如何在 30 歲前，存到人生第一個 100 萬

雷立芬 台灣大學農業經濟學系 教授

2007 年 10 月 26 日

Contents• Objectives• Time Value of Money

– Future value– Present value

• Investment instruments– Stocks– Bonds

• Ehrhardt, M.C. and E. F. Brigham, Corporate Finance: A Focused Approach

Objectives

• To dream up for the future– What do you want to be?

– What do you want to do with $1 million?

• To use simple tools– Are there simple tools?

• To have lots of fun

Time lines

CF0 CF1 CF3CF2

0 1 2 3i%

Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

What’s the FV of an initial $100 after 3 years if i = 10%?

FV = ?

0 1 2 310%

Finding FVs (moving to the righton a time line) is called compounding.

100

After 1 year:

FV1 = PV + INT1 = PV + PV (i)= PV(1 + i)= $100(1.10)= $110.00.

After 2 years:

FV2 = FV1(1+i) = PV(1 + i)(1+i)= PV(1+i)2

= $100(1.10)2

= $121.00.

After 3 years:

FV3 = FV2(1+i)=PV(1 + i)2(1+i)= PV(1+i)3

= $100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.

Spreadsheet Solution

• Use the FV function:

– = FV(Rate, Nper, Pmt, PV)

– = FV(0.10, 3, 0, -100) = 133.10

FV of $100 at a 12% nominal rate for 5 years with different compounding

FV(Annual)= $100(1.12)5 = $176.23.

FV(Semiannual)= $100(1.06)10=$179.08.

FV(Quarterly)= $100(1.03)20 = $180.61.

FV(Monthly)= $100(1.01)60 = $181.67.

FV(Daily) = $100(1+(0.12/365))(5x365)

= $182.19.

10%

What’s the PV of $100 due in 3 years if i = 10%?

Finding PVs is discounting, and it’s the reverse of compounding.

100

0 1 2 3

PV = ?

Solve FVn = PV(1 + i )n for PV:

PV =

FV

1+ i = FV

11+ i

nn n

n

PV = $100

11.10

= $100 0.7513 = $75.13.

3

Spreadsheet Solution

• Use the PV function:

– = PV(Rate, Nper, Pmt, FV)

– = PV(0.10, 3, 0, 100) = -75.13

Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

What’s the difference between an ordinary annuity and an annuity due?

PV FV

What’s the FV of a 3-year ordinary annuity of $100 at 10%?

100 100100

0 1 2 310%

110 121FV = 331

FV Annuity Formula• The future value of an annuity with n periods

and an interest rate of i can be found with the following formula:

.33110.

100

0.10

1)0(1

i

1i)(1PMT

3

n

Spreadsheet Solution

• Use the FV function:

– = FV(Rate, Nper, Pmt, Pv)

– = FV(0.10, 3, -100, 0) = 331.00

What’s the PV of the ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13248.69 = PV

PV Annuity Formula• The present value of an annuity with n

periods and an interest rate of i can be found with the following formula:

69.24810.

100

0.10)0(1

11-

ii)(1

11-

PMT

3

n

Spreadsheet Solution

• Use the PV function:

– = PV(Rate, Nper, Pmt, Fv)

– = PV(0.10, 3, 100, 0) = -248.69

What is the PV of this uneven cash flow stream?

0

100

1

300

2

300

310%

-50

4

90.91247.93225.39-34.15

530.08 = PV

Spreadsheet Solution

Excel Formula in cell A3:

=NPV(10%,B2:E2)

A B C D E

1 0 1 2 3 4

2 100 300 300 -50

3 530.09

滿期本利和

每月存入金額（假設年利率為 10 ％）$100 $500 $1,000 $5,000 $10,000 $15,000

5 年 7,908 39,541 79,082 395,412 790,824 1,186,236

10 年 20,755 103,776 207,552 1,037,760 2,075,520 3,113,280

30 年 228,033 1,140,163 2,280,325 11,401,627

22,803,253

34,204,880

插入→函數→ 財務→FV(10%/12,5*12,-100,-100,0)

滿期本利和

每月存入金額（假設年利率為 4 ％）$100 $500 $1,000 $5,000 $10,000 $15,000

5 年 6,752 33,760 67,520 337,600 675,200 1,012,800

10 年 14,874 74,370 148,741 743,703 1,487,406 2,231,200

30 年 69,736 348,681 697,363 3,486,815 6,973,629 10,460,444

插入→函數→財務→FV(4%/12,5*12,-100,-100,0)

