Table of Contents

I. The definition and the importance of time value of money......................2

1. What is the time value of money?..........................................................2

2. How is it important?...............................................................................3

II. Future values and Present values of a single cash flow, an annuity and

perpetuity........................................................................................................ 3

1. Motivation of calculation of future values and present values of a

single cash flow............................................................................................ 3

2. Formula of future and present of a single cash flow..............................4

3. Annuity and Perpetuity...........................................................................7

4. Inflation and the Time value of money.................................................10

III. Implications of time value of money in financing and investment.........11

1. Implications of TVM in investment:........................................................11

2. Implications of TVM in financing decisions:............................................14

Summary....................................................................................................... 16

References:.................................................................................................... 17

I. The definition and the importance of time value of money

1. What is the time value of money?

The time value of money (TVM), also referred to as "present discounted value", is

the premise that an investor prefers to receive a payment of a fixed amount of money

today, rather than an equal amount in the future, all else being equal. It is based on the

concept that a dollar that you have today is worth more than the promise or expectation

that you will receive a dollar in the future due to its potential earning capacity. Money

that you hold today is worth more because you can invest it and earn interest. After all,

you should receive some compensation for foregoing spending. For instance, if you won

a cash prize, you would have had two payment options: receiving $50,000 now or

receiving $50,000 in five years. Which option would you choose? If you're like most

people, you would choose to receive the $50,000 now. Five years is a long time to wait.

Why would any rational person defer payment into the future when he or she could have

the same amount of money now? For most of us, taking the money in the present is just

plain instinctive. So at the most basic level, the time value of money demonstrates that,

all things being equal, it is better to have money now rather than later.

Present Value is an amount today that is equivalent to a future payment, or series

of payments, that has been discounted by an appropriate interest rate.  The future amount

can be a single sum that will be received at the end of the last period, as a series of

equally-spaced payments (an annuity), or both.  Since money has time value, the present

value of a promised future amount is worth less the longer you have to wait to receive it.

Future Value is the amount of money that an investment with a fixed,

compounded interest rate will grow to by some future date. The investment can be a

single sum deposited at the beginning of the first period, a series of equally-spaced

payments (an annuity), or both.  Since money has time value, we naturally expect the

future value to be greater than the present value. The difference between the two depends

on the number of compounding periods involved and the going interest rate.

2. How is it important?

Your preferences indicate how important this factor is to you. The time value of

money is fundamental to money management. It is the key to understanding stocks,

bonds, financing your loans, and making good business investments because they are

many times valued by determining the present value of an expected cash flow.

With money we have the opportunity to invest and gain interest. In other words,

we can measure the opportunity cost of getting the money later. The time value of money

concept can save an individual or business significant and measurable dollar in interest

payments if the concept is integrated into their debt payoff plan or budget. Common

loans such as mortgages and automobiles can be paid of significantly quicker if the

borrower understands that small additional principle payments now shorten the life, and

interest paid to, the loan. This is even more important to understand if a loan is amortized

because larger down-payments or additional principle payments in the early stages of

borrowing will make economical sense to the borrower. Individual borrowers may be

tempted to take on a part-time job to make extra contributions to paying off current debt

when they see the value their dollar has today, versus how many dollars they are actually

going to have to pay in the future.

II. Future values and Present values of a single cash flow, an

annuity and perpetuity

1. Motivation of calculation of future values and present values of a single

cash flow

a) The future value of a single cash flow serves as a means of valuation, which

shows how much an asset or a liability will be worth at a future date. Any cash flow that

is scheduled to occur sometime later than today is referred to as a “future value.”

The future worth of a current sum of money is the amount of money that an

investment with a fixed, compounded interest rate will grow to by some future date. For

example, if we deposit 1 million VND into a bank account today, how much will it be

worth in 1 year? Or if we borrow 500,000 VND today and don't make any payments until

the loan comes due two years from now, how much will we owe at that time?

b) The present value of a single cash flow formula is used as a valuation

mechanism. It tells us how much an amount to be transacted in the future is worth today

(or some date prior to the receipt or payment date). The above mentioned amount can be

either a liabilities or an assets.

