eng econ cash flow l4 -mme 4272

Engineering Economy Factors

Outline Compound amount factor and

Present worth factor for single payment

Uniform series present worth and capital recovery factors

Uniform series compound amount and sinking fund factors

Engineering Economy Factors

Single Payment Factor (F/P and P/F) is a fundamental factor, determines the amount of money, F accumulated after n years from a

single present worth, P with compound interest.

Single Payment Factor (F/P and P/F)

Recall that compound interest refers to interest paid on top of interest.

Thus, if an amount P is invested at time t = 0, the amount F1 accumulated over 1 year at an interest rate i will be

F1 = P + Pi = P(1 + i)

Single Payment Factor (F/P and P/F)

At the end of the second year, the amount accumulated F2 is the amount after year 1 plus the interest from the end of year 1 to the end of year 2 on the entire F1. F2 = F1 + F1i = P(1+i) + P(1+i)i= P(1+i)2

Similarly, the amount of money accumulated at the end of year 3,

F3 = P(1+i)3

Single Payment Factor (F/P and P/F) It is evident by mathematical induction

that the formula can be generalized for n years to

F = P( 1 + i )n The factor,(1+i)n is called the single-

payment compound amount factor (SPCAF). It is usually referred to as the F/P

factor. This is the conversion factor, when

multiplied by P, yields the future amount F of an initial amount (P) after n years at interest rate i.

Single Payment Factor (F/P and P/F)

Use the formula to evaluate the value of the factor for i=0.06 and n=12

F = P( 1 + i )n Verify the validity of rule of 72

Single Payment Factor (F/P and P/F)

In a reverse situation, the value P can be determined for a stated future amount F that occurs after n periods in the future, by solving the following equation:

P = F[1/(1+i)n]

The factor [1/(1+i)n] is known as the single payment present worth factor (SPPWF), or the P/F factor.

Relevant cash flow diagrams are shown in Figure 2-1.

Single Payment Factor (F/P and P/F)

0 n-1n-221

i=givenP=Given

n

F=?

0 n-1n-221

i=givenP=?

n

F=GivenFigure 2-1Cash flow diagram for single-payment factors: (a) find F and (b) find P

(a)

(b)

Single Payment Factor (F/P and P/F)

Note: The two factors derived here are for single payments.

They are used to find the present or future amount when only one payment or receipt is involved.

Single Payment Factor (F/P and P/F)

Factor Find/Given

Standard Notation Equation

Equation with factor formula

Excel functions

Notation Name

(F/P,i,n) Single-payment compound amount

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

F=P(1+i)n FV(i%,n,,P)

(P/F,i,n) Single-payment present worth

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

P=F[1/(1+i)n] PV(i%,n,,F)

Uniform-series Present Worth Factor and Capital Recovery Factor (P/A and A/P)

The equivalent present worth P of a uniform series A of end-of-period cash flows is shown:

0 n-1n-221

i=given

P=?

n

A=Given

Fig 2-5 (a)Cash flow diagram used to determine P of a uniform series

Uniform-series Present Worth Factor and Capital Recovery Factor (P/A and A/P)

To reverse the situation, the present worth P is known and the equivalent uniform-series amount A is sought [Fig 2-5 (b)]

0 n-1n-221

i=given

P=given

n

A= ?Fig 2-5 (b)Cash flow diagram used to determine A for a present worth

Uniform-series Present Worth Factor and Capital Recovery Factor (P/A and A/P)

An expression for the present worth [Fig 2-5 (a)] can be determined by following the steps mentioned below: Consider each A value as a future

worth F, Calculate its worth with the P/F

factor and Sum up the results.

Uniform-series Present Worth Factor and Capital Recovery Factor (P/A and A/P)

The equation is

nn iA

iA

iA

iA

iAP

)1(1

)1(1

.....)1(

1)1(

1)1(

1

1

321

Uniform-series Present Worth Factor and Capital Recovery Factor (P/A and A/P)

The terms in brackets are the P/F factors for years 1 through n, respectively. By factoring out A, we can write

nn iiiiiAP

)1(1

)1(1......

)1(1

)1(1

)1(1

1321

To simplify the above equation and obtain the P/A factor, multiply the n-term geometric progression in brackets by the (P/F, i%,1) factor which is 1/(1+i).

