2 l2_time value of money

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Engineering Economics

Time Value of Money


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Time Value of Money

! Engineering decisions frequently involve tradeoffs among costsand benefits occurring at different times

Typically, invest in project today to gain benefits in future! For example, if you borrow money on your credit card, you

have to pay (exorbitant) interest

i.e. it costs you extra to buy your Eminem album now,

rather than waiting until pay-day! Conversely, if you invest the same money now, you can

afford more than just the album at a future date

! In engineering projects, the sums of money can be large,

so it makes a big difference how something is paid for 2


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Time Value of Money (Contd)

"The time value of money discusses economic methodsused to compare benefits and costs occurring at different

times"

The essence of the problem is therefore!how do wecompare options when we consider the time value ofmoney?

"

The key to making these comparisons is the use of interestrates.

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Interest and Interest Rate

Why are there interest rates?

because the lender could have done something of value

with the money that you now have so it costs them to lend you the money

interest is therefore the compensation that the borrower

pays to the lender for the loss of use of their money

"

The value lost may be investments, new equipment, pleasure,whatever

" Hence money has both a present worth and a future worth

or a dollar today is worth more than a dollar at a future time

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Present WorthFuture Worth

An amount of money today, P, can be related to a futureamount, F, by the interest amount I, or interest rate i:

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F = P + I = P + Pi = P(1+ i)

i #interest rate, P !presentworthofF

F #future worthofP, baseperiod#interest period

Source: Engineering Economics, 5thedition, by Fraser Pearson


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Example 2.1

Samuel bought a one-year guaranteed investment certificate(GIC) for $5000 from a bank on 15 May 2002. The bank

was paying 10% on one year GICs at that time. On 15 May2003, he will cash the GIC for!!!.?

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Interest Period

The dimension of an interest rate is currency/currency/time period.For example: a 9% interest rate means that for every dollar lent,

0.09 dollars (or other unit of money) is paid in interest for each timeperiod.

" The value of the interest rate depends on the length of the timeperiod. Period over which interest calculated is interest period.

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Compound Interest

If number of periods is greater than one, interest usuallycompounded(at the end of each period, interest is added to

principal that there was at the beginning of that period).

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Example 2.2: If you were to lend $100 to your friend for three yearsat 10% per year, how much would you get back?


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Compound Interest Computation

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F = P(1+ i)N IC= F - P = P(1 + i)N- P

From the previous example: F=$133.1 and Ic = $100(1 + 0.1)3- $100 = $33.10


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Compound InterestSimple Interest

" Simple Interest interest without compounding (interest isnot added to principal at end of each period). i.e.

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Compound and simple interest amounts equal if N = 1!

IS= P i N IC= P(1 + i)N- P

Linear Exponential

As Nincreases, difference between accumulated interestamounts for the two methods increases exponentially.


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Compound InterestSimple Interest

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In 1993, a couple in Nevada presentedthe government with a state bondfor $1000 with a demand that it be honored

The bond carried an annual interest of24% And was issued in 1865! The compounded interest would be $1000(1 + 0.24)128= $9.07 x 1014

In contrast, simple interest would have

made the bond worth a mere$31,720


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NominalEffective Interests Rates

Nominal Interest Rate: The standardmethod for expressing an annualinterest rate. e.g. a nominal interest rate of 18% per year, compoundedmonthly, is the same as 18/12 = 1.5% per month

" Generally, the sub-period interest rate iscan be given by is= r/m, where

ris the nominal rate and mis the number of sub-periods.

" Note that 18% compounded annually is not the same as 1.5% permonth compounded. The real interest rate cab be given as:

(1 + 0.015)12= 1.196, so this gives ~20% real interest rate. This is knownas the effective interest rate

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Effective Interest Rate

" We know that F = P(1 + is)m

If we want the effective interest rate for the whole period then

P(1 + is)m= P(1 + ie)

" Giving

ie= (1 + is)m 1 = (1 + r/m)m 1

" So what, for example, is the annual rate of interest compounded

annually is equivalent of 1% per month, compounded monthly?

is= 0.01, m = 12, so

ie= (1 + 0.01)12 1 = 12.7%

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Example 2.6

$Cardex Credit Card Co. charges a nominal 24 percentinterest on overdue accounts, compounded daily. What is

the effective interest rate?

