1. Combining Factors – Shifted Uniform Series
Question: What is P for the following cash flow, a shifted uniform series of n equal installments? The first installment occurs at the end of period 5. $P
n+4

2. Combining Factors – Shifted Uniform Series$P
n+4
Approaches for finding P:
Use (P/F) for each of the n payments.
Use (F/P) for each of the payments to find FT, then use FT(P/F,i%,n+4) to find P.
Use (P/A) to find the P4, then use P4(P/F,i%,4) to find P.

3. Combining Factors – Shifted Uniform Series$P
$200
Example: What is P for a computer you purchase in which installments of $200 are paid for 10 months, with the first payment deferred until the 5th month after purchase. Assume i = 5%.
A = $200
P4 = $200(P/A,5%,10) = $200 x = $
P = P4(P/F,5%,4) = $ x = $
P = $200(P/A,5%,10) (P/F,5%,4)

4. Combining Factors – Shifted Uniform Series$P
$2000
Example: How much will be in an account in which you invest $2000 beginning now and at the end of each year for 10 years? The account pays interest at 6%.
P = $ $2000(P/A,6,10)
P = $ $2000(7.3601) = $

5. Combining Factors – Shifted Uniform Series
$200
Example: What is F for the following cash flow. Installments of $200 are paid for periods 5 through 14. Assume i = 5%.
A = $200
F = $200(F/A,5%,10)
F = $200( ) = $
What is the account worth in period 20 (no installments made after period 14)?

6. Combining Factors – Single Amounts and Uniform Series$P
$200
How might you approach the above cost flow? $200 paid in periods 1,2,3,4,8,9,10; and $400 paid in periods 5.
$P1
$P2
$200
$200

7. Combining Factors – Single Amounts and Uniform Series$P
$200
P = $200(P/A,i%,10) + $200(P/F,i%,5)
$P1
$P2
$200
$200

8. Combining Factors – Multiple Uniform Series
$200
How might you approach the above cost flow? $200 paid in periods 1,2,3,4,8,9,10; and $400 paid in periods 5,6,7.
$P1
$P2
$200
$200

9. Combining Factors – Multiple Uniform Series
$200
P = $200(P/A,i%,10) + $200(P/A,i%,3)(P/F,i%,4)
$P1
$P2
$200
$200

10. Combining Factors – Shifted Gradients
$175
$150
$125
$100 $P

11. Combining Factors – Shifted Gradients
$175
$150
$125
$100 $P
P = P1+ P3
$75
$100
$50
$25
$P3
$P2
$P1

12. Combining Factors – Shifted Gradients
$75
$100
$50
$25
$P3
$P2
$P1
P = P1+ P3
P = $100(P/A,i%,8) + 25(P/G,i%,4)(P/F,i%,4)
Note, P2 = $25(P/G,i%,4) term is for 4 periods.

13. Combining Factors – Shifted Decreasing Gradients
$1000
$850
$700
$550 $P

14. Combining Factors – Shifted Decreasing Gradients
$1000
$1000
$850
$700
$550
=>
$450 $P
$P1
$300
$150
$P3
$P2

15. Combining Factors – Shifted Decreasing Gradients
$1000
$450
$300 -
$150
$P3
$P2
$P1
P = P1- P3
P = $1000(P/A,i%,7) - 150(P/G,i%,4)(P/F,i%,3)
Note, P2 = $150(P/G,i%,4) term is for 4 periods.