Assignment

Introduction

 

As an essential analysis tool in financial risk measurement, portfolio construction effectively helps investors to evaluate risky assets more accurately. By integrating the historical information and market value information related to the returns of risky assets, investors can build a portfolio that produces the maximum expected return with the minimum risk. The optimal portfolio is the effective selection of various portfolio stocks. There are some models for optimal portfolio development: Mean-Variance Model, Capital Asset Pricing Model (CAPM) and Black-Litterman Model [1].  We will focus on the BL model in this article. In order to design a mean-variance optimal (MVO) portfolio that is more stable and more accurately reflects investment views than the MVO portfolio, Black and Litterman (1991, 1992) proposed the expected return model named after them (1991, 1992). BL hedges the problem in the MOV portfolio by combining the “equilibrium” expected return implied in the market relative to the benchmark portfolio with the expected return relative to the portfolio reflecting the “view” of the active investment manager. BL has replaced the traditional MVO portfolio on a large scale and has been adopted by more and more investors [2].

The extent to which the BL portfolio differs from the benchmark depends on:

  1. the relative size of positive views,
  2. the degree of confidence in each individual’s positive views,
  3. the overall level of confidence in positive views and equilibrium expected earnings.

In this report, we will work as fund managers investing in U.S. stocks to create an optimal portfolio of five known stocks. They are Catterpeter, Johnson & Johnson, Wal-Mart, Amgen and Procter & Gamble. We will start building a portfolio of these five stocks on December 31, 2019. In this report, we plan to analyze and compute mainly through Black-Litterman model. The Black-Litterman model integrates the Bayesian method, takes the subjective viewpoint of investors into consideration. In other words, Black-Litterman model combines the implied equilibrium return of the market with the executive viewpoint of investors and forms the estimated value of the expected return. Moreover, it effectively solves the problem that the Markowitz model is sensitive to parameters. 

In this report, we will compute the covariance matrix and average return vector by using the first four years of return data and risk-free rate. Then our five members in our groups as investors, and we will give our views on the future performance of these five stocks. Compute the adjust expected returns through the method of Black-Litterman model and finally compute the optimal active portfolio containing the adjusted expected returns.

This report will be divided into three parts. The first part is the introduction, it is going to talk about the task, the motivation, the models, and the construction of the report. In the second part, we will give more details about our method to build a portfolio. And in the last part, we will use the data of these five stocks from Yahoo Finance to construct the optimal active portfolio and give the list of reference.

 

Method

In this part, the calculation is divided into four steps: constructing the tangency portfolio, market-weighted portfolio, optimal active portfolio, and then comparing their performance in the following 12 months.

  1. Tangency portfolio

There are two methods to calculate the tangency portfolio. 

  1. The first method

According to the lecture notes, the minimum frontier is the set of portfolios with the same expected returns but the lowest standard deviation. And investors usually choose the upper part of it which is called the efficient frontier so that they can get higher expected return. When there is a risk-free asset, the efficient frontier is the straight line that connects the risk-free asset with the tangency portfolio, P*, and is known as the capital market line. 

Since tangency portfolio is the tangent point of the minimum variance frontier and the capital market line, it has the highest slope on the curve. We can obtain the tangency portfolio by maximizing the Sharpe Ratio:

SR  P =Erprfp

where  Erp=PTE(R)  and   p2=PTVP. And we can implement this by Solver.

  1. The second method

Based on the first method, we know that the tangency portfolio, P*, has the largest sharpe ratio on the efficient frontier. We can differentiate the sharpe ratio with respect to the portfolio weight vector and set that to zero, and we will get the weights of tangency portfolio. In addition, we need to force these weights to add up to one by dividing the sum of them. Here is the formula:

P=V−1ERRFV−1ERRF

where P is the portfolio weight vector, V is the sample variance-covariance matrix, E(R) is the expected return vector, RF is the risk-free rate. Mean-variance optimization is based on historical data in practice, which will cause a main problem: portfolio optimization will give very high weights to stocks with high expected returns and very small (and often negative) weights to the stocks with low expected returns. This problem can make the calculated weight of the optimal portfolio unreliable. However, it is useful to improving our estimates of expected returns through using this formula. 

The optimal portfolio is the solution to a straightforward minimization problem. When there are no short-selling constraints, we can use the following formula to construct the optimal portfolio and calculate the tangential portfolio.

