Chapter 5 – Unit 1
Annual Amount and Gradient
Functions
IET 350
Engineering Economics
Learning Objectives – Chapter 5
Upon completion of this chapter you should understand:
Calculating future values from annual amounts.
Calculating present values from annual amounts.
Calculating future and present values from gradient amounts.
Calculating present value of a future perpetual amounts.
Calculating deferred annuities.
2
Learning Objectives – Unit 1
Upon completion of this unit you should understand:
Calculating future values from annual amounts.
Calculating present values from annual amounts.
Calculating future and present values from gradient amounts.
Calculating present value of a future perpetual amounts.
Calculating deferred annuities.
3
Introduction
The prior chapter covered single‐payment functions where a cash inflow occurred at one point in time and a cash outflow occurred at a second point in time.
Many financial transactions have elements that occur at y multiple points in time. These can include:
Equal annual cash flow.
Linear gradient cash flow.
Non‐linear gradient cash flow.
Mixed annual cash flow.
4
Introduction
This chapter covers three types of multiple‐payment situations:
Equal annual amounts (A) – equal dollar amounts flow into or out of an investment or project each year.p j y
Linear gradient amounts (G) – dollar amounts flowing into or out of an investment or project increase/decrease each year by a constant amount (linear).
Mixed annual amounts – differing dollar amount flow into and/or out of an investment or project each year.
5
Equal Annual Amounts
Assumptions for equal annual amount analysis include:
Cash flow occurs at the end of each year.
All cash flows are equal and occur each year.
Note that most interest table such as those in appendix B of the Bowman text are based on end of year transactions.
Interest table are available that use the beginning or middle of time periods.
If using a time value of money function on your calculator, check the manual to determine if the time basis is end of period (year) or some other basis.
6
Equal Annual Amounts
Notations used for time value of money calculation
Future Value (one‐time occurrence) → F
Present Value (one‐time occurrence) → P
Equal Annual Amount → A
Cash flow diagrams represent annual amounts as equal length lines as illustrated in Figure 5‐1:
7
Future Value Calculations
Future value for an equal annual amount is determined by the following equation:
( ) ⎥ ⎤
⎢ ⎡ + 1 i
n‐ 1
8
( ) ⎥
⎦
⎢ ⎤
⎣
× ⎡ +
= i
1
i
1
A
F
Where: F = Future Value ($) A = Annual Amount ($) n = Time (years) i = Interest (% per year)
Future Value Calculations
Solution methods for finding future values:
Use the F/A column on a Interest Factors table (Bowman text appendix B, page 580).
Notation F/A is interpreted as → Find F given A
Notation F/A is interpreted as → Find F given A.
Notation (F/A, n, i) is interpreted as → Find F given A for n years at i interest rate.
Use the Excel function1→ FV(rate, nper, pmt, pv, type)
Use the formula and calculator.
9 1Note that the cash outflows are entered as a negative number.
(/ )
Future Value/Annual – Example
Your plan is to save $100 at the end of each year at 8% interest.
What will be the size of the account in 10 years?
( )
( )
$1,448.70 F
14.487
$100 8%
10, F/A, A F
=
×
=
×
=
10
FValue/Annual – Example
(continued) Solution using Excel®:
Note that the annual amount
11
was entered as a negative number which indicates a cash outflow.
( )
1
‐ 0.08)
$ (1 i
1 ‐ i 1 A F
10 n
⎤
⎡ +
⎥⎦
⎢ ⎤
⎣
×⎡ +
=
FValue/Annual – Example
(continued)Solution using formula:
F = Future Value = ? A = Annual Amount = $100
[ ]
$1,448.66 F
14.48656
$100
0.08 1 ‐ 2.158925
$100
0.08 1 0.08) (1
$100
=
×
= ⎥⎦⎤
⎢⎣⎡
×
=
⎥⎦
⎢ ⎤
⎣
×⎡ +
=
12
n = Time = 10 years i = Interest = 8% per year The slight difference between this amount and the amount determined by the factor from the tables is due to rounding.
