chapter 3 interest and equivalence -...
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Cash Flow Diagrams (CFD)
Used to model the positive and negative cash flows.
At each time at which cash flow will occur, a vertical
arrow is added, point down for costs and up for
revenues.
Cash flow are drawn to relative scale
Rent and insurance are beginning-of-period cash
flows; i.e. just put an arrow in where it occurs.
O&M, salvages, and revenues are assumed to be end-
of-period cash flows.
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Example 3-1Purchase a new $30,000 mixing machine. The machine
may be paid for in one of two ways
A. Pay the full price now minus a 3% discount
B. Pay $5000 now, $8000 at the end of 1st yr, and $6000 at
end of each following year
List the alternatives in the form of a table of cash flows
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Continue … Example 3-1
End of year
0 (now)
1
2
3
4
5
Pay in Full Now Pay over 5 Yrs
-$29,100 -$5000
0 -$8000
0 -$6000
0 -$6000
0 -$6000
0 -$6000
Cash flow table:
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Example 3-2A man borrowed $1000 from a bank at 8% interest.
At the end of 1st yr: Pay half of the $1000 principal amount
plus the interest.
At the end of 2nd yr: Pay the remaining half of the principal
amount plus the interest for the second year.
Compute the borrower’s cash flow
End of Year Cash Flows
0 (Now) +$1000
1 -580
2 -540
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Time Value of Money
If monetary consequences occur in a short period of
time → Simply add the various sums of money
What if time span is greater?
$100 cash today vs. $100 cash a year from now?
Money is rented. The rent is called the interest
If you put $100 in the bank today, and interest rate is
9% → $109 a year from now
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Simple Interest
Interest that is computed only on the original sum and
not on accrued interest.
e.g. if you loaned someone the amount of P at a simple
interest rate of i for a period of n years:
Total interest earned = P× i× n = P i n
The amount of money due after n years:
F = P + P i n
Or F = P(1+ i n)
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Example 3-3You loaned a friend $5000 for 5 years at a simple
interest rate of 8% per year.
How much interest you receive from the loan?
How much will your friend pay you at the end of 5 yrs.
Total interest earned = P i n = (5000)(0.08)(5) = $2000
Amount due at the end of loan = P + P i n = 5000 + 2000
= $7000
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Compound Interest
This is the interest normally used in real life
Interest on top of interest
Next year’s interest is calculated based on the unpaid
balance due, which includes the unpaid interest from
the preceding period.
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… Compound Interest
Compound interest is interest that is charged on the original sum and un-paid interest.
You put $500 in a bank for 3 years at 6% compound interestper year.
At the end of year 1 you have (1.06) 500 = $530.
At the end of year 2 you have (1.06) 530 = $561.80.
At the end of year 3 you have (1.06) $561.80 = $595.51.
Note:
$595.51 = (1.06) 561.80
= (1.06) (1.06) 530
= (1.06) (1.06) (1.06) 500 = 500 (1.06)3
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Single Payment Compound Amount Formula
If you put P in the bank now at an interest rate of i% for n years,
the future amount you will have after n years is given by
F = P (1+ i )n
i = interest rate per interest period (stated as decimal)
n = number of interest periods
P = a present sum of money
F = a future sum of money
The term (1+i)n is called the single payment compound factor.
F = P (1+i)n = P (F/P,i,n)
Also P = F (1+i)-n = F (P/F,i,n)
The factor (F/P,i,n) is used to compute F, given P, and given i and n.
The factor (P/F,i,n) is used to compute P, given F, and given i and n.
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Present Value
Example 3-6
If you want to have $800 in savings at the end of four years, and 5% interest is paid annually, how much do you need to put into the savings account today?We solve F = P (1+i)n for P with i = 0.05, n = 4, F = $800.
P = F/(1+i)n = F(1+i)-n
P = 800/(1.05)4 = 800 (1.05)-4 = 800 (0.8227) = $658.16.
Alternate Solution
Single Payment Present Worth Formula
P = F/(1+i)n = F(1+i)-n
P = F (P/F,i,n) , i = 5% and n = 4 periods
From tables in Appendix B, (P/F,i,n) = 0.8227
P = 800 x 0.8227 = $658.16
F = 800
P = ?
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Present ValueExample: You borrowed $5,000 from a bank at 8%
interest rate and you have to pay it back in 5 years. The debt can be repaid in many ways.
Plan A: At end of each year pay $1,000 principal
plus interest due.
Plan B: Pay interest due at end of each year and
principal at end of five years.
Plan C: Pay in five end-of-year payments.
Plan D: Pay principal and interest in one payment
at end of five years.
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…Example (cont’d)You borrowed $5,000 from a bank at 8% interest rate and you have to pay
it back in 5 years.
Plan A: At end of each year pay $1,000 principal plus interest due.
a b c d e f
Year Amnt.
Owed
Int. Owed Total OwedPrincip.
