Feedback and Warm-Up Review

1

Feedback and Warm-Up ReviewFeedback and Warm-Up Review

• Feedback of your requests• Cash Flow • Cash Flow Diagrams• Economic Equivalence

2

FeedbackFeedback

• Feedback 1: Power point on line to save toner $$$ -- done; background changed;

• PPT: there is a non-background option• Feedback 2: More examples in class-------------

yes, we also have tutorial class;• Feedback 3: Arrange projects

early----------------yes, quiz review changed to project and quiz review, starting this Friday.

• Important: Homepage updates……

3

Cash FlowsCash Flows• The expenses and receipts due to

engineering projects.

4

Cash Flow DiagramsCash Flow Diagrams

• The costs and benefits of engineering projects over time are summarized on a cash flow diagram.

• Cash flow diagram illustrates the size, sign, and timing of individual cash flows

5

Cash Flow DiagramsCash Flow Diagrams

1 2 3 4 5

0Time (# of interest periods)

Positive net Cash flow(receipts)

Negative net Cash Flow(payments)

$15,000

$2000

$13,000 is net positive cash flow

6

Economic EquivalenceEconomic Equivalence

•We need to compare the economic worth of $.

•Economic equivalence exists between cash flows if they have the same economic effect.

Convert cash flows into an equivalent cash flow atany point in time and compare.

7

• Single Sum Compounding • Annuities• Conversion for Arithmetic Gradient Series• Conversion for Geometric Gradient Series

Topics Today

8

Simple InterestSimple Interest• The interest payment each year is found by multiplying the

interest rate times the principal, I = Pi. After any n time periods, the accumulated value of money owed under simple interest, Fn, would be:

• For example, $100 invested now at 9% simple interest for 8 years would yield

• Nobody uses simple interest.

Fn = P(1 + i*n)

F8 = $100[1+0.09(8)] = $172

9

Compound InterestCompound Interest• The interest payment each year, or each period, is found

by multiplying the interest rate by the accumulated value of money, both principal and interest.

End of Period (EOP)

Accumulated EOP Value or Amount

Owed (1)Interest for Period (2)

Amount Owed or Value Accumulated Next Period (3) = (1) + (2)

0 P Pi P + Pi = P ( 1 + i )

1 P ( 1 + i )1 [P ( 1 + i )1]i P ( 1 + i ) + P ( 1 + i )i = P ( 1 + i )2

2 P ( 1 + i )2 [P ( 1 + i )2]i P ( 1 + i )2 + P ( 1 + i )2i = P ( 1 + i )3

3 P ( 1 + i )3 [P ( 1 + i )3]i P ( 1 + i )3 + P ( 1 + i )3i = P ( 1 + i )4

10

Compound InterestCompound Interest• Consequently, the value for an amount P invested for n

periods at i rate of interest using compound interest calculations would be:

• For example, $100 invested now at 9% compound interest for 8 years would yield:

• Compound interest is the basis for practically all monetary transactions.

Fn = P( 1 + i )n

F8 = $100( 1 + 0.09 )8 = $199

11

Future/Present ValueFuture/Present Value

• FV = PV(1 + i)n.

• PV = FV / (1+i)n.

• Discounting is the process of translating a future value or a set of future cash flows into a present value.

12

Calculating Present ValueCalculating Present ValueIf promised $500,000 in 40 years, assuming 6% interest, what is the value today? (Discounting)

FVn= PV(1 + i)n

PV = FV/(1 + i)n

PV = $500,000 (.097)PV = $48,500

13

The Rule of 72The Rule of 72• Estimates how many years an investment

will take to double in value

• Number of years to double =

72 / annual compound interest rate• Example -- 72 / 8 = 9 therefore, it will

take 9 years for an investment to double in value if it earns 8% annually

• Challenge: Prove it!!!!!!!!!!!!!!!!!!!!!

14

Example: Double Your Money!!!Example: Double Your Money!!!

Quick! How long does it take to double $5,000 at a compound rate of 12% per year?

Key ““Rule-of-72Rule-of-72”.”.

15

Approx. Years to Double = 7272 / i%

7272 / 12% = 6 Years 6 Years

[Actual Time is 6.12 Years]

Quick! How long does it take to double $5,000 at a compound rate of 12% per year?

Example: Double Your Money!!!Example: Double Your Money!!!