Future Value Calculations
Equal annual amounts for a future value is determined by the following equation:
⎤
⎡ i
Note that this formula is the
13
( ) ⎥ ⎦ ⎤
⎢ ⎣
⎡
× +
= 1 i ‐ 1
i
F
A
nWhere: F = Future Value ($) A = Annual Amount ($) n = Time (years) i = Interest (% per year) inverse of the
formula to find F given A.
Future Value Calculations
Solution methods for finding annual amounts:
Use the A/F column on a Interest Factors table (Bowman text appendix B, page 580).
Notation A/F is interpreted as → Find A given F
Notation A/F is interpreted as → Find A given F.
Notation (A/F, n, i) is interpreted as → Find F given A for n years at i interest rate.
Use the Excel function1→ PMT(rate, nper, pv, fv, type)
Use the formula and calculator.
14 1Note that the cash outflows are entered as a negative number.
Annual/Future Value – Example
Your goal is to save $7,500 for a car down payment in 4 years by investing part of your end‐of‐year bonus.
How much to you need to save annually at 4% interest?
( )
( )
year
$1,766.25/
A
0.2355
$7,500 4%
4, A/F, F A
=
×
=
×
=
15
Annual/FValue – Example (continued)
Solution using Excel®:
Note that the function returns
ti
16
a negative number which indicates a cash outflow.
( )
1 i ‐1 iF
A n ⎥
⎦
⎢ ⎤
⎣
⎡
× +
=
Annual/FValue – Example (continued)
Solution using the formula:
F = Future Value = $7,500
A = Annual Amount = ?
( )
[ ]
$1,766.17 A
0.235490
$7,500
1 ‐ 1.169859
$7,500 0.04
1 ‐ 0.04) (1 0.04
$7,500 1 i 1
4
=
×
=
⎥⎦⎤
⎢⎣⎡
×
=
⎥⎦
⎢ ⎤
⎣
⎡
× +
=
⎦
⎣ +
17
n = Time = 4 years i = Interest = 4% per year
The slight difference between this amount and the amount determined by the factor from the tables is due to rounding.
End Unit 1 Material
Additional Reading Ö Financial Functions:
http://www.functionx.com/excel/Lesson12.htm
Go to Unit 2 Present Value Amounts
18
Chapter 5 – Unit 2
Present Value Amounts
IET 350
Engineering Economics
Learning Objectives – Unit 2
Upon completion of this unit you should understand:
Calculating future values from annual amounts.
Calculating present values from annual amounts.
Calculating future and present values from gradient amounts.
Calculating present value of a future perpetual amounts.
Calculating deferred annuities.
20
Present Value Calculations
Present value for an equal annual amount is determined by the following equation:
( ) ⎥ ⎤
⎢ ⎡ + 1 i
n‐ 1
21
( ) ( ) ⎥ ⎦ ⎤
⎢ ⎣
⎡
+
× +
=
ni
1
i
1
i
1
A
P
Where: P = Present Value ($) A = Annual Amount ($) n = Time (years) i = Interest (% per year)
Present Value Calculations
Solution methods for finding present values:
Use the P/A column on a Interest Factors table (Bowman text appendix B, page 580).
Notation P/A is interpreted as → Find P given A
Notation P/A is interpreted as → Find P given A.
Notation (P/A, n, i) is interpreted as → Find P given A for n years at i interest rate.
Use the Excel function1→ PV(rate, nper ,pmt, fv, type).
Use the formula and calculator.
22 1Note that the cash outflows are entered as a negative number.
Present Value/Annual – Example
You scheduled to receive $15,000 at the end of the next 7 years.
If the current interest rate is 6%, what is the equivalent amount today?
( )
( )
$83,736 P
5.5824
$15,000 6%
7, P/A, A P
=
×
=
×
=
23
PValue/Annual – Example (continued)
Solution using Excel®:
Note that the
24
function returns a negative number which indicates a cash outflow.
