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Markowitz (1952, 1956) introduced a quantitative approach that considers the benefits of diversification in portfolio allocation. This work led to the development of modern portfolio theory, which focuses on optimizing portfolios. Ideally, a mean-variance optimization model should consider all assets in the investment opportunity set. However, in practice, investors typically categorize assets into different classes. Our analysis adopts a four-step approach to asset allocation, similar to Bodie, Kane and Marcus (2005).

The process encompasses choosing asset classes, establishing market expectations, and creating the efficient frontier to identify the optimal asset mix.

The primary objective is to maximize wealth for an investor solely focused on assets, without taking liabilities into account. The investment timeframe spans 5 - 10 years and there are twelve available asset classes. The aim is to enhance returns using mean-variance optimization by considering excess returns.

The objective is to optimize the Sharpe ratio, a risk-adjusted return measure, for the portfolio while ensuring that exposure to any risky asset class is non-negative and the sum of weights adds up to one.

The main emphasis lies on allocating risky assets in the optimal portfolio using mean-variance analysis based on arithmetic excess returns.

Geometric returns are not appropriate within a mean-variance context as the weighted average of geometric returns does not match the geometric return of a simulated portfolio with identical composition. The discrepancy can be attributed to the diversification advantages of portfolio allocation. The arithmetic returns are derived from the geometric returns and volatility.

The CIO has shared the aforementioned results with the IPC.

Upon reviewing these results, some members of the IPC requested an explanation from the CIO regarding the underperformance of the equal-weighted portfolio compared to the mean-variance optimal portfolio during the studied periods. Please justify this discrepancy to the CIO, solely based on the overall period results.

First, we will examine the values of the fields that are employed in creating the capital allocation line. As an illustration to clarify my explanation, we will examine 2 potential capital allocation lines originating from the risk-free rate (rf = 3.5%).

The first CAL is produced for a portfolio that is naively diversified over the entire period, with a risk-free rate of 3.5%. The expected return for this portfolio is 0.006224053, and its standard deviation is 0.025002148. The slope of the CAL, which represents the reward-to-volatility ratio, is 0.132284095.

The second Capital Allocation Line (CAL) is drawn for the Optimal portfolio for the entire period with a risk-free rate (rf) of 3.5%. The expected return for this portfolio is 0.009508282, and its standard deviation is 0.00734826. The reward-to-volatility ratio of this portfolio is 0.897030832.

By observing the numbers, it becomes apparent that the optimal portfolio outperforms the naively diversified portfolio due to its higher RTV. This can be attributed to the fact that we have determined the optimal portfolio of risky assets by identifying the portfolio weights that yield the steepest CAL. The CAL supported by the optimal portfolio coincides with the efficient frontier.

In conclusion, we have selected the optimal portfolio with weights lying on the capital allocation line and being tangent to the efficient frontier. This portfolio consists of risky assets that offer the lowest risk for the expected return. Therefore, this chosen portfolio is expected to outperform the naively diversified portfolio.

The IPC has observed a significant disparity in the optimal allocations of sub-period 1 and sub-period 2, depending on various scenarios of target returns and investment limits. They have requested an explanation for this discrepancy, specifically in relation to the set of results for a risk-free rate of 3.5%. This analysis will involve examining the economic conditions during different periods.

In a diversified portfolio, the contribution to portfolio risk of a specific security is determined by the covariance between the return of that security and the returns of other securities.

If you examine the correlation matrix for the two sub periods, you will notice that the economic-wide risk factors have caused positive correlations among the stock returns for Sub Period 2 (03 – 10). This period coincided with an economic crisis (08-10), and as a result of the predominant economic risk, the optimal portfolio consists of less risky assets.

During the sub period 1, which lasted from 95 to 03, there was a period of healthy growth characterized by mostly firm-specific risk and a lower level of economic risk.

The CIO wants to suggest investment limits for certain asset classes to the IPC. However, it is possible that the CIO is not aware of the potential impact on the Fund's performance. I have conducted analysis on the proposed limits and would like to present the findings. Based on historical arithmetic average (target) return and the 6% p.a. target return, I will provide a recommendation regarding investment limits.

The core concept of MPT is that when building an investment portfolio, asset selection should not solely rely on their individual qualities. Rather, it is essential to assess how each asset's price fluctuates in comparison to the price movements of all other assets within the portfolio.

By producing tangency portfolios, the analysis generates portfolios that provide the highest expected return for a specific level of risk. However, altering investment limits can result in suboptimal portfolios.

It is easy to compare the tables of naïve allocation to optimal allocation.

When drawing the CAL and efficient frontier with the given values, it is evident that the weights in the optimal portfolio lead to the highest slope of the CAL. This is demonstrated by the portfolios' improved reward-to-volatility ratio.

