Chapter 2: Factors: How Time and Interest Affect Money

Engineering Economy

ENMG 400

Outline

• Single Amount Factors

• Uneven Cash Flow Series

• Even Cash Flow Series (A)

• Arithmetic (Linear) Gradient (G)

• Geometric Gradient (g)

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Single-Amount Factors

• One cash flow at one time

• Goal: to determine the cash flow at a different time

• P F or F P

• Equation needed: F = P(1 + i)N

N (years)

i = 4%

0

1 2 3 4 5

$1000

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Example: Single Amount Factor

• You put $5,000 into a bank CD (certificate of deposit) for 3 years at 5% interest rate (interest is credited annually). How much money will you have at the end of 3 years, assuming you make no withdrawals?

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Factor Notation

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• We can express the compound interest formula as

• We can rewrite the formulas as:

• Where 𝐹 𝑃 , 𝑖, 𝑁 = (1 + 𝑖)𝑁

𝑃 𝐹 , 𝑖, 𝑁 = (1 + 𝑖)−𝑁

(1 )

(1 )

N

N

F P i

P F i

( / , , )

( / , , )

F P F P i N

P F P F i N

Look up on chart in back of book.

Compound Amount Factor

Present Worth Factor

Chart Table

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Example using Chart

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Excel Formulas: P and F

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To find… Formula

Present Value (P) =PV(rate, nper, pmt, [fv], [type])

Future Value (F) =FV(rate, nper, pmt, [pv], [type])

Rate: interest rate (i) Nper: number of periods (time, N) Pmt: payment (annuity) Pv: present value Fv: future value Type: specifies if payments occur at the beginning (1) or end (0) of the period. Default is 0. Note: brackets indicate optional variables (the formula will compute without them). [fv] and [pv] are assumed to be zero. Note: PV and FV functions change the sense of the sign. Place a minus in front of the function to retain the same sign.

Excel (Same Example)

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Uneven Cash Flow Series

• Basic concept: treat each cash flow as a single cash flow diagram, and add together

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N (years)

F1 F5 F4

F2

F3

0

1 2

3

4 5

?

Example: Uneven Cash Flow Series

• Basic concept: treat each cash flow as a single cash flow diagram, and add together

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N (years)

$400

$350 $400 $400

$600

$400

i = 4%

0

1 2

3

4 5

?

Excel

• Solve the last problem using excel.

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Excel: Net Present Value

• NPV function: calculates the net present value of a series of future cash flows at a stated interest rate.

=NPV(rate,value 1,value 2,value 3, etc.)

• NPV does not change the sign on the final answer like the PV, FV, and PMT functions.

• Any year with a 0 cash flow must have a 0 entered to ensure a correct result.

• Work Last Example.

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Summary for P/F and F/P

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• Net Present Value: for a series of future cash flows at a stated interest rate.

=NPV(rate,value 1,value 2,value 3, etc.)

Example

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Equal payment series

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• Scenario: pay or receive an equal amount every year, month, etc.

• Examples: loan repayment, automatic payroll deductions for retirement

A

0 1 2 3 4 5

?

Equal payment series formulas

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• Uniform Series Compound Amount

• Sinking Fund

• Uniform-Series Present Worth

• Capital Recovery

(1 ) 1NiF A

i

(1 ) 1N

iA F

i

(1 ) 1

(1 )

N

N

iP A

i i

(1 )

(1 ) 1

N

N

i iA P

i

Equal payment series

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• Example: You receive $400 per year for five years. What is the equivalent worth of these annual receipts in the present?

A=$400

i = 4%

0 1 2 3 4 5

?

Example

• How much money must carol deposit every year starting 1 year from now at 5.5% per year in order to accumulate $6000 seven years from now?

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Example: Equal payment series

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• IRAs (Individual Retirement Accounts) can be setup by anyone earning income to save for retirement in a tax-advantaged account. The current IRS limits are $5000 per year when you are younger than 55 (pending income restrictions). If you contribute $5000 to an IRA every year at age 24 until you expected retire age (67), how much will you have if the account earns 8% annually?

– Draw a cash flow diagram

– Calculate the amount you will have in the future

Equal payment series charts

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• Uniform Series Compound Amount 𝐹 = 𝐴(𝐹 𝐴 , 𝑖, 𝑁)

• Sinking Fund 𝐴 = 𝐹(𝐴 𝐹 , 𝑖, 𝑁)

• Uniform-Series Present Worth 𝑃 = 𝐴(𝑃 𝐴 , 𝑖, 𝑁)

• Capital Recovery 𝐴 = 𝑃(𝐴 𝑃 , 𝑖, 𝑁)

Chart Table

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Example

• How much money should you be willing to pay now for a guaranteed $600 per year for 9 years, starting next year, at a rate of return of 16% per year?

