40%
35%
30%
Return
25%
20%
15%
10%
Implied_EF
5%
Black-Litterman EF
0%
10%
15%
20%
25%
30%
35%
40%
Risk
Figure 4. Efficient Frontier: Black-Litterman versus implied return.
15
16. REFERENCES
[1]
The intuition behind Black-Litterman model portfolios - Guangliang He,
[2]
A step-by-step guide to the Black-Litterman model - Thomas M. Idzorek (2005),
[3]
Exercises in Advanced Risk and Portfolio Management – A. Meucci (with code)
[4]
Optimization Methods in Finance - Gerard Cornuejols (2005),
[5]
An equilibrium approach for tactical asset allocation: Assessing Black-Litterman
model to Indian stock market - Alok Kumar Mishra (2011),
16
