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- 1. FEC 512 Financial Econometrics -About the Course- • What is Financial Econometrics? – Science of modeling and forecasting financial time series. • Who is this course for? – Students in finance, practitioners in the financial services sector. • How is the presentation of the lectures? – Begins with review of necessary statistics and probability theory, continues with basics of econometrics reaches up to the most recent theoretical results – Uses computer applications, Eviews. – Course materials are online at online.bilgi.edu.tr – Students are supposed to choose a data set from the list at the beginning of the term in order to do assignments – Attendence is not required. 1
- 2. FEC 512 Financial Econometrics -About the Course- • Textbooks: 1. (for Statistics part only) Groebner D.F. et al.(2008) Business Statistics. 2. Ruppert D. (2004), Statistics and Finance, Springer. 3. Brooks, C. “Introductory Econometics for Finance” 4. Stock J.H. and Watson M.W. (2003), Introduction to Econometrics (first edition), Addison-Wesley. • Method of Evaluation: – Assignments (50%) – Final Examination (50% ) 2
- 3. Overview Before getting into applications in financial econometrics we will first define • returns on assets Then we will review • Probability – probability density functions, cumulative distribution functions – expectations, variance, covariance and correlation • Statistics – Testing – Estimation • We’ll also be studying some new areas of statistics: – Regression • interesting connections with portfolio analysis. – Probit, Logit Analysis – Time series models 3
- 4. I. Preliminary: Asset Return Calculations Istanbul Bilgi University FEC 512 Financial Econometrics-I Asst. Prof. Dr. Orhan Erdem FEC 512 Preliminaries and Review Lecture 1-4
- 5. Background How do the prices of stocks and other financial assets behave? We will start by defining returns on the prices of a stock. Lecture 1-5 FEC 512 Preliminaries and Review
- 6. Prices and Returns Main data of financial econometrics are asset prices and returns. Almost all empirical research analyzes returns to investors rather than prices. Why? Investors are interested in revenues that are high r.t. size of the initial invstmnt. Returns measure this: Changes in prices expressed as a fraction of the initial price. Lecture 1-6 FEC 512 Preliminaries and Review
- 7. Asset Return Calculations Pt is the price of a stock at time t. Stock pays no dividends. Simple return ( Pt − Pt −1 ) = P Rt = −1 t P−1 P−1 t t Simple gross return Pt Rt + 1 = Pt −1 Lecture 1-7 FEC 512 Preliminaries and Review
- 8. Multi-period returns e.g. Pt Pt Pt −1 R t (2) = −1 = −1 Pt − 2 Pt −1 Pt − 2 = (1 + R t )(1 + R t −1 ) − 1 In general, k-month gross return is defined as 1 + R t ( k ) = (1 + R t )(1 + R t −1 )....(1 + R t − k + 1 ) Note: For small values of Rt 1 + Rt (k ) ≅ 1 + Rt + ... + Rt − k +1 or k −1 Rt (k ) ≅ ∑ Rt −i i =0 Lecture 1-8 FEC 512 Preliminaries and Review
- 9. Example 1 Suppose that the price of Arçelik stock on January is 100YTL, and on February is 105YTL, and that you sell the stock now(on March) at Pt=110YTL. Assume no dividends,then Rt=(110-105)/105=0.0476 Rt-1=(105-100)/100=0.05 Rt(2)=(110-100)/100=0.10 Check also that 1+Rt(2)=(1+ Rt)(1+ Rt-1) 1.0476*1.05=1.1 Lecture 1-9 FEC 512 Preliminaries and Review
- 10. Annualizing Returns If investment horizon is one year 1+RA =1+R(12) =(1+R1) (1+R2)... (1+R12) One month inv. with return Rt, (assume Rt=R) 1+RA=(1+R)12 Two month inv. with return Rt(2), (assume Rt(2)=R(2)) 1+RA=(1+R(2))6 Lecture 1-10 FEC 512 Preliminaries and Review
- 11. Cont. to Example 1 In the first example the one month return was 4.76%. If we assume that we can get this return for 12 months then the annualized return is RA=(1.0476)12-1=1.7472-1=0.7472 or 74.72% Lecture 1-11 FEC 512 Preliminaries and Review
- 12. Log-Returns The log-return is rt = log( Pt ) − log( Pt −1 ) == log( Pt / Pt −1 ) = ln(1 + Rt ) The log return in the previous example is rt=ln(0.0476)=0.0465 or 4.65% The above return measures are very similar numbers since daily returns are very rarely outside the range of -10% to 10%. Lecture 1-12 FEC 512 Preliminaries and Review
- 13. Log returns are approximately equal to net returns: x small ⇒ log(1 + x) ≅ x Therefore, rt = log(1 + Rt) ≅ Rt Examples: * log(1 + 0.05) = 0.0488 * log(1 -0.05) = -0.0513 Lecture 1-13 FEC 512 Preliminaries and Review
- 14. log(1+x) and x when x is small Lecture 1-14 FEC 512 Preliminaries and Review
