APPEND I X NOTATION. The product of the values produced by a function f by inputting all n from n=o to n=n

Transcription

1 APPEND I X NOTATION In order to be able to clearly present the contents of this book, we have attempted to be as consistent as possible in the use of notation. The notation below applies to all chapters of this book. Generally, plain text indicates scalars, and bold text indicates vectors or matrices. CALCULUS N Lf(n) n O N n J(n) II Indicates standard multiplication Indicates vector or matrix multiplication Indicates 'for all observations' within a set of data The sum of the values produced by a function f by inputting all n from n=o to n=n The product of the values produced by a function f by inputting all n from n=o to n=n RETURN, RISK AND EXPOSURES RA,11 r ~., The total return on an asset A at time period n The rate of return on an asset A at time period n

2 APPENDIX NOTATION E;,t W; -+ w -+T w -+ r u f3a -+ p The risk-free rate of return The rate of excess return on an asset A at time period n The expected rate of excess return on an asset A at time period n The non-factor related or asset specific (residual) return on asset i The return on risk factor k at time t The residual return on asset i at timet The volatility of asset i The portfolio weight of asset A A vector of portfolio asset weights The transpose of the vector of portfolio asset weights A vector of portfolio asset active weights The transpose of the vector of portfolio asset active weights A vector of portfolio returns The diagonal matrix of portfolio volatilities The transpose of the diagonal matrix of portfolio volatilities The beta of asset A The vector of sector (or portfolio asset) betas Portfolio active risk (tracking error) Portfolio residual risk PORTFOLIO CORRELATION AND COVARIANCE -+ e -+T e I The correlation coefficient between asset A and asset B The portfolio's Correlation Matrix The portfolio's Variance-Covariance Matrix The inverse of the portfolio's Variance-Covariance Matrix The unit vector of ls. The transpose of the unit vector of ls. The Identity Matrix of diagonal Is.

3 GLOSSARY OF TERMS Active Portfolio Management Active Return Active Risk Active Weight Agent AIMR Alpha Arbitrage Arbitrage Pricing Theory (APT) Asset Allocation Asset Manager Autoregressive Conditional Heteroscedastidty The pursuit of returns above that of a predefined benchmark, by actively managing a portfolio. The portfolio's return above the return of the benchmark. The volatility of the portfolio's active returns. The portfolio holding or weight (in percentage terms) exceeding the holding or weight of the portfolio's benchmark. The legal entity (person or company) that is employed by a Principal to perform a task. Association for Investment Management and Research. The statistical outperformance of a portfolio relative to a benchmark. Determined as the intercept, when regressing observed benchmark returns onto observed portfolio returns. The ability to make a profit on trading an asset while incurring no risk. A multi-factor version of the CAPM, which includes a number of explanatory variables. The intentional distribution of a portfolio across the available range of investable assets. A person or a firm that has as its business the investment and management of client assets. A statistical forecasting technique that builds on the assumption that future returns are to some extent a function of past returns.

4 GLOSSARY OF TERMS Bayesian Priors Behavioural Finance Benchmark Benchmark Related Risk Benchmark Risk Bet Beta Bivariate Return Distribution CAC40 Capital Asset Pricing Model (CAPM) Characteristic Line Compounding Effect Compound Interest Consensus Efficient Frontier Constraints Continuously Compounded Interest Correlation Correlation Coefficient Values or numbers that are applied to an estimation technique in order to serve as an anchor for the end result. Usually reflects some a priori knowledge about the variable being estimated. The science of explaining the behaviour of financial agents, based on human nature. An index against which portfolio return and risk are measured, providing the basis for calculating Active Weights, Active Return and Active Risk. The proportion of Active Risk or Tracking Error related to the market (or benchmark). The volatility (annualised standard deviation) of the return on the benchmark. See Active Weight. Mathematical measure of the co-variation of the return on an asset with the return on the market (or benchmark). Determined by dividing the covariance of the asset and the market with the variance of the market. Non-normal return distribution that is a weighted sum of two normal distributions. French Equity Index. A single factor model for explaining asset returns as a function of the return on the market. The empirical relationship between the return on an asset and the return on a portfolio or a benchmark. The arithmetic procedure of calculating interest on payable interest. Interest calculated for each period on interest accrued in previous periods. The efficient frontier that is implied by the consensus expectations about return, risk and correlations for a portfolio of assets. Limitations imposed on a quantitative optimisation in order to restrict the resultant asset allocation to a desired sub-space of results. Interest calculated continuously on interest accrued in previous periods. The degree to which two variables move in similar direction over time. The coefficient measuring the degree of correlation between two variables. By definition the correlation coefficient ranges between -1 and + 1.