年 1% 4% 6% 8% 10% 15% 20% 30%

1 9,901 9,615 9,434 9,259 9,091 8,696 8,333 7,692

5 9,515 8,219 7,473 6,806 6,209 4,972 4,019 2,693

10 9,053 6,756 5,584 4,632 3,855 2,472 1,615 725

15 8,613 5,553 4,173 3,152 2,394 1,229 649 195

20 8,195 4,564 3,118 2,145 1,486 611 261 53

25 7,798 3,751 2,330 1,460 923 304 105 14

30 7,419 3,083 1,740 994 573 151 42 4

35 7,059 2,534 1,301 676 356 75 17 1

40 6,717 2,083 972 460 221 37 7 0

插入→函數→財務→PV(1%,1,0,10000,0)

每月存入金額

年報酬率4% 8 ％ 10 ％ 15 ％ 20 ％ 30 ％

$1,000 71 年 11月

44 年 4 月 37 年 8月

27 年 10月

22 年 4月

16 年 3月

$2,000 50 年 11月

36 年 30 年 11月

23 年 3月

18 年 11月

14 年

$5,000 36 年 8 月 25 年 6 月 22 年 5月

17 年 5月

14 年 5月

10 年 11月

$10,000 24 年 7 月 18 年 4 月 16 年 5月

13 年 3月

11 年 3月

8 年 9 月

插入→函數→財務→

NPER(4%/12,-1000, 0, 5000000,0)

• Represents ownership.

• Ownership implies control.

• Stockholders elect directors.

• Directors hire management.

• Since managers are “agents” of shareholders, their goal should be: Maximize stock price.

Common Stock

Initial Public Offering (IPO)?

• A firm “goes public” through an IPO when the stock is first offered to the public.

• Prior to an IPO, shares are typically owned by the firm’s managers, key employees, and, in many situations, venture capital providers.

ssss r

D

r

D

r

D

r

DP

1. . .

111ˆ

33

22

11

0

One whose dividends (Dn) are expected togrow forever at a constant rate, g.

rs: rate of return on stock

Stock Value = PV of Dividends

For a constant growth stock,

D D gD D gD D gt t

t

1 01

2 02

111

gr

D

gr

gDP

ss

100

1ˆ

If g is constant, then:

What’s the stock’s market value? D0 = 2.00, rs = 13%, g = 6%.

Constant growth model:

gr

D

gr

gDP

ss

100

1ˆ

= = $30.29.0.13 - 0.06

$2.12 $2.12

0.07

gr

ED

E

P

s

/10

E: earning per share, $2.5

D1/E: payout ratio, 60%

P0 = 2.5X(0.6/0.13-0.06) = 21.43

P0/E =0.6/(0.13-0.06) = 8.57

Key Features of a Bond•Par value: Face amount; paid at maturity. Assume $1,000.•Coupon interest rate: Stated interest rate. Multiply by par value to get dollars of interest. Generally fixed.•Maturity: Years until bond must be repaid. Declines.•Issue date: Date when bond was issued.•Default risk: Risk that issuer will not make interest or principal payments.

What’s the value of a 10-year, 10% coupon bond if rd = 10%?

V

rB

d

$100 $1,000

11 10 10 . . . +

$100

1+ r d

100 100

0 1 2 1010%

100 + 1,000V = ?

...

= $90.91 + . . . + $38.55 + $385.54= $1,000.

+++1 r+ d

Suppose the bond was issued 20 years ago and now has 10 years to maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%, or at 7%?

M

Bond Value ($)

Years remaining to Maturity

1,372

1,211

1,000

837

775

30 25 20 15 10 5 0

rd = 7%.

rd = 13%.

rd = 10%.

You are offered a note which pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank which pays a 6.76649% nominal rate, with 365 daily compounding, which is a daily rate of 0.018538% . You plan to leave the money in the bank if you don’t buy the note. The note is riskless.

Should you buy it?

1. Greatest Future Wealth

Find FV of $850 left in bank for15 months and compare withnote’s FV = $1,000.

FVBank = $850(1.00018538)456

= $924.97 in bank.

Buy the note: $1,000 > $924.97.

2. Greatest Present Wealth

Find PV of note, and comparewith its $850 cost:

PV = $1,000/(1.00018538)456

= $918.95.

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