Life, Investment, and Wealth Management:

Everything is Mathematics?生活˙投資˙理財，源來是數 ?

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

Expectation of Life in Hong Kong

010

2030

4050

6070

8090

1960 1970 1980 1990 2000 2010

male

female

Long Life

• Some researchers say that 50% of baby girls born today will reach age 100.

• The average life expectancy in 1900 was 49 years.

• In ten years midlife crises will start at age 64, according to a recent survey.

Economic Problems of Long Life

• To maintain the standard of living

• Declining employment opportunities

• Relatively small individual savings

The cause of Pension Planning problems

• Retirement has been a problem since the ancient time.

• Either in agriculture, primitive or natural economy, taking care of the olds is one of the instinctive functions among families. Therefore, the idea of pension planning never exists.

How much you have?

• Assume that you start work at the age of 21. Your earning is $120,000 per year. You and your employer both contribute 5% of your pay of MPF, and your retirement age is 65. How much you have when you retired?

How much you have

Pay grow

ROI

3%

ROI

5%

ROI

8%

ROI

10%

2%

$0.28M

$3.89M $5.22M $9.25M $14.82M

5%

$1.03M

$1.95M $3.29M $7.68M $14.01M

10%

$7.95M

$2.09M $3.58M $8.53M $15.71M

How much you neededfor retirement?

$1,000,000?

$10,000,000?

$100,000,000?

Components of Personal Savings

• Bank Accounts • Individual Retirement Accounts• Life Insurance• Mutual Funds• Stocks and Bonds

Retirement Plan

• Assume that you are 60 year old now.

• Yearly spending is $200,000

• Inflation rate 1%

• Expected life is 85 year old

• How much you need?

How much you need

Investment Return Money needed

1% $5,150,000

2% $4,520,000

3% $4,000,000

5% $3,200,000

10% $1,990,000

15% $1,380,000

How much money required?

• If your spending is $100,000 per year• Yearly spending increment is 2%• Holding Government Bond with 3% return• $2,000,000 enable you to live for 22 years and

$5,000,000 enable you to live for 71 years• If you invest in stock market with 10% return• $1,000,000 enable you to live for 21 years and

$1,500,000 enable you to live for whole life.

ReturnMoney 3% 5% 8% 9% 10% 15%

.5M 5 5 6 7 6 8

1M 10 12 16 18 21 >100

1.5M 16 20 40 >100 >100

2M 22 31 >100

2.5M 29 47

3M 36 79

3.5M 44 >100

4M 52

4.5M 61

5M 71

Investment Plan

• Assume that you are 21 year old now.

• Your earning is $120,000 per year

• Your pay grow at 3% per year

• Your retirement age is 60

• Your saving + MPF is 20% of your earning

• How much you have for the retirement?

How much you have

Investment Return Money needed

1% $3,300,590

2% $3,868,780

3% $4,582,090

5% $6,632,880

10% $19,816,090

15% $70,738,570

The Art of Investment is a Combination of

Psychology & Statistics

Psychology

• Human Weaknesses

• Self-analysis

• Patience

• Adventurous

• Pressure

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

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 of

investment, the concept of risk has slowly changed

over many years.• Risk is debt and variability of net asset value• Risk is the beta value of CPAM• 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

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