I have exam on Saturday. Questions will be released on the same day only. Is there anyone who can contact me over the phone to give more details. This way of communicating is not effective to explain the assignments.
AFIN-8090: Financial Modelling & Forecasting Topic-6: Introduction on Market Risk Contents 1 Portfolio with Risk Free Asset 4 1.1 Basic Portfolio- Two Assets, One Risky and One Risk Free . . . . . . . . . . . . 4 2 Single Index Model and Portfolio Return/Risk 6 2.1 Revisiting Single Index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Single Index Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 To do . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Introduction to Value at Risk 11 3.1 Financial Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Risk Measures and Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Historical or Non Parametric VaR . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 Financial Modelling & Forecasting Topic-6 3.5 Parametric VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5.1 Parametric VaR using Normal Distribution . . . . . . . . . . . . . . . . . . 28 References 32 Contents Page 2 Readings Readings Grover, J., & Lavin, A. M. (2007). Modern Portfolio Optimization. A Practical Approach Using an Excel Solver Single-Index Model, 10(1), 60-72. doi:10.3905/jwm.2007.684880 Chapter 19 from (Ruppert, 2015) Ruppert, David. 2015. Statistics and data analysis for financial engineering. 2 edn. Vol. 13. Springer. https://multisearch.mq.edu.au/permalink/f/1lmkbbh/TN_springer_s978-1-4939-2614-5_214296 Chapter-7 from An Introduction to Analysis of Financial Data with R (Reuy S. Tsay, 2013) https://multisearch.mq.edu.au/permalink/f/i7uiug/MQ_ALMA51144916730002171 3 Part 1 Portfolio with Risk Free Asset 1.1 Basic Portfolio- Two Assets, One Risky and One Risk Free • Let’s assume that we invest in one risky asset portfolio, e.g., a mutual fund and a risk-free asset, e.g., 90 day T-bill. • Suppose you have a w fraction of total wealth invested in the risky asset with an average return of rp and variance (risk) of σ2p and (1 − w) fraction invested in a risk free asset with average return of rf and variance (risk) of 0. • The total average (expected) return is E(rp) = w(rp) + (1− w)(rf) (1.1) • The total variance of the returns is σ2p = w 2(σ2p) + (1− w)(0) = w2σ2p (1.2) 4 Financial Modelling & Forecasting Topic-6 • Taking the square root σp = w1σp (1.3) • Rearranging and substituting E(rp) = rf + ( E(r1)− rf σ1 ) σp (1.4) • Finding an optimal portfolio can be achieved in two steps: – Finding the optimal portfolio of risky assets, called the tangency portfolio, and – Finding the appropriate mix of the risk free asset and the tangency portfolio. • One Fund Theorem – Most investment requirements can be satisfied using the Capital Market Line (line of the tangency portfolio). “There is a single fund M of risky assets such that any efficient port- folio can be constructed as a linear combination of the fund M and the risk-free asset.” Part 1. Portfolio with Risk Free Asset Page 5 Part 2 Single Index Model and Portfolio Return/Risk 2.1 Revisiting Single Index Model • Casual observation of stock prices reveals that when the market goes up (as measured by any of the widely available stock market indexes), most stocks tend to increase in price, and when the market goes down, most stocks tend to decrease in price. • This suggests that one reason security returns might be correlated. Ri = (αi + βiRm) + ei (2.1) Systematic part + Unsystematic Part • By construction – Mean of ei = 0 (for large N) 6 Financial Modelling & Forecasting Topic-6 • Assuming – Index unrelated to unique return E[ei(Rm − R̄m)] – Securities relate only through market return (common): E(ei, ej) = 0 • Variance of ei = E(ei)2 = σ2ei • Variance of Rm = σ2m • Mean returns: R̄ = αi + βiRm • Variance of security return: σ2i = β 2 i σ 2 m + σ 2 ei • Covariance of security returns between i and j σi,j = βiβjσ2m – In contrast, the covariance depends only on market risk. The single-index model implied that the only reason securities move together is a common response to market move- ments. 2.2 Single Index Portfolio If the single-index model holds. See chapter-6 to 8 from Francis & Kim (2013) Part 2. Single Index Model and Portfolio Return/Risk Page 7 Financial Modelling & Forecasting Topic-6 • Sharpe (1963,7) developed a simplified model of portfolio analysis called the single-index model. This simplified model requires fewer input data, thus simplifying both the parameter estimation process and the derivation of the efficient frontier. • Return on a portfolio rp = n∑ i=1 wiri (2.2) According to single index model rp = n∑ i=1 wi(αi + βirm + ei) (2.3) = αp + βprm + ep (2.4) where αp = ∑n i=1wiαi βp = ∑n i=1wiβi ep = ∑n i=1wiei Part 2. Single Index Model and Portfolio Return/Risk Page 8 Financial Modelling & Forecasting Topic-6 Sharpe suggested that, portfolio can be viewed as having two components • An investment in the basic characteristics of the security :αp + ep • An investment in the index: βprm • Portfolio risk premium Rp = E(rp) = αp + βpRM (2.5) • The portfolio Variance is σ2p = β 2 pσ 2 M + ∑ i w2iσ 2 e,i (2.6) =β2pσ 2 M + σ 2 ep Systematic + Unsystematic where σ2ep = ∑ iw 2 iσ 2 ei, is the residual variance of the portfolio. Part 2. Single Index Model and Portfolio Return/Risk Page 9 Financial Modelling & Forecasting Topic-6 2.3 To do • Calculate optimal weight in the 10 stocks portfolio • Calculate α, β and ei for each stock using linear regression • Calculate portfolio return and risk using single index model Part 2. Single Index Model and Portfolio Return/Risk Page 10 Part 3 Introduction to Value at Risk 3.1 Financial Risk • Three main categories – Reading: Chapter-7 from An Introduction to Analysis of Financial Data with R (Reuy S. Tsay) Tsay (2012) – Chapter-19 from Statistics and Data Analysis for Financial EngineeringRuppert (2015) – Market Risk: Loss arising from changes in stock prices, interest rates, foreign exchange rates, and commodity prices. It includes equity risk, interest rate risk, currency risk, com- modity risk, and volatility risk. – Credit Risk: Also known as default risk or counterparty risk. It occurs when a borrower fails to make a payment as promised. It covers consumer credit risk, concentration risk, securitization, and credit derivatives. – Operational Risk: Operational risk is concerned with risk of loss resulting from inadequate or failed internal processes, people and systems, or external events. Legal and political risks are examples of operational risk. 11 Financial Modelling & Forecasting Topic-6 3.2 Risk Measures and Coherence • Distributions of losses are random variables. • As the distributions of losses are unknown and it is hard to adequately estimate them based on the available data, we often employ some summary statistics to quantify the loss distri- butions in real applications. • A risk measure is simply one of these summary statistics. • A sensible risk measure in finance must be consistent with the basic theory in finance. Let η be a risk measure. η is coherent if it satisfies the following four conditions for any two random variables X and Y – Subadditivity: η(X + Y ) ≤ η(X) + η(Y ). Think of this in terms of variance of stock and variance of portfolio. Will this be true? – Monotonicity: If X ≤ Y for all X andY then η(X) ≤ η(Y ) – Positive homogeneity: For any positive constant c, η(cX) = cη(X) – Translation invariance: For any positive constant c, η(X + c) = η(X) + c Part 3. Introduction to Value at Risk Page 12 Financial Modelling & Forecasting Topic-6 3.3 Value at Risk • It is a single estimate of the amount by which an institution’s position in a risk category could decline because of general market movements during a given holding period • Value at Risk (Var) is the most widely used market risk measure in financial risk management and it