Economic System Analysis

January 15, 2002Prof. Yannis A. Korilis

2-2Topics

Compound Interest Factors Arithmetic Series Geometric Series Discounted Cash Flows Examples: Calculating Time-Value Equivalences

Continuous Compounding

2-3Compound Amount Factor

An amount P is invested today and earns interest i% per period. What will be its worth after N periods?

Compound amount factor (F/P, i%, N)

Formula:

Proof:

0 1 2 N

iP

F=?

(1 )NF P i

1

22

2 33

1

(1 )

(1 )(1 ) (1 )

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )

(1 )

N NN

NN

F P Pi P i

F P i i P i

F P i i P i

F P i i P i

F F P i

2-4Present Worth Factor

What amount P if invested today at interest i%, will worth F after N periods?

Present worth factor (P/F, i%, N)

Formula:

Proof:

(1 ) NP F i

(1 ) (1 )N NF P i P F i

0 1 2 N

iP=?

F

2-5Sinking Fund Factor

Annuity: Uniform series of equal (end-of-period) payments A

What is the amount A of each payment so that after N periods a worth F is accumulated?

Sinking fund factor(A/F, i%, N)

Formula

Proof:

0 1 2 N

?A

F

i

(1 ) 1N

iA F

i

1 2

1

0

(1 ) (1 ) ...

(1 ) 1(1 )

(1 ) 1

N N

NNn

n

N

F A i A i A

iA i A

i

iA F

i

2-6Series Compound Amount Factor

Uniform series of payments A at i%. What worth F is accumulated after N periods?

Series compound amount factor(F/A, i%, N)

Formula

Proof:

0 1 2 N

A

?F i

(1 ) 1NiF A

i

1 2

1

0

(1 ) (1 ) ...

(1 ) 1(1 )

N N

NNn

n

F A i A i A

iA i A

i

2-7Capital Recovery Factor

What is the amount A of each future annuity payment so that a present loan P at i% is repaid after N periods?

Capital recovery factor(A/P, i%, N)

Formula

Proof:

0 1 2 N

?A

Pi

(1 )

(1 ) 1

N

N

i iA P

i

2

1

0

...1 (1 ) (1 )

1 1/(1 ) 1

1 (1 ) 1 1/( 1) 1

(1 ) 1

(1 )

(1 )

(1 ) 1

N

NN

nn

N

N

N

N

A A AP

i i i

A A i

i i i i

iA

i i

i iA P

i

( / , , )

( / , , )( / , , )

(1 )(1 ) 1

NN

A F A F i N

P F P i N A F i N

iP i

i

EZ Proof:

2-8Series Present Worth Factor

What is the present worth P of a series of N equal payments A at interest i%?

Series present worth factor(P/A, i%, N)

Formula

Proof:

0 1 2 N

A

?P i

(1 ) 1

(1 )

N

N

iP A

i i

2

1

0

...1 (1 ) (1 )

1 1/(1 ) 1

1 (1 ) 1 1/( 1) 1

(1 ) 1

(1 )

N

NN

nn

N

N

A A AP

i i i

A A i

i i i i

iA

i i

2-9Arithmetic Series

Series increases (or decreases) by a constant amount G each period

Convert to equivalent annuity A

(A/G, i%, N): Arithmetic series conversion factor

Formula:

i

'AG

0 1 2 3 4 N

' ( 1)A N G

?A

0 1 2 3 4 N

', ' ,..., ' ( 1)A A G A N G

' ( / , , )A A G A G i N

0

1

(1 ) 1N

NA G

i i

2-10

Proof of Arithmetic Series Conversion Future worth

Multiplying by (i+1):

2 3(1 ) 2 (1 ) ... ( 1)N NF G i G i N G

1 2( 1) (1 ) 2 (1 ) ... ( 1) (1 )N NF i G i G i N G i

1 2

1 2

( 1)

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

[(1 ) (1 ) ... (1 ) 1]

( / , , )

( / , , )