- 15. Advantage of Log-Returns Simplicity of multiperiod returns. Simply the sum: rt (k ) = log{ + Rt (k )} = log{( + Rt )...(1 + Rt −k +1 )} 1 1 = log( + Rt ) + ... + log( + Rt −k +1 ) 1 1 = rt + rt −1 + ...rt −k +1. Lecture 1-15 FEC 512 Preliminaries and Review
- 16. Returns are scale-free, meaning that they do not depend on monetary units (dollars, cents, etc.) not unit-less, unit is time; they depend on the units of t (hour, day, etc.) Lecture 1-16 FEC 512 Preliminaries and Review
- 17. Portfolio Return N R p = ∑ wi Ri i =1 where wi is the weight of each asset in the portfolio. Example: Lecture 1-17 FEC 512 Preliminaries and Review
- 18. About Returns Returns cannot be perfectly predicted, they are random. This randomness implies that a return might be smaller than its expected value and even negative, which means that investing involves RISK. It took quite some time before it was realized that risk could be described by probability theory Lecture 1-18 FEC 512 Preliminaries and Review
- 19. II. Review of Probability & Statistics FEC 512 Preliminaries and Review Lecture 1-19
- 20. Probability and Finance Because we cannot build purely deterministic models of the economy, we need a mathematical representation of uncertainty in finance (probability, fuzzy measures etc…) In economic and finance theory, probability might have 2 meanings: As a descriptive concept 1. As a determinant of the agent decision 2. making theory. Lecture 1-20 FEC 512 Preliminaries and Review
- 21. Probability as a Descriptive Concept The probability of an event is assumed to be approx. equal to the rel.freq. of its occurrence in a large # experiments. There is one difficulty with this interpretation: Empirical data have only one realization. Every estimate is made on a single time-evolving series. If stationarity(!) is not assumed, performing statistical estimation is impossible. Lecture 1-21 FEC 512 Preliminaries and Review
- 22. Probability Concepts Experiment – a process of obtaining outcomes for uncertain events Outcome – the possible results of an observation, such as the price of a security at t. However, probability statements are not made on outcomes but on events, which are sets of possible outcomes. The Sample Space is the collection of all possible outcomes Lecture 1-22 FEC 512 Preliminaries and Review
- 23. Examples Event Example 1: The probability that the price of a security be in a given range, say (10,12)YTL Example 2: Outcome Sample Space= The Set of Odd numbers is an Event Probabilities are defined on events. Lecture 1-23 FEC 512 Preliminaries and Review
- 24. Mutually Exclusive Events If E1 occurs, then E2 cannot occur E1 and E2 have no common elements E2 A die cannot be E1 Odd and Even at Even the same time. Odd Numbers Numbers Lecture 1-24 FEC 512 Preliminaries and Review
- 25. Independent and Dependent Events Independent: Occurrence of one does not influence the probability of occurrence of the other Dependent: Occurrence of one affects the probability of the other Lecture 1-25 FEC 512 Preliminaries and Review
- 26. Independent vs. Dependent Events Independent Events E1 = heads on one flip of fair coin E2 = heads on second flip of same coin Result of second flip does not depend on the result of the first flip. Dependent Events E1 = rain forecasted on the news E2 = take umbrella to work Probability of the second event is affected by the occurrence of the first event Lecture 1-26 FEC 512 Preliminaries and Review
- 27. Assigning Probability Classical Probability Assessment Number of ways Ei can occur P(Ei) = Total number of elementary events Relative Frequency of Occurrence Number of times Ei occurs Relative Freq. of Ei = N Subjective Probability Assessment An opinion or judgment by a decision maker about the likelihood of an event Lecture 1-27 FEC 512 Preliminaries and Review
- 28. Rules of Probability Rules for Possible Values and Sum Individual Values Sum of All Values k 0 ≤ P(Ei) ≤ 1 ∑ P(e ) = 1 i For any event Ei Rule 1 Rule 2 i=1 where: k = Number of individual outcomes in the sample space ei = ith individual outcome Lecture 1-28 FEC 512 Preliminaries and Review
- 29. Addition Rule for Elementary Events The probability of an event Ei is equal to the sum of the probabilities of the individual outcomes forming Ei. That is, if: Ei = {e1, e2, e3} then: P(Ei) = P(e1) + P(e2) + P(e3) Rule 3 Lecture 1-29 FEC 512 Preliminaries and Review
- 30. Complement Rule The complement of an event E is the collection of all possible elementary events not contained in event E. The complement of event E is represented by E. E Complement Rule: P( E ) = 1 − P(E) E P(E) + P( E ) = 1 Or, Lecture 1-30 FEC 512 Preliminaries and Review
