1. CTC 475 Review Uniform Series Find F given A
Find P given A (and deferred withdrawal)
Find A given F
Find A given P (and deferred withdrawal)
Rules:
P occurs one period before the first A
F occurs at the same time as the last A
n equals the number of A cash flows

2. Gradient Series and Geometric Series
CTC 475
Gradient Series and
Geometric Series

3. Objectives Know how to recognize and solve gradient series problems
Know how to recognize and solve geometric series problems

4. Gradient Series
Cash flows start at zero and vary by a constant amount G
G=?
EOY
Cash Flow 1 $0 2
$200 3
$400 4
$600 5
$800

5. Gradient Series Tools Find P given G Find A given G
Converts gradient to uniform
There is no “find F given G”
Find “P/G” and then multiply by “F/P” or
Find “A/G” and then multiply by “F/A”

6. Gradient Series Rules (differs from uniform/geometric)
P occurs 2 periods before the first G
n = the number of cash flows +1
(or---the first n cash flow is zero)

7. Find A given G (n=???) EOY Cash Flow 1 2 G 3 2G 4 3G 5 4G EOY1 2 G 3 2G 4 3G 5 4G
EOY
Cash Flow 1 A 2 3 4 5

8. Find P given G (Pure Gradient)
How much must be deposited in an account today at i=10% per year compounded yearly to withdraw $100, $200, $300, and $400 at years 2, 3, 4, and 5, respectively?
P=G(P/G10,5)=100(6.862)=$686

9. Find P Uniform +Gradient
How much must be deposited in an account today at i=10% per year compounded yearly to withdraw $1000, $1100, $1200, $1300 and $1400 at years 1, 2, 3, 4, and 5, respectively?
This is not a pure gradient (doesn’t start at $0) ; however, we could rewrite this cash flow to be a gradient series with G=$100 added to a uniform series with A=$1000
P=1000(P/A10,5)+100(P/G10,5)

10. Uniform + Gradient EOY Cash Flow 1 A=$1000 2 3 4 5 EOY Cash Flow 11
A=$1000 2 3 4 5
EOY
Cash Flow 1
G=$0 2
G=$100 3
G=$200 4
G=$300 5
G=$400

11. Combinations Uniform + a gradient series Uniform – a gradient series
(like previous example)
Uniform – a gradient series
(see next slide for example)

12. Find P Uniform–Gradient
What deposit must be made into an account paying 8% per yr. if the following withdrawals are made: $800, $700, $600, $500, $400 at years 1, 2, 3, 4, and 5 years respectively.
P=800(P/A8,5)-100(P/G8,5)

13. Uniform-Gradient EOY Cash Flow 1 A=$800 2 3 4 5 EOY Cash Flow 1 G=0 21
A=$800 2 3 4 5
EOY
Cash Flow 1
G=0 2
A=$100 3
A=$200 4
A=$300 5
A=$400

14. Example
What must be deposited into an account paying 6% per yr in order to withdraw $500 one year after the initial deposit and each subsequent withdrawal being $100 greater than the previous withdrawal? 10 withdrawals are planned.
P=$500(P/A6,10)+$100(P/G6,10)
P=$3,680+$2,960
P=$6,640

15. Example
An employee deposits $300 into an account paying 6% per year and increases the deposits by $100 per year for 4 more years. How much is in the account immediately after the 5th deposit?
Convert gradient to uniform
A=100(A/G6,5)=$188
Add above to uniform
A=$188+$300=$488
Find F given A
F=$488(F/A6,5)=$2,753

16. Geometric Series
Cash flows differ by a constant percentage j. The first cash flow is A1
Notes:
j can be positive or negative
geometric series are usually easy to identify because there are 2 rates; the growth rate of the account (i) and the growth rate of the cash flows (j)

17. Tools
Find P given A1, i, and j
Find F given A1, i, and j

18. Geometric Series Rules
P occurs 1 period before the first A1
n = the number of cash flows

19. Geometric Series Equations (i=j)
P=(n*A1) /(1+i)
Note: inside of the front cover of your book shows equation as A1*(n/(1+i))
F=n*A1*(1+i)(n-1)
Note: inside of the front cover of your book does not have this equation but F=P(1+i) so the above equation can be derived

20. Geometric Series Equations (i not equal to j)
P=A1*[(1-((1+j)n*(1+i)-n)/(i-j)]
F=A1*[((1+i)n-(1+j)n)/(i-j)]

21. Geometric Series Example
How much must be deposited in an account in order to have 30 annual withdrawals, with the size of the withdrawal increasing by 3% and the account paying 5%? The first withdrawal is to be $40,000?
P=A1*[(1-((1+j)n*(1+i)-n)/(i-j)]
A1=$40,000; i=.05; j=.03; n=30
P=$876,772

22. Geometric Series Example
An individual deposits $2000 into an account paying 6% yearly. The size of the deposit is increased 5% per year each year. How much will be in the fund immediately after the 40th deposit?
F=A1*[((1+i)n-(1+j)n)/(i-j)]
A1=$2,000; i=.06; j=.05; n=40
F=$649,146

23. Next lecture Changing interest rates
Multiple compounding periods in a year
Effective interest rates