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

CTC 475 CTC 475 Gradient Series and Gradient Series and

Geometric SeriesGeometric Series

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)

ToolsTools

• Find P given A1, i, and j

• Find F given A1, i, and 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

Next lectureNext lecture• Changing interest

rates• Multiple

compounding periods in a year

• Effective interest rates