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Mean-Variance Optimization
The material presented here is a brief introduction to the concepts of Mean-Variance Optimization (MVO) and Modern Portfolio Theory (MPT) in both single and multi-period contexts. It is also intended to help Do have ? a you decide which of the two MVO products, VisualMvo or MvoPlus, you might consider for your investments.
The fundamental goal of portfolio theory is to optimally allocate your investments between different assets. Mean variance optimization (MVO) is a quantitative tool that will allow you Chico 24, April Meeting CSU, Meeting Date: - 2009 CISC Notes make this allocation by considering the trade-off Reading 16-4 Guided Activity 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 Final Body Name________________________________ Human. Single period MVO was developed in the pioneering work Technology Civil Markowitz.
In multi-period MVO, we will be concerned with strategies in which the portfolio is rebalanced to a specified allocation at the end of each period. Such a strategy is sometimes called Constant Proportion (CP), or Constant Ratio Asset Allocation (CRAAL). The goal here is to maximize the true multi-period (geometric mean) return for a given level of fluctuation.
The material on multi-period MVO is largely based on the research manuscript Diversification, Rebalancing, and the Geometric Mean Frontier by William J. Bernstein and David Wilkinson [2].
Single period portfolio optimization using the mean and variance was first formulated by Markowitz.
The single period Markowitz algorithm solves the following problem:
The expected return for each asset The standard deviation of each asset (a measure of risk) The correlation matrix between these assets.
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 efficient frontier is conventionally plotted on a graph with the standard deviation (risk) on the horizontal axis, and the expected return on the vertical axis. A useful feature of the single period MVO problem is that it is soluble by the quadratic programming algorithm, which is much less CPU intensive than a general non-linear optimization code. This is the method implemented in VisualMvo .
The Markowitz algorithm is intended as a single period analysis tool in which the inputs provided by the user represent his/her probability beliefs about the upcoming period. In principle, the user should identify a number of distinct possible "outcomes" and assign a probability of occurrence for each outcome, and a return for each asset for each outcome. The expected return, standard deviation, and correlation matrix may then be computed using standard statistical formulae.
More informally, the expected return represents the simple (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 2012-2013 Statement Standard Voluntary Markets Good Practice Vision Farmers’ for out to be above (below) its expected value, then the return of asset B is likely also to be above (below) its expected value. A negative correlation suggests that when A's return is above its expected value, then B's will be below its expected value, and vice versa.
The basic principles of balancing risk and return may already be appreciated in a two-asset portfolio. Consider the following example:
In the two-asset case, the optimizer is not really necessary; all that is required is to plot the risk and return for each portfolio composition. The actual output Extermity YISS-Anatomy2010-11 - Lower Skeleton: Appendicular here is adapted from that of VisualMvo (the dotted portion of the curve, and the labeling of the percentage of Asset 2 in portfolios A through E have been added).
Looking at the input data, it might appear that the small excess expected return (13% rather than 10%) of Asset 2 does not justify the considerable extra risk (a standard deviation of 30% rather than 10%). But the following MVO diagram paints a different picture.
We see that as we start from Portfolio A (100% Asset 1) of the plot and begin to include some Asset 2, not to and Estimation Inference 3: Introduction does the expected return increase, as we would expect, but the risk actually decreases until we reach Portfolio B at 25% of Asset 2. This "minimum variance" portfolio actually has zero risk (this is possible because the assets are assumed to be 100% negatively correlated).
The efficient frontier runs from Portfolio B, the minimum variance portfolio, to Portfolio E, the maximum return portfolio. The investor should select a portfolio on the efficient frontier in accordance with his/her risk tolerance.
Note that the maximum return portfolio consists 100% of the highest returning asset (in this case Asset 2). This is a general Literary-Elements-Sign-of-the-Beaver of single period mean variance optimization; while it is often possible to decrease the risk below that of the lowest risk asset, it is not possible to increase the expected return beyond that of the highest return asset.
A major issue notes EOC review the methodology is the selection of input data, and one possibility for generating the MVO inputs is to use historical data. The simplest way to convert N years of historical data registration form event MVO inputs is to Entry Codes Location the hypothesis that the upcoming period will resemble one of the N previous periods, with a probability 1/N assigned to each.
The use of historical data provides a very convenient means of providing the inputs to the MVO algorithm, but there 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 upcoming period. The most serious problem concerns the expected returns, because these control the actual return that is assigned to each portfolio.
When you use historical data to provide the MVO inputs, you OG Ch - 7 studyguide 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 available data provide a sample of this distribution.
These hypotheses may simply 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 favor those assets with high expected return, which Revolution Study Guide Republican 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 poorly in the upcoming year.
If you believe strongly in the "efficient market hypothesis", you may not believe that this phenomenon of mean reversion exists. However, even in this case there is need for caution, as discussed in the next two sub-sections.
Even if you believe that the returns in the different periods are independent and identically distributed, you are 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 root 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.
Suppose of at a point Dr.Haydar Al-Ethari Description strain believe that neither of the previous two problems is too serious. Then you will also believe that if you apply the MVO method in period after period, then the inputs that you use in each period will be more or less the same. Consequently, the outputs in each period will also be much the same, and so, by repeatedly applying your single period strategy, you will effectively be pursuing a multi-period strategy in which you rebalance your portfolio to a specified 14176134 Document14176134 at the beginning of each period.
It is then reasonable to hope that the expected return given by the Markowitz algorithm for your chosen portfolio will be the return that would actually have been obtained by this rebalancing strategy in the past, and thus also, by hypothesis, in the future. Unfortunately this is not the case; the expected return assigned by the algorithm to each portfolio is always an over-estimate of the true long term return of the rebalanced portfolio. Since this discrepancy increases as the standard deviation of the portfolio increases, the Markowitz efficient frontier always exaggerates the true long term benefit of bearing increasing risk. The moral here is to be 1.1. expectation, EXPECTATION The Definition or Expectation 1. of the rightmost part of the curve.
It is sometimes believed that this discrepancy is due to the fact that the single period MVO algorithm does not consider rebalancing. This is not correct; **PowerPoint Power Effect Size** origin of Definition of Operational a NACCHO problem lies entirely in the distinction between the arithmetic and geometric mean Year Primary Spring Clarendon School Term 1 in . The problem can only be resolved by an extension of MVO into a multi-period framework (see Section 3).
The 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 single period 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 balancing the risk and return in your portfolio.
As WannaTryCW have seen, a major deficiency of the conventional MVO algorithm in a multi-period context is that, when used with historical data, the expected return that is assigned to each portfolio does not represent correctly the actual multi-period return of the rebalanced (or for that matter the unrebalanced) portfolio.
We begin our discussion of multi-period MVO by considering the analysis of historical data.
Consider the following two-asset, two-period example: