Mean-Variance

Question 1

Overview

Answer 1

Mean-Variance Portfolio was developed by Markowitz (1952). The mean-variance optimisation looks to maximise the expected return of a portfolio with respect to the risk tolerance of the investors. These investors select portfolios that maximize E(R) for a given level of risk.

The Markowitz efficient frontier represents all efficient portfolios in the sense that all other portfolios have less expected return for a given level of risk or, equivalently, more risk for a given level of expected return

From a theoretical point of view, only an MV optimization framework can optimally use active forecast information (Sharpe, 1985)

Question 2

Asset Allocation Overview

Answer 2

• An important use of Markowitz’s mean-variance efficiency is its ability to differentiate between optimal amounts of stocks, bonds, real estate, commodities ect in asset allocation.

Question 3

• Strategic asset allocation

Answer 3

• which is the long-term optimal mix of assets based primarily on return or

Question 4

Tactical asset allocation

Answer 4

which is a short-term strategy that changes the optimal weights due to changing inputs for u and V.

Question 5

Equity Optimisation

Answer 5

• In addition, another use of mean-variance is equity optimisation. It allows for comparison of proposed portfolio to a benchmark.
• This highlights possible abnormal returns relative to benchmark (r=Ri-Rb-mrk)
• Furthermore, one is able to use analysis for large scale portfolio optimisation, which is usually done in relation to a benchmark.
• All of the mean-variance analysis can be done using residual returns, which are asset returns minus the benchmark return.

Question 6

Index Tracking

Answer 6

can use mean-variance optimization to minimize the tracking error relative to a specified index. This is useful when not able to replicate an index, face liquidity and trading cost constraints, and when faced with ethical portfolio restrictions.

Question 7

Variance - poor measure of risk

Answer 7

• In MV efficiency the variance or standard deviation of return is the measure of risk. This means the variance both above and below the mean is considered risk. However, investors typically only view downside risk, ie the variance below the mean as ‘risk’ and thus would not typically count the variance above the mean. An alternative to this would be using alternative risk measures like such as semi-variance which is an example of a downside risk measure.
  It also does not account for time-varying covariance so as time increases, risk estimate becomes less accurate.

Using different measures of risk often lead to similar portfolios as mv efficiency over short return horizons

Question 8

Only consistent with expected utility maximisation under either of 2 conditions

Answer 8

1. Normally distributed asset returns
   - This is unacceptable as a realistic hypothesis as although diversified equity portfolio and capital market index returns are often reasonably symmetric, their distribution is not normal.
   
   2. Quadratic utility
      
      • This implies that investors can reach a point where more wealth actually reduces their utility, again this is not accepted as true.

• When the return interval is short, e.g. monthly or quarterly, optimal portfolio from maximizing utility functions are often similar to mean-variance portfolios (Kroll, Levy and Markowitz(1984), and Best and Gruaer(1993)).

Question 9

Single-period analysis

Answer 9

Markowitz MV efficiency is a single-period model for investment behaviour. Many institutional investors, however, such as endowment and pension funds, have long-term investment horizons on the order of 5, 10, or 20 years.

Question 10

Estimation Risk

Answer 10

Michaud argues that the estimation risk problem (oversensitivity to estimation error) is the biggest limitation of the mean-variance efficiency and not the previous criticisms mentioned in chapter 3 of their book (2008).

Risk-return estimates are highly uncertain in investment practice and sensitivity to changes in optimization inputs leads to portfolio optimality ambiguity. The problem of uncertain estimates is compounded by how investment information is represented in digital computer optimizations

The main problem is very unstable and extreme portfolio weights and poor out-of-sample performance (Jobson and Korkie(1981), DeMiguel, Garlappi and Uppal(2009)).

MV Optimisation is overly sensitive to estimation error in risk-return estimates and have poor out-of-sample performance characteristics. The Resampled Efficiency™ (RE) techniques presented in Michaud (1998) introduce Monte Carlo methods to properly represent investment information uncertainty in computing MV portfolio optimality and in defining trading and monitoring rules

The ambiguity of traditional MV optimization as a result of estimation error opens the door to a fundamentally new statistical perception of MV efficiency - one such procedure is the resampled efficiency.

Question 11

Resampled Efficiency

Answer 11

RE technology introduces Monte Carlo resampling and bootstrapping methods into MV optimization to more realistically reflect the uncertainty in investment information. The end result is generally more stable, realistic, and investment effective MV optimized portfolios. RE technology also includes statistically rigorous portfolio trading and monitoring rules and tests for significant assets avoiding the often ineffective and costly rebalancings typical of the MV optimization asset management process.
The REF frontier is relevant when there is estimation uncertainty in the inputs and moderates the extreme weights in the mean-variance portfolios.
Michaud and Michaud(2008) argue that the REF portfolios, as they are averages across simulated frontiers, are less affected by the characteristics of the inputs.
They provide more intuitive optimal portfolios and are more stable especially for the higher risk portfolios.

In a simulation performed by Michaud and Michaud (2007) following Jobson and Korkie (1981) and Michaud (1998, chpt 6 ) it proved that the RE optimizer, on average, achieves roughly the same return with less risk, or alternatively more return with the same level of risk, relative to the Markowitz optimizer.
This illustrates that the RE optimized portfolios are, on average, more effective at improving risk-adjusted investment performance.

When properly used, a new framework emerges that provides a far more reliable and productive route for research and effective asset manageme