- 31. Addition Rule for Two Events Addition Rule: ■ P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2) + = E1 E2 E1 E2 P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2) Don’t count common elements twice! Lecture 1-31 FEC 512 Preliminaries and Review
- 32. Addition Rule Example P( Even or Asal)= P(Even) +P(Asal) - P(Even and Asal) 3/6 + 3/6 - 1/6 = 5/6 2 2,4,6 2,3,5 Lecture 1-32 FEC 512 Preliminaries and Review
- 33. Addition Rule for Mutually Exclusive Events If E1 and E2 are mutually exclusive, then E1 E2 P(E1 and E2) = 0 So 0 utualvlye = if m lusi P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2) c ex = P(E1) + P(E2) Lecture 1-33 FEC 512 Preliminaries and Review
- 34. Conditional Probability Conditional probability for any two events E1 , E2: P(E1 and E 2 ) P(E1 | E 2 ) = P(E2 ) P(E2 ) > 0 where Lecture 1-34 FEC 512 Preliminaries and Review
- 35. Conditional Probability Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. What is the probability that a car has a CD player, given that it has AC ? i.e., we want to find P(CD | AC) Lecture 1-35 FEC 512 Preliminaries and Review
- 36. Conditional Probability Example (continued) Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. CD No CD Total .2 .5 .7 AC .2 .1 No AC .3 .4 .6 1.0 Total P(CD and AC) .2 P(CD | AC) = = = .2857 P(AC) .7 Lecture 1-36 FEC 512 Preliminaries and Review
- 37. Conditional Probability Example (continued) Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is about 28.57%. CD No CD Total .2 .5 .7 AC .2 .1 No AC .3 .4 .6 1.0 Total P(CD and AC) .2 P(CD | AC) = = = .2857 P(AC) .7 Lecture 1-37 FEC 512 Preliminaries and Review
- 38. For Independent Events: Conditional probability for independent events E1 , E2: P(E1 | E 2 ) = P(E1 ) P(E2 ) > 0 where P(E2 | E1 ) = P(E2 ) P(E1 ) > 0 where Lecture 1-38 FEC 512 Preliminaries and Review
- 39. Multiplication Rules Multiplication rule for two events E1 and E2: P(E1 and E 2 ) = P(E1 ) P(E2 | E1 ) Note: If E1 and E2 are independent, then P(E2 | E1 ) = P(E2 ) and the multiplication rule simplifies to P(E1 and E2 ) = P(E1 ) P(E2 ) Lecture 1-39 FEC 512 Preliminaries and Review
- 40. Bayes’ Theorem P(Ei )P(B| Ei ) P(Ei | B) = P(B) P(Ei )P(B| E i ) = P(E1 )P(B| E1 ) + P(E2 )P(B| E 2 ) + K + P(Ek )P(B| E k ) where: Ei = ith event of interest of the k possible events A = new event that might impact P(Ei) Events E1 to Ek are mutually exclusive and collectively exhaustive Lecture 1-40 FEC 512 Preliminaries and Review
- 41. More Simply, P( B | A)P(A) P(A | B) = P( B) Bayes Theorem allows one to recover the probability of the event A given B from the probability of the individual events A,B, and the probability of B given A. Lecture 1-41 FEC 512 Preliminaries and Review
- 42. Bayes’ Theorem Example Suppose that the probability that the price of a stock will rise on any given day, is 0.5. Thus, we have the prior probabilities P(Rise)=0.5 and P(No rise)=0.5. When it actually rises, the brokers correctly forecasts the rise 30% of the time. When it does not rise, they incorrectly forecast rise 6% of the time. What is the probability that the prices will rise if the brokers forecasted that it will rise tomorrow? Lecture 1-42 FEC 512 Preliminaries and Review
- 43. Bayes’ Theorem Example (cont.) Let A: the event that brokers forecast that the price of the stock will rise. P(ARise)=30% P(ANo Rise)= 6% P( Rise)P(A | Rise) P( Rise)P(A | Rise) P( Rise | A) = = P( Rise)P(A | Rise) + P(No Rise)P(A | No Rise) P( A) 0.30 * 0.5 0.15 = = = 0.83 0.30 * 0.5 + 0.06 * 0.5 0.15 + 0.03 As it can be seen we updated the probability of a rise (0.5) to 0.83 after we heard the brokers’s forecast of rise. Lecture 1-43 FEC 512 Preliminaries and Review
- 44. Bayes’ Theorem Example (continued) P(Rise) = .5 , P(U) = .5 (prior probabilities) Conditional probabilities: P(ARise)=30% P(ANo Rise)= 6% Revised probabilities Prior Conditional Joint Revised Event Prob. Prob. Prob. Prob. Rise .5 .30 .5*.30 = .15 .15/.18 = .833 No Rise .5 .06 .5*.06 = .03 .03/.18 = .166 Sum = .18 Lecture 1-44 FEC 512 Preliminaries and Review
- 45. Importance of Bayes Law We update our beliefs in light of new information. Revising beliefs after receiving additional info is smth that humans do poorly without the help of mathematics. There is a tendency to put either too little or too much emphasis on new info This problem can be mitigated by using Bayes’ Law. Lecture 1-45 FEC 512 Preliminaries and Review
- 46. For Further Study For Topic I:Ruppert D. (2004), Statistics and Finance, Springer. For Topic II:Groebner D.F. et al.(2008) Business Statistics. Lecture 1-46 FEC 512 Preliminaries and Review

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