time value of money

FHU 3213

TIME VALUE OF MONEY

Learning Goals

1. Discuss the role of time value in finance.

2. Understand the concept of future value and

present value, their calculation for single

amounts, and the relationship between them.

3. Find the future value and the present value of

both an ordinary annuity and an annuity due,

and the present value of a perpetuity.

Learning Goals (cont.)

4. Calculate both the future value and the present value of a mixed stream of cash flows.

5. Understand the effect that compounding interest more frequently than annually has on future value and the effective annual rate of interest.

6. Describe the procedures involved in (1) determining deposits needed to accumulate to a future sum, (2) loan amortization, (3) finding interest or growth rates, and (4) finding an unknown number of periods.

The Role of Time Value in Finance

• Financial managers and investors are always confronted

with opportunities to earn positive returns on their funds

– investments in projects or interest-bearing securities or

deposits.

• The time value of money is based on the believe that a

dollar today is worth more than a dollar that will be

received at some future date.

• From a company stand point, money that the firm has in

its possession today is more valuable than future

payments because the money can be invested now and

earn positive returns

The Role of Time Value

in Finance (cont.)

• Most financial decisions involve costs &

benefits that are spread out over time.

• Time value of money allows comparison

of cash flows from different periods.

Interest: the concept• Interest is an important concept that cannot be ignored

when dealing with significant sums of money.

• Interest is the cost of using somebody else’s money.

When you borrow money, you pay interest. When you

lend money, you earn interest.

• The interest rate can be interpreted as the cost of

capital because the use of capital (money for

investment) always has opportunity costs (you can

always invest it somewhere). Put another way, the

interest rate is really just the price of using money.

1. Time preference: People want to have things today

rather wait until a later time they are willing to pay

extra to avoid waiting.

2. People also borrow money so that they can invest it.

In this case people believe they can use the money to

make more money than what it costs to borrow it.

Why are people willing to pay interest

to use other people’s money?

There are two types of interest that can be applied to

loans:

1. Simple interest

- Calculated only on the principal amount

1. Compound interest

- Included interest earned on the interest which was

previously accumulated

Breaking Down Interest

Simple Interest

With simple interest, you don’t earn interest on interest.

• Year 1: 5% of $100 = $5 + $100 = $105

• Year 2: 5% of $100 = $5 + $105 = $110

• Year 3: 5% of $100 = $5 + $110 = $115

• Year 4: 5% of $100 = $5 + $115 = $120

• Year 5: 5% of $100 = $5 + $120 = $125

Compound Interest

With compound interest, a depositor earns interest on interest!

• Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00

• Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25

• Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76

• Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55

• Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63

Basic Concepts– A present value is a value that is expressed in terms of

dollars received immediately.

– A future value is a value that is expressed in terms of

dollars received at some future time.

– Discounting is the process of converting future values to

present values.

– Compounding is the reverse process: converting present

values to future values.

– Single cash flows & series of cash flows can be considered

– Time lines are used to illustrate these relationships

Time Line

A time line can be used to depict the cash flows

associated with a given investment

outflows at

time zero

inflows

Time line

Basic Patterns of Cash Flow

• The cash inflows and outflows of a firm can be

described by its general pattern.

• The three basic patterns include a single amount, an

annuity, perpetuity, or a mixed stream.

• Single amount: A lump sump either currently held or

expected at some future date. Example include $1,000

today and $50 to be received at the end of 10 years

• Annuity: A level periodic stream of cash flow. For our

purpose, we will work primarily with annual cash flows.

Example include either paying out or receiving at the

end end of each of the next 7 years

Basic Patterns of Cash Flow

• Perpetuity: The periodic annuity or cash flow stream

continues forever.

• Mixed stream: A stream of cash flow that is not an

annuity; a stream of unequal periodic cash flows that

reflect no particular pattern. Example is as follows:

Finding Future and Present Values of

Single Amounts, Annuities, Perpetuities,

and Mixed Streams

• Use the Equations

• Use the Financial Tables

• Use Financial Calculators

• Use Electronic Spreadsheets

Finding Future and Present Values of

Single Amounts, Annuities, Perpetuities,

and Mixed Streams

• Use the Equations

• Use the Financial Tables

• Use Financial Calculators

• Use Electronic Spreadsheets

Terms Used

• PV0 = present value or beginning amount

• i = interest rate

• FVn = future value at end of “n” periods

• n = number of compounding periods

• A = an annuity (series of equal payments

or receipts)

Future Value of a Single Amount

• Future Value techniques typically measure cash flows at the end of a project’s life.

