Exam I ReviewExam I ReviewExam is closed text book, no loose sheet paper

Materials to bring:

Engineering Notebook with FE reference materials inside

Calculator with good batteries

Mechanical pencil with lead & eraser – NO PEN

Process:

Wait until exams are all laid out and instructed to start

Exam starts about 1:30 – lasts until 3:00 (at maximum)

Exams must be in folder when proctor leaves

Engineering Notebooks will be handed in with exam

Exam I CoverageExam I CoverageCash Flow Diagrams:

Solid arrows are cash flows, dashed are unknown flows

Horizontal axis is time, units should be noted for n

Arrow direction is important: inflow (up), outflow (down)

Time Value of Money:Simple Interest (only if asked … two formulas to

know)

Compound Interest (for everything else)

End of period convention for inter-period payments

Table Factors:Write solution equations in table factor form

Substitute either table values or closed-form equations

Identify solution and units at the end (double lines / box)

Exam I CoverageExam I CoverageCash Flow Patterns:

Single flow patterns (Present / Future values)

Annual flow patterns (equal, periodic values)

Gradient flow patterns (Linear / Geometric incr / decrease)

Complex Cash Flows:Decompose into P, F, A, G, or g patterns

Convert to P or F values to move across time:

Combine cash flows only at the same point in time!

Convert to other flows from P or F values, if required!

Effective Interest Rates:Nominal rate (APR) adjusted for compounding

period

Must match effective rate to pay period for factors / eqns!

Future Given Present

P is the present value at Time 0

F is the future value at Time n (n periods in the future)

i is the effective interest rate

0 n

P

F ?

1 2 3

F = P(F/P,i,n)

Present Given Future

P is the present value at Time 0

F is the future value at Time n (n periods in the future)

i is the effective interest rate for each period

0 n

P ?

F

1 2 3

P = F(P/F,i,n)

Future Given AnnualA is the equal annual value over the time period

(time period: Time 0 to Time n, 1st flow at Time 1)

F is the future value at Time n (n periods in the future)

i is the effective interest rate for each period

Note: cash flow A does not have to be annual, just periodic

0 n

A

F ?

1 2 3

F = A(F/A,i,n)

Annual Given FutureA is the equal annual value over the time period

(time period: Time 0 to Time n, 1st flow at Time 1)

F is the future value at Time n (n periods in the future)

i is the effective interest rate for each period

Note: cash flow A does not have to be annual, just periodic

0 n

A ?

F

1 2 3

A = F(A/F,i,n)

Present Given AnnualA is an equal annual flow over the time period

(time period: Time 0 to Time n, 1st flow at Time 1)

P is the present value at Time 0 (n periods in the past)

i is the effective interest rate for each period

Note: cash flow A does not have to be annual, just periodic

0 n

A

P ?

1 2 3

P = A(P/A,i,n)

Annual Given PresentA is the equivalent annual flow over the time period

(time period: Time 0 to Time n, 1st flow at Time 1)

P is the present value at Time 0 (n periods in the past)

i is the effective interest rate for each period

Note: cash flow A does not have to be annual, just periodic

0 n

A ?

P

1 2 3

A = P(A/P,i,n)

Present Given Gradient (Linear)

G is the linear gradient over the time period(time period: Time 0 to Time n, 1st flow at Time 2)

P is the present value of the flow at Time 0 (n periods in the past)

i is the effective interest rate for each period

Note: cash flow is periodic, no flow at Time 1, flow of G at Time 2

0 n

G = $/yr

P ?

1 2 3

P = G(P/G,i,n)

Future Given Gradient (Linear)

G is the linear gradient over the time period(time period: Time 0 to Time n, 1st flow at Time 2)

F is the future value of the flow at Time n (n periods in the future)

i is the effective interest rate for each period

Note: cash flow is periodic, no flow at Time 1, flow of G at Time 2

0 n

G = $/yr

F ?

1 2 3

F = G(F/G,i,n)

Annual Given Gradient (Linear)

G is the linear gradient over the time period(time period: Time 0 to Time n, 1st flow at Time 2)

A is the annual equivalent of the gradient flow(annual flow starts at Time 1, goes through Time n)

i is the effective interest rate for each period

Note: cash flow of G starts at Time 2, flow of A starts at Time 1

0 n

G=$/yr

A ?

1 2 3

A = G(A/G,i,n)

Present Given Gradient (Geometric)

g is the geometric gradient over the time period(time period: Time 0 to Time n, 1st flow at Time 1)

P is the present value of the flow at Time 0 (n periods in the past)

i is the effective interest rate for each period

Note: cash flow starts with A1 at Time 1, increases by constant g%

0 n

A1

P ?

