Introduction to Probability and Risk in Financial Investment

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 Hs and
  half will be Ts

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 www.hkjc.com 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, PrOtherwise35/36, 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
200.5 - 100.5 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
(No Transcript)
22
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

23
Where to Get More Information

• Books to read to know more
• 1. ltThe Book of Riskgt by Dan Borge
• 2. ltPrinciples of Risk Management and
  Insurancegt By George E. Rejda
• 3. ltA Brief Introduction to Probability and
  Statisticsgt 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

24
END of Lecture

• March 2006