16

Given:• Amount of deposit today (PV):

$50,000• Interest rate: 11%• Frequency of compounding: Annual • Number of periods (5 years): 5

periodsWhat is the future value of this single sum?FVn = PV(1 + i)n

$50,000 x (1.68506) = $84,253

Single Sum Problems: Future ValueSingle Sum Problems: Future Value

17

Given:• Amount of deposit end of 5 years:

$84,253• Interest rate (discount) rate: 11%• Frequency of compounding: Annual • Number of periods (5 years): 5 periodsWhat is the present value of this single sum?• FVn = PV(1 + i)n

$84,253 x (0.59345) = $50,000

Single Sum Problems: Present ValueSingle Sum Problems: Present Value

18

AnnuitiesAnnuities

• Definition -- a series of equal dollar payments coming at the end of a certain time period for a specified number of time periods.

• Examples -- life insurance benefits, lottery payments, retirement payments.

19

An annuity requires that:• the periodic payments or receipts

(rents) always be of the same amount,

• the interval between such payments or receipts be the same, and

• the interest be compounded once each interval.

Annuity ComputationsAnnuity Computations

20

If one saves $1,000 a year at the end of every year for three years in an account earning 7% interest, compounded annually, how much will one have at the end of the third year?

Example of AnnuityExample of AnnuityExample of AnnuityExample of Annuity

$1,000 $1,000 $1,000

0 1 2 3 3 4

$3,215 = FVA$3,215 = FVA33

End of Year

7%

$1,070

$1,145

FVA3 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0 = $3,215

21

Derivation of EquationDerivation of EquationA A A A A A A A

?1 2 3 4 n-2 n-1 n

Year

n

n-1

n-2

.

.

1

Future Value of Annuity

A

A(1+i)

A(1+i)2

.

.

A(1+i)n-1

Total Future Value (F) = A + A(1+i) + A(1+i)2 + ... + A(1+i)n-1

22

Derivation (cont.)Derivation (cont.)

F = A + A(1+i) + A(1+i)2 + ... + A(1+i)n-1 :Eqn 1

Multiply both sides by (1+i) to get:

F(1+i) = A(1+i) + A(1+i)2 + ...+ A(1+i)n :Eqn 2

Subtract Eqn 2 from Eqn 1 to get:

F = A[(1+i)n - 1] / i = A (F/A,i,n)

23

Given:• Deposit made at the end of each

period: $5,000• Compounding: Annual• Number of periods: Five• Interest rate: 12%What is future value of these deposits?F = A[(1+i)n - 1] / i

$5,000 x (6.35285) = $ 31,764.25

Annuities: Future ValueAnnuities: Future Value

24

Given:• Rental receipts at the end of each

period: $6,000• Compounding: Annual• Number of periods (years): 5• Interest rate: 12%

What is the present value of these receipts?

F = A[(1+i)n - 1] / i$6,000 x (3.60478) = $ 21,628.68

Annuities: Present Annuities: Present ValueValue

25

Given: Deposit made at the beginning of each

period: $ 800

• Compounding:Annual

• Number of periods: Eight• Interest rate 12%

What is the future value of these deposits?


26

First Step:Convert future value of ordinary annuity factor to future value for an annuity due:

• Ordinary annuity factor: 8 periods, 12%: 12.29969

• Convert to annuity due factor: 12.29969 x 1.12: 13.77565

Second Step:Multiply derived factor from first step by the amount of the rent:

• Future value of annuity due: $800 x 13.77565 =

$11,020.52


27

Given:• Payment made at the beginning of each

period: $ 4.8• Compounding: Annual• Number of periods: Four• Interest rate 11%

What is the present value of these payments?

Annuities: Present Annuities: Present ValueValue

28

First Step:Convert future value of ordinary annuity factor to future value for an annuity due:

• Ordinary annuity factor: 4 periods, 11%: 3.10245

• Convert to annuity due factor: 3.10245 x 1.11 3.44372

Second Step:Multiply derived factor from first step by the amount of the rent:

• Present value of annuity due: $4.8M x 3.44372: $16,529,856


29

Key of Annuity CalculationKey of Annuity Calculation

Fv = Pv[(1+i)n - 1] / i

30

SummarySummary

• Single Sum Compounding

• Annuities

• Key: Compound Interests Calculation

Feedback and Warm-Up Review

Documents

Transcript of Feedback and Warm-Up Review