wk 5 time value of money 1

8/18/2019 Wk 5 Time Value of Money 1

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TIME VALUE OF MONEY


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Chapter objectives

At the end of this topic, you should able to:

1. Describe the return to capital concept in the form of

interest.

2. Illustrate how basic equivalence calculations are madewith respect to the time value of capital in engineering

economy studies.


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Introduction

The term capital refers to wealth in the form of money or property that can be used to produce more wealth.

The majority of engineering economy studies involve

commitment of capital for extended periods of time, so

the effect of time must be considered. In this regard, it is recognized that a dollar today is worth

more than a dollar one or more years from now because

of the interest (or profit) it can earn. Therefore, money

has a time value.


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Interest rate and interest

Interest rate is the ratio of the gain received from aninvestment and the investment over a period usually one

year. Also, an interest rate may be expressed as a ratio

between the amount paid for the use of funds and the

amount of funds use.

Interest is an amount of money received as a result of

investing funds either by loaning it or by using it in thepurchase of materials, labor or facilities. Interest received

in this connection is a gain or income.


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Example 1

An employee at LaserKinetics.com borrows $10,000 onMay 1 and must repay a total of $10,700 exactly 1 year later.

Determine the interest amount and the interest rate paid.

SolutionInterest paid =

Interest rate =


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Time value of Money: Simple Interest

The payment of interest at a fix rate based on the principalvalue, P . The total interest I for the principal value P at

interest rate i per year for n years is given by the following

equation:

Where P = principal amount lent or borrowed

i = interest rate per interest period

n = number of interest periods

The total amount repaid at the end of n interest periods is,


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Example 2

P = RM100,n = 3 years,

i= 10% per year

Find the total amount of payback.

Solutions

Figure shows the cash flow:


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Time value of Money: Compound Interest

For compound interest, the interest accrued for eachinterest period is calculated on the

principal plus the total amount of interest accumulated in

all previous periods. Thus,

the compound interest means interest on top of interest.


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Example 3

The effect of compounding of interest can be seen in thefollowing table for $1000 loaned for three periods at an

interest rate of 10% compounded each period.


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Equivalence

In engineering economy, when considered together, the timevalue of money and the interest rate help develop the

concept of economic equivalence, which means that

different sums of money at different times would be equal

in economic value.


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Example 4

If the interest rate is 6% per year, $100 today (present time)is equivalent to $106 one year from today and $94.34 one

year ago ($100/1.06).

Solution

Amount in one year Amount one year ago


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Activity 1

a) Calculate the amount deposited 1 year ago to have$1000 now at an interest rate of 5% per year.

b) Calculate the amount of interest earned during this

time period.


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Activity 2

Compare the interest earned by $500 for 10 years at 8%simple interest with that earned by the same amount for 10

years at 8% compounded annually.


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Interest Factor Derivations

a. Single-Payment Compound-Amount Factor This factor may be used to find the compound amount, F,

of a present principal amount, P.

)%,,/(

)1(

ni P F P F

or

i P F N


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Example 5

X borrowed RM1000 for 8 years with 10% interest per year.How much should he pay after the 8th years?

Solutions

Or by referring to the table of

discrete compounding; i= 10%


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b. Single-Payment Present-Worth Factor This factor is used to find the present worth, P of a future

amount, F, for the investment.

)%,,/(

1

1

ni F P F P

or

i F P

n


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Example 6

Y wants to have RM 1000 in 6 years’ time from now. Howmuch should Y invest from now, if the interest rate per year is

10%.

SolutionsOr by referring to the table of

discrete compounding; i = 10%


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Activity 3

What is the present value of these future payments?

1. $5500, 6 years from now at 9% compounded annually.

2. $1700, 12 years from now at 6% compounded annually.


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c. Equal-Payment-Series Compounded-Amount Factor This factor used to find the single future value that would

accumulate from a series of equal payments occurring at the

end of succeeding annual interest periods.

)%,,/(

11

ni A F A F

or

i

i A F

n


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Example 7

For 3 yearly investment of RM2000 continuously, how muchmoney will it gain after the last investment, given i is 10%?

Solutions

Or by referring to the table of

discrete compounding; i = 10%


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Activity 4

A future amount, F, is equivalent to RM1500 now when sixyears separate the amounts and the annual interest rate is

12%. What is the value of F?


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d. Equal-Payment-Series Present-Worth Factor To find what single amount must be deposited now so that

equal end-of-year payments can be made, P must be found

in terms of A.

)%,,/(

1

11

,

111

ni A P A P

or

ii

i A P

Therefore

i

i Ai P F

Given

n

n

nn


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e. Deferred Annuities (Uniform Series) All annuities (uniform series) discussed to this point

involve the first cash flow being made at the end of the

first period, and they are called ordinary annuities.

If the cash flow does not begin until some later date, theannuity is known as deferred annuity.


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Example 9

X wants to buy a present worth RM2000 on her daughter’s 18th, 19th, 20th and 21st birthdays. How much money should

she invest on the day of her child birth, given i =5%?

Solutions


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f. Uniform Gradient A cash flow diagram of a sequence of end-of-period cash

flows increasing by a constant amount, G, in each period. The

G is known as the uniform gradient amount.

The timing of cash flows on which the derived formulas andtabled values are based is as follows:


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Notice that the first cash flow occurs at the end ofperiod two. The direct use of gradient conversion factors

applies when there is no cash flow at the end of period

one. There may be an A amount at the end of period one,

but it is treated separately (see example later).

Finding A when given G

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

Finding P when given G

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


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Example 10

Suppose that certain end-of-year cash flows are expected to be

$1,000 for the second year, $2,000 for the third year, and

$3,000 for the fourth year, and that if interest is 15% per year, it

is desired to find the (a) present equivalent value at the

beginning of the first year, and (b) uniform annual equivalent at

the end of each of the four years.

Solution

(a)The present equivalent

can be calculated as

(b) The annual equivalent

can be calculated as


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Example 11

Suppose that one has cash flows as follows:

One wishes to calculate their present equivalent at i =15% per

year using arithmetic gradient interest formulas.


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Example 11 (continued)

wk 5 time value of money 1

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