• Future value is cash you will receive at a given future date.

• The future value technique uses compounding to find the future value of each cash flow at the end of an investment’s life and then sums these values to find the investment’s future value.

• We speak of compound interest to indicate that the amount of interest earned on a given deposit has become part of the principal at the end of the period.

Future Value of Single Amount Formula

• The general equation for future value at

the end of period n is:

𝐹𝑉𝑛 = 𝑓𝑢𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑 𝑛𝑃𝑉 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒, 𝑜𝑟 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒𝑖 = 𝑎𝑛𝑛𝑢𝑎𝑙 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑎𝑖𝑑𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑦𝑒𝑎𝑟𝑠 𝑜𝑟 𝑚𝑜𝑛𝑡ℎ𝑠

FVn = 𝑃𝑉 (1 + 𝑖)𝑛

• If Fred Moreno places $100 in a savings

account paying 8% interest compounded

annually, how much will he have in the

account at the end of one year?

Future Value at end of year 1 = PV0(1+i)n

= $100 x (1.08)1

= $100 x 1.08

= $108

Future Value of a Single Amount –

Example 1

• If Fred were to leave this money in the

account for another year, the future value

at the end of 2nd year is:

Future Value at end of year 2 = PV0(1+i)n

= $100 x (1.08)2

= $100 x 1.1664

= $116.64

Future Value of a Single Amount –

Example 1

FV5 = $800 X (1 + 0.06)5 = $800 X (1.338) = $1,070.40

Future Value of a Single Amount:

Example 2

• Jane Farber places $800 in a savings account paying

6% interest compounded annually. She wants to know

how much money will be in the account at the end of five

years.

• This analysis can be depicted on a time line as follows:

A Graphical View of Future Value

• Figure above shows the relationship among various interest rate, the

number of periods interest is earned, and the future value of one dollar

• The higher the interest rate, the higher the value,

• The longer the period of time, the higher the future value.

• Interest rate of 0%, the FV equals PV ($1.00).

Present Value of a Single Amount

• Present value is the current dollar value of a future amount of money.

• It is based on the idea that a dollar today is worth more than a dollar tomorrow.

• It is the amount today that must be invested at a given rate to reach a future amount.

• Calculating present value is also known as discounting.

• The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, or the cost of capital.

Present Value of Single Amount Formula

• The general equation for present value of

a future amount to be received n periods

from now is:

𝐹𝑉𝑛 = 𝑓𝑢𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑 𝑛𝑃𝑉 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒, 𝑜𝑟 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒𝑖 = 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑟𝑎𝑡𝑒 𝑜𝑟 𝑜𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑖𝑡𝑦 𝑐𝑜𝑠𝑡𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑦𝑒𝑎𝑟𝑠 𝑜𝑟 𝑚𝑜𝑛𝑡ℎ𝑠

PVn =𝐹𝑉𝑛

(1 + 𝑖)𝑛= 𝐹𝑉𝑛 ×

(1 + 𝑖)𝑛

• Paul Shorter has an opportunity to receive

$300 one year from now. If he can earn

6% on his investments, what is the most

he should pay now for this opportunity?

PV0 = FVn/(1+i)n

= $300 / (1.06)1

= $283.02

Present Value of a Single Amount –

Example 1

• Pam Valentino wishes to find the present value of

$1,700 that will be received 8 years from now. Pam’s

opportunity cost is 8%.

• The following time line shows the analysis:

PV = $1,700/(1 + 0.08)8 = $1,700/1.851 = $918.42

Present Value of a Single Amount -

Example 2

A Graphical View of Present Value

• Figure above shows the relationship among the discount rates, time

periods, and present value of one dollar.

• The higher the discount rate, the lower the present value

• The longer the period of time, the lower the present value.

• Given a discount rate of 0%, the value always equal the future value

($1.00)

Annuities

• Annuities are equally-spaced cash flows of equal size.

• Annuities can be either inflows or outflows.

• An ordinary (deferred) annuity has cash flows that occur at the end of each period.

• An annuity due has cash flows that occur at the beginning of each period.

• An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.

