By the diagram, the introduction of the risk-free asset as a possible component of the portfolio has improved the range of risk-expected return combinations available, because everywhere except at the tangency portfolio the half-line gives a higher expected return than the hyperbola does at every possible risk level.

Hire Writer The goal is to maximize the Sharpe ratio risk-adjusted return of the portfolio, bounded by the restriction that the exposure to any risky asset class is greater than or equal to zero and that the sum of the weights adds up to one.

The focus is on the relative allocation to risky assets in the optimal portfolio. In the mean-variance analysis, we use arithmetic excess returns. Geometric returns are not suitable in a mean-variance framework. The weighted average of geometric returns does not equal the geometric return of a simulated portfolio with the same composition.

The observed difference can be explained by the diversification benefits of the portfolio allocation. We derive the arithmetic returns from the geometric returns and the volatility. After the members of the IPC perused the results, some of them asked the CIO to explain why the equal-weighted portfolio underperformed the mean-variance optimal portfolio for the periods studied.

Explain to the CIO using only the whole period results. The expected return for this portfolio is 0. We can see from the numbers that the optimal portfolio does better than the naively diversified portfolio because the RTV is higher for the optimal portfolio.

The CAL that is supported by the optimal portfolio is tangent to the efficient frontier. The bottom line is that we have chosen the optimal portfolio that has the portfolio weights that lie on the capital allocation line that is tangent to the efficient frontier.

Which means a portfolio of risky assets that provides the lowest risk for the expected return and thus this selected portfolio is bound to outperform the naively diversified. Would you please explain using the set of results for 3.

This entails an analysis of the economic conditions for different periods. If you see the correlation matrix for the 2 sub periods, we can see that the economic-wide risk factors have imparted positive correlations among the stock returns for Sub Period 2 03 — This was the time of economic crisis and since most of the risk was economic, the optimal portfolio incorporates less risky assets.

While the sub period 1 95 — 03 went through a healthy growth period, had mostly firm specific risk and lesser economic risk. The fundamental concept behind MPT is that the assets in an investment portfolio should not be selected individually, each on their own merits. Rather, it is important to consider how each asset changes in price relative to how every other asset in the portfolio changes in price.

The optimal portfolios derived from the analysis are tangency portfolios and represents the combination offering the best possible expected return for given risk level. If we change the investment limits it could result in sub-optimal portfolios.

When we draw the CAL and the efficient frontier using the above values, we see that the weights in the optimal portfolio result in the highest slope of the CAL. We can see this with the improved reward-to-volatility ratio of the portfolios.

We have tested the sensitivity of the mean-variance analysis to the input parameters.

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Table below shows the impact on the optimal portfolio of an increase and a decrease in the expected volatility of an asset, all other things being equal. Note that a change in volatility affects both the arithmetic return and the covariance matrix.

Again, this table demonstrates the sensitivity of a mean-variance analysis to the input parameters. An increase in expected volatility leads to a lower allocation to that asset class.

High yield even vanishes completely from the optimal portfolio. It is noteworthy that commodities are hardly affected by a higher standard deviation.Mean variance optimization (MVO) is a quantitative tool that will allow you to make this allocation by considering the trade-off between risk and return.

In conventional single period MVO you will make your portfolio allocation for a single upcoming period, and the goal will be to maximize your expected return subject to a selected level of risk. Portfolio optimizer supporting mean variance optimization to find the optimal risk adjusted portfolio that lies on the efficient frontier, and optimization based on minimizing cvar, diversification or maximum drawdown.

Optimization Solutions - Investment and Portfolio Management Examples An investor wants to put together a portfolio, drawing from a set of 5 candidate stocks. What is the best combination of stocks to achieve a given rate of return with the least risk?

Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type.

Markowitz (, ) pioneered the development of a quantitative method that takes the diversification benefits of portfolio allocation into account.

Modern portfolio theory is the result of his work on portfolio optimization. In this paper we present the Markowitz Portfolio Theory for portfolio selection. There is also a reading guide for those who wish to dug deeper into the world of portfolio optimization. Both of us have contributed to all parts .

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Markowitz Portfolio Optimization Case Study Solution and Analysis of Harvard Case Studies