1 2 3

P = A1(P/A,g,i,n)

11

1

1

( )( )

( / , , , )

( )

ngi

when i gi g

P A g i n

nwhen i g

i

g = %/yr

Table Factors Listed(P/F, i, n) Present given Future

(P/A, i, n) Present given Annual

(P/G, i, n) Present given Gradient (linear)

(F/P, i, n) Future given Present

(F/A, i, n) Future given Annual

(A/P, i, n) Annual given Present

(A/F, i, n) Annual given Future

(A/G, i, n) Annual given Gradient

Note: There is NO P/g. Present given Geometric Gradient is:

(P/A, g, i, n)

Complex Cash FlowsComplex Cash Flows

Complex Cash Flows – Break apart (or separate) complex cash flows into component cash flows in order to use the standard formulas.

Remember: You can only combine cash flows if they occur at the same point in time.

CRITICAL POINTCRITICAL POINT

When using the factors,

n and i must always match!Use the effective interest rate

formulas to make sure that i matches the period of

interest

Interest Rate Terms…Interest Rate Terms…● Compounding Period (cp) – the time between

points when interest is computed and added to the initial amount.

● Payment Period (pp) – the shortest time between payments. Interest is earned on payment money once per period (cost of money)

● Nominal Rate ( r ) – is a simplified expression of the annual cost of money. It means nothing, unless the compounding period is stated along with it.

● Annual Percentage Rate (APR) – is the nominal interest rate on a yearly basis (credit cards, bank loans, …). It, too, should have a compounding period stated.

● Effective Rate ( i ) – is the rate that is used with the table factors or the closed form equations, and it converts the nominal rate taking into account both the compounding period and the payment period so that the blocks match.

Summary of Effective Rates

Summary of Effective RatesAn “APR” or “% per year” statement is a Nominal

interest rate – denoted r – unless there is no compounding period stated

The Effective Interest rate per period is used with tables & formulas

Formulas for Effective Interest Rate:

If continuous compounding, usey is length of pp, expressed in decimal years

If cp < year, and pp = 1 year, usem is # compounding periods per year

If cp < year, and pp = cp, usem is # compounding periods per year

If cp < year, and pp > cp, useme is # cp per payment period

11

m

a m

ri

m

ri

1ei )y(r

11

em

e m

ri

Remember…Remember…

As the compounding period gets smaller, the effective interest rate will increase.

CONTINUOUS COMPOUNDING:

i = e( r )(years) – 1

NOTE: In this continuous formula, the r value will be specified as a nominal annual rate (% per year) so you have to use the length of pay period in years, not number of periods, for which you are looking for the effective interest rate.

(Note: r and years are expressed as decimals!)

IF CP < 1 Year, and PP = CP …IF CP < 1 Year, and PP = CP …

EFFECTIVE INTEREST RATE r nominal interest rate

i = __ = _________________________________ m # of compounding periods per nominal

NOTE: In this formula, the interest rate you want (i ) is for a smaller period than the interest rate that you have (interest rate

per compound period). So… (m) is the number of those smaller

periods in the larger, nominal period for which you have the

nominal interest rate (r).

If CP < 1 Year and PP = 1 Year…

If CP < 1 Year and PP = 1 Year…EFFECTIVE INTEREST RATE

ia = ( 1 + interest rate per cp) (#

of cp per year) – 1

REMEMBER: In this formula, the interest you want (ia ) is for a larger period than the interest rate that you have (interest rate per compound period). So… the “number of compound periods per year” is the number of those smaller periods in the annual period for which you want to find ia . (And interest rate per cp = r / m !)

EFFECTIVE INTEREST RATE

ie = ( 1 + interest rate per cp) (#

of cp per pp) – 1

REMEMBER: In this formula, the interest you want (ie ) is also for a larger period than the interest rate that you have (interest rate per compound period). So… the “number of cp per pp” is the number of those smaller compounding periods in the payment period for which you want to find ie . (And interest rate per cp = r / m !)

If CP < 1 Year and PP > CP …If CP < 1 Year and PP > CP …

Other Terms to Know…Other Terms to Know…● Sunk Costs – money that was already spent in

the past that has no effect on the financial decision to be made now.

● Equity – financing funds provided by someone with an ownership stake in the project.

● Debt – financing funds provided by someone without an ownership stake in the project. These funds will depend on the owner’s credit worthiness and possibly on collateral assets that may be sold to pay off the remainder owed.

● Loan – Funds provided upfront (at Time 0) to finance a project that will be paid back, with interest, over the life of the project loan (starting one payment period later, and continuing through Time n).

● Lease – Use of an asset over the life of the project lease, provisioned upon periodic payments, with interest, that occur ahead of each period of use. (So the 1st payment begins at Time 0, and continues through Time n – 1)

Final Hints …Final Hints …

RTDP

especially the bold, italic, underlined words

Show your work – partial credit helps

All point values are shown & the problems are numbered – so they can be done in any order…but do sub-steps (2a, 2b, 2c, …) in

order to save time

RTDP