Additionally, when we enforced a constraint on the portfolio return of 6% pa, we observed that the weights of the optimal portfolio adjusted accordingly. As a result, the Risk Tolerance Value (RTV) was reduced compared to the optimal portfolio without constraints.

The CIO is interested in testing the impact of changing the portfolio target return on the mean-variance optimization. You should update the CIO with what you have learned from your analysis, which utilized the historical arithmetic average return and a 6% p.a. target return.

We conducted sensitivity tests on the input parameters for mean-variance analysis. The following table illustrates the effect on the optimal portfolio when the expected volatility of an asset is increased or decreased, while keeping all other factors constant. It is important to note that changes in volatility impact both the arithmetic return and the covariance matrix. This table further highlights the sensitivity of mean-variance analysis to input parameters, as an increase in expected volatility results in a reduced allocation to the respective asset class.

High yield might completely disappear from the optimal portfolio. It is worth noting that commodities are rarely affected by increased standard deviation. A decrease in volatility generally results in a higher allocation. Despite their expected risk-free premium, government bonds contribute value due to strong diversification benefits. They seem unaffected by changes in expected volatility in this analysis. Credits and bonds are similar asset classes and, in a mean-variance context, the optimal portfolio tends to lean towards one or the other. In summary, the mean-variance analysis indicates that including real estate, stocks, and high yield with the traditional asset mix of stocks and bonds offers the 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|

The question is whether the optimal weights from a previous period, such as sub-period 1, sub-period 2, or the entire period, can be used as the recommended asset allocation for the next 5 or 10 years. Please provide an explanation along with the out-of-sample test results you have conducted.

We cannot suggest asset allocation using the out-of-sample test results. The in-sample MV efficient frontiers overestimate portfolio optimization returns, not only in terms of resampled efficiency but also in terms of out-of-sample investment performance. Even with reliable inputs, MV efficiency error increases risk and return inputs, causing upward biased estimates of future performance and significantly lower efficiency compared to resampled efficiency.

f) What lessons and implications can be learned from the analysis on mean-variance portfolio optimization?

Portfolio theory aims to effectively distribute investments among various assets. Mean variance optimization (MVO) is a quantitative tool that facilitates this allocation by assessing the risk-return trade-off.

- 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 designed for single period analysis. The user provides inputs that represent their probability beliefs for the next period. The algorithm then uses standard statistical formulas to calculate the expected return, standard deviation, and correlation matrix.

The average possible returns for each asset, weighted by probability, are represented by the expected return. The uncertainty associated with these returns is indicated by the standard deviation. The correlation matrix is a symmetric matrix where the diagonal elements have a value of unity, and all other elements range between -1 and +1. If assets A and B have a positive correlation, it means that if asset A's return exceeds or falls below its expected value, asset B's return is also likely to exceed or fall below its expected value. On the other hand, if assets A and B have a negative correlation, it suggests that when asset A's return surpasses its expected value, asset B's return will be lower than its expected value, and vice versa.

A crucial concern regarding the methodology is the choice of input data. Utilizing historical data is an efficient approach for supplying inputs to the MVO algorithm. However, several factors may suggest that this approach might not be the most ideal method to adopt. All these factors revolve around the question of whether this method genuinely presents an accurate statistical representation of the future period. The primary issue lies in the expected returns, as these determine the actual return allocated to each portfolio.

- 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 not be true as the most significant inaccuracies occur due to mean reversion. This phenomenon implies that a period or periods of superior (inferior) performance of an asset are typically followed by a period or periods of inferior (superior) performance.

If you have used 5 years of historical data as inputs for the upcoming year in an MVO algorithm, it will favor assets with high expected return - meaning those that have performed well in the past 5 years. However, if mean reversion is happening, these assets may actually perform poorly in the upcoming year.

Even if you believe that the returns in different periods are independent and identically distributed, you still use the available data to estimate the properties of this statistical distribution. For instance, you determine the expected return for a specific asset as the simple average R of the N historical values, and the standard deviation as the root mean square deviation from this average value. According to elementary statistics, the one standard deviation error in the estimation of the mean value R is determined by dividing the standard deviation by the square root of N. However, if N is not sufficiently large, this error can significantly distort the results of the MVO analysis.

While it is not suggesting that the Markowitz algorithm is wrong, the above discussion emphasizes the risks associated with utilizing historical data as inputs for an optimization strategy. However, if one develops their own estimates of the inputs for Mean-Variance Optimization (MVO) based on their own predictions for the future, using single period MVO can be a suitable approach to balancing the risk and return in a portfolio.

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