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CTC 475 Review

CTC 475 Review. Uniform Series Find F given A Find P given A Find A given F Find A given P 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. CTC 475. Gradient Series and Geometric Series. Objectives.

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CTC 475 Review

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  1. CTC 475 Review Uniform Series • Find F given A • Find P given A • Find A given F • Find A given P 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. 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

  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

  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/G10,5)=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

  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/A8,5)-100(P/G8,5)

  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 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

  15. 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)

  16. Tools • Find P given A1, i, and j • Find F given A1, i, and j

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

  18. Geometric Series Equations (i=j) • P=(n*A1) /(1+i) • F=n*A1*(1+i)n-1

  19. 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)]

  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=A1*[(1-(1+j)n*(1+i)-n)/(i-j)] • A1=$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 40th deposit? • F=A1*[((1+i)n-(1+j)n)/(i-j)] • A1=$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

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