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Introduction to Probability and Risk in Financial Investment

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  1. Introduction to Probability and Risk in Financial Investment Professor Gu Ming Gao Department of Statistics CUHK For New Asia General Education Course

  2. Introduction • Example 1: I Brought China Mobile (0941) Stock two years ago at $50.0 per share, now it only worth $26.85; • Example 2: On the other hand, I brought Shanghai Pechem (0338) the day after 911 at $0.53 per share, now it worth $3.725 per share • Why there are so much uncertainty? • Uncertainty is in the nature of investment

  3. My Stock Portfolio

  4. Risk and Probability • Those uncertainties about the future bad outcomes (possible losses) in financial investment are called risks • The stock is not the risk, nor is the loss the risk. • Risk is unavoidable • Risk is measured through Probability and Expectation

  5. Risk Management Risks (of other type) are everywhere in our life; Taking a train, taking a bus, walking in Causeway bay, etc., Avian Flu, SAS, 911 types of events, etc… Risk management means finding the best possible decision to make (through buy or sell stocks in the case of financial investment) when faced with uncertainty Increasing the odds (probabilities) of a good outcome and reducing the odds of a bad outcome.

  6. What we need to manage risk? • The basic tools for managing risk are: Probability, Expectation and Utility • We need establish (mathematical) models and make assumptions • The model should capture the essence of the problem but should be as simple as possible • In depth understanding of the nature of the uncertainty is a must

  7. Probability 1 • In many real life events, the final outcome is uncertain, they are Random Outcomes • Toss fair a coin, they are two possible random outcomes: { Head, Tail }---All possible outcome from a random event, which is also called sample space • We cannot know in advance whether we got a Head or a Tail • However, we are not completely ignorant about the matter: we know that if we toss a coin many times, about half the time it will be H’s and half will be T’s

  8. Probability 2 • In fact, we know that the probability of getting a Head is ½; or Pr{ Head } = ½ and Pr{ Tail } = ½ (p and 1-p for biased coin) • Other probability of a particular outcome from a random event might be more complicate to imagine, but not unsolvable. • The probability of getting a double in toss a pair of dice is 1/6; • The probability of getting (6,6) is 1/36 • The probability of getting ace of spade in a poker hand is 5/52

  9. Probability 3 • We can assign probabilities to all possible outcomes of a random event • Those Probabilities add up to 1 • The probability of each possible outcome represents the Odds or the likeliness of that outcome to happen. For example, Getting a double is 6 times more likely than getting a (6,6) in tossing a pair of dice

  10. A real life Example, probability in horse racing

  11. Horse racing competitions assemble many real life competitions. Success and failure are only differ by a fraction of second

  12. Hong Kong Cup, December 15, 2002

  13. Win Odds are inverse to amount bet by the public, or

  14. Table 1 Accuracy of Public Probability Estimates Ppub

  15. Hong Kong Horse Racing • Hong Kong horse racing wagering market is the largest per race in the world • HKJC is the largest tax income for the government. For each dollars invested, about 10 cents went to the SAR government. • HKJC is behind many social programs • We need to know HOW HKJC made their money • Is it true that HKJC overall does not contribute to Hong Kong society? • Visit to know more

  16. Expectation 1 • For any decision facing uncertain outcomes, we can evaluate it by Expectation of the decision • Suppose I offer you a game: You pay $3 for a ticket to toss a pair of dice, if it is a (6,6), you win $100, otherwise you win nothing. • Because Pr{(6,6)}=1/36, Pr{Otherwise}=35/36, the decision of playing the can be evaluated by the formula at the bottom. You lose one third of a dollar every time you play. (100-3) (1/36) + (-3) (35/36) = -1/3

  17. Expectation 2 • Compare to the expectation of the decision of not playing the game: 0. You are better off not playing the game • On the other hand, if I offer you $2 a ticket to play the same game, you should jump on it since Expectation (playing) = 2/3, you making 2/3 of a dollar every time you play. • If you are not an expert on horse racing, then the expectation of betting on any horse to win is negative. For every $10 ticket, you expected to lose $1.75

  18. Expectation of Betting on Horse Racing If Public win probability is accurate, then Exp[ Bet on horse I to win ] = (Win Odds – 10) Prpub{ Horse I won} + (-10) Pr{ Loss} = - 10 = - 1.75 Where is the Government tax and Jockey Club ‘s take percentage, which is 17.5% for win, quinella … And 20 % (was 19 %) for 3T, 6 up, … Compare to the decision of not betting (Exp = 0), you should not bet on horse racing, unless you have better probability estimates than the public estimates.

  19. A Coin Toss Example • Suppose that we are allowed to bet on the outcomes of a coin toss. This game is similar to some financial investment situation. • The rule of the game are: • We start with $1000 • We always bet that heads come up • We can bet any amount that we have left • If tails comes up, we lose our bet • If heads come up, we win twice as much as we bet • The coin is fair so the probability of heads is 50%

  20. Expectation of Betting on Heads The expectation of betting $10 dollars is Exp[ Bet on head with 10 dollars ] = (20) Pr{ coin turn up head} + (-10) Pr{ turn up tail} =20*0.5 - 10*0.5 = 5 So this is a winning investment. But how much should we bet? Should we bet $100, $200 dollars or all our many $1000 dollars? A good risk manager would know how much to bet in each instance and maximize long term profitability.

  21. Summary • We have learn what is risk in financial investment • Two major tools to manage such risk are Probability and Expectation • Other topic concerning the risk management such as Statistics, Utility function and Mathematical model are beyond this class

  22. Where to Get More Information • Books to read to know more: 1. <The Book of Risk> by Dan Borge 2. <Principles of Risk Management and Insurance> By George E. Rejda 3. <A Brief Introduction to Probability and Statistics> By Mendenhall, Beaver & Beaver • Search on the internet • Pop up in my office to ask any questions concerning the topics we have discussed • Well, you can always take some courses in the department of Statistics

  23. END of Lecture March 2006