  1. Market-weighted portfolio

When the market is efficient (all investors have the same information about the inputs), investors will choose the same weights of individual portfolios which is the market portfolio. In practice, market portfolio constructed by fund managers will be narrowly track a broad based benchmark, such as the S&P 500.

The market weighted portfolio is formed based on the market capitalization of the underlying asset. The market capitalization is what we can obtain from Yahoo Finance. Then the market-weighted portfolio can be easily calculated by the following formula:

Wi=Market ValueTotal Market Value

  1. Optimal active portfolio

Active portfolio management is based on the premise that there exist, at any point in time, stocks that are either undervalued or overvalued. Active portfolio management attempts to identify these stocks and include them in an otherwise passive portfolio, with increased weight (if they are undervalued) or decreased weight (if they are overvalued).

According to the lecture notes, in the real financial market, investors usually have divergent opinions from the market-based expected returns, which means they believe that stocks tend to outperform or underperform their expected return. The Black-Litterman model incorporates these opinions into the procedure. Therefore, while we are constructing the optimal active portfolio, we need to incorporate the views about the future performance of the five stocks either which is given by analysts within our company.

In this part, we are going to use δ to represent analysts’ direct views about the five stocks future performance. Besides, we know that these five stocks are correlated with each other, and that will also affect differences between the return we believe and the market-implied return. Here we are using θ to show that affection. Therefore, we can change the expected return by using the following formula:

E(R)Adjusted=E(R)Implied+θδ

In this formula, θ can show the sensitivity of one stock to each of the other four stocks and can be calculated as follow:

i,j=Covri,rjVarrj

To obtain the adjusted expected return, we still need to know the market-implied expected return. Due to the historical mean portfolio return is a very noisy estimate of the actual expected return, we cannot use it. Thus, we are introducing λ to represent the market’s coefficient to risk aversion. We can have the following expression for the expected return implied by market:

E(R)Implied=λVP+RF

where  λ=ERp)RFp2 , P is the market weighted portfolio. 

According to the discussion before, we know that optimal portfolio weights can be computed as:

P=V−1ERRF1TV−1ERRF

However, since the stocks are correlated, assigning a view for one of the five stocks will affect the expected return for the others, and the other four are the same. To find the optimal active portfolio, we need to set up two columns of values for the deltas. One is “Analyst Delta” that can satisfy analysts’ views of the five stocks on Excel, the other is “Solver Delta” that can make the adjusted expected return achieve the analysts’ expected return after combining the coefficient among the five stocks. To achieve that, we are going to create an objective function which gathers absolute difference between the adjusted expected returns and analysts’ expected returns. We can get the results using Solver by minimizing the objection function.

After all of that, we can construct the optimal active portfolio of the five stocks by using the optimal portfolio weights formula.

  1. Comparison

After constructing the tangency portfolio, the market- weighted portfolio and the optimal active portfolio, we are going to get three groups of weights for each stock.

By using the following 12 months’ simple returns multiplied by portfolio weights, we can have these three groups of portfolio returns, with which we can calculate mean return, standard deviation and sharp ratio of portfolios of these three groups. We can compare their performance over the following 12 months. 

Data

The five years of data by monthly from Yahoo Finance. Constructing the optimal active portfolio using the first four years of data (the in-sample period). It contains the adjusted close prices of the five stocks and the historical data of S&P 500 index. The sample period covers five years which is from 31 December 2015 to 31 December 2020 and the sample frequency is monthly. Assuming that the currency risk is perfectly hedged and there is no short sale constraint. The first four years of data will be used to construct the optimal portfolio while the last year will be used to evaluate the portfolio.

 

Reference

[1] Meilina Pudjiani, Yusman Syaukat and Tony Irawan (2020) ‘Optimum Portfolio Analysis of Black-Litterman Model in The Indonesian Stock Exchange on Consumer Goods Industrial Sector’, Journal the Winners, 21(1), pp. 27–33. doi: 10.21512/tw.v21i1.5954.

[2] O’Toole, R. (2017) ‘The Black-Litterman Model: Active Risk Targeting and the Parameter Tau’, Journal of Asset Management, 18(7), pp. 580–587. Available at: http://search.ebscohost.com.uoelibrary.idm.oclc.org/login.aspx?direct=true&db=ecn&AN=1688274&site=eds-live&scope=site (Accessed: 4 March 2021).

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