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The effective interest rate is 27.1%


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Continuous Compounding

Suppose that the nominal interest rate is 12% and interest iscompounded semi-annually.

We compute the effective interest rate as follows: where r= 0.12,m= 2

is= r/m = 0.12/2 = 0.06

ie= (1+ iS)m- 1 = (1 + 0.06)2 - 1= .1236 (12.36%)

What if interest were compoundedmonthly?

ie= (1 + iS)m- 1 = (1 + 0.01)12- 1 = 0.1268 (12.68%)

Daily? ie= 0.127475 or about 12.75%

More than daily?!Continuous Compounding

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Continuous Compounding (Contd)

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" The effective interest rateunder continuouscompounding is

Growth of $1 at 30% interest for variouscompounding periods

" The effective interest rate for anominal interest rate of 12%

under continuous compounding:

ie= er- 1 = e0.12- 1 = 0.12750 = 12.75%



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Cash Flow Diagram

Cash flow diagram is a graphical summary of the timing andmagnitude of a set of cash flows. This displays graphically

the inflow and outflow of cash as a function of time period

17Source: Engineering Economics, 5thedition, by Fraser Pearson


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Cash Flow Diagram (Contd)

Beginning and Ending of Periods:

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Assumptions:

$

Cash flows occur at the ends of periods.$ End of time period 1 = beginning of time period 2,

etc.$ Time 0 = now



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Equivalence

" The purpose of all this future/present worth calculation is to allow a faircomparison of the costs of various options

" Hence we need to establish an equivalence of values occurring at different

times

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!" $% &'(% ' )*%+%,- $.*-& /-'- 01% - 2

2 ',3 ' "4-4*% $.*-& 5- 6 7 '- 01% - 6 7

-&%, -&%+% '*% %849(':%,-; $9-& *%+)%? 6 9@7 6 B

A,3 5- 6 7 6 B

= 5- 6 7

>? 6 9@B

$100,000 now

Interest rate = 6% peryear

01

$106,000 in one year


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Example 1.17 (Blank)

A father wants to deposit an unknown lump-sum amount intoan investment opportunity 2 years from now that is large

enough to withdraw $4000 per year for university tuition for5 years starting 3 years from now. If the rate of return isestimated to be 7% per year, construct the cash flowdiagram.

20Source: Engineering Economy, 2ndedition, by Leland Blank. Copyright McGraw-Hill


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Example 1.20 (Blank)

An electrical engineer wants to deposit an amount P now suchthat she can withdraw an equal annual amount of A1=

$2000 per year for the first five years starting 1 year afterthe deposit, and a difference annual withdraw of A2=$3000per year for the following 3 years. How would the cash flowdiagram appear if i=5% per year?

21Source: Engineering Economy, 2ndedition, by Leland Blank. Copyright McGraw-Hill


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Review Problem 2.1

Atsushi has had $800 stashed under his matters for 30 years.How much money has he lost by not putting it in a bank

account at 8% annual compound interest all these years?

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Review Problem 2.2

You want to buy a new computer, but you are $1000 short of the amountyou need. Your aunt has agreed to lend you the amount you needprovided you pay her $1200 two years from now. She compounds

interest monthly. Or you may go to a bank and get a loan. However, aloan processing fee of $20 is applied, which will be included in the loanamount. The bank is expecting $1220 two years from now on monthlycompounded interest.

(a) What monthly rate is your aunt charging for the loan? What is the bank

charging?(b)

What effective annual rate is your aunt/ bank charging?

(c)

Which option is better for you?

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Readings and End of ChapterProblems

Reading: Chapters 1 and 2 (Fraser) or Chapter 1 (Blank)

Chapter-End Problems (Fraser):

2.1, 2.4, 2.8, 2.10, 2.16, 2.22, 2.31, 2.37,2.38, 2.40, 2.43

Green: Key concept Blue: Applications Red: Challenging

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Problem # 2.1 2.4 2.8 2.10 2.16 2.22 2.31

Answer 120 10% 18.75%

Draw 9.14%,8.76%

14.06%,13.98%,13.98%

Draw

Problem # 2.37 2.38 2.40 2.43

Answer $600 982 B2 $51 less 16.08%,$58 038,

13.75%

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2 l2_time value of money

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