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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

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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 Black-

Litterman efficient portfolios achieve a significantly better return-to-risk performance

than the mean-variance optimal approach/strategy.

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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 Black-

Litterman 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 Black-

Litterman 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

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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

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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+

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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

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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 Market

capitalization (billion VND)

Proportion

1 Baoviet Holdings BVH 46,272 17.51%

2 PetroVietnam Fertilizer and Chemicals Company

DPM 12,160 4.61%

3 Vietnam export import Bank EIB 15,523 5.89% 4 FPT Corporation FPT 10,610 4.02% 5 Hoang Anh Gia Lai JSC HAG 13,645 5.17% 6 Masan Group Corporation MSN 64,409 24.39% 7 Saigon Securities Inc SSI 7,549 1.79% 8 Sai Gon Thuong Tin Bank STB 14,040 5.31% 9 Vingroup VIC 35,595 13.48%

10 Vinamilk Corp VNM 47,095 17.84%

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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. Stocks Risk (%) Return (%) 1 BVH 52.61% 10%

2 DPM 35.56% 12%

3 EIB 18.52% 15%

4 FPT 29.26% 15%

5 HAG 41.33% 16%

6 MSN 48.04% 15%

7 SSI 40.62% 18%

8 STB 25.00% 20%

9 VIC 42.67% 15%

10 VNM 27.32% 25%

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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.

Figure 2: Efficient frontier, historical return versus risk.

BVHDPM

EIB FPT HAG

MSN

SSISTB

VIC

VNM

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10%

15%

20%

25%

30%

0% 10% 20% 30% 40% 50% 60%

Re

turn

Risk

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10%

15%

20%

25%

30%

35%

10% 20% 30% 40% 50% 60%

Ret

urn

Risk

Assets

Implied_EF

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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 Black-

Litterman 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 Black-

Litterman 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

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BVH DPM EIB FPT HAG MSN SSI STB VIC VNM

BVH 0.2768 0.0858 0.0126 0.0367 0.0927 0.1449 0.0761 -0.0044 0.0527 0.0210

DPM 0.0858 0.1264 0.0213 0.0517 0.0892 0.0704 0.0948 0.0120 0.0281 0.0352

EIB 0.0126 0.0213 0.0343 0.0114 0.0177 0.0095 0.0245 0.0141 0.0125 0.0040

FPT 0.0367 0.0517 0.0114 0.0856 0.0468 0.0323 0.0586 0.0078 0.0151 0.0217

HAG 0.0927 0.0892 0.0177 0.0468 0.1708 0.0702 0.0935 0.0059 0.0613 0.0276

MSN 0.1449 0.0704 0.0095 0.0323 0.0702 0.2308 0.0527 0.0001 0.0336 0.0283

SSI 0.0761 0.0948 0.0245 0.0586 0.0935 0.0527 0.1650 0.0132 0.0313 0.0326

STB -0.0044 0.0120 0.0141 0.0078 0.0059 0.0001 0.0132 0.0625 0.0089 0.0080

VIC 0.0527 0.0281 0.0125 0.0151 0.0613 0.0336 0.0313 0.0089 0.1821 0.0135

VNM 0.0210 0.0352 0.0040 0.0217 0.0276 0.0283 0.0326 0.0080 0.0135 0.0747

Table 2. The covariance matrix of our stocks.

To compute the weighted market capitalization of each of stocks, we have considered the

closing prices and shares outstanding of each member constituents on 1st November,

2011, as this is the end date of the study period. The market capitalization weights are

represented in the following Table 3.

Table 3. Implied returns, market capitalization weights ()

No. Stocks Market capitalization*

(billion VND) weight

(%) 1 BVH 46,272 17.51%

2 DPM 12,160 4.61%

3 EIB 15,523 5.89%

4 FPT 10,610 4.02%

5 HAG 13,645 5.17%

6 MSN 64,409 24.39%

7 SSI 7,549 1.79%

8 STB 14,040 5.31%

9 VIC 35,595 13.48%

10 VNM 47,095 17.84% * Dated 1st November, 2011

Finally, the implied excess returns of the sectoral indices have been computed by

considering their corresponding risk aversion coefficient (λ) and market capitalization

weights (). This is represented in Table 4.

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No. Stocks Risk (%) Total implied return* (%)

1 BVH 52.61% 44.84%

2 DPM 35.56% 24.51%

3 EIB 18.52% 5.25%

4 FPT 29.26% 12.84%

5 HAG 41.33% 27.04%

6 MSN 48.04% 42.38%

7 SSI 40.62% 22.22%

8 STB 25.00% 3.13%

9 VIC 42.67% 21.51%

10 VNM 27.32% 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

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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: = 0.0065

