# Black Litterman Model

Last Updated :

21 Aug, 2024

Blog Author :

Edited by :

prarthana Khot

Reviewed by :

Dheeraj Vaidya, CFA, FRM

Table Of Contents

## Black Litterman Model Definition

The Black Litterman model is a mathematical financial model developed for portfolio allocation incorporating views of investors and market equilibrium. It ensures an optimized asset allocation in a portfolio using the Bayesian theory to integrate subjective forecasts. In addition, it permits the investors to use their personal opinions to optimize the asset allocation in the portfolio.

This model helps relieve the load of entities involved in the asset investment process and asset allocation. It enables fund managers to handle any error when allocating portfolio assets. The model enhances the portfolio returns by staying within the mean-variance optimization framework. Moreover, it solves the issue of return sensitivity and the provision of a stable-efficient portfolio.

##### Table of contents

- Black Litterman Model takes the help of mathematical models and investors' personalized views on asset performance to build an optimized portfolio asset allocation.
- Investors, portfolio managers, and analysts use it to get higher returns through Bayesian theory for the portfolio's integrative subjective forecast of asset performance.
- It is based on two major assumptions, i.e., assets' normal distribution and unknown returns of the verified actual mean.
- If the view confidence of the investor exceeds the implied equilibrium returns difference, the portfolio will tilt towards the outperforming asset or vice versa.

### Black Litterman Model Explained

The Black Litterman Model refers to a financial model or analytical tool to enhance the allocation of assets in a portfolio to enhance the returns for the investors. In __1990__, Goldman Sachs economists Robert Litterman and Fisher Black developed this model and published it in 1992. For the implementation of this model, analysts use matrix algebra and softwares. The software is usually very expensive, so they use excel, which also requires complex calculations.

Thus, analysts and portfolio managers can implement the Black Litterman model asset allocation by using this model in python or excel by taking the help of its formula. Moreover, it improves and adds a step to the Markowitz Model. That step involves incorporating investors' personal views concerning the allocation of assets in a portfolio.

The model starts with the expected returns based on market implications. Then, the said returns based on market implications get adjusted to reflect the fund manager's or investor's personal opinions. Furthermore, the two most important assumptions for applying this model without which the model cannot function properly are:

**Normal Distribution Of**- Although it could use other varieties of distributions, the normal distribution is the simplest to use.**Verified True Mean Returns Are Unknown**- The variation of preceding and provisional distributions regarding actual means gets recognized.

### Black Litterman Model Formula

Let us understand the working of the black litterman model formula.

For determining the high-performing portfolio using this model, one needs to have the data on the following –

- The outperformance of one security over other.
- View the confidence of the investor.
- Implied equilibrium returns of all the assets.
- The difference in the implied equilibrium returns.

As per the Litterman model, if the view confidence of the investor:

- Exceeds implied equilibrium returns difference, making the portfolio tilt toward the outperforming asset.
- Exceeds implied equilibrium returns difference, making the portfolio tilt toward the non-performing asset.

This model is based on a new Combined Return Vector that uses investors' knowledge to develop stable and mean-variance effective portfolios. So, one can represent the model mathematically as:

Where:

- E = Combined Return Vector.
- Τ = A Scalar (From 0 To 1).
- Σ = The covariance matrix related to excess returns.
- P = Matrix that identifies the assets involved in the views.
- Ω= Diagonal covariance matrix of error representing the uncertainty in each view.
**Π =**Implied Equilibrium Return Vector.- Q = View Vector.

### Examples

Let us go through the following examples of the Black Litterman Model to understand the concept.

#### Example #1

Suppose investor A has to find the best asset allocation for the portfolio of energy sectors using this model. Then, as per the statistical model, 70% of manufacturing sector securities will outperform the securities of the firms in energy sectors in the range of 8% to 13%.

However, as per the Black Litterman model, the investors will also add their personal views to the above results. And their views regarding the securities performance of the energy and manufacturing sector will be that- the energy sector has the potential to outperform the manufacturing sector by almost 11% percent, having a variance of 0.35 ^ 2.

Thus, from the assessment of the above information, one gets a clear picture that the portfolio allocation for investor A must have greater security in the energy sector compared to the manufacturing sector.

#### Example #2

Let us assume the following conditions regarding securities A & B as follows:

Views of investor X for security:

A is that it will outperform security B by 4%

The view confidence of the investor = 70 %

Implied equilibrium returns of A = 50%

Implied equilibrium returns of B = 70%

One can see that A will be better than B by four average percentages. Knowing the positive/negative impact of the above on the portfolio implied equilibrium will also be required.

The difference in the implied equilibrium returns of A & B = (70 -50) %= 20%

Here, the investor's view confidence (70%) exceeds the implied equilibrium returns difference (20%). As a result, some will make the portfolio tilt towards outperforming asset A by this model.

#### Example #3

A financial accounting company, Wealthfront, provides asset allocation services. They started by using Capital Asset Pricing Model (CAPM) model to find the values of expected returns on investments in market equilibrium. Then in CAPM, to analyze the values of long-term return on assets based on macroeconomic variables like interest rates, gross domestic product growth, dividend yields, and other such variables, they use the Black Litterman Model.

According to them, this highly technical mathematical model does not help eliminate fudging. But, it helps the users to ensure a mathematically valid consistency between non-fudged and fudged numbers and thus implement fudging effectively.

### Frequently Asked Questions (FAQs)

**1. What is Tau in the Black Litterman model?**

Tau (T) is a parameter in the black-litterman model that determines the all-embracing weight attributed to investment related to active versus passive views. Tau finds its origin in the seminal Bayesian derivation of the model. Nevertheless, Tau is the most controversial facet of the model.

**2. How does the Black Litterman model work?**

The black litterman model requires three inputs to work properly - vector of expected/personal returns, capital asset pricing model (CAPM) derived weights, and confidence level. Moreover, the mean-variance optimizer situated at the posterior mean plus covariance gets utilized to extract the final weights. First of all, CAPM gets utilized to derive the returns. Then, reverse optimization gets utilized to determine the implied excess equilibrium returns vector. After this, the BL model formula gets used to calculate the optimization value of the portfolio.

**3. What are the advantages of the Black Litterman model over the Markowitz model?**

The greatest advantages of the black litterman model over the Markowitz model are as follows -:

· This model has an integrative approach as compared to the Markowitz model.

· It is more flexible and,

· It has no restrictions on the views compared to the Markowitz model.

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