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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