Example 1:
In the following cash flow diagram, A8=A9=A10=A11=5000, and
starting with A12, the deposits start decreasing in the amount
of 50 (that is, A12=4950, A13=4900). Find the
equivalent worth of the following cash flow at t=60. The
interest rate is 10%.
8
9
10
11
12
13

60
A60
A11
Example 1:
F60  5000( F | A,10%, 53)  50( P | G,10%, 50) ( F | P,10%, 50)  7205406.83
Example 2:
Set up an equation to find the value of Z on the left-hand
cash-flow diagram that establishes equivalence with the
right-hand cash-flow diagram. Both diagrams are drawn on
a yearly scale. The interest rate is 12% compounded quarterly.
Note that you are only asked to set up an equation, not to
find the actual value of Z.
1000
0
1
Z
Z
2
Z
3
Z
4
Z
5
6
years
0
1
2
3
4
5
6
5000
Example 2:
4
 0.12 
Effectiveannualrate  1 
  1  12.55%
4 

Z  Z ( P | A,12.55%, 4)  5000( P | F ,12.55%, 6)  1000( P | F ,12.55%, 2)
Example 3:
Find the equivalent worth of the following cash-flow at t=0
assuming an inflation rate of 2% per month.
a)if the market interest rate is 12% compounding monthly
b)if the market interest rate is 12% compounding continuously
Note the timings of cash flows and that the cash flows
are shown on a monthly scale. Also all amounts are in actual
dollars.
0 1 2
9 10

13
16
37

40
45
50
75

A1=1000
A2=2000
A3=3000
80
months
Example 3:
(1  0.01) 3  1  3.03%
(a) Effective rate for three months:
(1  0.01) 5  1  5.10%
Effective rate for five months:
P  1000( P | A,1%,10)
 2000( P | A, 3.03%,10) ( P | F ,1%,10)
 3000( P | A, 5.10%, 8) ( P | F ,1%, 40)
(b) Effective rate for one month:
e
Effective rate for three months:
0.12
12
 1  1.01%
0.12
 1  3.05%
e
0.12
Effective rate for five months:
e
4
12
5
 1  5.13%
P  1000( P | A,1.01%,10)
 2000( P | A, 3.05%,10) ( P | F ,1.01%,10)
 3000( P | A, 5.13%, 8) ( P | F ,1.01%, 40)
Example 4:
Suppose that a bank is paying 8% nominal interest,
compounded semiannually.
a) What is the semiannual effective rate?
b) What is the effective rate per year?
c) What is the effective rate per month?
d) Find the equivalent nominal interest rate, compounding
continuously.
Example 4:
0.08
 4%
2
2
 0.08 
1 
  1  8.16%
2 

1
 0.08  6
1 
  1  0.656%
2 

e r  1  0.0816 r  7.84%
Example 5:
Suppose that some years ago your grandfather has created an account
which will provide you with a certain amount of cash over a 20 year period.
He arranged such that you will receive your first payment one year from now
in the amount of 3000. Also suppose that you will receive 6% net increase
every year over the next 20 years. That is, your first payment will be 3000,
second will be 3000(1.06), third will be 3000(1.06)2, and so on in terms of
constant dollars. Suppose that over the next 20 years, the inflation rate will be
5% per year and the inflation-free interest rate will be 3% per year.
a)How much will be your last payment (20th year) in terms of constant dollars?
b)How much will be your last payment (20th year) in terms of actual dollars?
c)Using constant dollar analysis and the geometric gradient formula, find the
present worth of your total earnings over 20 years.
d)Using actual dollar analysis and the geometric gradient formula, find the
present worth of your total earnings over 20 years.
Example 6:
The following equation describes the conversion of a cash flow into an equivalent
equal payment series of amount A for 8 years with an interest rate of 10% compounded
annually. Draw the original cash flow diagram.
A = [-1000 - 1000(P / F, 10%, 1) - 200(P / A1, 5%, 10%, 3) (P / F, 10%, 2)] (A / P, 10%, 8)
+ [3000+500(A / G, 10%, 4)] (P / A, 10%, 4) (P / F, 10%, 1) (A / P, 10%, 8)
+750(F / A, 10%, 2) (A / F, 10%, 8)
Example 6:
The following equation describes the conversion of a cash flow into an equivalent
equal payment series of amount A for 8 years with an interest rate of 10% compounded
annually. Draw the original cash flow diagram.
A = [-1000 - 1000(P / F, 10%, 1) - 200(P / A1, 5%, 10%, 3) (P / F, 10%, 2)] (A / P, 10%, 8)
+ [3000+500(A / G, 10%, 4)] (P / A, 10%, 4) (P / F, 10%, 1) (A / P, 10%, 8)
+750(F / A, 10%, 2) (A / F, 10%, 8)
3000
3500 4000 4500
750
750
0
1
1000 1000
2
3
4
200
210
5
220.5
6
7
8
Example 7:
A man borrows a loan of $50,000 from a bank. According to the agreement between
the bank and the man, the man will pay nothing for two years and starting at year 3,
he will pay equal amounts of A in every 3 months for the next 5 years. If the interest
rate is 10% compounded monthly and the inflation rate is 8%,
a) Find the payment amount A
b) Find the interest payment and principal payment for the 10th payment.
c) Find the total interest paid to the bank
d) Suppose you want to pay off the remaining loan in lump sum right after making the
15th payment. How much would this lump be? How much would this decrease your
total interest payment to the bank?
e) Now, assume that, instead of paying equal amounts of A in actual dollars, you want
to pay equal amounts of B in constant dollars. Find this constant dollar payment
amount B.
Example 7:
a) i3 = (1 +0.1/12)3 – 1
A = 50000 (F | P, i3, 8) * (A | P, i3, 20)
b) P9 = A(P | A, i3, 11)
Interest = P9* i3
Principal Payment = A – Interest
c) 20A-50000
d) P = A(P | A, i3, 5)
Decrease in total interest: 5A - P
e) f3 = (1 +0.08)˄1/4 - 1
i3’ = (i3 - f3) / (1 + f3)
B = 50000 (F | P, i3’, 8) * (A | P, i3’, 20)
Example 8: Assume that you have the following two options for a new machine.
Assuming that the salvage value of each option is constant at any time after the
machine is bought and MARR is 10%, answer the following using
PW, AE and IRR analysis.
a)
b)
c)
d)
OPTION 1
OPTION 2
Initial Cost
$20,000
$40,000
Annual Savings
12,000
25,000
Salvage Value
10,000
20,000
Life
3 yrs.
2 yrs.
Assuming that you will only operate the facility for 2 years, which option will you choose?
Assuming that you will close the facility after 5 years, which option will you choose?
Which option will you choose if you plan to operate your facility for an indefinite period
and each option will be available forever.
Assume that you are planning to operate your facility for an indefinite period and option 1 is
available forever. However, option 2 is available only now and if you choose option 2 now,
you will have to replace it by option 1 at the end of its service life. Which option will you
choose now?
Example 9:
Referring to the accompanying cash-flow diagram, answer
the following questions:
2000
1000
0
1
2
3
4
5
6
1000
3000
a)If i = 0, what is the PW?
b)What is the PW if i
∞?
c)If MARR is 10%, what is the discounted payback period?