N N

N N

Fi F i F

G i G i G i N G

G i i i NG

G F A i N NG

GF F A i N N

i

0 ( / , , ) ( / , , ) ( / , , )

11 ( / , , )

(1 ) 1N

GA F A F i N F A i N N A F i N

i

G NN A F i N G

i i i

2-11 Growing Annuity and Perpetuity

Growing Annuity: series of payments that grows at a fixed rate g

Series present value:

Growing Perpetuity: infinite series of payments that grows at a fixed rate g

Series present value:

12 )1(,...,)1(),1(, NgAgAgAA

igi

NAP

igi

g

gigiAP

N

if ,1

if ,1

111

,...)1(),1(, 2gAgAA

igAP

iggi

AP

if ,

if ,1

0 1 2 3 4

10001100

1210

1464

0 1 2 3 4

. . .1000

1100

1210

1464

2-12

Growing Annuity and Perpetuity: Proofs

Payment at end of period n:

PV of payment at end of period n (discounted at rate i):

Growing Annuity PV:

Growing Perpetuity PV:

Results follow using:

For g=i, proof is easy

1)1( nn gAA

n

n

n i

gAniFPA

)1(

)1(),,/(

1

N

n

n

N

nn

n

i

g

i

A

i

gAP

1

1

1

1

1

1

1

)1(

)1(

N

N

N

n

n

i

g

gi

i

ig

ig

i

g

1

11

1

11

1

11

1

1

1

1

1

1

1

1

1

1 n

n

i

g

i

AP

gi

i

igi

g

n

n

1

11

1

1

1

1

1

1

2-13 Example: Effects of Inflation

When Marilyn Monroe died, ex-husband Joe DiMaggio vowed to place fresh flowers on her grave every Sunday as long as he lived. Assume that he lived 30 yrs. Cost of flowers in 1962, $5. Interest rate 10.4% compounded weekly. Inflation rate 3.9% compounded weekly. 1 yr = 52 weeks. What is the PV of the commitment1. Not accounting for inflation2. Considering inflation

%2.052

%4.10i %075.0

52

%9.3g

weeks15603052 N

2389$

)002.1(

11

002.0

5$

)1(

11

1560

1

Nii

APV

3429$

002.1

00075.11

00125.0

5$

1

11

1560

2

N

i

g

gi

APV

2-14 Examples on Discounted Cash Flows

Goal: Develop skills to evaluate economic alternatives

Familiarity with interest factors Practice the use of cash flow diagrams Put cash-flow problems in a realistic setting

2-15Cash Flow Diagrams

Clarify the equivalence of various payments and/or incomes made at various times

Horizontal axis: times Vertical lines: cash

flows0 1 2 3 4 5

%10i1000$

51.1610$

2-16 Ex. 2.3: Unknown Interest Rate

At what annual interest rate will $1000 invested today be worth $2000 in 9 yrs?

0 9

1000$

2000$

?i

08.0121/

1/

)1(),,|(

9

N

N

N

PFi

PFi

iNiPFP

F

2-17 Ex. 2.4: Unknown Number of Interest Periods

Loan of $1000 at interest rate 8% compounded quarterly. When repaid: $1400. When was the loan repaid?

0 (Quarters) N

1000$

1400$

%2i

17

)02.1log(

4.1log

)1log(

/log

)1log(

/log

)1log(log)1(

%24

%8

i

PFN

i

PFN

iNP

Fi

P

Fm

ri

N

2-18 Ex. 2.5: More Compounding Periods than Payments

Now is Feb. 1, 2001. 3 payments of $500 each are to be received every 2 yrs starting 2 yrs from now. Deposited at interest 7% annually. How large is the account on Feb 1, 2009?

01 03 05 07 09

500$A

%7i?F

1978$

])07.1()07.1()07.1[(500$

)2%,7,/(500$

)4%,7,/(500$)6%,7,/(500$

246

PF

PFPFF

2-19 Ex. 2.6: Annuity with Unknown Interest

Cost of a machine: $8065. It can reduce production costs annually by $2020. It operates for 5 yrs, at which time it will have no resale value. What rate of return will be earned on the investment?

