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Introduction to Probability and Introduction to Probability and Risk in Financial InvestmentRisk in Financial Investment

Professor Gu Ming Gao

Department of Statistics

CUHK

For New Asia General Education Course

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Introduction 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

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My Stock PortfolioMy Stock Portfolio

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Risk and ProbabilityRisk 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 unavoidableRisk is measured through Probability and

Expectation

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Risk ManagementRisk 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.

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What we need to manage risk?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

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Probability 1Probability 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

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Probability 2Probability 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

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Probability 3Probability 3

We can assign probabilities to all possible outcomes of a random event

Those Probabilities add up to 1The 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

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A real life Example, probability A real life Example, probability in horse racingin horse racing

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

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Hong Kong Cup, December 15, 2002

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)1(*OddsWin 1

i

Ii B

BB

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

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Table 1 Accuracy of Public Probability Table 1 Accuracy of Public Probability Estimates P Estimates Ppubpub

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Hong Kong Horse RacingHong 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

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Expectation 1Expectation 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

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Expectation 2Expectation 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

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Expectation of Betting on Expectation of Betting on Horse Racing 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.

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A Coin Toss ExampleA 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%

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Expectation of Betting on HeadsExpectation 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.

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SummarySummary

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

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Where to Get More InformationWhere 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

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END of LectureEND of LectureMarch 2006