5 GLOSSARY OF TERMS Covariance DAX30 Debt to Equity (D/E) Discount Discount Factor Discounted Free Cash Flow Discounting Diversification Dividend Dividend Discount Model (DDM) Dividend Yield Dow Jones Industrial Average Earnings Yield EBIT Efficient Frontier Efficient Portfolios Efficient Ridge The degree and magnitude to which two variables move together over time. German Equity Index. The ratios of debt to equity on a company's balance sheet. Applied along with other measures to measure the financial quality of a company. The amount less that an investor is willing to pay for an asset which is believed to be inferior relative to an appropriate peer group. The factor used to discount future payments back to the present, to adjust for the opportunity cost of receiving the payment later as opposed to immediately. The cash flow available in a company after operating, investing and financial activities. Used to assess the value of a company or other cash generating (portfolios of) assets. The mathematical procedure of determining the present value of future payments using the discount factor. Also, the inclusion of the effects of relevant events into the assessment of the value of a stock, on the part of investors. The property of a portfolio of assets exhibiting less risk than the (weighted) average of asset volatilities. A result of correlation coefficients being less than 1. The payment by a company to its shareholders as a form of return on their investment. A series of valuation models using the discounted future stream of dividends paid, to determine the value of a company. The ratio of dividends to current share price. Comparable to the yield on a bond. US Equity Index. The ratio of earnings to current share price. Comparable to the yield on a bond. Earnings Before Interest and Tax. The two-dimensional representation of the return/risk space in which efficient portfolios are plotted. Portfolios or asset allocations that generate the highest attainable return for a given level of risk, or the lowest attainable level of risk for the highest level of return. The three-dimensional representation of the return/risk/probability space, in which efficient portfolios are plotted. &;.3&1. ~~

6 GLOSSARY OF TERMS Equal Weighting Equity Risk Premium EVA EV/EBITDA Exponential Weighting Factor Exposures Factor Models Factors Fund Manager Gearing Geometric Compounding GIPS Hang Seng Historical VaR Simulation Identity Matrix Implied Growth Implied Volatility Information Coefficient Weighting scheme applied to the calculation of standard deviation. Also, the action of ensuring that all types of assets in a portfolio are of equal value. The required additional return on equity over the riskfree return (approximated by the return on a long-term government bond). Reflect the premium the investor requires to hold a risky asset such as equity. Economic Value Added. A corporate management tool for assessing the efficiency or profitability of resources employed. The ratio of Enterprise Value (market capitalisation less debt) to Earnings Before Interest, Tax, Depreciation and Amortisation. Applied along with other measures to measure the value of a company. The process of calculating a weighted average while placing an exponentially increasing weight on observations the more recent they are. Exponentially Weighted Moving Average (EWMA) volatility is an example. The factor model exposure of a company's stocks to a specific factor. Quantitative multiple regression models that attempt to explain return and risk characteristics of individual assets based on each asset's exposure to a number of predefined global or universal factors. The explanatory variables that enter into factor models. Examples are interest rates, GOP growth, sector classification, balance sheet leverage and so on. See Portfolio Manager. When a fund or portfolio has borrowed to invest more than the value of its assets. See Compounding Effect. Global Investment and Performance Standards. Hong Kong Equity Index. Risk assessment technique that employs past return distributions to infer future portfolio risk. A matrix that has 1 s along the diagonal and Os in all other cells. The short-term growth implied by a share price as a function of its current level, an assumed long term growth rate and an assumed discount rate. The volatility implied by the price of an option. A measure of portfolio manager skill, this is the correlation between forecast and realised returns.

7 GLOSSARY OF TERMS Information Ratio Investable Universe Investment Manager LaGrange Estimator Leverage Long Position Marginal Contribution to Risk Market Efficiency Maximum Variance Portfolio Mean Mean-Variance Efficiency Mean-Variance Optimisation Minimum Variance Portfolio Modern Portfolio Theory Modified Dietz MSCI The ratio of active return to active risk, or the ratio of residual return to residual risk. The universe of stocks or other assets from which the asset manager is not restricted when investing. See Asset Manager. A technique for determining a model optimum by use of partial derivatives. The degree to which a company has borrowed to finance its operational activities. A net bought position in an asset. The net increase in either total portfolio or active risk, as a result of a marginal change in the (active) weight of a portfolio asset. The property of assets reflecting completely all the information available about them. The portfolio which exhibits the greatest attainable degree of risk. Average of a sample. See Efficient Portfolios. See Quantitative Portfolio Optimisation. The portfolio which exhibits the lowest attainable degree of risk. Mathematical framework for describing and assessing the return and risk of a portfolio of assets, using only returns, volatilities and correlations. Return calculation method designed to account for intertemporal errors that may occur if an asset is traded between two time measurements. Morgan Stanley Capital International. Multivariate Density Estimation Mathematical forecasting technique, whereby a forecast is made based on the similarity of the present state with previous states, across a range of arbitrarily chosen variables. NASDAQ Composite Net Present Value OLS Opportunity Cost Optimal Portfolios US Equity Index. The present value of a discounted future stream of cash flows. Ordinary Least Squared. A statistical technique for assessing the fit of a model with a set of observed data. The unrealised cost forgone by not participating in an investment or similar. See Portfolio Efficiency.