( )i ( )1 i 1 i 1 A
P n
n
⎥⎦
⎢ ⎤
⎣
⎡ +
−
× +
=
PValue/Annual – Example (continued)
Solution using the formula:
P = Present Value = ? A = Annual Amount = $15 000
[ ]
$83,735.72 P
5.582381
$15,000
(1.50363) (0.06)
1 1.50363
$15,000
0.06) (1 0.06
1 0.06) (1
$15,000 7
7
=
×
=
⎥⎦
⎢ ⎤
⎣
⎡
×
× −
=
⎥⎦
⎢ ⎤
⎣
⎡ +
×
−
× +
=
25
A = Annual Amount = $15,000 n = Time = 7 years i = Interest = 6% per year
Present Value Calculations
Equal annual amounts for a present value is determined by the following equation:
( ) ⎥ ⎤
⎢ ⎡ i 1 + i
nNote that this formula is the
26
( ) ( ) ⎥ ⎦
⎢ ⎤
⎣
⎡
+
× +
= 1 i ‐ 1
i
1
i
P
A
nWhere: P = Present Value ($) A = Annual Amount ($) n = Time (years) i = Interest (% per year) inverse of the
formula to find P given A.
Present Value Calculations
Solution methods for finding present values:
Use the A/P column on a Interest Factors table (Bowman text appendix B, page 580).
Notation A/P is interpreted as → Find A given P
Notation A/P is interpreted as → Find A given P.
Notation (A/P, n, i) is interpreted as → Find A given P for n years at i interest rate.
Use the Excel function1→ PMT(rate, nper, pv, fv, type).
Use the formula and calculator.
27 1Note that the cash outflows are entered as a negative number.
(A/P83%)
P
A ×
Annual/Present Value – Example
You $5,000 invest in an account that returns 6% annual interest.
How much can you withdraw each semester (twice/year) over the next 4 years for books and supplies?
( )
( )
mester
$712.50/se A
0.1425
$5,000 3%
8, A/P, P A
=
×
=
×
=
28
Time periods other than a year can be used, however, the tabulated interest rate is a yearly rate so it must be adjusted to match the number of periods/ year → 6% per year/2 periods per year = 3% per period . Also the total number of periods is used → 4 yrs x 2 periods/yr = 8 periods.
Annual/PValue – Example (continued)
Solution using Excel®:
Remember that i and n must be
29
and n must be adjusted for time periods other than yearly.
1 ‐ m r 1
m r m 1
r P m A
nm nm
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎣
⎡
⎟⎠
⎜ ⎞
⎝⎛ +
⎟⎠
⎜ ⎞
⎝⎛ +
×
=
Annual/PValue – Example (continued)
Solution using the formula (see page 187 Bowman text):
P = Present Value = $5 000
[ ]
mester
$712.28/se m A
0.142456
$5,000
1 ‐ 1.26677
1.26677
$5,000 0.03
1 ‐ 2 ) 0.06 (1
2 ) 0.06 (1 2 0.06
$5,000
2 4
2 4
=
×
=
⎥⎦⎤
⎢⎣⎡ ×
×
=
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡ +
+
×
×
=
×
×
30
P = Present Value = $5,000 A = Annual Amount = ? n = Time = 4 years M = #Periods/year = 2 r = Annual Interest = 6%
Finding Unknown n or i Values
Occasionally an engineering economic analysis will occur when the number of years (n) or the interest rate (i) is unknown.
Like single‐payment calculations, if three of the four factors g p y , are known, we can solve for the unknown.
Future value factors → F, A, i, n.
Present value factors → P, A, i, n.
31
Unknown i and n Calculations
Solution methods for finding interest or time period values:
Interpolate using the appropriate column on a Interest Factors table (Bowman text appendix B, page 580).