The current worth of a future sum of money or stream of cash flows given a

specified rate of return. Future cash flows are discounted at the discount rate. The higher

the discount rate, the lower the present value of the future cash flows. Determining the

appropriate discount rate is the key to properly valuing future cash flows, whether they be

earnings or obligations.

For example, if we want to buy a 1 million VND bond that matures in 10 years,

how much should we pay for that bond today? Another instance can be quoted as follow:

if we owe a debt that obligates us to pay 5 million VND in 3 years time, what is a fair

amount we could offer to settle that debt today?

Note: Interest is a charge for borrowing money, usually stated as a percentage

of the amount borrowed over a specific period of time.  

Simple interest is computed only on the original amount borrowed. It is the

return on that principal for one time period.  

In contrast, compound interest is calculated each period on the original

amount borrowed plus all unpaid interest accumulated to date.  Compound interest is

always assumed in Time Value Money problems.

2. Formula of future and present of a single cash flow

a) Future value:

Where: FV = future value;

PV = present value;

i = interest rate;

n = number of compounding period.

FV = PV (1+i)n

The following example illustrates the basic operation of the future value of a single

cash flow formula: How much will I receive at the end of 3 years if I invest a single sum

of 50,000 VND today at 10 % interest compounded annually?

The problem is illustrated below.

After identifying the initial deposit, the term of the investment, and the interest rate, we

can summarize our inputs to the future value of a single cash flow equation as:

FV = 50,000 i = 0.10 n = 3

and plugging these values into the equation, we have:

FV = PV (1 + i)n = 50,000 ( 1+ 0.10)3 = 66,550 VND

The mechanics of the calculation are illustrated below...

b) Present value of a single cash flow

Where: PV = present value

FV = future value

i = discounted rate

n = number of compounding period

The following simplified example illustrates the basic operation of the present value

of a single cash flow formula: If the prevailing interest rate is 5% compounded annually,

how much do I need to deposit to have 1million VND in 3 years?

The problem is illustrated below.

After identifying how much we will receive at maturity, the term of the investment, and

the interest rate, we can summarize our inputs to the present value of a single cash flow equation

as:

FV = 1,000,000 i = 0.05 n = 3

and plugging these values into the equation, we have:

The mechanics of the calculation are illustrated below.

Note: Solving other variables

While the equation discussed above allows us to calculate the future value and

present value of a single cash flow, there are times when we need to know the value of

other variables like n, or i.

Variables Formula Example

Compounding

periods (n)

Interest rate (i)

In reality, there are other factors that need to be taken into consideration like

taxes, default risk, cash flow, etc. before we can really declare "financing equivalence."

Still, Time Value of Money theory and its associated calculations provide a powerful tool

for analyzing financial alternatives by providing a mechanism for placing cash flows at

different time periods on a comparable basis.

3. Annuity and Perpetuity

The term annuity is used in finance theory to refer to any terminating stream of

fixed payments over a specified period of time. In other words an annuity is an equally

spaced level stream of cash flows.

A perpetuity is an annuity that has no definite end, or a stream of cash payments

that continues forever. It is a stream of level cash payments that never ends.

The motivation for these financial terms is that, frequently, we may need to value

a stream of equal cash flows. For example, a home mortgage might require the

homeowner to make equal monthly payments for the life of the loan, or in the case of any

buyer required to pay equally and annually… These payment streams will result in equal

payments.

3.1 How to value perpetuity

PV of perpetuity = C / r = cash payment / interest rate

If the rate of interest is 10 percent and the aim is to provide $100,000 a year

forever, the amount that must be set aside today is PV = $100,000/0.1 = $1,000,000.