Thus P = 0 0 0 0 0 0 0 0 0 10 0 0 0 −1 1 0 0 0 0

, q = 0.1

0.08 and Ω =

0.0019 00 0.0065

Applying formula (1) to compute E[R], we get

BVH DPM EIB FPT HAG MSN SSI STB VIC VNM E[R] 43.56% 23.97% 5.25% 12.55% 27.77% 39.43% 22.01% 3.04% 21.60% 11.46%

Set up the quadratic problems for portfolion optimization:

min ¸

μx ≥ 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

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4. The results

Solving for R = 3.0% to R = 32% with increments of 1% we now get the optimal

portfolios and the effcient frontier depicted in Table 5 and Figure 3

Table 5: Black-Litterman Efficient Portfolios

Return Variance Weights

BVH DPM EIB FPT HAG MSN SSI STB VIC VNM

3% 14.43% 1.05% 0.00% 44.77% 9.84% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62%

4% 14.43% 1.05% 0.00% 44.77% 9.84% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62%

5% 14.43% 1.05% 0.00% 44.77% 9.84% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62%

6% 14.43% 1.05% 0.00% 44.77% 9.84% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62%

7% 14.43% 1.05% 0.00% 44.77% 9.84% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62%

8% 14.43% 1.05% 0.00% 44.77% 9.84% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62%

9% 14.43% 1.53% 0.00% 43.91% 9.83% 0.00% 2.70% 0.00% 18.67% 4.79% 18.56%

10% 14.53% 2.72% 0.00% 41.80% 9.82% 0.00% 3.94% 0.00% 17.84% 5.48% 18.41%

11% 14.72% 3.91% 0.00% 39.70% 9.80% 0.00% 5.18% 0.00% 17.00% 6.16% 18.26%

12% 15.01% 5.10% 0.00% 37.59% 9.78% 0.00% 6.43% 0.00% 16.16% 6.84% 18.11%

13% 15.39% 6.11% 0.00% 35.45% 9.55% 0.68% 7.56% 0.00% 15.40% 7.36% 17.88%

14% 15.85% 7.10% 0.00% 33.32% 9.28% 1.47% 8.69% 0.00% 14.66% 7.84% 17.65%

15% 16.38% 8.09% 0.00% 31.18% 9.01% 2.25% 9.81% 0.00% 13.92% 8.33% 17.42%

16% 16.98% 9.07% 0.00% 29.05% 8.74% 3.04% 10.93% 0.00% 13.17% 8.81% 17.19%

17% 17.64% 10.06% 0.00% 26.91% 8.47% 3.83% 12.05% 0.00% 12.43% 9.30% 16.96%

18% 18.35% 11.05% 0.00% 24.78% 8.21% 4.61% 13.17% 0.00% 11.68% 9.79% 16.72%

19% 19.11% 12.03% 0.00% 22.64% 7.94% 5.40% 14.29% 0.00% 10.94% 10.27% 16.49%

20% 19.91% 12.95% 0.51% 20.46% 7.55% 6.01% 15.35% 0.00% 10.20% 10.79% 16.18%

21% 20.75% 13.82% 1.31% 18.25% 7.09% 6.52% 16.39% 0.00% 9.48% 11.32% 15.83%

22% 21.61% 14.68% 2.08% 16.04% 6.61% 7.01% 17.43% 0.08% 8.75% 11.85% 15.47%

23% 22.51% 15.52% 2.70% 13.80% 6.07% 7.43% 18.48% 0.47% 8.03% 12.38% 15.11%

24% 23.42% 16.37% 3.33% 11.56% 5.53% 7.85% 19.53% 0.85% 7.32% 12.91% 14.75%

25% 24.36% 17.22% 3.95% 9.32% 4.99% 8.27% 20.58% 1.24% 6.60% 13.45% 14.39%

26% 25.32% 18.06% 4.57% 7.08% 4.45% 8.69% 21.63% 1.62% 5.89% 13.98% 14.03%

27% 26.30% 18.91% 5.20% 4.83% 3.90% 9.11% 22.68% 2.01% 5.17% 14.52% 13.67%

28% 27.29% 19.75% 5.82% 2.59% 3.36% 9.53% 23.73% 2.39% 4.46% 15.05% 13.31%

29% 28.29% 20.60% 6.44% 0.35% 2.82% 9.95% 24.78% 2.78% 3.74% 15.59% 12.95%

30% 29.31% 21.52% 6.98% 0.00% 1.86% 10.41% 25.94% 3.07% 1.93% 15.99% 12.28%

31% 30.34% 22.48% 7.49% 0.00% 0.78% 10.89% 27.13% 3.35% 0.00% 16.36% 11.51%

32% 31.41% 23.98% 7.54% 0.00% 0.00% 11.60% 28.66% 3.26% 0.00% 16.23% 8.73%

33% 32.51% 25.58% 7.37% 0.00% 0.00% 12.32% 30.24% 2.97% 0.00% 16.03% 5.50%

34% 33.65% 27.18% 7.20% 0.00% 0.00% 13.03% 31.83% 2.68% 0.00% 15.82% 2.27%

35% 34.82% 29.23% 6.26% 0.00% 0.00% 13.87% 33.47% 2.02% 0.00% 15.16% 0.00%

36% 36.07% 32.36% 3.51% 0.00% 0.00% 15.01% 35.23% 0.48% 0.00% 13.41% 0.00%

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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.

Figure 4. Efficient Frontier: Black-Litterman versus implied return.

0%

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10% 15% 20% 25% 30% 35% 40%

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Implied_EF

Black-Litterman EF

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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),