Portfolio Theory and Behavioral Finance Chapter 4b PDF

Portfolio Theory and Behavioral Finance w.MA.XX.IN.19HS.2024-HS Chapter 4b: Value at Risk Building Competence. Crossing Borders. Dr.-Ing. Martin Schnauss, CFA, FRM / Dr. Jan-Alexander Posth [email protected], [email protected], 20...

Portfolio Theory and Behavioral Finance w.MA.XX.IN.19HS.2024-HS Chapter 4b: Value at Risk Building Competence. Crossing Borders. Dr.-Ing. Martin Schnauss, CFA, FRM / Dr. Jan-Alexander Posth [email protected], [email protected], 2024 / Vers. 1.1.3 Topics Value at Risk: Specific and systematic risk – Example Nestlé Share – Specific and systematic risk -2- Value at Risk: Example Nestlé Share Value at Risk: Example Nestlé Share -3- Value at Risk: Example Nestlé Share Nestlé share Price at 12.Jan. 2017: 76.50 CHF Quelle: http://www.boerse.de/technische-analyse/Nestlé-Aktie/CH0038863350 -4- Value at Risk: Example Nestlé Share Challenge: Determine the “Value-at-Risk” for a portfolio of 10 Nestlé stocks! For answering this question we need: a) A given confidence level, for instance: One wants not to lose more than a certain amount with a probability of 99%. b) The time horizon: it is common sense to use in general 10 days as holding period. -5- Value at Risk: Example Nestlé Share Assumptions: The return distribution of the Nestlé Stock is normal distributed. The average return over 10 days equals 0% Current price is 76.50 CHF Determine the value at risk (VaR) for the Nestlé Stock with a confidence level of 99%. -6- Value at Risk: Example Nestlé Share Step 1: Calculation of the 10 days volatility of Nestlé : 10 days volatility = (14.71%) x Sqrt(10/252) = 2.89% -7- Value at Risk: Example Nestlé Share Step 2: Calculation of the Nestlé Value at Risk for 1 CHF: VaR(Nestlé, 99%, 10d) = 1 CHF x 2.89% x 2.33 = 0.0672 CHF Question: Why do we use the factor 2.33? Answer: 2.33 standard deviations from the mean we find the 99%- percentile of the normal distribution. -8- Value at Risk: Example Nestlé Share Step 3: Calculation of Value at Risk for a portfolio of 10 Nestlé shares: VaR(Portfolio) = 10 * 76.50 * VaR(Nestlé, 99%, 10d) = 10 * 5.14 CHF = 51.43 CHF -9- Value at Risk: Hypothesis Value at Risk is by is nature a hypothesis with a certain confidence Hypothesis 𝑯𝑯𝟎𝟎 : A portfolio of 10 Nestlé stocks should not loose in 10 days more than 51.43 CHF with a probability of 99%. Hypothesis 𝑯𝑯𝟏𝟏 : A portfolio of 10 Nestlé stocks could loose in 10 days more than 51.43 CHF with a probability of 1%. Hypothesis 𝑯𝑯𝟐𝟐 : Once in a 1’000 days the 10-day-lost of portfolio of 10 Nestlé stocks will exceed 51.43 CHF. - 10 - Value at Risk: Specific and systematic risk Value at Risk: Splitting the risk - systematic und specific Value at Risk - 11 - Value at Risk: Specific and systematic risk Remember: Systematic und specific risk are uncorrelated. Therefore: σ 2total = β 2 σ systematic 2 2 + σ specific ⇒ σ specific = σ 2total − β 2 σ systematic 2 - 12 - Value at Risk: Specific and systematic risk Value at Risk Specific and Systematic VaR Example: Nestlé Stock - 13 - Value at Risk: Specific and systematic risk VaR 2Nestle(total) = VaR systematic 2 + VaR 2Nestle(specific) As specific VaR, one gets: VaR Nestle(specific) = VaR 2Nestle(total) − VaR systematic 2 - 14 - Value at Risk: Specific and systematic risk Step 1: Calculation of the Nestlé Beta with regard to the SMI market As average Beta, one gets: Beta = 0.96 Remark: Beta is varying a lot Beta is related to the past, BUT is used for forcasting future developments. - 15 - Value at Risk: Specific and systematic risk Step 2: Calculation of the SMI 10 day Vola SMI 1 Woche 2 Wochen 1 Monat 3 Monate 6 Monate 1 Jahr 3 Jahre 5 Jahre 10 Jahre Volatilität 0.75% 1.56% 2.42% 5.71% 7.88% 15.79% 28.69% 33.79% 60.54% Volatilität ann. 5.41% 7.95% 8.38% 11.42% 11.14% 15.79% 16.56% 15.11% 19.14% Vola ann. Im Durchschnitt 12.32% Vola 10-Tage 2.42% As 10 day volatility, we receive: Vola = 2.42% - 16 - Value at Risk: Specific and systematic risk Step 3: Calculation of the systematic Value at Risk for a portfolio of 10 Nestlé stocks: VaR(systematic) = 10*SharePrice(Nestlé)*VaR(SMI)*Beta = 10*76.50 CHF*2.33*2.42%*0.96 = 41.41 CHF - 17 - Value at Risk: Specific and systematic risk We had calculated (slide 69): Total: VaR for Nestlé direct: 51.43 CHF Systematic: VaR for Nestlé in relation to the SMI and Beta: 41.41 CHF With: σ 2Nestle(total) = β 2 σ SMI 2 + σ 2Nestle(specific) we get: VaR 2Nestle(total) = VaR systematic 2 + VaR 2Nestle(specific) - 18 - Value at Risk: Specific and systematic risk According to the formula: VaR Nestle(specific) = VaR 2Nestle(total) − VaR systematic 2 this results in: 𝑉𝑉𝑎𝑎𝑅𝑅𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁é(𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐) = 51.43 2 − 41.412 CHF 𝑉𝑉𝑉𝑉𝑉𝑉𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) = 30.50 CHF (Remember: 10 Nestlé shares was the size of our risky position) - 19 - Value at Risk: Specific and systematic risk Summary: – Systematic VaR = 41.41 CHF – Specific VaR = 30.50 CHF – Total VaR = 51.43 CHF - 20 - Value at Risk: Specific and systematic risk Interpretation of the 30.50 CHF? The appropriate question would be: How would the VaR of the 10 Nestlé share position change if we would hedge it to 100% against the risks of the SMI-market. (Hedging with shortening SMI Futures.) 10 Nestlé Aktien 100% Hedge mit SMI - Residual risk Nestlé - Futures = Net VaR = 30.50 CHF VaR = 51.43 CHF Remaining risk after hedge: 30.50 CHF as VaR. - 21 - Value at Risk: Specific and systematic risk Historic Simulation of General Riskofactors VaR Aggregate Stock Volatilität Portfolio + Residual total Index Volatility and VaR VaR Stock Beta Aggregate Residual Portfolio VaR residual VaR Residual VaR To aggregate does not mean simple addition! - 22 -