Empirical properties of stock prices and returns Κωνσταντίνος Δράκος Αναπληρωτής Καθηγητής Τμήμα Λογιστικής & Χρηματοοικονομικής Οικονομικό Πανεπιστήμιο Αθηνών Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Stock Prices (e.g ASE, 8/2/2011-8/2/2016 Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Are stock prices Random Walks: evidence I, correlogram Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Evidence II: running an AR(1) model Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

conclusions Stock prices exhibit very strong autocorrelation Lasting for many time periods Hence stock prices are very persistent Stock prices contain a unit root They are best described by a Random Walk Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Returns Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Conclusions Stock returns are volatile They show little persistence They are stationary Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

The empirical distribution of returns Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Up and down movements are almost equally likely Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Comparison of actual data with a normal generating function interval % of obs under normality % outside % of obs in actual sample μ +/-σ 68,20% 31,80% 75,22% 24,78% μ +/- 2σ 95,40% 4,60% 95,49% 4,51% μ +/- 3σ 99,70% 0,30% 98,83% 1,17% Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Focus on Losses: VaR notion “What loss level is such that we are X% confident it will not be exceeded in N business days?” So VaR summarizes the: expected maximum loss (or worst loss) over a target horizon within a given confidence level, where extremely adverse scenarios are excluded 1-day value-at-risk at 95% confidence level, is: The maximum loss that will occur tomorrow, if the worst 5% situations are not considered The minimum loss that will occur tomorrow, if only the worst 5% situations are considered Value at Risk is an estimate of the worst possible loss an investment could realize over a given time horizon, under normal market conditions (defined by a given level of confidence). Normal market conditions – the returns that account for 95% of the distribution of possible outcomes. Abnormal market conditions – the returns that account for the other 5% of the possible outcomes. Ειδικά Θέματα Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

Parametric VaR The starting point is that the return distribution is normal with a distribution function: It can be easily shown that the probability of extracting a value in the range (μ – κ*σ , μ + κ*σ) centered on the mean μ whose size is a multiple κ of the standard deviation depends only on the multiple κ, and is equal to: Where it becomes clear that the probability does not depend either on μ or on σ, and can de derived by analyzing a standard normal distribution only. K. Drakos, Risk Management, MSc 2010

For instance the probability of extracting a value in the range between (μ-σ) and (μ+σ) is approximately 68%, irrespectively of the values of μ and σ. Since the normal distribution is symmetric, this implies that the remaining 32% is divided into a 16% probability to observe values lower than (μ-σ) and a 16% probability to observe values greater than (μ+σ) Suppose that a trader holds a position with daily volatility of 2.46% and daily mean - 0.076% The trader wants to estimate the maximum potential loss in 95% of possible cases There is a 90% probability of extracting a value in the range (μ-1.645*σ , μ + 1.645*σ) for any normal distribution Therefore since for the particular position μ-1.645*σ = -4.13% and μ + 1.645*σ = 3.98%, there is a 90% probability of extracting a value between -4.13% and +3.98%, while only a 5% probability of facing a return lower than -4.13% Thus, -4.13% is the percentage VaR at 95% confidence level for the position K. Drakos, Risk Management, MSc 2010

Non-parametric VaR: Historical Simulation Estimate VaR without making strong assumptions about distributions Let the data ‘speak for themselves’ as much as possible Based on the assumption that the near future will be sufficiently like the near past, so we can use the data from the recent past to forecast risks The most popular non-parametric method is the Historical Simulation Advantages: intuitive and conceptually simple, easy to implement, easy to report and communicate, can accommodate fat tails and other deviations from normality Disadvantages: complete dependence on the dataset, dataset maybe dominated by extreme unusual events, we cannot extrapolate from the largest historical loss to anything larger that might occur in the future K. Drakos, Risk Management, MSc 2010

Historical Simulation Example with a single asset K. Drakos, Risk Management, MSc 2010

Advantages of VaR It captures an important aspect of risk in a single number It is easy to understand VaR translates portfolio volatility into a dollar/euro value. Measure of Total Risk rather than Systematic (or Non-Diversifiable Risk) measured by Beta. VaR is useful for monitoring and controlling risk within the portfolio. VaR can measure the risk of many types of financial securities (i.e., stocks, bonds, commodities, foreign exchange, off-balance-sheet derivatives such as futures, forwards, swaps, and options, and etc.) K. Drakos, Risk Management, MSc 2010

VaR: caveats VaR does not describe the worst loss. In fact we would expect the VaR to be exceed with a frequency of p, that is, 5 days out of 100, for a 95% confidence level. VaR is measured with some error. It is itself subject to normal sampling variation (i.e. different sample periods, or different lengths would lead to different VaR). VaR does not describe the losses in the left tail. VaR does not offer any information about the distribution of losses in its left tail, it just indicates the probability of such a value occurring. For instance, in our example the average value of losses worse than 52.87 is 71.49 which is about 35% worse than the VaR. K. Drakos, Risk Management, MSc 2010

VaR parameters: confidence level The higher the confidence level, the greater the VaR measure. Varying the confidence level provides useful information about the return distribution and potential extreme losses. It is not clear, however, whether one should stop at 99%, or 99.9% and so on. Each of these values will create an increasingly larger, but less likely, loss. Another problem is that as one increases the confidence level, the number of occurrences below VaR shrinks, leading to poor measurement in the non-parametric case. For example 99.99% and 10 business days will require history of 100*100*10 = 100,000 days in order to have only 1 point. Regulators use 99% for market risk and 99.9% for credit/operational risk. A bank wanting to maintain a AA credit rating will often use 99.97% for internal calculations K. Drakos, Risk Management, MSc 2010

VaR parameters: Horizon The longer the horizon T, the greater the VaR measure. This extrapolation is driven by two factors: the behavior of the risk factors and the position itself. To extrapolate from a one-day horizon to a longer one, we need to assume that returns are iid. As we have seen this allows the transformation of daily volatility to any ‘multiple-day’ volatility by simply multiplying by the square root of time. So we have: Regulators in effect use 1-day for bank market risk and 1-year for credit/operational risk. Fund managers often use one month K. Drakos, Risk Management, MSc 2010