8 GLOSSARY OF TERMS Orthogonal Portfolio Efficiency Portfolio Manager Portfolio Optimisation Portfolio Risk Portfolio Tracking Error Premium Price to Book Ratio Price to Cash Ratio Price to Earnings Ratio Principal Principal Component Analysis (PCA) Quantitative Portfolio Optimisation R Squared Regression Residual Return Residual Risk Reverse Optimisation Geometrically, the diagonal. Implies zero correlation between two variables. See Efficient Portfolios. A person employed by an asset manager to manage individual portfolios of assets, often relative to a benchmark. See Quantitative Portfolio Optimisation. The volatility (annualised standard deviation) of the return on the portfolio. The annualised standard deviation of active returns on a portfolio relative to a benchmark. The additional amount an investor is willing to pay for an asset which is believed to be superior relative to an appropriate peer group. The price of a stock relative to its book value. The price of a stock relative to the company's free cash flow per share. The price of a stock relative to the company's earnings per share. The legal entity (person or company) that employs an Agent to perform a task. A statistical technique for decomposing and characterising correlation between factors. The process of maximising the expected return on a portfolio relative to the expected volatility, using asset returns, volatilities and correlations. A statistical measure of the explanatory power of a variable in a regression analysis. Conveys how much of the observed behaviour of the variable under examination can be explained by the explanatory variable. A statistical data analysis technique which optimally fits a model based on the squared differences between data points and the model fitted points. Portfolio return independent of the benchmark. Defined as the return relative to Beta times the benchmark return. The volatility (annualised standard deviation) of the residual return. A mathematical technique whereby the implied asset or sector returns in a portfolio can be computed using volatilities and correlations.

9 GLOSSARY OF TERMS Risk-Free Asset Risk-Free Rate Risk Management Risk Measurement Risk Spectrum Security Market Line Sharpe Ratio Short Position Short Selling Standard&Poors Standard&Poors 500 Standard Deviation Stein Estimators Strategic Asset Allocation Stress Testing Tactical Asset Allocation Target Return Level Target Volatility Level Time Weighted Return To pix Tracking Error Variance Volatility An asset assumed to have zero risk attached to it, typically a long term government bond, although a market implied risk-free rate can be determined. The return on the risk-free asset. The process of actively managing the risk of a portfolio along all its dimensions. The process of attaining the level and nature of risk of an asset or in a portfolio. The spectrum from zero to maximum volatility, in an absolute asset allocation framework. The spectrum from zero to maximum tracking error, in a relative asset allocation framework. The line through the plot of asset returns against their covariances or Betas with respect to the benchmark. The ratio of return to risk (volatility). A net sold position in an asset. Selling assets that are borrowed and not owned. A credit rating agency. US Equity Index. A statistical measure of variation around a sample mean. The square root of variance. Techniques applying Baysian priors in order to shrink observed returns towards a mean determined a priori. The long-term asset allocation of an asset manager. The procedure of subjecting a portfolio to a number of severe scenarios, possibly reflecting previously observed extreme events, in order to gauge its stability. The short-term asset allocation of an asset manager. The desired level of return set prior to a quantitative portfolio optimisation. The desired level of risk or volatility set prior to a quantitative portfolio optimisation. See Modified Dietz. Japanese Equity Index. See Active Risk. A statistical measure of variation around a sample mean. The square of volatility. The annualised standard deviation of asset or portfolio returns.

10 INDEX A Active portfolio, 279 return, 279 return efficient frontier, 282 risk, 280, risk management, 364 risk minimisation, 280 weight vector, 279 Agent, 4 AIMR, 20 Alpha, 317 Arbitrage Pricing Theory (APD, 68, 143 Asset allocation, 165, 177 allocation line, 141 allocation policy, 167 allocation topography, 189 management objectives, 4 pricing, 38 returns, 9 risk, 23 Autoregressive Conditional Heteroscedasticity (ARCH), 147, 150 B Balance sheet, Bayesian priors, 240 Behavioural finance, 5 Benchmark choice, 167, 171 hedge ratio, 173 weight vector, 277 Beta, 32, 63, 319, 343, 356 Bivariate return distribution, 405 Bootstrapping, 39 Business cycle, 293 c Capital asset pricing model (CAPM), 40, 63, Cash flow statement, 48-9 Characteristic line (CL), 66 Communication procedures, 169 Compounding effect, Consensus efficient frontier, 275 Corner portfolios, 114 Correlation analysis, 337, 367 coefficient, 7 6 coefficient distributions, 160 matrix, 83 structure, 158 Covariance, 32, 75-6 matrix, 85 shrinkage, 255, 260 Cross overs, 21 Currency effects, D DAX30, 16 Day trader, 30 Direct cash flow method, 44 Discipline, 6 ( 44

