Transcript Examples with Geometric Gradient, Complex Cash Flows

Concepts from the
example problems
( F / P, i, N)
( P / F, i, N)
( F / A, i, N)
( A / F, i, N)
( P / A, i, N)
( A / P, i, N)
( P / G, i, N)
( A / G, i, N)
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Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
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Example 1
How much would you have to deposit
today to have $2000 in 4 years if you can
get a 12% interest rate compounded
annually?
GIVEN:
F4 = $2 000
i = 12%
DIAGRAM:
$2 000
0
1
2
3
n=4
FIND P:
P = F4(P/F,i,n)
= 2 000(P|F,12%,4)
P?
= 2 000(0.6355) = $1 271
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Different Ways
of Looking at P/F
•
•
•
•
•
From previous example, if you can earn 12%
compounded annually, you need to deposit
$1271 to have $2000 in 4 years.
You are indifferent between $1271 today and
$2000 in 4 years, assuming you can earn a
return on your money of 12%.
$1271 today and $2000 in 4 years are
equivalent, if you can earn 12% on your money.
The present worth of $2000 in 4 years is $1271
(i = 12% cpd annually).
If you could get 13% on your money, would you
rather have the $1271 today, or $2000 in 4 years
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Example 2
Tuition costs are expected to inflate at the
rate of 8% per year. The first year’s
tuition is due one year from now and will
be $10,000. To cover tuition cost for 4
years, a fund is to be set up today in an
account that will earn interest at the rate
of 5% per year, compounded annually.
How much must be deposited into the
fund today in order to pay the 4 years of
tuition expenses?
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Present Given Gradient (Geometric)
• g is the geometric gradient over the time period
 (time period: Time 0 to Time n, 1st flow at Time 1)
• P is the present value of the flow at Time 0
 (n periods in the past)
• i is the effective interest rate for each period
P?
0
1
A1
2
3
n
   (1  g )  n 
 1  
 
(
1

i
)
 
 
when i  g


i g
 

( P / A, g , i , n )   




n

when i  g
(
1

i
)

g=%
P = A1(P/A,g,i,n)
Note: cash flow starts with A1 at Time 1, increases by constant g%
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Example 2
Tuition costs are expected to inflate at the rate of 8% per
year. The first year’s tuition is due one year from now and
will be $10,000. To cover tuition cost for 4 years, a fund is
to be set up today in an account that will earn interest at
the rate of 5% per year, compounded annually. How much
must be deposited into the fund today in order to pay the 4
years of tuition expenses?
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Example 2 - Concept
If your rich Aunt Edna wanted to put a
sum of money in the bank today to pay
for your next four years of tuition, that
sum would be $39,759 assuming 5%
return on investment and tuition that
begins at $10,000 increasing by 8% per
year. This problem assumes tuition is
due at the end of the year.
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Complex Cash Flows
Complex Cash Flows – Break apart
(or separate) complex cash flows
into component cash flows in order
to use the standard formulas.
Remember: You can only combine
cash flows if they occur at the same
point in time.
(This is like building with LEGOs!)
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Problem 3
A construction firm is considering the purchase of an
air compressor.
The compressor has the following expected end of
year maintenance costs:
Year 1 $800
Year 2 $800
Year 3 $900
Year 4 $1000
Year 5 $1100
Year 6 $1200
Year 7 $1300
Year 8 $1400
What is the present equivalent maintenance cost if the
interest rate is 12% per year compounded annually?
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Problem 3 – Alt Soln 1
GIVEN:
MAINT COST1-8 PER DIAGRAM
i = 12%/YR, CPD ANNUALLY
FIND P: P = PA + PG + PF = A(P/A,i,n) + G(P/G,i,n) + F(P/F,i,n)
= $700(P/A,12%,8) + $100(P/G,12%,8) + $100(P/F,12%,1)
DIAGRAM:
P?
1
2
3
= $700(4.9676) + $100(14.4715) + $100(0.8929) = $5014
PA ?
1
2
3
4
n=8
n=8
4
$700
0
PF ?
n=1
0
$700
$100 $100 $200
$300
PG ?
1
$700
0
NOTE: CAN BREAK INTO 3 CASH FLOWS:
ANNUAL, LINEAR GRADIENT, AND FUTURE
2
3
$100 $200
4
n=8
$300
0
$100
$700
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Problem
3
–
Alt
Soln
2
GIVEN:
MAINT COST1-8 PER DIAGRAM
i = 12%/YR, CPD ANNUALLY
FIND P: P = PA + PG(PPG) = A(P/A,i,n) + G(P/G,i,n-1)(P/F,i,1)
= $800(P/A,12%,8) + $100(P/G,12%,7)(P/F,12%,1)
DIAGRAM:
P?
1
2
= $800(4.9676) + $100(11.6443)(0.8929) = $5014
3
4
PA ?
1
n=8
0
2
3
4
n=8
0
$800
$100
$800
PPG ?
$200
PG ?
0
$600
0
NOTE: PG MUST BE OFFSET ONE YEAR – SO
BRING THE OFFSET YEAR BACK TO TIME ZERO
1
2
1
PG ? $100
3
n=7
$200
$600
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Problem 4
A young couple has decided to make advance plans for
financing their 3 year old daughter’s college education.
Money can be deposited at 8% per year, compounded
annually.
What annual deposit on each birthday, from the 4th to the
17th (inclusive), must be made to provide $7,000 on each
birthday from the 18th to the 21st (inclusive)?
DIAGRAM:
4
0
$7 000
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GIVEN:
WITHDRAWALS18-21 = $7 000
i = r = 8%/YR, CPD YEARLY
FIND A4-17:
18 21 yrs
A?
STRATEGY: CAN BREAK INTO 2 CASH FLOWS,
SO PICK A CONVENIENT POINT IN TIME AND SET
DEPOSITS EQUAL TO WITHDRAWALS…
P17 = A(F/A,i,n) = A(P/A,i,n)
= A(F/A,8%,14) = 7 000(P/A,8%,4)
= A(24.2149) = 7 000(3.3121)
 A = $957
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