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1 Chapter 2 Factors: How Time and Interest Affect Money 2-1

2 LEARNING OBJECTIVES 1. F/P and P/F factors 2. P/A and A/P factors 3. Interpolate for factor values 4. P/G and A/G factors 5. Geometric gradient 6. Calculate i 7. Calculate n 8. Spreadsheets 2-2

3 Sct 2.1 Single-Payment Factors (F/P and P/F) Objective: Derive factors to determine the present or future worth of a cash flow Cash Flow Diagram basic format i% / period n-1 n F n P 0 P 0 = F n 1/(1+i) n (P/F,i%,n) factor: Excel: =PV(i%,n,,F) F n = P 0 (1+i) n (F/P,i%,n) factor: Excel: =FV(i%,n,,P) 2-3

4 Sct 2.2 Uniform-Series: Present Worth Factor (P/A) and Capital Recovery Factor(A/P) Cash flow profile for P/A factor i% per interest period n-2 n-1 n Find P $A per interest period Required: To find P given A Cash flows are equal, uninterrupted and flow at the end of each interest period 2-4

5 (P/A) Factor Derivation Setup the following: P= A (1 ) (1 ) (1 ) n n i i i (1 i) Multiply by to obtain a second equation (1+i) (1) P = A i (1 + i) (1 + i) (1 + i) (1 + i) 2 3 n n+ 1 Subtract (1) from (2) to yield i 1 1 P = A + i + i + i n+ 1 1 (1 ) (1 ) 2-5 (3) (2)

6 (P/A) and (A/P) Factor Formulas Simplify (3) to yield n (1 + i) 1 P= A fori n i(1 + i) 0 (4) (P/A,i%,n) factor Excel: =PV(i%,n,A) Solve (4) for A to get (A/P) factor n i(1 + i) A= P n (1 + i) 1 (5) (A/P,i%,n) factor Excel: =PMT(i%,n,P) 2-6

7 ANSI Standard Notation for Interest Factors Standard notation has been adopted to represent the various interest factors Consists of two cash flow symbols, the interest rate, and the number of time periods General form: (X/Y,i%,n) X represents what is unknown Y represents what is known i and n represent input parameters; can be known or unknown depending upon the problem 2-7

8 Notation - continued Example: (F/P,6%,20) is read as: To find F, given P when the interest rate is 6% and the number of time periods equals 20. In problem formulation, the standard notation is often used in place of the closed-form equivalent relations (factor) Tables at the back of the text provide tabulations of common values for i% and n 2-8

9 Sct 2.3 Sinking Fund Factor and Uniform Series Compound Amount Factor (A/F and F/A) Cash flow diagram for (A/F) factor i% per interest period.... F = given n-2 n-1 n Start with what has already been developed n 1 i(1 + i) A= F n n (1 + i) (1 + i) 1 A=? per interest period Find A, given F i A= F n (1 + i) 1 2-9

10 Given: (F/A) factor from (A/F) i A= F n (1 + i) 1 (A/F,i%,n) factor Excel: =PMT(i%,n,,F) Solve for F in terms of A to yield (1 + i ) n 1 F = A i (F/A,i%,n) factor Excel: =FV(i%,n,A) 2-10

11 Sct 2.4 Interpolation in Interest Tables When using tabulated interest tables one might be forced to approximate a factor that is not tabulated Can apply linear interpolation to approximate See Table 2-4 Factors are nonlinear functions, hence linear interpolation will yield errors in the 2-4% range Use a spreadsheet model to calculate the factor precisely 2-11

12 Sct 2.5 Arithmetic Gradient Factors (P/G) and (A/G) A 1 +(n-1)g Cash flow profile A 1 +(n-2)g Find P, given gradient cash flow G A 1 +2G Base amount = A 1 A 1 +G n-1 n CF n = A 1 ± (n-1)g 2-12

13 $100 Gradient Example $200 $300 $400 $500 $600 $ Gradients have two components: 1. The base amount and the gradient 2. The base amount (above) = $100/time period 2-13

14 Gradient Components Find P of gradient series 1G 2G (n-3)g (n-2)g (n-1)g 0G Base amount = A / period n-2 n-1 n Present worth point is 1 period to the left of the 0G cash flow For present worth of the base amount, use the P/A factor (already known) For present worth of the gradient series, use the P/G factor (to be derived) 2-14

15 Gradient Decomposition As we know, arithmetic gradients are comprised of two components 1. Gradient component 2. Base amount When working with a cash flow containing a gradient, the (P/G) factor is only for the gradient component Apply the (P/A) factor to work on the base amount component P = PW(gradient) + PW(base amount) 2-15

