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ISE 211

Engineering Economy

Chapter 4

More Interest Formulas (cont’d)

Continuous Compounding

 As

m

, the number of compounding sub-periods per year, grows, to  , the interest compounds continuously:  Under continuous compounding, the

effective

interest rate per year is given as:

i

a = e r – 1



Single Payment – continuous compounding:

 Compound Amount:

F = P[e r*n ] = P[F/P,r,n]

 Present worth:

P = F[e -r*n ] = F[P/F,r,n]

Examples

1)

What is the amount of interest earned on $

2000

for two years earning

5%

nominal interest rate compounded continuously?

2)

A bank offers to sell savings certificates that will pay the purchaser $

5000

at the end of

10

years but will pay nothing to the purchaser in the meantime. If interest is computed at

6%

compounded continuously, at which price is the bank selling the certificates?

Examples (cont’d)

3)

How long will it take for money to double at

10%

nominal interest, compounded continuously?

4)

What is the effective annual interest rate of

6

%, compounded continuously?

Uniform Payment Series-Continuous Compounding at Nominal Rate r per Period

Substitute

i = e r – 1

into the equations for periodic compounding.

Continuous Compounding Sinking Fund:

[A/F,r,n] = e r – 1 e r*n – 1

Continuous Compounding Capital Recovery:

[A/P,r,n] = e r*n (e r – 1) e r*n – 1

Continuous Compounding Series Amount:

[F/A,r,n] = e r*n – 1 e r – 1

Continuous Compounding Series Present Worth:

[P/A,r,n] = e r*n – 1 e r*n (e r – 1)

Example 1

How much money will accrue in an account earning

5%

compounded continuously for

5

years, if $

500

is deposited each year?

Solution:

Example 2

Jim wished to save a uniform amount each month so he would have $

1000

at the end of one year. Based on

6%

nominal interest, compounded monthly, he had to deposit $

81.10

per month. How much would he have to deposit if his credit union paid

6%

nominal interest, compounded continuously?

Solution:

Continuous Uniform Cash Flow (One Period) With Continuous Compounding at Nominal Interest Rate r



This is the situation when a continuous uniform cash flow occurs during one period only, with continuous compounding.



Examples: Amount of money out of an ATM, amount of money into a credit card company in the form of payments, etc. P 1 F=?

Continuous Uniform Cash Flow (One Period) With Continuous Compounding at Nominal Interest Rate r (cont’d)

 The amount of money,

F

, at the end of one period can be obtained as follows:

F = P ((e r – 1) / r)

 To find the amount for any future time,

n

: P

F = P (e r – 1) e rn

1 2 3 … n

re r

 1 For any present time: 1 2 3 …

F

n

F = ?

P = F (e r – 1) re rn

P=

?

Example A self-service gasoline station has been equipped with an ATM. Customers may obtain gasoline simply by inserting their ATM card into the machine and filling their car with gasoline. When they have finished, the ATM unit automatically deducts the gasoline purchase price from the customer’s bank account and credits it to the gas station’s bank account. The gas station receives $40,000 per month in

1

this manner with the cash flowing uniformly throughout the month. If the bank pays 9% nominal interest, compounded continuously, how much will be in the gasoline station bank account at the end of the month?

Homework # 04 (Chapter # 4)

1 6 13 16 20 34 42 48 49 52 9 54 17 66 50 70

Homework # 04 (Chapter # 4)

– cont’d

73 75 77 81 85 86 96 97 101 110 102 116 104 108