Lecture No7 Equal-Payment -Series - Jett Notes

7/31/2019 Lecture No7 Equal-Payment -Series - Jett Notes

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Lecture No.7

Contemporary Engineering Economics, 5th edition, 2010


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A repayment series for a loan.


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Equal Payment Series


0 1 2 N

0 1 2 N

A A A

F

P

0 N


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Equal-Payment Series Compound Amount Factor

Formula



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An Alternate Way of Calculating the

Equivalent Future Worth,F

0 1 2 N 0 1 2 N

A A A

F

A(1+i)N-1

A(1+i)N-2

A


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There are interest factors for a series of

end-of-period cash flows.

How much will you have in 40 years if you

save $3,000 each year and your account

earns 8% interest each year?


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Practice Problem 1a:

Find F, Given i,A, and

NGiven:A = $3,000, N= 10

years, and i= 7% per year

Find: F

Excel Solution:



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Practice Problem 1b

Handling Time Shifts:

Find F, Given i,A, and NGiven:A = $3,000, N= 10


Find: F

Excel Solution:


oEach payment has been shifted to one yearearlier, thus each payment would be compounded

for one extra year


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Practice Problem 2

Sinking-Fund Factor:

FindA, Given i,A,

andF

Given: F= $5,000, N= 5


Find:A

Excel Solution:


Formula Sinking Fund Factor

$5,000

A

0

5

1

=PMT(7%,5,0,5000)


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Comparison of Three

Different Investment

Plans

Given: Three investment

plans and i= 8%

Find: Balance on the 65th

birthday



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Cash flow diagrams for three

investment options


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How Long

Would It

Take to Save1 Million?



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Finding A when given P.

If you had $500,000 today in an account

earning 10% each year, how much could you

withdraw each year for 25 years?


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Practice Problem 3a

Uniform Series: Find

A, Given

P,i, and

N

Given: P= $250,000, N= 6


Find:A

Excel Solution:


Capital Recovery Factor


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Practice Problem 3b

Deferred Loan Repayment

Given: P= $250,000, N= 6

years, and i= 8% per year, but

the first payment occurs at the

end of year 2

Find:AStep 1: Find the equivalent

amount of borrowing at the end

of year 1:

Step 2: Use the capital

recovery factor to find the sizeof annual installment:



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Practice Problem 4

Uniform Series: Find

P, Given

A,i, and

N

Given:A = $10,576,923, N=

26 years, and i= 5% per year

Find: P

Excel Solution:


Present Worth Factor


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It can be challenging to solve for N or i.

We may know P, A, and i and want to find N.

We may know P, A, and Nand want to find i.

These problems present special challenges that arebest handled on a spreadsheet.


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Finding NAcme borrowed $100,000 from a local bank, which

charges them an interest rate of 7% per year. If Acme

pays the bank $8,000 per year, now many years will it

take to pay off the loan?

So,

This can be solved by using the interest tables and

interpolation, but we generally resort to a computer

solution.


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Finding iJill invested $1,000 each year for five years in a local

company and sold her interest after five years for

$8,000. What annual rate of return did Jill earn?

So,

Again, this can be solved using the interest tables

and interpolation, but we generally resort to a

computer solution.


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There are specific spreadsheet functions

to find N and i.

The Excel function used to solve forNis

NPER(rate, pmt, pv), which will compute the number

of payments of magnitude pmtrequired to pay off a

present amount (pv) at a fixed interest rate (rate).

One Excel function used to solve fori is

RATE(nper, pmt, pv, fv), which returns a fixed interestrate for an annuity ofpmtthat lasts fornperperiods to

either its present value (pv) or future value (fv).


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We need to be able to handle

cash flows that do not occuruntil some time in the future.

Deferred annuities are uniform series that do not

begin until some time in the future. If the annuity is deferred Jperiods then the first

payment (cash flow) begins at the end of periodJ+1.


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Finding the value at time 0 of

a deferred annuity is a two-

step process.1. Use (P/A, i%, N-J) find the value of the deferred

annuity at the end of period J(where there areN-Jcash flows in the annuity).

2. Use (P/F, i%, J) to find the value of the deferredannuity at time zero.


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Lecture No7 Equal-Payment -Series - Jett Notes