Markowitz Portfolio Optimization Essay

IntroductionMarkowitz (1952, 1956) pi angiotensin-converting enzymeered the development of a decimal method that takes the diversification benefits of portfolio assignation into account. Modern portfolio theory is the conduct of his break d let on portfolio optimization. Ideally, in a mean-variance optimization model, the complete coronation opport hotshot amaze, i.e. all additions, should be considered simultaneously. However, in practice, nigh investors oppositeiate in the midst of different addition mobes inwardly their portfolio- apportioning frameworks. In our let online, we view the process of summation allocation as a four-step execute like Bodie, Kane and Marcus (2005).It consists of choosing the asset classes under conside proportionalityn, piteous forward to establishing neat market expectations, followed by deriving the efficient frontier until decision the best asset mix. We take the panorama of an asset-only investor in search of the optimum port folio. An asset-only investor does non take liabilities into account. The investiture horizon is 5 10 years and the opportunity set consists of twelve asset classes. The investor pursues wealth maximisation and no another(prenominal) particular investiture goals are considered. We solve the asset-allocation problem employ a mean-variance optimization ground on excess elapses.The goal is to maximize the Sharpe ratio ( stake-adjusted croak) of the portfolio, bounded by the restriction that the picture show to any raging asset class is greater than or satisfactory to postcode and that the sum of the weights adds up to one. The focus is on the relative allocation to uns control panel assets in the optimum portfolio. In the mean-variance outline, we drop arithmetic excess pop offs.Geometric increases are non suitable in a mean-variance framework. The plodding average of geometric refunds does non equal the geometric take of a delusive portfolio with the said(p renominal) composition. The observed difference sight be explained by the diversification benefits of the portfolio allocation. We arrive the arithmetic returns from the geometric returns and the excitability.a) The CIO has sent or so of the results you set out through high up to the IPC. after the members of the IPC per wasting diseased the results, some of them asked the CIO to explain wherefore the equal-weighted portfolio underperformed the mean-variance optimal portfolio for the degrees studied. Explain to the CIO employ only the entirely period results.First, lets right away look at some of the determine of the fields that are used to wind the capital allocation line. As an eccentric to my explanation lets go by 2 literalizable capital allocation lines from the jeopardize- loosen rate (rf = 3.5%).The initiatory possible CAL is retracen for naively modify portfolio for the social unit period with rf = 3.5%. The anticipate return for this portfolio is 0.00 6224053, and its commonplace excursus is 0.025002148, the reward-to- excitableness ratio, which is the slope of the CAL is 0.132284095.The second CAL is drawn for the optimal portfolio for the whole period with rf = 3.5%. The evaluate return for this portfolio is 0.009508282, and its exemplification release is 0.00734826, the re- reward-to- unpredictability ratio is 0.897030832.We leave behind cast off from the numbers that the optimal portfolio does disclose than the naively diversified portfolio because the RTV is high for the optimal portfolio. The reason for that is that weve identified the optimal portfolio of put on the liney assets by finding the portfolio weights that result in steepest CAL. The CAL that is supported by the optimal portfolio is tan to the efficient frontier.The bottom line is that we founder chosen the optimal portfolio that has the portfolio weights that lie on the capital allocation line that is tangent to the efficient frontier. Which mean s a portfolio of risky assets that erects the lowest risk for the pass judgment return and thus this selected portfolio is bound to outperform the naively diversified.b) The IPC has noticed that the optimal allocations of sub-period 1 and sub-period 2 are precise different (based on different scenarios of direct returns and investment limits). They asked why. Would you delight explain ( victimisation the set of results for 3.5% risk free rate)? This entails an psychoanalysis of the economic conditions for different periods.The most important insight we get is that in a diversified portfolio, the contri just nowion to portfolio risk of a particular security bequeath depend on the covariance of that securitys return with those of other securities.If you see the coefficient of coefficient of correlativity matrix for the 2 sub periods, we rat see that the economic-wide risk factors have imparted irresponsible correlations among the stock returns for Sub Period 2 (03 10). Th is was the time of economic crisis (08-10) and since most of the risk was economic, the optimal portfolio incorporates less risky assets. season the sub period 1 (95 03) went through a healthy growth period, had more often than not firm specific risk and lesser economic risk.c) The CIO wants to propose investment limits on certain asset classes to the IPC for consideration, but the CIO whitethorn not be aware of the apparent mend on the consummation of the Fund. Since you have run some analysis in a higher place based on the proposed limits, present your analysis and make a recommendation regarding investment limits for the historical arithmetic average (target) return and the 6% p.a. target return.The fundamental judgment 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 channelises in price relative to how every other asset in the portfolio channelises in price.The optimal portfolios derived from the analysis are contact 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.This can be easily from the tables from (comparing nave allocation to optimal allocation)Optimal PortfolioWhen 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 overly motto from the analysis where we constrained the portfolio return to 6% pa, the weights of the optimal portfolio changed and the RTV was lower than the un constrained optimal portfolio.ConstrainedUnconstrainedd) The CIO would like to analyse the sensitivity of the mean-variance optimization to a change in the portfolio target return. Since you have done some runs using the historic al arithmetic average return and 6% p.a. target return, present what youve well-read from your analysis to the CIO using your results.We have judgeed the sensitivity of the mean-variance analysis to the scuttlebutt parameters. 