Markowitz Portfolio Optimization

Categories: FinanceInvestment


Markowitz (1952, 1956) 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. Ideally, in a mean-variance optimization model, the complete investment opportunity set, i.e. all assets, should be considered simultaneously. However, in practice, most investors distinguish between different asset classes within their portfolio-allocation frameworks. In our analysis, we view the process of asset allocation as a four-step exercise like Bodie, Kane and Marcus (2005).

It consists of choosing the asset classes under consideration, moving forward to establishing capital market expectations, followed by deriving the efficient frontier until finding the optimal asset mix. We take the perspective of an asset-only investor in search of the optimal portfolio. An asset-only investor does not take liabilities into account. The investment horizon is 5 – 10 years and the opportunity set consists of twelve asset classes. The investor pursues wealth maximization and no other particular investment goals are considered.

We solve the asset-allocation problem using a mean-variance optimization based on excess returns.

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.

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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.

a) The CIO has sent some of the results you have done above to the IPC. 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.

First, let’s quickly look at some of the values of the fields that are used to draw the capital allocation line. As an example to my explanation let’s go through 2 possible capital allocation lines from the risk-free rate (rf = 3.5%).

The first possible CAL is drawn for naively diversified portfolio for the whole period with rf = 3.5%. The expected return for this portfolio is 0.006224053, and its standard deviation is 0.025002148, the reward-to-volatility 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 expected return for this portfolio is 0.009508282, and its standard deviation is 0.00734826, the re- reward-to-volatility ratio is 0.897030832.

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 reason for that is that we’ve identified the optimal portfolio of risky assets by finding the portfolio weights that result in steepest CAL. 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.

b) The IPC has noticed that the optimal allocations of sub-period 1 and sub-period 2 are very different (based on different scenarios of target returns and investment limits). They asked why. Would you please explain (using the set of results for 3.5% risk free rate)? This entails an analysis of the economic conditions for different periods.

The most important insight we get is that in a diversified portfolio, the contribution to portfolio risk of a particular security will depend on the covariance of that security’s return with those of other securities.

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 – 10). This was the time of economic crisis (08-10) 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.

c) The CIO wants to propose investment limits on certain asset classes to the IPC for consideration, but the CIO may not be aware of the likely impact on the performance of the Fund. Since you have run some analysis above 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 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.

This can be easily from the tables from (comparing naïve allocation to optimal allocation):

Optimal Portfolio:

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 also saw 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.

d) The CIO would like to test the sensitivity of the mean-variance optimization to a change in the portfolio target return. Since you have done some runs using the historical arithmetic average return and 6% p.a. target return, present what you’ve learned from your analysis to the CIO using your results.

We have tested the sensitivity of the mean-variance analysis to the input 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 completely from the optimal portfolio. It is noteworthy that commodities are hardly affected by a higher standard deviation. A decrease in volatility mostly leads to a higher allocation. Government bonds, despite their expected zero risk premium, add value due to the strong diversification benefit. In this analysis, they appear 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 tends 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 Index| 0| 0|
RTY Index| 0| 0|
MXEA Index| 0| 0.747626014|
MXEU Index| 0| 0|
MXEF Index| 0| 0|
SPGSCITR Index| 0| 0|
FNCOTR Index| 0.862665445| 0.179140105|
H15T3M Index| 0| 0.05|
WOG1| 0| 0|
C0A0| 0| 0|
H0A0| 0| 0|
G0Q0| 0.137334555| 0.023233881|

e) Could we use the optimal weights from a previous period, say 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-sample test results. The in-sample MV efficient frontiers overestimate the return associated with portfolio optimization not only with respect to resampled efficiency but importantly with respect to out-of-sample investment performance. Even with good inputs, MV efficiency error maximizes the risk and returns inputs, creates upward biased 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?

Key lessons:

The fundamental goal of portfolio theory is to optimally allocate your investments between different assets. Mean 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 Optimization

The single period Markowitz algorithm solves the following problem:

  • Single Period Problem
  • Inputs
  • 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 intended as a single period analysis tool in which the inputs provided by the user represent his/her probability beliefs about the upcoming period. The expected return, standard deviation, and correlation matrix are computed using standard statistical formulae.

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 turns 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, and then B’s will be below its expected value, and vice versa.

Input Data Issues

A major issue 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 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 which is assigned to each portfolio.

Failure of underlying hypothesis

  • When 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 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 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 poorly in the upcoming year.

Error in the estimated mean

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.


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 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.

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Markowitz Portfolio Optimization. (2017, Mar 30). Retrieved from

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