Investment and Decision Making: Everything is Mathematics?

投資與決策 ── 一切皆數 ?

Dr Chan Yan Chong 曾淵滄博士City University of Hong Kong

The Art of Investment is a Combination of

Psychology & Statistics

Psychology

• Human Weaknesses

• Self-analysis

• Patience

• Adventurous

• Pressure

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55001996 20012000199919981997

NASDAQNASDAQ IndexIndex

Source: Bloomberg ， Jan 1, 1996 to Apr 30, 2001

Greedy

Afraid

Emotion of Investors

An investor wants to make a week basis investment on Stock A with the information below.

Price change of Stock A after one week ($) 2 1 0 +1 +2

Probability 0.20 .40 .20 .10 .1

From the principle of expected value, the expected price change in

Stock A (in one week)

= $[2 × 0.2 + (1) × 0.4 + 0 × 0.2 + (+ 1) × 0.1 + (+ 2) × 0.1]

= $ 0.5

A negative expected price change indicates that the price of Stock A is

likely to decrease after one week. In other words, the investor is likely to

lose money if he makes the investment on Stock A.

Difference between Investment and Gambling

3 Dice Game

Probability of BIG or SMALL

= 1/2 - 1/36 = 17/36

Expected Value for $300 game

= $300 X 17/36 - $300 X 19/36

= -$16.67

Expected Value for 500 $300 games

=-$16.67 X 500

= -$8333.30

3 Dice GamePay Win/Loss Cash

1. $300 L -$3002. $600 L -$9003. $1200 W $3004. $300 L 05. $600 L -$6006. $1200 W $6007. $300 W $9008. $300 L $6009. $600 W $120010. $300 L $90011. $600 W $1500

Expected Value for 500 games = $300 X 17/36 X 500 = $70833.33

Risk of 3 Dice Game

Pay Gain/Loss Cash1. $300 L -$3002. $600 L -$9003. $1200 L -$21004. $2400 L -$45005. $4800 L -$93006. $9600 L -$189007. $19200 L -$381008. $38400 L -$765009. $76800 L -$15330010. $153600 L -$30690011. $307200 L -$614100

Assessing Risk

Risk has many connotations. In the field ofinvestment, the concept of risk has slowly changedover many years.• Early 1900: Risk is debt and variability of net

asset value• Markowitz(1952): Risk is the beta value of

CPAM• Sharp(1981): Risk is the variability of rate of

return which can be measured by standard deviation

Beta Value: CAPM

• CAPM: Capital Asset Pricing Model

• Y = A + B X

• A: Alpha value

• B: Beta value

• Y: portfolio`s return

• X: market`s return

Standard Deviation

n

i

i

n

ERR

1

2

1

= standard deviation of single asset

n

i

n

jjiijP WWCov

1 1

p : standard deviation of a portfolio

Wi : weight of stock i in the portfolio

Avoiding Bankruptcy Companies

• Z = 1.2X1 + 1.4X2 + 3.3X3 + X4 + 0.6X5

• X1: working capital / assets• X2: retained earnings / assets• X3: pretax earnings / assets• X4: sales / assets• X5: market value of equities / liabilities

• If Z < 1.81 is classified as troubled.

How to use logarithm (log) application for analyzing stock

market tendency ？

Y = logb X equal to by = X

b = base

Y = exponent

What is log?

X Log2 X

1 0 20=1

2 1 21=2

4 2 22=4

8 3 23=8

16 4 24=16

32 5 25=32

Linear Regression LineHow this method is used to analyze the

stock market tendency and the cycle?

y = a+bx

b = nΣxy – ΣxΣy

nΣx2 - (Σx)2

a = Σx2Σy - ΣxΣxy

nΣx2 - (Σx)2

Moving Average

n

• MA(t,n) = P(t-k+1) /n

k=1

• MA(t,n): n day moving average on period t• P(t): price on period t

Derived Securities

• Future

• Option: Call and Put