Payment
Total
Paymentint*b b+c
1 5,000 400 5,400 1,000 1,400
2 4,000 320 4,320 1,000 1,320
3 3,000 240 3,240 1,000 1,240
4 2,000 160 2,160 1,000 1,160
5 1,000 80 1,080 1,000 1,080
SUM 1,200 5,000 6,200
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…Example (cont'd)You borrowed $5,000 from a bank at 8% interest rate and you have to pay
it back in 5 years.
Plan B: Pay interest due at end of each year and principal at end of five years.
a b c d e f
Year Amnt.
Owed
Int. Owed Total OwedPrincip.
Payment
Total
Paymentint*b b+c
1 5,000 400 5,400 0 400
2 5,000 400 5,400 0 400
3 5,000 400 5,400 0 400
4 5,000 400 5,400 0 400
5 5,000 400 5,400 5,000 5,400
SUM 2,000 5,000 7,000
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… Example (cont'd)You borrowed $5,000 from a bank at 8% interest rate and you have to pay
it back in 5 years.
Plan C: Pay in five equal end-of-year payments.
a b c d e f
Year Amnt.
Owed
Int. Owed Total OwedPrincip.
Payment
Total
Paymentint*b b+c
1 5,000 400 5,400 852 1,252
2 4,148 332 4,480 920 1,252
3 3,227 258 3,485 994 1,252
4 2,233 179 2,412 1,074 1,252
5 1,160 93 1,252 1,160 1,252
SUM 1,261 5,000 6,261
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… Example (cont'd)You borrowed $5,000 from a bank at 8% interest rate and you have to pay
it back in 5 years.
Plan D: Pay principal and interest in one payment at end of five years.
a b c d e f
Year Amnt.
Owed
Int. Owed Total OwedPrincip.
Payment
Total
Paymentint*b b+c
1 5,000 400 5,400 0 0
2 5,400 432 5,832 0 0
3 5,832 467 6,299 0 0
4 6,299 504 6,802 0 0
5 6,802 544 7,347 5,000 7,347
SUM 2,347 5,000 7,347
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The four plans were
Year Plan 1 Plan 2 Plan 3 Plan 4
1 $1400 $400 $1252 0
2 1320 400 1252 0
3 1240 400 1252 0
4 1160 400 1252 0
5 1080 5400 1252 7347
Total $6200 $7000 $6260 $7347
How do we know whether these plans are equivalent or not?
→We won’t be able to know by simply looking at the cash flows,
therefore some effort should be made.
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Equivalence
In the previous example, four payment plans were
described.
The four plans were used to accomplish the task of
repaying a debt of $5000 with interest at 8%.
All four plans are equivalent to $5000 now.
i.e. all four plans are said to be equivalent to each
other and to $5000 now.
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Present Value
Example 3-8
If you want to have $800 in savings at the end of four years, and 5% interest is paid annually, how much do you need to put into the savings account today?We solve F = P (1+i)n for P with i = 0.05, n = 4, F = $800.
P = F/(1+i)n = F(1+i)-n
P = 800/(1.05)4 = 800 (1.05)-4 = 800 (0.8227) = $658.16.
Alternate Solution
Single Payment Present Worth Formula
P = F/(1+i)n = F(1+i)-n
P = F (P/F,i,n) , i = 5% and n = 4 periods
From tables in Appendix B, (P/F,i,n) = 0.8227
P = 800 x 0.8227 = $658.16
F = 800
P = ?
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In 3 years, you need $400 to pay a debt. In two more years, you need $600 more to pay a second debt. How much should you put in the bank today to meet these two needs if the bank pays 12% per year?
Interest is compounded yearly
P = 400(P/F,12%,3) + 600(P/F,12%,5)
= 400 (0.7118) + 600 (0.5674)
= 284.72 + 340.44 = $625.16
$400
0 1 2 3 4 5
$600
Alternate Solution
P = F(1+i)-n
P = 400(1+0.12)-3
+ 600(1+0.12)-5
P = $625.17
Example 3-8
P
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In 3 years, you need $400 to pay a debt. In two more years, you need $600 more to pay a second debt. How much should you put in the bank today to meet these two needs if the bank pays 12% compounded monthly?
Interest is compounded yearly
P = 400(P/F,12%,3) + 600(P/F,12%,5)
= 400 (0.7118) + 600 (0.5674)
= 284.72 + 340.44 = $625.16
$400
0 1 2 3 4 5
$600
Interest is compounded monthly
P = 400(P/F,12%/12,3*12) + 600(P/F,12%/12,5*12)
= 400(P/F,1%,36) + 600(P/F,1%,60)
= 400 (0.6989) + 600 (0.5504)
= 279.56 + 330.24 = $609.80
Example 3-8 (Interest Compounded monthly)
P
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Borrower point of view:You borrow money from the bank to
start a business.
Investors point of view:You invest your money in a bank and
buy a bond.
Year Cash flow
0 - P 1 0 2 0 3 +400 4 0 5 +600
Year Cash flow
0 + P 1 0 2 0 3 -400 4 0 5 -600
Points of view
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