Cash Flow Patterns
The “LEGO” blocks of Engineering
Economics
1
Future Given Present
 P is the present value at Time 0
 F is the future value at Time n
 (n periods in the future)
 i is the effective interest rate
F?
0
1
2
3
n
P
F = P(F/P,i,n)
2
Present Given Future
 P is the present value at Time 0
 F is the future value at Time n
 (n periods in the future)
 i is the effective interest rate for each period
F
0
1
2
3
n
P?
P = F(P/F,i,n)
3
Future Given Annual
 A is the equal annual value over the time period
 (time period: Time 0 to Time n, 1st flow at Time 1)
 F is the future value at Time n
 (n periods in the future)
 i is the effective interest rate for each period
F?
0
1
2
3
n
A
F = A(F/A,i,n)

Note: cash flow A does not have to be annual, just periodic
4
Annual Given Future
 A is the equal annual value over the time period
 (time period: Time 0 to Time n, 1st flow at Time 1)
 F is the future value at Time n
 (n periods in the future)
 i is the effective interest rate for each period
F
0
1
2
3
n
A?
A = F(A/F,i,n)

Note: cash flow A does not have to be annual, just periodic
5
Present Given Annual
 A is an equal annual flow over the time period
 (time period: Time 0 to Time n, 1st flow at Time 1)
 P is the present value at Time 0
 (n periods in the past)
 i is the effective interest rate for each period
P?
0
1
2
3
n
A
P = A(P/A,i,n)

Note: cash flow A does not have to be annual, just periodic
6
Annual Given Present
 A is the equivalent annual flow over the time period
 (time period: Time 0 to Time n, 1st flow at Time 1)
 P is the present value at Time 0
 (n periods in the past)
 i is the effective interest rate for each period
P
0
1
2
3
n
A?
A = P(A/P,i,n)

Note: cash flow A does not have to be annual, just periodic
7
Present Given Gradient
(Linear)
 G is the linear gradient over the time period
 (time period: Time 0 to Time n, 1st flow at Time 2)
 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
2
3
n
G=$/pd
P = G(P/G,i,n)

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Note: cash flow is periodic, no flow at Time 1, flow of G at Time 2
Future Given Gradient
(Linear)
 G is the linear gradient over the time period
 (time period: Time 0 to Time n, 1st flow at Time 2)
 F is the future value of the flow at Time n
 (n periods in the future)
 i is the effective interest rate for each period
F?
0
1
2
3
n
G=$/pd
F = G(F/G,i,n)

9
Note: cash flow is periodic, no flow at Time 1, flow of G at Time 2
Annual Given Gradient
(Linear)
 G is the linear gradient over the time period
 (time period: Time 0 to Time n, 1st flow at Time 2)
 A is the annual equivalent of the gradient flow
 (annual flow starts at Time 1, goes through Time n)
 i is the effective interest rate for each period
A?
0
1
2
3
n
G=$/pd
A = G(A/G,i,n)

10
Note: cash flow of G starts at Time 2, flow of A starts at Time 1
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
g = %/pd
P = A1(P/A,g,i,n)
n
   (1  g )  n 
 1  
 
(
1

i
)
 
 
when i  g


i g
 

( P / A, g , i , n )   




n

when i  g
(1  i )

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Note: cash flow starts with A1 at Time 1, increases by constant g%
Table Factors Listed
 (P/F, i, n) Present given Future
 (P/A, i, n) Present given Annual
 (P/G, i, n) Present given Gradient (linear)
 (F/P, i, n) Future given Present
 (F/A, i, n) Future given Annual
 (A/P, i, n) Annual given Present
 (A/F, i, n) Annual given Future
 (A/G, i, n) Annual given Gradient

Note: There is NO P/g. Present given Geometric Gradient is:
(P/A, g, i, n)
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