Future Value Annuities- Example 1

Note that the amount of both annuities total $5,000.

• Fran Abrams is choosing which of two annuities

to receive. Both are 5-year $1,000 annual cash

flows or payment (A) ; annuity A is an ordinary

annuity, and annuity B is an annuity due. Fran

has listed the cash flows for both annuities as

shown in Table 4.1 on the following slide.

Future Value of an Ordinary Annuity

and Annuity Due

Future Value of an Ordinary Annuity

• Fran Abrams wishes to determine how much money she

will have at the end of 5 years if he chooses annuity A,

the ordinary annuity and it earns 7% annually. Annuity

A is depicted graphically below:

• The calculation in the preceding example can be expressed

as follows:

FV annuity at end of year 5 = [$1,000 x (1.07)4 ]+ [$1,000 x (1.07)3 ] +

[$1,000 x (1.07)2 ]+ [$1,000 x (1.07)1 ] +

[$1,000 x (1.07)0 ]

= [$1,000 x1.311] + [$1,000 x1.225] +

[$1,000 x1.145] + [$1,000 x1.070] +

[$1,000 x 1.000]

= $1,311+$1,225+$1,145+$1,070+$1,000

= $5,751

• It’s a time consuming methods, let’s use the FV of ordinary annuity

formula 4-34

Future Value of an Ordinary Annuity –

The long method

Finding Future Value of an Ordinary

Annuity Using the Formula

• The formula for finding ordinary annuity:

𝐹𝑉 𝑜𝑓 𝑜𝑟𝑑𝑖𝑛𝑎𝑟𝑦 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =𝐴 (1 + 𝑖)𝑛−1

Where; 𝐴 = 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑒𝑟𝑖𝑜𝑑𝑖 = 𝐴𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑛 = 𝑁𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑦𝑒𝑎𝑟𝑠 𝑜𝑟 𝑚𝑜𝑛𝑡ℎ𝑠

Using FV Ordinary Annuity Formula

for - Fran Abram's Example

Finding Future Value of an Annuity Due

Now, lets calculate how much money Fran Abrams will

have at the end of 5 years if she chooses annuity B, the

annuity due and it earns 7% annually.

1. Draw the time line

2. Calculate using the future value formula.

3. How much Fran Abram will have for the annuity due,

Finding Future Value of an Annuity Due

using formula

• The formula for finding FV annuity due:

𝐹𝑉 𝑜𝑓 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑑𝑢𝑒 = 1 + 𝑖 ×𝐴 (1 + 𝑖)𝑛−1

Where; 𝐴 = 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝑎𝑡 𝑡ℎ𝑒 𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑒𝑟𝑖𝑜𝑑𝑖 = 𝐴𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑛 = 𝑁𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑦𝑒𝑎𝑟𝑠 𝑜𝑟 𝑚𝑜𝑛𝑡ℎ𝑠

Using the FV Annuity Due Formula- Fran

Abram’s Example

• Remember from earlier example that Fran Abrams wanted to choose between ordinary and an annuity due, both offering similar terms except for the timing of the cash flows.

• Using the FV of annuity due below:

𝐹𝑉 𝑜𝑓 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑑𝑢𝑒 = 1 + 𝑖 ×𝐴 (1 + 𝑖)𝑛−1

𝑖= 1 + 0.07 × 5751= $6154

Comparison of an Annuity Due with

an Ordinary Annuity Future Value

• Annuity A, ordinary annuity: $ 5,751

• Annuity B, annuity due: $ 6,154

• As noted earlier, the FV of annuity due is always

greater than the FV of an ordinary annuity.

• Because the annuity due’s cash flow occurs at

the beginning of the period rather at the end

which can earn interest one more year.

• Nonetheless, ordinary annuity are more

frequently used in finance.

Present Value of an Ordinary Annuity –

Braden Company Example

• Braden Company, a small producer of plastic toys,

wants to determine the most it should pay to purchase a

particular annuity. The annuity consists of cash flows of

$700 at the end of each year for 5 years. The required

return is 8%.