16 Derivation Summary for (P/G) Start with: P= G( P / F, i,2) + 2 G( P / F, i,3) + 3 G( P / F, i, 4) [(n-2)g](p/f,i,n-1)+[(n-1)g](p/f,i,n) (1) Multiply (1) by (1+i) 1 to create a second equation Subtract (1) from the second equation and simplify Yields G (1 + i) 1 n (1 + i) in 1 i + i + i i + i n n P= = i (1 ) n (1 ) n 2 (1 ) n (P/G,i,n) factor No Excel relation exists 2-16

17 Use of the (A/G) Factor A = G(A/G,i,n) (n-1)g Find A, given gradient cash flow G (n-2)g A A A... A A 2G G Equivalent A of gradient series n-1 n CF n = (n-1)g 2-17

18 Sct 2.6 Geometric Gradient Series Factor Geometric Gradient Cash flow series that starts with a base amount A 1 Increases or decreases from period to period by a constant percentage amount This uniform rate of change defines A GEOMETRIC GRADIENT Notation: g = the constant rate of change, in decimal form, by which future amounts increase or decrease from one time period to the next 2-18

19 Typical Geometric Gradient A 1 (1+g) n-1 Given A 1, i%, and g% A 1 A 1 (1+g) A 1 (1+g) n-2 n-1 n Required: Find a factor (P/A,g%,i%,n) that will convert future cash flows to a single present worth value at time t =

20 Basic Derivation: Geometric Gradient Start with: P Factor out A 1 out and re-write g A A (1 + g) A (1 + g) A (1 + g) = (1 + i) (1 + i) (1 + i) (1 + i) 2 n n 1 (1 + g) (1 + g) (1 + g) Pg = A (1 + i) (1 + i) (1 + i) (1 + i) Multiply by (1+g)/(1+i) to obtain Eq. (3 ) 1 2 n 1 n (1) (2) P g ( 1 + g ) ( 1 + g ) 1 (1 + g ) (1 + g ) (1 + g ) = A (1 + i) ( 1 + i) (1 + i ) (1 + i ) (1 + i ) (1 + i ) 1 2 n 1 n (3) Subtract Eq. (2 ) from Eq. (3 ) to yield P g 1 + g (1 + g ) i (1 + i ) 1 + i n 1 = A 1 n + 1 Solve for P g and simplify to yield n 1+ g 1 1+ i Pg = A 1 g i i g

21 Two Forms to Consider n 1+ g 1 1+ i Pg = A 1 g i i g P g na = 1 (1 +i ) Case: g = i Case: g = i To use the (P/A,g%,i%,n) factor A 1 is the starting cash flow There is NO base amount associated with a geometric gradient The remaining cash flows are generated from the A 1 starting value No tables available to tabulate this factor too many combinations of i% and g% to support tables 2-21

22 Sct 2.7 Determination of Unknown Interest Rate Class of problems where the interest rate, i%, is the unknown value For simple, single payment problems (i.e., P and F only), solving for i% given the other parameters is not difficult For annuity and gradient type problems, solving for i% can be tedious Trial and error method Apply spreadsheet models 2-22

23 The IRR Spreadsheet Function Define the total cash flow as a column of values within Excel Apply the IRR function: =IRR(first_cell:last_cell, guess value) If the cash flow series is an A value then apply the RATE function: =RATE(number_years, A,P,F) See examples 2.12 and

24 Sct 2.8 Determination of Unknown Number of Years Class of problems where the number of time periods (years) is the unknown In single payment type problems, solving for n is straight forward In other types of cash flow profiles, solving for n requires trial and error or spreadsheet In Excel, given A, P, and/or F, and i% values apply: =NPER(i%,A,P,F) to return the value of n 2-24

25 Sct 2.9 Spreadsheet Application Basic Sensitivity Analysis Sensitivity Analysis is a process of determining what input variables really matter in a given problem formulation Sensitivity analysis aids in evaluating certain what-if scenarios Spreadsheet modeling is the best approach to formulate sensitivity analysis for a given problem 2-25

26 Sensitivity Analysis See Example 2.15 Illustratesa what-if situation for receiving money in three different time periods Tabulates the associated rate of returns for the three situations See Example 2.16 The evaluation of non-sequential cash flows 2-26

27 Summary Interest factors exist to aid in determining economic equivalence of various cash flow patterns Notation is introduced that is applied throughout the remainder of the text Introduction of important Excel spreadsheet financial functions to aid in evaluation of engineering economy problems 2-27

28 End of Slide Set 2-28