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Excel Formulas (Appendix A)

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To find… Formula

Present Value (P) =PV(rate, nper, pmt, [fv], [type])

Future Value (F) =FV(rate, nper, pmt, [pv], [type])

Annuity (A) =PMT(rate, nper, pv, [fv], [type])

Rate: interest rate (i) Nper: number of periods (time, N) Pmt: payment (annuity) Pv: present value Fv: future value Type: specifies if payments occur at the beginning (1) or end (0) of the period. Default is 0. Note: brackets indicate optional variables (the formula will compute without them). [fv] and [pv] are assumed to be zero. Note: PV, FV, and PMT functions change the sense of the sign. Place a minus in front of the function to retain the same sign.

Example using Excel

• Work last problem in Excel

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Summary for P/A and A/P

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Summary for F/A and A/F

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Linear or (Arithmetic) Gradient Series

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• In some cases cash flows increase by a fixed amount in each time period

• G: gradient; amount of increase each period

• Note: starts at n=2 for formulas

G

0 1 2 3 4 5

2G

3G

4G

(N-1)G

P=?

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Linear Gradient Series

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Linear Gradient (G) Present Worth (P)

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• Where Pg is the equivalent at time 0 of the linear cash flow series with gradient G (i.e. the series having G, 2G, 3G,…, [n-1]G).

G=$100

0 1 2 3 4 5

2G

3G

4G

(N-1)G

N=6

i=5%

P=?

Linear Gradient (G) Future Worth (F)

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• Where Fg is the future equivalent at time n of the linear cash flow series with gradient G.

G=$100

0 1 2 3 4 5

2G

3G

4G (N-1)G

N=6

i=5%

F = ?

Linear Gradient (G) Equal Payment (A)

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G=$100

i=5%

0 1 2 3 4 5 6 7

A=?

i=5%

0 1 2 3 4 5 6 7

Note: To use the formulas “as is” -Gradient series starts with n=2 -Equal payment series starts with n=1

Linear Gradient Series

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Increasing Gradient Series

G=$50 A1=$100

P=?

i=5% i=5% i=5%

Note: P (n=0) MUST match for both series

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Decreasing Gradient Series

G=$100 A1=$600

P=?

What is the present worth? What if we were looking for future worth?

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i=5%

Example: Linear Gradient

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this

Geometric Gradient (g) to Present Worth (P)

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• Percentage increasing/decreasing cash flow

Geometric Gradient (g) to Annuity (A) or Future (F)

• It is possible to derive factors for the equivalent A and F values; however, it is easier to determine the Pg amount and then multiply by the A/P or F/P factor.

– Ag = Pg (A/P, i, n)

– Fg = Pg (F/P, i, n)

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Example: Geometric Gradient

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Excel for Linear and Geometric Gradient

• There are no direct spreadsheet functions for linear and geometric gradient series. Once the cash flows are entered, P and A are determine using the NPV and PMT functions, respectively.

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i

Factor Values for Untabulated i or n Values

• How to determine a factor value (e.g. P/F, F/A, A/P, etc.) with an i or n that is not listed in the charts in the back of the book.

• Given i and n, there are several ways to obtain any factor value:

– Use the Factor Formula (in previous slide)

– Use an Excel Function (next slide)

– Use Linear Interpolation (second next slide)

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Excel: Determining Factor Values

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Interpolation: Determining Factor Values

𝑓 − 𝑓1𝑓2 − 𝑓1

= 𝑥 − 𝑥1𝑥2 − 𝑥1

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Example

• Determine the P/A factor value for i = 7.75% and n = 10 years, using the three methods described previously.

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Finding i or n

• Determining i or n for known cash flow values – Using Excel (below)

– Using the tables (find the factor value and interpolate in the tables)

– Using the factor formulas

Using Excel to find:

• Interest (i)

=IRR(first_cell:last_cell)

=RATE(n,A,P,F)

• Number of periods (n)

=NPER(i%,A,P,F)

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Example

• If Laurel made a $30,000 investment in a friend’s business and received $50,000 5 years later, determine the rate of return using: – Excel

– Tables in the back of the book (round up)

– Factor Formulas

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Example

• A cement factory makes an investment of $200 million and expects an annual revenue of $50 million for future years. Determine the number of years required to generate 10%, 15%, and 20% per year returns on the investment.

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