Portfolio Theory - ACC2034 - PDF

ACC2034 Portfolio Theory Reading List Chapter Chapter Chapter 3 7. Learning Objectives 1. Explain the concept of portfolio theory and understand the mathematical measures and framework. 2. Identify the optimal p...

ACC2034 Portfolio Theory Reading List Chapter Chapter Chapter 3 7. Learning Objectives 1. Explain the concept of portfolio theory and understand the mathematical measures and framework. 2. Identify the optimal portfolio by using the investment forecasts and the risk appetite of the investor. 3. Recognise the advantages and disadvantages of using portfolio theory. Portfolio Management Process 1. Preparing policy statement The Investment Policy Statement (IPS). Specify investment objectives and constraints. 2. Examine current/projected financial and economic conditions Determine investment strategies to meet goals. Requires monitoring and updating. 3. Construct the portfolio Asset allocation based on the Investment Policy Statement. Minimise investment risk and achieve investment objectives. 4. Monitor and update Measure and evaluate performance. Monitor and adjust accordingly. Types of Investors B Risk-Loving Investor: An investor who has a preference for risk. A Risk-Averse Investor: An investor that will only take on increased risk if there is sufficient prospective return to compensate. C Risk-Neutral Investor: An investor that, given the same expected return, will be indifferent towards the degree of risk. How do we know what risk profile an investor is? See GWM Investment Policy Statement (IPS) A typical format will include the client’s investment objectives and the constraints that apply to the client’s portfolio. The IPS should state clearly the risk tolerance of the client. Risk objectives are specifications for portfolio risk that reflect the client’s risk tolerance. Quantitative risk objectives can be absolute, relative, or a combination of the two. Examples of an absolute risk objective would be a desire not to suffer any loss of capital or not to lose more than a given percentage of capital in any 12-month period. Challenge to a Rational Investor In practice, an investor is presented with different assets/ securities with different levels of return basing on the underlying risk. The challenge is how to best select the asset or combination of assets or securities investments that present an optimal balance between risk and return on the investment and maximizes their utility. Mordern Portfolio Theory – Some concepts Risk: the uncertainty of future outcomes => measure: the variance or deviation of the investment return from the level expected. Return: the reward an investor earns from investing/committing their capital to a given asset/security. Portfolio: A portfolio is a collection of assets such as stocks, property, bonds, currencies, etc. Opportunity set: This is the set of available portfolios that an investor can choose based on their combinations of risk and return. Diversification: Diversification is the process of mixing different assets within a portfolio to ensure that unsystematic risk is smoothed out. In this case, the negative performance of a given asset/security within the portfolio is balanced out by the positive performance of other assets within the portfolio. Mordern Portfolio Theory Initiated by Harry Markowitz in the early 1950s (later awarded the Nobel prize in 1990).  Quantitative trade-off between return and risk of the investment.  Quantify the concept of diversification.  Foundation of many asset allocation models: mean-variance optimisation model (MV model).  MV model provides the “optimal portfolio”.  Requirements: expected performance of the investments and the Portfolio Theory Assumptions: Mean Expected Return 1. Investors consider each investment alternative as being represented by a probability distribution of expected returns over some holding period. – True or False? 2. Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth. Utility Explanation 3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns. – True or False? 4. Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only. ? 5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected return,  Markowitz Portfolio Theory Using these five assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk or lower risk with the same (or higher) expected return.  Alternative Measures of Risk Variance or standard deviation of expected return Range of returns Returns below expectations – Semivariance – a measure that only considers deviations below the mean – These measures of risk implicitly assume that investors want to minimise the damage from returns less than some target rate  Alternative Measures of Risk The advantages of using standard deviation of returns – This measure is somewhat intuitive – It is a correct and widely recognised risk measure – It has been used in most of the theoretical asset pricing models  Expected Rates of Return For an individual asset - Revison – It is equal to the sum of the potential returns multiplied with the corresponding probability of the returns.  