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 in all from the optimal portfolio. It is noteworthy that commodities are only affected by a higher standard deviation. A decrease in volatility mostly leads to a higher allocation. Government bonds, despite their expected energy risk premium, add value repayable to the strong diversification benefit. In this analysis, they count to be insensitive to a change in their expected volatility. Credits and bonds are quite similar asset classes and, in a mean-variance context, the optimal portfolio run fors to incline towards one or the other. In short, the mean-variance analysis suggests that adding real estate, stocks and high yield to the traditional asset mix of stocks and bonds creates most value for investors.Assets Optimal Portfolio Optimal Portfolio (6%)SPTR exponent 0 0RTY mogul 0 0MXEA index number 0 0.747626014MXEU Index 0 0MXEF Index 0 0SPGSCITR Index 0 0FNCOTR Index 0.862665445 0.179140105H15T3M Index 0 0.05WOG1 0 0C0A0 0 0H0A0 0 0G0Q0 0.137334555 0.023233881e) Could we use the optimal weights from a previous period, differentiate sub-period 1 or sub-period 2 or the whole period, as the recommended asset allocation for the next 5 or 10 years? Explain your answer with the out-of-sample test results you have done.No, we cannot recommend asset allocation based on the out-of-s ample test results. The in-sample MV efficient frontiers overestimate the return associated with portfolio optimization not only with respect to resampled susceptibility but importantly with respect to out-of-sample investment performance. Even with good inputs, MV talent hallucination maximizes the risk and returns inputs, creates upward aslant estimates of future performance, and substantially underperforms resampled efficiency.f) Based on the above analyses, what lessons and implications can be learned from your analysis on the mean-variance portfolio optimization? learn lessonsThe fundamental goal of portfolio theory is to optimally allocate your investments amongst different assets. think active variance optimization (MVO) is a quantitative tool which allows you to make this allocation by considering the trade-off between risk and return.Markowitz Portfolio optimizationThe single period Markowitz algorithm solves the next problemSingle Period puzzle* commentarys* The expected return for each asset* The standard deviation of each asset (a measure of risk) * The correlation matrix between these assets* Output* The efficient frontier, i.e. the set of portfolios with expected return greater than any other with the same or lesser risk, and lesser risk than any other with the same or greater return. The Markowitz algorithm is mean as a single period analysis tool in which the inputs provided by the user represent his/her probability beliefs about the approaching period. The expected return, standard deviation, and correlation matrix are computed using standard statistical formulae.The expected return represents the childlike (probability weighted) average of the possible returns for each asset, and the standard deviation represents the uncertainty about the outcome. The correlation matrix is a symmetric matrix, with unity on the diagonal, and all other elements between -1 and +1. A positive correlation between two assets A and B indicates that when the return of asset A turns out to be above (below) its expected value, and then the return of asset B is likely also to be above (below) its expected value. A negative correlation suggests that when As return is above its expected value, and then Bs will be below its expected value, and vice versa.Input Data IssuesA major turn off for the methodology is the selection of input data. The use of historical data provides a very convenient means of providing the inputs to the MVO algorithm, but in that location are a number of reasons why this may not be the optimal way to proceed. All these reasons have to do with the question of whether this method really provides a valid statistical picture of the future period. The most full problem concerns the expected returns, because these control the actual return which is assign to each portfolio.Failure of underlying guessworkWhen you use historical data to provide the MVO inputs, you are implicitly assuming that * The returns in the different periods are independent.* The returns in the different periods are drawn from the same statistical distribution. * The N periods of operable data provide a sample of this distribution. These hypotheses may only not be true. The most serious inaccuracies arise from a phenomenon called mean reversion, in which a period, or periods, of superior (inferior) performance of a particular asset tend to be followed by a period, or periods, of inferior (superior) performance.Suppose, for example, you have used 5 years of historical data as MVO inputs for the upcoming year. The outputs of the algorithm will party favor those assets with high expected return, which are those which have performed well over the past 5 years. Yet if mean reversion is in effect, these assets may well turn out to be those that perform most seedy in the upcoming year.Error in the estimated meanEven if you believe that the returns in the different periods are independent and identically distributed, you ar e of necessity using the available data to estimate the properties of this statistical distribution. In particular, you will take the expected return for a given asset to be the simple average R of the N historical values, and the standard deviation to be the go under mean square deviation from this average value. Then elementary statistics tells us that the one standard deviation error in the value R as an estimate of the mean is the standard deviation divided by the square root of N. If N is not very large, then this error can distort the results of the MVO analysis considerably.SummaryThe above discussion does not mean to imply that the Markowitz algorithm is incorrect, but simply to point out the dangers of using historical data as inputs to a optimization strategy. If you make your own estimates of the MVO inputs, based on your own beliefs about the upcoming period, single period MVO can be an entirely appropriate means of fit the risk and return in your portfolio.