Mean-Variance Portfolio Optimisation

Mean-Variance Portfolio Optimisation

Hey guys,

Below is the link to download my MV Portfolio Optimisation spreadsheet.

MV Portfolio Optimisation

Basically, this spreadsheet determines how much of each stock an investor should hold for the next investment period (year/month/week/day) to minimise the variance (risk) of the portfolios returns for that period given a target return. The spreadsheet also plots the Efficient Frontier for the portfolio, and the required weight of each asset against portfolio target return.

Check out the instructional video below for a brief demonstration on how to use the spreadsheet.

Once the user has input the ticker symbol for each stock within their portfolio on the “Assets” sheet, they can initiate the download of all historical pricing data from Yahoo Finance. This data is then used to calculate the Variance-Covariance matrix based on a user-defined Holding Period (Day, Week, Month or Year). The inverse of the Variance-Covariance matrix can then be estimated, and the expected returns of each asset calculated. The expected return for each asset is the arithmetic mean of the historical returns for that asset based on the previously defined Holding Period, i.e. the Holding Period used to calculate the Variance-Covariance matrix. Now the user can optimise their portfolio, i.e. calculate the asset weights to minimise portfolio variance for a given target return, based on the theory below. The user will be prompted to input their desired target return as an annual figure when they click the “Optimise” button on the “Results” sheet. The target return will be appropriately scaled to coincide with the previously defined Holding Period. Finally, now the ‘a’ and ‘b’ vectors (refer below theory) have been calculated, the user can plot the Efficient Frontier of the portfolio, and each asset weight against target return.

The spreadsheet adopts Mean-Variance Portfolio theory, which is a mathematical technique that determines asset weights within a portfolio to minimise the variance (risk) of the portfolios returns given a target return for that portfolio. This spreadsheet assumes the portfolio is composed of only risky assets, specifically stocks listed on the Australian Stock Exchange (ASX). To minimise the variance for a given target return, MV Portfolio theory adopts the Method of Lagrange Multipliers, and after a bunch of tedious matrix algebra, the following relationship for portfolio weights is derived:

Where represents the Variance-Covariance matrix of the portfolio and µ represents a vector of each assets expected return. The ‘a’ and ‘b’ vectors shown in the above relationship are calculated and displayed in the “Results” sheet.

As can be seen from the above formula, the inverse of the Variance-Covariance matrix is required to compute the asset weights vector. Since the Variance-Covariance matrix is always square and generally non-singular, MS Excel’s “MInverse” function can normally calculate the inverse without issue. In the instance where there exists near-singular entries (values smaller than 1e-16) in the Variance-Covariance matrix, or if “MInverse” returns an error for some reason, the Moore-Penrose Pseudoinverse of the matrix will be estimated. Refer to my previous blog post, Moore-Penrose Pseudoinverse for more information on the algorithm used in this scenario.

So the reason why I went to all the trouble of building a VBA algorithm for calculating the Moore-Penrose Pseudoinverse was because the first Variance-Covariance matrix I tried to invert was singular (non-invertible). I stupidly assumed that this would be the case for all Variance-Covariance matrices… it isn’t. The Variance-Covariance matrix was only singular in that instance because I input the same stock twice (to represent the purchase of shares of the same company on two separate occasions) leading to multiple entries of the same value within the matrix, and the matrix to be non-invertible. In any case, the “Calculate Inverse” button on the “Results” sheet will produce at least an estimate of the inverse of the Variance-Covariance matrix. So the user can input the same ticker symbol any number of times and the portfolio can still be optimised but I would recommend the user only input each ticker symbol once, the calculation will be quicker this way and the result more simple.

I hope you find this spreadsheet useful. Let me know if you have any trouble or if you come up with any improvements I might be able to incorporate.

Enjoy!

Cheers,

Nick

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