Need to solve the equation for i. numerically; using tables for (P|A,i,N); using program provided with textbook.

Question: What if the company could invest at interest rate 10% annually?

993.32020

8065

)1(

1)1()5,,|(

5

5

ii

iiAP

A

P

%8i

0 1 2 3 4 5

2020$A

8065$P

?i

2-20Annuity Due

Definition: series of payments made at the beginning of each period

Treatment:1. First payment translated

separately2. Remaining as an ordinary

annuity

0 1 2 4 6 8 10 12 14

1000$A %5i1000$

?P

2-21Example 2.7: Annuity Due

What is the present worth of a series of 15 payments, when first is due today and the interest rate is 5%?

0 1 2 4 6 8 10 12 14

1000$A %5i1000$

?P

N

N

ii

iAA

NiAPAAP

)1(

1)1(

),,|(

64.898,10$

)05.1(05.0

1)05.1(1000$1000$

14

14

P

2-22Deferred Annuity

Definition: series of payments, that begins on some date later than the end of the first period

Treatment:1. Number of payment periods2. Deferred period3. Find present worth of the

ordinary annuity, and4. Discount this value through

the deferred period

2001 06 11

70.317$A

%6i%6i

2-23Ex. 2.8: Deferred Annuity

With interest rate 6%, what is the worth on Feb. 1, 2001 of a series of payments of $317.70 each, made on Feb. 1, from 2007 through 2011?

2001 06 11

70.317$A

%6i%6i

06P

01P

27.1338$)06.1(06.0

1)06.1(70.317$

)5%,6,|(70.317$

5

5

06

APP

1000$)06.1(

127.1338$

)5%,6,|(

5

0601

FPPPP

2-24 Ex. 2.9: Present Worth of Arithmetic Gradient

Lease of storage facility at $20,000/yr increasing annually by $1500 for 8 yrs. EOY payments starting in 1 yr. Interest 7%. What lump sum paid today would be equivalent to this lease payment plan?[Can be compared to present cost for expanding existing facility]

000,20$'A

G=$1500

%7i

0 1 2 3 4 5 6 7 8

?P

2-25 Ex. 2.9: Present Worth of Arithmetic Gradient

Convert increasing series to a uniform

Find present value of annuity

8

( | , , )

1

(1 ) 1

1 8$20,000 $1500

0.07 (1.07) 1

$24,719.81

N

A A G A G i N

NA G

i i

37.609,147$

)07.1(07.0

1)07.1(81.719,24$

)1(

1)1(

),,|(

8

8

N

N

ii

iA

NiAPAP

2-26Example: Income and OutlayBoy 11 yrs old. For college education, $3000/yr, at age 19, 20, 21, 22.1.Gift of $4000 received at age 5 and invested in bonds bearing interest 4% compounded semiannually2.Gift reinvested3.Annual investments at ages 12 through 18 by parentsIf all future investments earn 6% annually, how much should the parents invest?

5 11 12 15 18 20 22

3000$A

4000$20

%2

N

i

?A

%6i18F

18P

F

2-27Income and Outlay

Evaluation date: 18th birthday

P18: present worth of education annuity

F18: future worth of gift F=P18-F18

F will be provided by a series of 7 payments of amount A beginning on 12th birthday

5 11 12 15 18 20 22

3000$A

4000$20

%2

N

i

?A

%6i18F

18P

F

2-28Income and Outlay

395,10$)06.1(06.0

1)06.1(3000$

)4%,6,|(3000$

4

4

18

APP 5 11 12 15 18 20 22

3000$A

18P %6i

5 11 12 15 18 20 22

4000$20

%2

N

i

%6i18F

15F

7079$)06.1(5943$)3%,6,|(

5943$)02.1(4000$)20%,2,|(4000$3

1518

2015

PFFF

PFF

5 11 12 15 18 20 22

?A

F

395$1)06.1(

06.03316$

)7%,6,|(3316$

3316$

7

1818

FAA

FPF%6i