11 INDEX Discount rate definition, 38 determination of, 39 Discounted cash flow analysis, 39 Discounted cash flow model (DCF), 43 Discounting, 38, 49, 61 Distributions of returns, Diversification, 76, 87, 90, 125 Dividend discount model (DDM), 41 Downside risk, 34 Dow Jones Industrial Average, 1 3, 114 E Economic cycle, 293 Economic value added (EVA), 183 Efficient frontier, 111 markets, 9 portfolios, see Portfolio efficiency ridge, 205, 360 surface, 124 Estimation techniques, 138 Excess returns, Explainability, 365 Exponentially weighted moving average (EWMA), 146, 148 F Factor biases, 370 models, 143, 322 Forecasting horizon, 48-9 Frank Russel, 1 72 Free cash flow (FCF), 38 FTSE 100, 16 Fundamental factor models, see Factor models G Generalised Autoregressive Conditional Heteroscedasticity (GARCH), 147, 150 Geometric compounding, 1 0 returns, 9 GIPS, 20 Global Sector Model, 194 H Hang Seng, 16 Hedged benchmarks, 173 Herding, 6 Historical VaR simulation, 396 IBM, 19, 44-8 Identity matrix, 96 Implied growth rate, 59 Incentive scheme, 183 Income statement, 44 Indirect cash flow method, 44 Information coefficient, 330 overload, 4 ratio, 315, 324 International diversification, investing, 117 Investment objectives, 167 policy statement, 167 returns, 9-11 time horizon, 170 Irrational exuberance, 1 3, 292 J J. P. Morgan, 1 72 K Kurtosis, 35-6, 204, 398 l LaGrange optimisation, 101, 106 Liquidity requirements, 1 71 Long position, 21 Lower partial moment, 34 M Macro factor models, see Factor models Management premium, 183 procedures, 168 Manager -specific efficient frontier, 2 77 Marginal lending rate, 40

12 INDEX Marginal return and volatility analysis (MRVA), 157 Marginal sector contribution to active risk (MSCAR), 342, 347 Marginal sector contribution to total risk (MSCTR), 341, 347 Market return, 32 Markowitz, 37, 320 Matrix, 74, Matrix calculus, 74, 83-7, 92-6 Maximising portfolio return, 109 Maximum likelihood estimator, 150 Maximum-variance portfolio, 112, 211 Mean-variance analysis, 26, 73 Minimising portfolio volatility, 1 06 Minimum-variance portfolio, , 188, 208 Modern Portfolio Theory, 73 Monte Carlo Simulation, Morgan Stanley Capital International, 172 Multivariate Density Estimation (MOE), 147, 151 Multivariate normal distributions, 404 N NASDAQ Composite, 1 3, 114 Net present value (NPV), 40 Normal distribution, 35 0 Optimal covariance shrinkage, see Covariance shrinkage return shrinkage, see Return shrinkage Optimisation algorithms, 1 00 Ordinary least Squares, 37 Overconfidence, 6 p Parameter estimation, 1 39 Peer groups, 175, 371 Performance attribution, 1 77 Perpetual growth rate, 42, 43 Perpetuity, 49 Portfolio consistency, 367 construction techniques, 1 78 diversification, see Diversification efficiency, 97 optimisation, see Quantitative portfolio optimisation return, see Return risk, see Risk tracking error, see Active risk volatility, see Volatility weight vector, 74, 277 Present value, 39, 49 Principal, 4 Principal component analysis (PCA), 146 Pythagoras, 320 Q Quantitative asset allocation, 1 77 Quantitative portfolio optimisation, 99, 177 Quasi-Random Monte Carlo Simulated R Asset Allocation (QRMCSM), 201 Retention rate, 42 Return abnormal, 19 definition, distributions, 34-45, 153, 207 excess, 18 multiple periods, 1 0 one period, 1 0 portfolio, 74 residual, 19 series analysis, 334 shrinkage, 241 time-weighted returns, 20-1 Return on equity (ROE), 42 Risk assets, 23 attribution, 344 characterisation, 153, 333 definitions, 23, 26 downside, 34 geometry, 318 management, 332, 363 matching 373 portfolio, 74 premium, 68 residual, 319 shortfall, 34 systematic, 63 tolerance, 169 unsystematic, 63