» Single Payment ˃ Simplest case which involves the equivalence of a single present

amount and its future worth

˃ Formula deals with only two amounts: + P – present amount + F – future worth

( )1 NF P i= +

Single Payment Cash Flow

» Uniform Series ˃ Transactions are arranged as a series of equal cash flows at regular

intervals ˃ Also known as an equal-payment series ˃ Describes the cash flows of common installment loan contract which

arranges for the repayment of a loan in equal periodic installments ˃ Formula deals with the equivalence relations to P, F, and A (constant

amount of the cash flows in the series)

» Uniform Series ˃ Compound Amount Factor – Find F, Given A, i, N

+ Used when we are interested to compute for the future amount (F) of a fund to which we contribute (A) amount each period and earn interest (i) per period

+ Notation: A = F (A/F, i, N)

( )1 1NiF A

i

+ −=

Uniform-series compound amount factor

» Uniform Series ˃ Sinking-fund Factor – Find A, Given F, i, N

+ It is an interest bearing account where a fixed asset is deposited each interest period

+ Purpose: Replace fixed assets + Notation: F = A (F/A, i, N)

Sinking fund factor

( )1 1NiA F

i

=

+ −

(?) (?) (?) (?) (?)

()

» Uniform Series ˃ Capital Recovery Factor (Annuity Factor) – Find A, Given P, i, N

+ Notation: A = P (A/P, i, N)

( )( )

11 1

N

N

i iA P

i

+=

+ −

(?) (?) (?) (?) (?)

()

Capital Recovery Factor

» Uniform Series ˃ Present Worth Factor – Find P, Given A, i, N

+ Notation: P = A (P/A, i, N) + This answers the question, “What would you have to invest now

in order to withdraw A amounts at the end of N periods?”

( )( )

1 11

N

N

iP A

i i

+ −=

+

Equal-payment-series present worth factor

() () () () ()

(?)

1. What is the future worth of an equal payment series of $1,000 each for 5 years if the interest rate is 9% compounded continuously? Illustrate the transaction.

($ 1000) ($ 1000) ($ 1000) ($ 1000) ($ 1000)

(?)

i = 9% comp. cont.

Given:

( )1 1NiF A

i

+ −=

0.09

1

10.0942

rc

c

c

i ei ei

= −

= −=

( )51.0942 11,000

0.0942

$6,034.63

F

F

−=

=

Sol’n:

2. If you desire to withdraw the following amounts over the next 5 years from a savings account that earn a 6% interest compounded annually, how much do you need to deposit now?

1 2 3 4 5

$ 2,000 $ 3,000

$ 5,000 $ 3,000

P (?)

Given:

( )

( )

( )

( )

( )

2 3 4

2

3

4

5

52

2

33

44

55

$1,779.99

$2,518.86

$

1

2000 1.06

3000 1.06

5000 1.06

3000 1.

3,960.47

$2,241.7706

$1,779.99 $2,518.86 $3,960.47 $2,2$10,501

41.77.09

NP F i

P

P

P

P

P P P P P

P

P

P

PP

P

−

−

−

−

−

= +

= + + +

=

=

=

=

= +=

+ +

=

=

=

=

Sol’n:

3. A house and lot can be acquired a down payment of 500,000.00 PhP and a yearly payment of 100,000.00 PhP at the end of each year for a period of 10 years, starting at the end of 5 years from the date of purchase. If money is worth 14% compounded annually, what is the cash price of the property?

Given: DP = PhP 500,000 A = PhP 100,000 starting after 5 years i = 14% per year Req’d: P

( )( )

( )( )

( )

2

1

10

1 10

1

2 14

2

2

808,835.92

500,000

1 11

1.14 1100000

0.14 1.14

521,611.56

(1 )

521,611.56 1.14308,835.92

N

N

N

P P

iP A

i i

P

PP P i

P

PP

−

−

= +

+ −=

+ −

=

=

=

= +

=

=

A A A A A A A A A A

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

500,000 I I I I I I I I I I P1

P2

Sol’n:

Linear and Geometric Gradient Series

» Linear Gradient Series ˃ Cash flow which increase or decrease by a fixed amount ˃ In addition to P, F, and A, the formulas used in this problems involve

the constant amount, G (amount change per period), of the change in each cash flow.

» Linear Gradient Series ˃ Present Worth Factor – Linear Gradient: Find P, Given A, G, N, i ˃ Notation: P = G (P/G, i, N)

( )( )2

1 11

N

N

i iNP G

i i

+ − −=

+

Gradient-series present worth factor

» Linear Gradient Series ˃ Linear Gradient: Find P, Given, A1, G, i, N

Example 1 A textile mill has just purchased a lift truck that has a useful life of 5 years. The engineer estimates that the maintenance costs for the truck during the first year will be $1,000. Maintenance costs are expected to increase as the truck ages at a rate of $250 per year over the remaining life. Assume that the maintenance costs occur at the end of each year. The firm wants to set up a maintenance account that earns 12% interest. All future maintenance expenses will be paid out of this account. How much does the firm have to deposit in this account now?

