CTC 475 Review Uniform Series –Find F given A –Find P given A –Find A given F –Find A given...
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Transcript of CTC 475 Review Uniform Series –Find F given A –Find P given A –Find A given F –Find A given...
CTC 475 Review CTC 475 Review Uniform Series
– Find F given A– Find P given A– Find A given F– Find A given PRules:1. P occurs one period before the first A2. F occurs at the same time as the last A3. n equals the number of A cash flows
ObjectivesObjectives• Know how to recognize and solve
gradient series problems• Know how to recognize and solve
geometric series problems
Gradient Series Gradient Series • Cash flows start at
zero and vary by a constant amount G
EOY Cash Flow
1 $0
2 $200
3 $400
4 $600
5 $800
Gradient Series Tools 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”
Gradient Series Rules Gradient Series Rules (differs from uniform/geometric)(differs from uniform/geometric)
• P occurs 2 periods before the first G
• n = the number of cash flows +1
Find A given GFind A given GEOY Cash Flow
0 0
1 0
2 G
3 2G
4 3G
5 4G
EOY Cash Flow
0 0
1 A
2 A
3 A
4 A
5 A
Find P given GFind P given GHow 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
Find P given GFind P given GHow 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
Gradient + UniformGradient + UniformEOY Cash Flow
0 0
1 0
2 G=$100
3 G=$200
4 G=$300
5 G=$400
EOY Cash Flow
0 0
1 A=$1000
2 A=$1000
3 A=$1000
4 A=$1000
5 A=$1000
CombinationsCombinations
• Uniform + a gradient series (like previous example)
• Uniform – a gradient series
Uniform–GradientUniform–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)
ExampleExample• 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
ExampleExample• 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 uniformA=100(A/G6,5)=$188
• Add above to uniform A=$188+$300=$488
• Find F given AF=$488(F/A6,5)=$2,753
Geometric SeriesGeometric Series
Cash flows differ by a constant percentage j. The first cash flow is A1
Notes:j can be positive or negativegeometric 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)
Geometric Series Rules Geometric Series Rules
• P occurs 1 period before the first A1
• n = the number of cash flows
Geometric Series Equations Geometric Series Equations (i=j)(i=j)
• P=(n*A1) /(1+i)
• F=n*A1*(1+i)n-1
Geometric Series Equations Geometric Series Equations (i not equal to j)(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)]
Geometric Series ExampleGeometric 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
Geometric Series ExampleGeometric 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