ENGINEERING ECONOMICS

Lecture # 2Time value of money •Interest•Present and Future Value•Cash Flow•Cash Flow Diagrams

Definitions• Project : an investment opportunity generating cash

flows over time

• Cash Flow: the movement of money (in or out) of a

project

• Interest: The rent for loaned money

• Cash Flow Diagram: Describes type, magnitude and

timing of cash flows over some horizon

• Time value of money: The change in the amount of

money over given time period is called the time value of

money. It is the most important concept in the

engineering economy

The Time Value of Money

Money NOW

is worth more than

money LATER!

Obviously, $10,000 today$10,000 today.

You already recognize that there is TIME VALUE TO MONEYTIME VALUE TO MONEY!!

Which would you prefer -- $10,000 $10,000 today today or $10,000 in 5 years$10,000 in 5 years?

Interest• Cost of Money

– Rental amount charged by lender for use of money

– In any transaction, someone “earns” and someone pays

• Interest is the difference between an ending amount and

beginning amount

• Interest is paid when money is borrowed (loan) and

repayment involves an additional amount

• Interest is earned when money is saved, invested and

rented for an additional return

o Interest rate, or the rate of capital growth, is the rate of

gain received from an investment

o When interest paid over specific time period is

expressed as %age of principal (original) amount, it is

called as interest rate

o Interest Rate = (Interest per unit time/original amount)

*100

o Time unit of interest rate is interest period

o Interest Rate = Rate of return (ROR) = Rate of

investment (ROI)

Interest Rate

Types of InterestTypes of Interest

• Compound InterestCompound InterestInterest paid (earned) on any previous

interest earned, as well as on the principal amount

Simple InterestSimple Interest

Interest paid (earned) on only the original amount, or principal amount

Simple Interest FormulaSimple Interest Formula

FormulaFormula SI = P0(i)(n)

SI: Simple Interest

P0: Deposit today (t=0)

i: Interest Rate per Period

n: Number of Time Periods

• SI = P0(i)(n)= $1,000(.07)(2)= $140$140

Simple Interest ExampleSimple Interest Example

• Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

Simple Interest

• Interest earned/paid is directly proportional to capital involved.

I = P * i * n

Ex. $ 1000 loan for 2 years at 10 % per year – no compounding

I = P * i * n = 1000 * .10 * 2 = $200

Payback = F = P + I

= 1000 + 200 = $1200

A sum of money today is called a present value

• We designate it mathematically with a subscript, as occurring in time period 0

• For example: P0 = 1,000 refers to $1,000 today

A sum of money at a future time is termed a future value

• We designate it mathematically with a subscript showing that it occurs in time period n.

• For example: Sn = 2,000 refers to $2,000 after n periods from now.

As already noted, the number of time periods in a time value problem is designated by n

• n may be a number of years• n may be a number of months• n may be a number of quarters• n may be a number of any defined time periods

The interest rate or growth rate in a time value problem is designated by i

• i must be expressed as the interest rate per period.

• For example if n is a number of years, i must be the interest rate per year.

• If n is a number of months, i must be the interest rate per month.

The first of the general type of time value problems is called future value and present value problems. The formula for these problems is:

• Sn = P0(1+i)n

An example problem:

• If you invest $1,000 today at an interest rate of 10 percent, how much will it grow to be after 5 years?

• Sn = P0(1+i)n

• Sn = 1,000(1.10)5

• Sn = $1,610.51

Another example problem:

• How long will it take for $10,000 to grow to $20,000 at an interest rate of 15% per year?

• Sn = P0(1+i)n

• 20,000 = 10,000(1.15)n

• n = 4.96 years (or, about 5 years)

One more example problem:• If you invest $11,000 in a mutual fund today, and it

grows to be $50,000 after 8 years, what compounded, annualized rate of return did you earn?

• Sn = P0(1+i)n

• 50,000 = 11,000(1+i)8

• i = 20.84 percent per year

The next two general types of time value problems involve annuities

• An annuity is an amount of money that occurs (received or paid) in equal amounts at equally spaced time intervals.

• These occur so frequently in business that special calculation methods are generally used.

For example:

• If you make payments of $2,000 per year into a retirement fund, it is an annuity.

• If you receive pension checks of $1,500 per month, it is an annuity.

• If an investment provides you with a return of $20,000 per year, it is an annuity.

A common mathematical symbol for an annuity amount is PMT

For the future value of an annuity:

• FV = PMT[(1+i)n - 1]/i

For the present value of an annuity:

• PV = PMT[(1+i)n -1]/[i(1+i)n]

An example problem:

• If you save $50 per month at 12 percent per annum, how much will you have at the end of 20 years?

• Note that since time periods are months, i = 12%/12 months = 1% per period, for 240 periods.

• FV = PMT[(1+i)n - 1]/i• FV = 50[(1.01)240 - 1]/.01• FV = $49,463

Another example problem:

• If you want to save $500,000 for retirement after 30 years, and you earn 10 percent per annum, how much must you save each year?

• FV = PMT[(1+i)n - 1]/i

• 500,000 = PMT[(1.1)30 - 1]/.1

• PMT = $3,040 per year

An example problem:

• If you borrow $100,000 today at 9 percent interest per annum, and repay it in equal annual payments over 10 years, how much are the payments?

• PV = PMT[(1+i)n -1]/[i(1+i)n]

• 100,000 = PMT[(1+.09)10 -1]/[.09(1.09)10]

• PMT = $15,582 per year

A last type of time value problem involves what are

called, perpetuities

• A perpetuity is simply an annuity that continues forever (perpetually).

• The formula for finding the present value of a perpetuity is:

• PV = PMT/i

A variation to perpetuity problems is the case of growing perpetuities

• If an annuity continues forever, and grows in amount each period at a rate g, then

• PV = PMT1/(i - g)

An example problem:

• If you invest in a stock that will pay a dividend of

$10 next year and grow at 5 percent per year, and

you require a 14 percent rate of return, how much

is the stock worth to you today?

• PV = PMT1/(i - g)

• PV = 10/(.14-.05)

• PV = $111.11

Thank You