» Linear Gradient Series ˃ Equal-Payment-Series Factor: Find A, Given A1, G, i, N ˃ Notation: A = G (A/G, i, N)

( )( )

1 1

1 1

N

N

i iNA G

i i

+ − − = + −

Gradient-to-equal payment series conversion factor

» Linear Gradient Series ˃ Linear Gradient: Find A, Given, A1, G, i, N

Example 2 John and Barbara have just opened two savings accounts at their credit union. The accounts earn 10% annual interest. John wants to deposit $1,000 in his account at the end of the first year and increase this amount by $300 for each of the following 5 years. Barbara wants to deposit an equal amount each year for the next 6 years. What should be the size of Barbara’s annual deposit so that the two accounts would have equal balances at the end of 6 years?

» Linear Gradient Series ˃ Future Worth Factor – Find F, Given A, G, i, N ˃ Notation: F = G (F/G, i, N)

( )1 1NiGF Ni i

+ −= −

» Linear Gradient Series ˃ Declining Linear Gradient: Find F, Given A1, G, i, N

Example 3 Suppose that you make a series of annual deposits into a bank account that pays 10% interest. The initial deposit at the end of the first year is $1,200. The deposit amounts decline by $200 in each of the next 4 years. How much would you have immediately after the fifth deposit?

» Geometric Gradient Series ˃ Usually applied to construction costs ˃ Involves cash flows that increase or decrease over time by a constant

percentage, g = the percent change in payment per period ˃ Growth by geometric gradient is sometimes called compound growth ˃ Geometric series can illustrate changes in price due to inflation

» Geometric Gradient Series ˃ Present Worth Factor – Find P, Given A1, g, i, N ˃ Notation: P = A1 (P/A1, g, i, N)

( ) ( )1

1

1 1 1, if

, if 1

N Ng iP A N i g

i g

NAP i gi

− − + += − ≠

−

= =+

Geometric-gradient-series present worth factor

» Geometric Gradient Series ˃ Geometric Gradient: Find P, Given A1, g, i, N

Example 4 A municipal power plant is expected to generate a net revenue of $500,000 at the end of its first year, and this annual amount will increase by 8% per year for the next 5 years. To finance a new construction project, the municipal government wants to issue a tax exempt bond that pays 6% annual interest. The amount of bond financing will deposit on the equivalent present worth of the expected future earnings from the power plant, which will be used to pay off the bonds. What would be the maximum amount of bond financing that could be secured? Note: A bond is a long-term promissory note issued by a business or governmental unit.

» Geometric Gradient Series ˃ Future Worth Factor – Find F, Given A1, g, N, i ˃ Notation: F = A1 (F/A1, g, i, N)

( ) ( )

( )

1

11

1 1, if

1 , if

N N

N

i gF A i g

i g

F NA i i g−

+ − += ≠

−

= + =

» Geometric Gradient Series ˃ Geometric Gradient: Find F, Given A1, g, i, N

Example 5 Jimmy Carpenter is opening a retirement account at a bank. His goal is to accumulate $1,000,000 in the account by the time he retires from work in 20 years time. A local bank is willing to open a retirement account that pays 8% interest, compounded annually throughout the 20 years. Jimmy expects his annual income will increase at a 6% annual rate during his working career. He wishes to start with deposit at year 1 (A1) and increase the deposit at a rate of 6% each year thereafter. What should be the size of his first deposit? The first deposit will occur at the end of year 1, and subsequent deposits will be made at the end of each year. The last deposit will be made at the end of year 20.

» Irregular Series ˃ Exhibits no regular overall pattern ˃ One or more of the patterns discussed may over portions of the

irregular series

Example 6: Cash flows with sub-patterns The two cash flows shown below are equivalent at an interest rate of 12% compounded annually. Determine the unknown value X.

1. Single Payment ˃ Deals with two amounts, P and F

2. Uniform Series ˃ Cash flows at regular intervals

3. Linear Gradient Series ˃ Cash flows in a series that increase or decrease by a fixed amount

4. Geometric Gradient Series ˃ Cash flow in a series that increase or decrease by some fixed rate

5. Irregular Series ˃ Cash flow that does not exhibit regular pattern

1. The cash flow transactions shown below are said to be equivalent at 10% interest compounded annually. Find the unknown X value that satisfies the equivalence.

2. Consider the cash flow shown below. What value of C makes the inflow series equivalent to the outflow series at an interest rate of 12%