Use the Excel functions1:
Use the Excel functions :
RATE(nper, pmt, pv, fv, type, guess) → returns the interest rate per period for a cash flow.
NPER(rate, pmt, pv, fv, type) → returns the number of periods for a cash flow with a constant interest rate.
Rearrange the appropriate formula and solve with your calculator.
1Note that the cash outflows are entered as a negative number. 32
Unknown Interest – Example
You have $5,000 to invest in an account and would like to withdraw $1,550 per year for the next four years..
What interest rate will be required to meet the needs?
33
You will need to invest the $5,000 at 9.2% annual interest rate.
Unknown Time – Example
You plan to invest $1,250 per year in a security with a 4.75%
annual return rate.
How many years before the account grows to $12,500?
34
Time Required 8 years 4 months
15 days
End Unit 2 Material
Go to Unit 3 Gradient Amounts
35
Chapter 5 – Unit 3
Gradient Amounts
IET 350
Engineering Economics
Learning Objectives – Unit 3
Upon completion of this unit you should understand:
Calculating future values from annual amounts.
Calculating present values from annual amounts.
Calculating future and present values from gradient amounts.
Calculating present value of a future perpetual amounts.
Calculating deferred annuities.
37
Gradient Amounts
Unlike equal annual amounts, gradient amounts increase or decrease each time period. Types:
Linear – change in cash flow is by an equal amount for each time period. Gradient factors tabulated in the p interest tables or determined by formula.
Non‐linear – change is cash flow varies between time periods. Non‐linear gradient functions must be calculated with a series of P/A or A/P for each time period.
38
Gradient Amounts
Assumptions for linear gradient amount analysis include:
Cash flow occurs at the end of each year.
Change in cash flow year to year is at a constant rate. The amount of change is designated → Gg g
Initial cash flow is designated → A’
39
Gradient Calculations
Solution methods for finding present values:
Use the A/G column on a Interest Factors table (Bowman text appendix B, page 580). Future or present values can then be determined using the annual amount (A):
Notation A/G is interpreted as → Find A given G.
Notation (A/G, n, i) is interpreted as → Find A given P for n years at i interest rate.
Use the Excel function1→ XNPV(rate, values, dates)
Use the formula and calculator.
40 1Note that the cash outflows are entered as a negative number.
Gradient Calculations
Equal annual amounts for a linear gradient values is determined by the following equation:
⎥ ⎤
⎢ ⎡
−
×
′ ±
= 1 n
G
A
A
41
( ) ⎥ ⎦
⎢ ⎣ − +
×
±
= 1 i ‐ 1
i
G
A
A
nWhere: A = Annual Amount ($) A′ = Initial Cash Flow($) G = Gradient Amount ($) n = Time (years) i = Interest (% per year)
When the change is increasing between time
periods, the gradient is added
(+) to the initial value and subtracted (‐) when decreasing.
Gradient – Example
Your 1styear’s salary is $45,000. Your contract states that your raise will be $5,000/year in years 2 through 6.
What is the present value of the contract at 5% interest?
Cash Flow Diagram:
Cash Flow Diagram:
42
(/ )
′
Gradient – Example (continued)
Solution method:
Find annual value (A) of the gradient (G).
Convert the annual amount (A) into the present value (P).
( )
( )
0
$288,246.5 P
(5.0757)
$56,789.50 5%) 6, A(P/A, P
/year
$56,789.50 A
2.3579
$5,000
$45,000 5%
6, A/G, G A A
=
×
=
=
=
× +
=
′+
=
43
Gradient – Example (continued)
Solution using Excel®. You must create a schedule of amounts with a date. The schedule must start at time = 0 (today).
Non‐linear gradients can be solved with this method.
44
( )
( )
6 1 ‐ 0.05 1 6 0.05
$5,000 1
$45,000
1 ‐ i 1 n i 1 G A A
6 n
⎤
⎡
⎥⎦
⎢ ⎤
⎣
⎡
− +
× +
=
⎥⎦
⎢ ⎤
⎣
⎡
− +
×
′±
=
Gradient – Example (continued)
Solution using the formulas:
P = Present value = ? A = Annual Amount = ?