Uniform-series Present Worth Factor and Capital Recovery Factor (P/A and A/P)

This results in an equation as given below:

1432 )1(

1)1(

1......)1(

1)1(

1)1(

11 nn iiiii

Ai

P

Subtracting the previous equation from this one i.e.

nn iiiiiAP

)1(1

)1(1......

)1(1

)1(1

)1(1

1321

1432 )1(

1)1(

1......)1(

1)1(

1)1(

11 nn iiiii

Ai

P

Uniform-series Present Worth Factor and Capital Recovery Factor (P/A and A/P)

Results in

1

)1(1

niiAP

11 )1(1

)1(1

1 iiAP

ii

n

Or

n

n

iiiAP)1(1)1(

Or

For, i 0

Uniform-series Present Worth Factor and Capital Recovery Factor (P/A and A/P)

n

n

iii)1(1)1(The term,

is the conversion factor referred to as the uniform-series present worth factor (USPWF).

It is the P/A factor used to calculate the equivalent P value in year 0 for a uniform end-of-period series of A values beginning at the end of period 1 and extending for n periods. Cash flow diagram is in Fig 2-5(a)

Uniform-series Present Worth Factor and Capital Recovery Factor (P/A and A/P) In a reverse situation, the present worth P

is known and the equivalent uniform-series amount A is sought (Fig 2-5 (b).

The first A value occurs at the end of period 1, that is, one period after P occurs. Solve previous equation for A to obtain:

1)1()1(

n

n

iiiPA

The term in brackets is called the capital recovery factor (CRF), or A/P factor. It calculates the uniform annual worth A over n years for a given P in year 0, when the interest rate is i.

Table 2-2: P/A and A/P Factors: Notations and Equations

Factor Find/Given

Standard Notation Equation

Equation with factor formula

Excel Function

Notation Name

(P/A,i,n) Uniform-series present worth

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

(A/P,i,n) Capital recovery

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

1)1(

)1(n

n

iii

n

n

iii)1(1)1(

Example 2.4

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?

SOLUTIONThe cash flow diagram is

75430 6 821

i=16%P=?

9

A=$600

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

The simplest way to derive the A/F factor is to substitute into factors already developed.

We already know

P = F[1/(1+i)n] and

1)1()1(

n

n

iiiPA

Now if P from the first equation is substituted in the second equation, we get the equation as follows: (next slide)

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

1)1(

)1()1(

1n

n

n iii

iFA

After simplification, the following equation is developed

1)1( ni

iFA

The expression in brackets in the above equation is the A/F or sinking fund factor. It determines the uniform annual series that is equivalent to a given future worth F.

The cash flow diagram can be shown as (ex. Investing in Pension Scheme)

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

0 n-1n-221

i = given

A = ?

n

F = Given

The uniform-series A begins at the end of period 1 and continues through the period ofthe given F.

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

1)1( ni

iFA

Refer to the following equation (already developed)

The term in brackets is called uniform-series compound amount factor (USCAF) or F/A factor.

This equation can be rewritten as

iiAF

n 1)1(

The cash flow diagram can be shown as

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

0 n-1n-221

i = given

A = ?

n

F = Given

When F/A factor is multiplied by A, it yields the future worth of the uniform series, F.

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

Remember that the future amount F occurs in the same period as the last A.

Table 2-3 summarizes the notations and equations.

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

Table 2-3: F/A and A/F Factors: Notations and Equations

Factor Find/Given

Standard Notation Equation

Equation with factor formula

Excel Function

Notation Name

(F/A,i,n)Uniform-series compound amount

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

(A/F,i,n) Sinking fund

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

ii n 1)1(

1)1( nii

i

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

Example 2.5 Formasa Plastics has major fabrication

plants in Texas and Hong Kong. The president wants to know the equivalent future worth of $1 million capital investment cash for 8 years, starting 1 year from now. Formasa capital earns at a rate of 14% per year.

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

Example 2.5 (continued) Solution: The cash flow diagram shows the

annual payments starting at the end of year 1 and ending in the year the future worth is desired.

1 5432

i = 14%

A = $1,000,000

6

F = ?

0 7 8

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

Example 2.5 (continued) Solution: The F value in 8 years is

1 5432

i = 14%

A = $1,000,000

6

F = ?

0 7 8

F = $1,000,000 (F/A,14%,8) = $1,000,000 (13.23281) = $13,232,810

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

Example 2.6How much money must Carol deposit every year

starting 1 year from now at 5% per year in order to accumulate $6000 seven years from now?

Solution: The cash flow diagram

1 20 543

i = 5%

A = ?