• The calculation in the preceding example are expressed

as follows:

PV ordinary annuity = [$700 /(1.08)1 ]+ [$700/(1.08)2 ] +

[$700/(1.08)3 ]+ [$700 /(1.08)4 ] +

[$700 /(1.08)5 ]

= $648.2 + $599.9 + $555.8 + $514.50

+ $476.70

= $2,795

Present Value of an Ordinary Annuity –

The Long Method

The Formula for Finding Present Value of

an Ordinary Annuity

• The formula for finding PV ordinary annuity:

𝑃𝑉 𝑜𝑓 𝑜𝑟𝑑𝑖𝑎𝑛𝑟𝑦 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =𝐴 (1 + 𝑖)𝑛−1

𝑖(1 + 𝑖)𝑛

Where; 𝐴 = 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑒𝑟𝑖𝑜𝑑𝑖 = 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑟𝑎𝑡𝑒 𝑜𝑟 𝑜𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑖𝑡𝑦 𝑐𝑜𝑠𝑡𝑛 = 𝑁𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑦𝑒𝑎𝑟𝑠 𝑜𝑟 𝑚𝑜𝑛𝑡ℎ𝑠

Using the Present Value Formula of an

Ordinary Annuity: Braden Company

Example

Finding Present Value of an Annuity Due

using the long method – Braden

Company Example

Steps:

1. Draw the time line

2. Calculate using the present value formula

3. How much Braden will have using the

annuity due?

The Formula for Finding Present Value of

an Annuity Due – Braden Company

Example

• The formula for finding PV annuity due:

𝑃𝑉 𝑜𝑓 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑑𝑢𝑒 = 𝐴 + 𝐴1 − (1 + 𝑖)−(𝑛−1)

Where; 𝐴 = 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝑎𝑡 𝑡ℎ𝑒 𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑒𝑟𝑖𝑜𝑑𝑖 = 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑟𝑎𝑡𝑒 𝑜𝑟 𝑜𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑖𝑡𝑦 𝑐𝑜𝑠𝑡𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑦𝑒𝑎𝑟𝑠 𝑜𝑟 𝑚𝑜𝑛𝑡ℎ𝑠

Present Value of an Annuity Due:

Using the formula

Comparison of an Annuity Due with an

Ordinary Annuity Future Value

• The present value of an annuity due is always greater

than the present value of an otherwise identical ordinary

annuity.

• We can see this by comparing the present values of the

Breden Company’s two annuity

Ordinary annuity = $2795

Annuity due = $3018

• Because the cash flow of the annuity due occurs at the

beginning of the period rather than the end, its present

value is greater.

Recap: Ordinary Annuity and

Annuity Due Formula

Application of Annuities Technique:

Payment to Achieve a Given Present Value

• Rearranging the formula of PV of ordinary

annuity we get:

𝐴 =𝑃𝑉𝑜𝑖(1 + 𝑖)𝑛

[(1 + 𝑖)𝑛 − 1]

• This formula is used to calculate the annual

payment (A) that would be required over a period

of n years in order to achieve a given present

value at a given interest rate i.

• Example:

Suppose you want to take out a loan for a car. You want

to borrow $10,000 over a 3-yr period at an annual interest

rate of 9.5%. How much will your payment be?

• Answer: first calculate the monthly interest rate:

• Example:

Suppose you want to take out a loan for a car. You want

to borrow $10,000 over a 3-yr period at an annual interest

rate of 9.5%. How much will your payment be?

• Answer: Next calculate the payment:

𝐴 =𝑃𝑉𝑜𝑖(1 + 𝑖)𝑛

[(1 + 𝑖)𝑛 − 1]

𝐴 =$10000 × 0.00759 × (1 + 0.00759)36

[(1 + 0.00759)36 − 1]= $318.15

• A perpetuity is a special kind of annuity.

• With a perpetuity, the periodic annuity or cash flow stream

continues forever.

• Present value interest factor for a perpetuity discounted at rate k

• The PVIFA is found by dividing the periodic annuity or cash flow by

the discount rate, k.

• Explanation: consider how much you should earn annually from an

investment if you take out the interest each year and keep the

original capital intact

• The future value of a perpetuity is infinite

Present Value of a Perpetuity

Present Value of a Perpetuity:Example

Mixed Stream

• A mixed stream of cash flows reflects no particular

pattern; an annuity, as showed earlier, is a pattern

of equal annual cash flows.

• Frey Company, a shoe manufacturer, has been

offered an opportunity to receive the following mixed

stream of cash flows over the next 5 years.

Future Value of a Mixed Stream

• If the firm earns at least 8% on its

investments annually, how much it will

earn at the end of 5 years?

