1 lecture 2: time value of money. 2 why learn this future value & present value analysis...

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Lecture 2: Time Value of Money

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Why learn this future value & present value analysis• Personal finance application

Home mortgage calculationcar loan amortization

• Business applicationsProject investment analysisstock and bond valuationfirm valuation

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Future value of a cash flow

• Assuming the interest rate for a period is r, then the future value (FV) of C dollars

one period from today is C(1+r),two periods from today is C(1+r)2, andn periods from today is C(1+r)n

FV=C(1+r) n

0 n

C C(1+r) n

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Future value of $100 at 10%.C=100, r=10%

Years Beginning amount Interest earned Ending amount Formula

1 100 10 110 100*(1+0.1)

2 110 11 121 100*(1+0.1)^2

3 121 12.1 133.1 100*(1+0.1)^3

4 133.1 13.31 146.41 100*(1+0.1)^4

5 146.41 14.641 161.051 100*(1+0.1)^5

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Future value of $100Years 1% 5% 10% 15%

0

1 101 105 110 115

2 102.01 110.25 121 132.25

3 103.0301 115.7625 133.1 152.0875

4 104.060401 121.550625 146.41 174.900625

5 105.101005 127.6281563 161.051 201.1357188

10 110.4622125 162.8894627 259.374246 404.5557736

20 122.019004 265.3297705 672.7499949 1636.653739

100 270.4813829 13150.12578 1378061.234 117431345.1

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Present value (PV) of a cash flow

• Assuming the interest rate for a period is r, then the PV of C dollars

one period from today is C/(1+r),two periods from today is C/(1+r)2, andn periods from today is C/(1+r) n

0 n

C/(1+r) n C

PVC

rn

( )1

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• The present value of cash flows {Ct} is

n

tt

t

nn

r

C

r

C

r

C

r

CPV

1

221

)1(

)1(......

)1()1(

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Present value of $100Years 1% 5% 10% 15%

1 99.00990099 95.23809524 90.90909091 86.956521742 98.02960494 90.70294785 82.6446281 75.614366733 97.05901479 86.38375985 75.13148009 65.751623244 96.09803445 82.27024748 68.30134554 57.175324565 95.14656876 78.35261665 62.09213231 49.71767353

10 90.52869547 61.39132535 38.55432894 24.7184706120 81.95444703 37.68894829 14.8643628 6.110027894

100 36.97112123 0.760449 0.007256572 8.51561E-05

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如果現金流量中的數額不等 ,如何計算現值 ?

利率 8%年 0 1 2 3 4 5

支付額 250 150 320 140 450現值利率因子 0.925926 0.857339 0.793832 0.73503 0.680583

折現值 231.4815 128.6008 254.0263 102.9042 306.2624現值 1023.28

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FVIF & PVIF

• FVIF: future value interest factor = (1+r)t

• r: interest rate, rate of return, discount rate, or discount factor

• PVIF: present value interest factor = 1/(1+r)t

• Basic present value equation:

tt

t

r

FVPV

rPVFV

)1(

)1(*

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Perpetuities

• A perpetuity is a constant payment of C dollars every period forever.

0 1 2 3 ….. t …..

C C C ….. C …..

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The present value of a perpetuity is

r

C

r

C

r

C

r

CPV

t

......)1(

......)1()1( 2

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Example

• An asset is promised to pay $500 every year forever. If we want to earn 10 percent on our money. How much would we pay for this asset?

000,51.0

500

......)1.01(

500......

)1.01(

500

)1.01(

5002

t

PV

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Growth perpetuities

• Assuming, the payment on a growth perpetuity grows at the rate of g:

0 1 2 3 ….. t …..

C C(1+g) C(1+g)2 C(1+g)t-1 …..

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The present value of a growth perpetuity is:

gr

C

r

gC

r

gC

r

CPV

t

t

......)1(

)1(......

)1(

)1(

)1(

1

2

1

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Annuities

• An annuity is a constant payment of C dollars every period until t=T.

0 1 2 3 ….. T

C C C ….. C

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The present value of an annuity is:

])1(

11[

)1(......

)1()1( 2

T

T

rr

C

r

C

r

C

r

CPV

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=FV(0.08,10,-1000,0)

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=PV(0.10,15,1500,10000)

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=PMT(0.10,15,-23579,0)

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-1523 365 543 458 951

=IRR(A2:E2)

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=NPV(0.12,A2:E2)-1523=166.14

-1523 365 543 458 951

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Example:How much would your annual end-of-year payments have to be on a $12,000 loan with a 15% interest rate that must be repaid in three years?

30.15

120000

=-PMT(b3,b2,b4,b5) $5,255.72

Loan payment Principal(beginning) Interest payments Principle payments Principal(end)

$5,255.72 12000 1800 $3,455.72 $8,544.28$5,255.72 $8,544.28 1281.64 $3,974.08 $4,570.19$5,255.72 $4,570.19 685.53 $4,570.19 -$0.00

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Exercise:Dick Tracy just turned 40. He has decided that he would like to retire when he is 65. He thinks that he will need $1,500,000 in his special retirement account at age 65 to maintain his current lifestyle. For the next 15 years he can afford to put $12,000 per year into the account. At age 55 he will need to withdraw $40,000 to purchase membership in the local country club. If his retirement account earns 11% compounded annually, how much will Dick need to deposit into it each year for the last ten years of his career to attain the $1,500,000 goal?