Expected Rates of Return If you want to construct a portfolio of n risky assets, what will be the expected rate of return on the portfolio is you know the expected rates of return on each individual assets? – The formula E(Rpo r t ) = where: Wi = the percent of the portfolio in asset i E(R i ) = the expected rate of return for asset i – See Exhibit 5.2 5-  Expected Rates of Return For a portfolio of investments – It is equal to the weighted average of the expected rates of return for the individual investments in the portfolio Individual Investment Risk Measure Variance is a statistical measurement of the spread between numbers in a data set. It measures how far each number in the set is from the mean (average), and thus from every other number in the set. Variance is often depicted by this symbol: σ2. It is used by both analysts and traders to determine volatility and market stability. Calculation 5-  Covariance of Returns Covariance – Direction of Movement A covariance refers to the measure of how two random variables will change when they are compared to each other. In a financial or investment context, though, the term covariance describes the returns on two different investments over a period of time when compared to different variables. These assets are usually marketable securities in an investor's portfolio, such as stocks. Calculation Correlation – degree of movement Positive Correlation vs. Inverse Correlation: An Overview In the field of statistics, correlation is a relationship between two variables. Variables are correlated if a change in one is accompanied by a change in the other. Correlation shows if the relationship is positive or negative and how strong the relationship is. Positive correlation is the relationship between two variables that change together. Inverse correlation is the relationship between two variables that change in opposite directions. Calculation Covariance vs Correlation Covariance Measures the direction of the relationship between two variables, and how much they vary together. Covariance is positive when both variables increase or decrease together, and negative when they move in opposite directions. Correlation Measures the strength of the relationship between two variables, and how one variable may impact the other. Correlation is a special case of covariance, and is calculated by dividing the covariance by the product of the standard deviations of the variables. Correlation values range from -1 to +1, with extreme values indicating a strong relationship. Asset Class Correlation Link to correlation  Standard Deviation of a The Portfolio formula where: 𝜎port = the standard deviation of the portfolio Wi = the weights of the individual assets in the portfolio, where weights are determined by the proportion of value in the portfolio 𝜎2 = the variance of rates of return for asset i i Covij = the covariance between the rates of return for assets i and j, where Link to calculation in practice  Standard Deviation of a Portfolio Calculations with a two-stock portfolio – Any asset of a portfolio may be described by two characteristics: ▪ The expected rate of return ▪ The expected standard deviations of returns – The correlation, measured by covariance, affects the portfolio standard deviation – Low correlation reduces portfolio risk while not affecting the expected return  Standard Deviation of a Portfolio – Assets may differ in expected rates of return and individual standard deviations – Negative correlation reduces portfolio risk – Combining two assets with +1.0 correlation will not reduces the portfolio standard deviation – Combining two assets with -1.0 correlation may reduces the portfolio standard deviation to zero  Standard Deviation of a Portfolio A three-asset portfolio – The results presented earlier for the two-asset portfolio can extended to a portfolio of n assets – As more assets are added to the portfolio, more risk will be reduced everything else being the same – The general computing procedure is still the same, but the amount of computation has increase rapidly – For the three-asset portfolio, the computation has doubled in comparison with the two-asset portfolio  Estimation Issues Results of portfolio allocation depend on accurate statistical inputs Estimates of – Expected returns – Standard deviation – Correlation coefficient ▪ Among entire set of assets ▪ With 100 assets, 4 950 correlation estimates Estimation risk refers to potential errors  The Efficient Frontier The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return Efficient frontier are portfolios of investments rather than individual securities except the assets with the highest return and the asset with the lowest risk The efficient frontier curves  Efficient Frontier and Investor Utility An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk The slope of the efficient frontier curve decreases steadily as you move upward The interactions of these two curves will determine the particular portfolio selected by an individual investor The optimal portfolio has the highest utility for a given investor  Efficient Frontier and Investor Utility The optimal lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility As shown in Exhibit 5.16, Investor X with the set of utility curves will achieve the highest utility by investing the portfolio at X As shown in Exhibit 5.16, with a different set of utility curves, Investor Y will achieve the highest utility by investing the portfolio at Y Diversificati on How many securities the investor should choose? A portfolio with a “sample” of securities that are less correlated to each other or a portfolio of “All” securities? diversification : The process of investing in a variety  Efficient of securities taking into account the correlations and variances of the securities to create optimal portfolios.  Naive diversification : The process of investing in a variety of securities in equal value-weighted proportions. * Naïve portfolio can outperform MV model (DeMiguel et al., Benefit of Diversification Number of Securities (N) Reduction in Specific Risk (%) 1 0 2 46 4 72 8 81 16 93 32 96 64 98 500 99 Source: Pike et al., 2015. Diversificati on Number of underlying assets Use of specific vehicles https://am.jpmorgan.com/gb/en/asset-management/adv/products/jpm-multi-asset- cautious-fund-c-net-accumulation-gb00bjrdjv84 Portfolio Theory - Concerns  Difficulty of having the correct and accurate input parameters.  Optimal allocation is very sensitive to small changes in the input parameters. Thank you. 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