[ ]
( )( )
[ ]
0
$288,246.9 P
5.075697
$56,789.61
0.05) (1 0.05
1 0.05) (1
$56,789.61 i 1 i
1 i 1 A P
/year
$56,789.61 A
2.357922
$5,000
$45,000
1 ‐ 1.340096 ‐ 6 20
$5,000
$45,000
6 6 n n
=
×
=
⎥⎦
⎢ ⎤
⎣
⎡ +
×
−
× +
=
⎥⎦
⎢ ⎤
⎣
⎡ +
−
× +
=
=
× +
=
⎥⎦⎤
⎢⎣⎡
× +
=
45
A′ = Initial Amount = $45,000 G = Gradient = $5,000 n = Time = 6 years i = Interest = 5% per year
End Unit 3 Material
Go to Unit 4 Perpetual Amounts and
Deferred Annuities
46
Chapter 5 – Unit 4
Perpetual Amounts and Deferred
Annuities
IET 350
Engineering Economics
Learning Objectives – Unit 4
Upon completion of this unit you should understand:
Calculating future values from annual amounts.
Calculating present values from annual amounts.
Calculating future and present values from gradient amounts.
Calculating present value of a future perpetual amounts.
Calculating deferred annuities.
48
Perpetual Gradient Amounts
Perpetual gradient amounts increase or decrease each time period. Assumptions:
Cash flow occurs at the end of each year with change at a constant rate. The amount of change is designated → Gg g
Project life is considered infinite: n → ∞
49
Perpetual Gradient Amounts
Present value for project with an infinite life and a gradient increase in cash flow is determined by the following equation:
Gradient Annual
0
Total
P P P
P = + +
50
0 2 Total
i
G
i
A
P
P = + +
Where: P0= Initial Project Cost ($) A = Initial Annual Amount ($) G = Gradient Amount ($) i = Interest (% per year)
Perpetual Gradient – Example
A warehouse was constructed at an initial cost of $500,000 and is expected to last forever. First year maintenance cost is
$5,000 and is expected to increase at a constant $500/year.
If the firm uses a 7.5% interest rate, what is the total present
$655,556 P
$88.888.89
$66,666.67
$500,000
0.075 $500 0.075
$5000
$500,000 i G i A P P
Total
2 0 2
Total
=
+ +
=
+ +
= + +
=
51
, p
value of the project?
P = Present value = ? P0= $500,000
A = Maintenance = $5000/yr G = Gradient = $500/yr i = 7.5% per year
Deferred Annuities
An annuity is equivalent to an annual amount.
A deferred annuity is a set of annual cash flows that will occur in the future (deferred) rather than immediately.
Figure 5 18 from the Bowman text illustrates an investment
Figure 5‐18 from the Bowman text illustrates an investment made a t0(present time) with annual disbursement beginning in the 4thyear.
52
Deferred Annuities
Solution steps for deferred annuities:
Determine the future value (F) of the initial investment at the beginning time point of the deferred annuity.
Set the future value (F) to the deferred present value (P′).( ) p ( )
Determine the annual amount (A) using the deferred present value (P′).
53
Deferred Annuity – Example
For your 8thbirthday your rich uncle gave you $25,000 for college which you invested in a deferred annuity at 4%.
You begin drawing on the annuity at age 18 for 4 years.
How much will you receive each year?
( )
( )
/year
$10,193.50 A
2755)
$37,000(0.
F P with 4%) 4, (A/P, P A
$37,000 F
1.480
$25,000 4%
10, F/P, P F
=
=
=
′
′
=
=
=
=
y y
54
End Chapter 5 Material
Student Study Guide Ö Chapter 5
Homework Assignment Ö Problem Set 5
55
