1. CTC 475 Review Uniform Series –Find F given A –Find P given A –Find A given F –Find A given P Rules: 1.P occurs one period before the first A 2.F occurs at the same time as the last A 3.n equals the number of A cash flows

2. 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 EOYCash 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

7. Find A given G EOYCash Flow 00 10 2G 32G 43G 54G EOYCash Flow 00 1A 2A 3A 4A 5A

8. Find P given G 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/G 10,5 )=100(6.862)=$686

9. Find P given G 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

10. Gradient + Uniform EOYCash Flow 00 10 2G=$100 3G=$200 4G=$300 5G=$400 EOYCash Flow 00 1A=$1000 2 3 4 5

11. Combinations Uniform + a gradient series (like previous example) Uniform – a gradient series

12. 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/A 8,5 )-100(P/G 8,5 )

13. 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

14. 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 5 th 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

15. Geometric Series Cash flows differ by a constant percentage j. The first cash flow is A 1 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)

16. Tools Find P given A 1, i, and j Find F given A 1, i, and j

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

18. Geometric Series Equations (i=j) P=(n*A 1 ) /(1+i) F=n*A 1 *(1+i) n-1

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

20. 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=A 1 *[(1-(1+j) n *(1+i) -n )/(i-j)] A 1 =$40,000; i=.05; j=.03; n=30 P=$876,772

21. 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 40 th deposit? F=A 1 *[((1+i) n -(1+j) n )/(i-j)] A 1 =$2,000; i=.06; j=.05; n=40 F=$649,146

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