6

F = $6000

7

Sinking fund factor and Uniform-series compound amount factor (A/F and F/A)

A = $6000(A/F,5%,7) = $6000(0.12096) = $725.76 per year.The A/F factor value of 0.12096 was

computed using the factor formula

1)1( ni

iFA

Example of interest Table (Partial)Discrete cash flows: Compound interest factors 7%)

n

Single Payment Uniform series Payments Arithmetic Gradients

Compound Amount F/P

Present worth P/F

Sinking fundA/F

Compound AmountF/A

Capital RecoveryA/P

Present worthP/A

Gradient Pres. worth P/G

Gradient uni. seriesA/G

91.8385 0.5439 0.08349 11.9780 0.15349 6.5152 23.1404 3.5517

10 1.9672 0.5083 0.07238 13.8164 0.14238 7.0236 27.7156 3.9461

11 2.1049 0.4751 0.06338 15.7836 0.13336 7.4987 32.4665 4.3296

Interpolation in interest tables

When it is necessary to locate a factor value for an i or n in the interest tables, the desired value can be obtained in one of the two ways:

(1) by using the formulas derived or (2) by linearly interpolating between

the tabulated values.However, the value obtained through linear

interpolation is not exactly correct, since the equations are nonlinear.

Interpolation in interest tables Nonetheless, interpolation is sufficient in most

cases as long as the values of i and n are not too distant from one another.

In linear interpretation it is necessary to set up the known (values 1 and 2) and unknown factors as shown in Table 2-4:

Table 2-4: Linear Interpretation setup

tabulated value 1 desired unlisted tabulated value 2

ab dc

Interpolation in interest tables A ratio equation is then set up and solved for

the value of unknown quantity, C

Or

Where a, b, c, and d represent the differences between the numbers shown in the interest tables.

dc

ba

dbac

Example 2.7

Determine the value of the A/P factor for an interest rate of 7.3% and n of 10 years, that is (A/P,7.3%,10)

SolutionThe values of the A/P factor for interest rate of 7 and 8% and n=10are listed in interest tables (Tables 12 and 13)

7%

7.3%

8%

ba

x

0.14903

0.14238c

d

Example 2.7

The unknown X is the desired factor value. For the ratio equation

= 0.00199Since the factor is increasing in value as the interest

rate increases from 7 to 8%, the value of c must be added to the value of the 7% factor. Thus

X = 0.14238+0.00199 = 0.14437.Compare this with the exact factor value (0.144358)

14238.014903.07873.7

dbac

Example 2.8

Find the value of the (P/F,7.3%,10) factor.

From the interest table, the values of the P/F factor for 45 and 50 years are found.

a45

48

50

b x

0.1407

0.1712c

Example 2.8

0183.01407.01712.045504548

dbac

From the equation,

Since the value of the factor decreases as n increases, c is Subtracted from the factor value for n = 45

X= 0.1712 – 0.0183 = 0.1529

Comment: Though it is possible to perform two-way linearInterpolation, it is much easier and more accurate to use the Factor formula or a spreadsheet function.

Table 12 (Partial): Discrete cash flows: Compound interest factors 7%)

n

Single Payment Uniform series Payments Arithmetic Gradients

Compound Amount F/P

Present worth P/F

Sinking fundA/F

Compound AmountF/A

Capital RecoveryA/P

Present worthP/A

Gradient Pres. worth P/G

Gradient uni. seriesA/G

91.8385 0.5439 0.08349 11.9780 0.15349 6.5152 23.1404 3.5517

10 1.9672 0.5083 0.07238 13.8164 0.14238 7.0236 27.7156 3.9461

11 2.1049 0.4751 0.06338 15.7836 0.13336 7.4987 32.4665 4.3296

Table 13 (Partial): Discrete cash flows: Compound interest factors (8%)

n

Single Payment Uniform series Payments Arithmetic Gradients

Compound Amount F/P

Present worth P/F

Sinking fundA/F

Compound AmountF/A

Capital RecoveryA/P

Present worthP/A

Gradient Pres. worth P/G

Gradient uni. seriesA/G

91.990 0.5002 0.08008 12.4876 0.16008 6.2469 21.8081 3.4910

10 2.1589 0.4632 0.06903 14.4866 0.14903 6.7101 25.9768 3.8713

11 2.3316 0.4289 0.06008 16.6455 0.14008 7.1390 30.2657 4.2395