Future Value of a Mixed Stream (cont.)

The calculation in the preceding example can be expressed

as follows:

FV at end of year 5 = [$11,500 x (1.08)4 ]+ [$14,000 x (1.08)3 ] +

[$12,900 x (1.08)2 ]+ [$16,000 x (1.08)1 ] +

[$18,000 x (1.08)0 ]

= $83,601.40

Present Value Mixed Stream

• Frey Company, a shoe manufacturer, has been

offered an opportunity to receive the following

mixed stream of cash flows over the next 5 years.

Present Value of a Mixed Stream

• If the firm must earn at least 9% on its

investments, what is the most it should pay for

this opportunity?

• This situation is depicted on the following

time line.

Present Value of a Mixed Stream

The calculation in the preceding example can be expressed

as follows:

PV of mixed stream = [$400/(1.09)1 ]+ [$800/(1.09)2 ] +

[$500/(1.08)3 ]+ [$400/(1.08)4 ] +

[$300/(1.08)5 ]

= $1,904.60

• Interest is often compounded more frequently than once

a year.

• Banks compound interest semi-annually, quarterly,

monthly, weekly, daily or even continuously.

• Compounding more frequently than once a year results

in a higher effective interest rate because you are

earning on interest on interest more frequently.

• As a result, the effective interest rate is greater than the

nominal (annual) interest rate.

• Furthermore, the effective rate of interest will increase

the more frequently interest is compounded.

Compounding Interest

More Frequently Than Annually

More Frequently Than Annually (cont.)

• Fred Moreno has decided to invest $100 in a saving

account paying 8% interest, compounded semiannually.

If he leaves the money in the account for 24 months (2

years), he will be paid 4% interest compounded over

four periods.

Fred Moreno has found an institution that will pay him

8% interest, compounded quarterly. If he leaves the

money in the account for 24 months (2 years), he will be

paid 2% interest compounded over eight periods.

As shown, the more frequently interest is compounded, the

greater the amount of money accumulated

• A General Equation for Compounding

More Frequently than Annually

Where;

FVn = Future value of n years

PV = Present value

i = interest rate

m = compounding frequency

n = year

k = discount rate

𝐹𝑉𝑛 = 𝑃𝑉 × (1 +𝑘

𝑚)𝑚×𝑛

• A General Equation for Compounding More

Frequently than Annually

– Recalculate the example for the Fred Moreno

example assuming (1) semiannual compounding and

(2) quarterly compounding.

• Interest can be compounded continuously

• With continuous compounding the number of

compounding periods per year approaches infinity.

• In this case, m in equation previously would approach

infinity. Through the use of calculus, the equation

thus becomes:

FVn (continuous compounding) = PV x (ekxn)

where “e” has a value of 2.7183.

Continuous Compounding

• Continuing with the previous example, find the Future

value of the $100 deposit after 2 years if interest (8%) is

compounded continuously.

FVn (continuous compounding) = PV x (ekxn)

where “e” has a value of 2.7183.

FVn = 100 x (2.7183)0.08x2 = $117.35

Continuous Compounding (cont.)

• The nominal interest rate is the stated or contractual rate

of interest charged by a lender or promised by a

borrower.

• The effective (true) interest rate is the rate actually paid

or earned.

• Effective annual rate (EAR), reflect the impact of

compounding frequency

• In general, the effective rate > nominal rate whenever

compounding occurs more than once per year

EAR = (1 + k/m) m - 1

Nominal & Effective

Annual Rates of Interest

Nominal & Effective Annual Rates of

Interest: Example

• Fred Moreno wishes to find the effective annual rate

associated with an 8% nominal annual rate (k = .08)

when interest is compounded (1) annually (m=1); (2)

semiannually (m=2); and (3) quarterly (m=4).

time value of money - wordpress.com · the role of time value in finance • financial managers and...

Documents

time value of time value of money

1 chapter 2 time value of money. 2 time value topics future...

1 chapter 4 time value of money. 2 time value topics future...

time value mk

time value of money increases in value over time/inflation...

economic principles in agribusiness: time value of money...

time value of money -...

time value

time & value

time value concepts

the value of time

time value analysis

time centric earned value

time value of money -...

time value questions

value of time for bus passengers - transport for...

quality | value | time

time value opportunity

time value of money dalam islam - direktori file...time...