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Factors: How Time and Factors: How Time and Money Affect Interest Money Affect Interest (CH2) (CH2)

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Objectives Objectives To understand and use factors to

account for TMV.◦ Single payment compound amount

(F/P)◦ Uniform Series Present Worth (P/A)◦ Uniform Series Sinking Fund (A/F)◦ Interpolation in interest tables◦ Arithmetic Gradient Factors (P/G , A/G)◦ Geometric Gradient Factors

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Single Payment factors (F/P Single Payment factors (F/P & P/F)& P/F)F/P : Determines the amount of

money F accumulated after n periods from a single present worth P with interest compounded one time per period

Only for one payment

Can you define P/F factor?

niPF )1(/

niFP )1(/

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Solving Factor Problems Solving Factor Problems Factor problems can be solved in

several ways ◦a) Using Equations

◦Using equations: F = P(1+i)n = 12,000(1+8%)24 = $76, 094.17

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Solving Factor Problems Solving Factor Problems b) Using tables: F=P(F/P,i,n)=12,000(F/P,8%,24)=12,000 x

(6.3412)=$76,094.40

c) Using Excel◦ (F/P,i, n) FV(i%, n,P)◦ (P/F,i,n) PV(i%, n,P)

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Uniform Series Present Worth Uniform Series Present Worth (P/A) & Capital Recovery (A/P) (P/A) & Capital Recovery (A/P) Factors Factors

P/A is calculated as ◦Uniform Series Present Worth Factor

Derive A/P equation◦Capital Recovery Factor

How P/A is related to P/F?

0)1(

1)1(

i

ii

iAP

n

n

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P/A ProblemHow much you should be willing to

pay for a project that brings $600 for the next 9 years starting next year, the rate of return is 16% per year.

P =600(P/A,16%,9) = 600(4,606.5)= 2763.90

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Sinking Fund Factor & Uniform –Series Compound Amount Factor (A/F & F/A)Sinking Fund Factor (A/F) determines

the uniform annual series that is equivalent to a given future worth F.

Uniform Series Compound Amount Factor (F/A) gives the future worth of a uniform series.

1)1( ni

iFA

i

iAF

n 1)1(

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F/A ProblemPlant x wants to know the

equivalent future worth of a $1 million capital investment each year for 8 yrs, starting 1 yr from now. Capital earns at a rate of 14% per yr.

In $1000 units, F=1000(F/A,14%,8) = $13,232.80

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Linear Interpolation & Interest TablesFor unlisted values in the tables

we can use Factors equations Linear interpolation

%7 0.142387.3 x8 0.14903

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Arithmetic Gradient Factors (P/G and G/P)

Arithmetic gradient is a cash flow series that increases or decreases by a constant amount.

Different from previous factorsGradient: constant arithmetic change in

the receipts or disbursements from one time period to the next, G can be positive or negative

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Arithmetic Gradient Factors (P/G and G/P)Cash Flow in yr n (CFn) = base

amount+(n-1)GThe relation to convert an arithmetic

gradient G (not including base amount) for n years into a present worth at yr 0

The equivalent uniform annual series (A value) for an arithmetic gradient G

nn

n

i

n

ii

i

i

GP

)1()1(

1)1(

1)1(

1ni

n

iGA

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Arithmetic Gradient Factors

F/G factor (future worth for arithmetic gradient) is calculated as

Base amount (A) and gradient (G) are considered separately

Total present worth ◦ Increasing PT=PA + PG

◦ Decreasing PT=PA - PG

Equivalent total annual series ◦ Increasing AT=AA + AG

◦ Decreasing AT=AA - AG

n

i

i

iGF

n 1)1(1

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Arithmetic Gradient Problem

Gradient problems are solved for 1) base & 2) G, then add two amounts.

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Arithmetic Gradient ProblemTotal present worth = PA+PG =

500(P/A,5%,10) +100(P/G,5%,10) Total annual series = AA+AG =

500 +100(A/G,5%,10)

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Geometric Gradient