Portfolio Analysis: Indices

MPT Background:

Modern portfolio theory (MPT), which was first introduced by Harry Markowitz in the 1950s, is a theoretical framework that provides a mathematical framework for analyzing and constructing investment portfolios. It is based on the idea that investors should be willing to accept higher levels of risk if they are compensated with higher expected returns, and that diversification can help reduce the overall risk of a portfolio.

Volatility is a measure of the risk or uncertainty of an investment. It is typically measured by the standard deviation of the investment’s returns over some time.

A higher volatility indicates that an investment’s returns are more uncertain and can fluctuate more widely from the expected return, while a lower volatility indicates that an investment’s returns are more predictable.

The efficient frontier is the set of portfolios that provide the highest expected return for a given level of risk, or the lowest level of risk for a given expected return. The purpose of the efficient frontier is to help investors identify the most efficient portfolio given their risk tolerance and investment goals.

The Sharpe ratio is a measure of the risk-adjusted return of an investment.

It is calculated by dividing the excess return of an investment (the difference between the investment’s return and the risk-free rate (e.g., U.S. T-bills) by the volatility of the investment.

A higher Sharpe ratio indicates a better risk-adjusted return, which indicates how much return an investment generates for each unit of risk taken.

OBS: The Sharpe ratio can be misleading because it is indifferent to upside vs downside volatility (investors want upside but limited downside). Therefore, the Sortino ratio might be better since it only considers the downside volatility.

Project: Finding the MEP & MVP for a Portfolio of Indices 

I want to analyze the optimal diversification allocation of the different indices I use in the Sunday Market Updated, namely: SP500, Nasdaq, EUROStoxx50, DAX, OMXS30, FTSE100, Nikkei225, and HangSeng.

First, I am going to define both the Most Efficient Portfolio and the Minimum Variance Portfolio for these indices. Then, I will compare the result of these diversified portfolios with the individual indices.

Conditions:

The metrics for comparison are (1) annualized return and volatility, (2) cumulative return, and (3) the Sharpe ratio.

To gather enough data and make sure the analysis covers at least the latest cycle, I decided to use 12 years (from “2010-01-01” to “2022-12-31”). This is also the furthest back it is possible to collect the same data range for all the indices from Yahoo Finance. The timeframe is going to be daily returns, which I am going to annualize when calculating the avg. return and volatility.

Below are the avg. return and volatility (standard deviation) for each index during this period, I also included a correlation (ρ) matrix for the indices during the period.

Something to keep in mind when it comes to diversification theory is that the lower the correlation (ρ) the better. To achieve optimal risk reduction (neutralize risk) we strive for negative correlation (best case: ρ = -1)

Rules for the simulated portfolios: (1) All the asset weights have to add up to 100% (i e no cash position), and (2) no short-selling.

To simulate the random movements of the market – I used a Monte Carlo simulation with 10,000 simulations. The result was 64 long-only portfolios with different asset weights consisting of all the indices. The weights are determined from the correlation between the indices to naturalize the volatility without compromising the returns.

Below are the results from the simulation.

Result (1): Most Efficient Portfolio (MEP)

Asset Weights

OBS: The returns are in nominal numbers and no consideration has been taken for fluctuations between local currencies. For example, the final return from an index in USD might differ if your home currency is EUR. 

Result (2): Minimum Variance Portfolio (MVP)

Asset Weights

OBS: The returns are in nominal numbers and no consideration has been taken for fluctuations between local currencies. For example, the final return from an index in USD might differ if your home currency is EUR. 

Summary:

Comments:

As expected the MEP performed better than the MVP during this period. MVP did have lower volatility, however, MEP generated more return given its volatility (which can be interpreted by the difference in the Sharpe ratio). 

Also, as expected, the MEP did have the highest Sharpe ratio (risk-to-reward) of all the assets, but Nasdaq was not far behind. 

What I thought was interesting is the difference in weights between MEP and MVP. MEP had a higher allocation towards Nasdaq and SP500, while MVP had a smaller Nasdaq position and a higher exposure towards HangSeng. Compare the correlations between the SP500 and HangSeng, you will see it is 0.21. This strengthens the concept that lower correlation is a key component to lower volatility. Regarding the European indices, it was fairly low in both portfolios.

Given the absolute return, Nasdaq performed the best with 180.58% cumulative return, the MEP came in 3rd place with 140.61%, and the MVP came in 7th place with 90.94%.

What this simulation confirms is that with diversification it is possible to lower the volatility without compromising the return, and if done right, it is also possible to increase the return given the same level of risk. Something to keep in mind here is that all these assets are relatively highly correlated (since they are the same asset class). The lowest correlation was 0.19, so still positive. If mixing assets with negative correlation it is possible to eliminate the expected volatility.

One last thing is that even though the MEP returns the highest risk-to-reward, 15.44% volatility might still be too much for some investors. A way to lower the risk but still utilize the optimal risk-to-reward ratio is to mix the MEP portfolio with a risk-free rate (eg. 10Y U.S. T-bills).

About: Portfolio Analysis Series

This portfolio analysis is part of a series where I will first analyze the different asset classes I use in the Sunday Market Update (Indices, Sectors, FX, and Commodities).

Later, I will combine these results into a market portfolio that consists of the asset allocations between the different simulated MEPs.

Finally, I am going to divide the data sets into different time series that are representative of the business cycle, to examine the optimal portfolio allocations given the economic conditions.

LINK TO THE PORTFOLIO ANALYSIS SERIES