The above perpetuity formula tells us the value of a regular stream of payments

starting one period from now, which means if an up front additional payment is

required at the very beginning then we will add one more C to PV: $1,000,000 +

$100,000 = $1,100,000

To calculate the value of a perpetuity that does not start to make payments for t

years: In year t-1, it will be an ordinary perpetuity. Therefore to find today’s value we

need to multiply it by the (t-1)-year discount factor. The formula is:

PV of delayed perpetuity = C

3.2 How to value annuity

PV of t-year annuity = C

The reason for this formula is, the present value of an annuity for t-year can be

simplified by considering it an ordinary perpetuity subtracted by a delayed perpetuity for

t years.

The expression in square brackets shows the present value of a t-year annuity of

$1 a year. It is generally known as the t-year annuity factor. Therefore, another way to

write the value of an annuity is: Present value of t-year annuity = payment * annuity

factor.

An example for this is, if we were required to pay an $1000 each year for next 7

years for an item, with an interest rate of 8%, the annuity formula would be:

PV of 7-year annuity = $1000* = $5206.37

The perpetuity and annuity formulas assume that the first payment occurs at the

end of the period. They tell you the value of a stream of cash payments starting one

period hence. However, streams of cash payments often start immediately. A level stream

of payments starting immediately is known as an annuity due.

It is easy to come to a conclusion that:

PV annuity due = 1 + PV ordinary annuity of t – 1 payments

Which means:

PV of t-year annuity due = C

Come back to the last example, the annuity due would be:

PV of 7-year annuity due = $1000* = $5622.88

3.3 Future value of an annuity

To calculate the future value of an annuity, first of all we calculate its present

value by the formula of PV of t-year annuity then multiply it by (1+r) :

Future value of annuity of $1 a year = present value of annuity of $1 a year * (1+r)

FV of annuity = C *(1+r) = C

4. Inflation and the Time value of money

Prices of goods and services continually change. Textbooks may become more

expensive while computers become cheaper. An overall general rise in prices is known as

inflation.

Economists track the general level of prices using several different price indexes.

The best known of these is the consumer price index, or CPI. This measures the number

of dollars that it takes to buy a specified basket of goods and services that is supposed to

represent the typical family’s purchases.3 Thus the percentage increase in the CPI from

one year to the next measures the rate of inflation.

Current dollar cash flows must be discounted by the nominal interest rate; real

cash flows must be discounted by the real interest rate.

The real rate of interest is calculated by

Here is a useful approximation. The real rate approximately equals the difference

between the nominal rate and the inflation rate:

Real interest rate ≈ nominal interest rate – inflation rate

III. Implications of time value of money in financing and investment

1. Implications of TVM in investment:

Any firm possesses a huge number of possible investments. Each possible

investment is an option available to the firm. Some options are valuable and some are

not. The essence of successful financial management, of course, is learning to identify

which are which. The capital budgeting (investment) question is probably the most

important issue in corporate finance. With this in mind, our goal in this part is to illustrate

how firms can apply the TVM to analyze potential business ventures to decide which are

worth undertaking.

Let us hope that the CFO is by now convinced of the correctness of the net present

value rule. However, it is possible that the CFO has also heard of some alternative

investment criteria and would like to know why you do not recommend any of them. Just

so that you are prepared, we will now look at three of the alternatives.

They are:

- The net present value

- The payback period

- The internal rate of return

1.1 Net present value

Net present value is Present value of cash flows minus initial investment. The net

present value rule states that managers increase shareholders’ wealth by accepting all

projects that are worth more than they cost. Therefore, they should accept all projects

with a positive net present value.

CF0 : initial investment

CFt : Cash flow for each period

r : the discount rate

The project can be accepted if NPV is positive

1.2 The Payback period

Some companies require that the initial outlay on any project should be

recoverable within a specified period. The payback period of a project is found by

counting the number of years it takes before the cumulative forecasted cash flow equals

the initial investment.

Consider the following three projects:

Project A involves an initial investment of $2,000 (C0 –2,000) followed by cash

inflows during the next three years. Suppose the opportunity cost of capital is 10 percent.

Then project A has an NPV of $2,624:

Project B also requires an initial investment of $2,000 but produces a cash inflow

of $500 in year 1 and $1,800 in year 2. At a 10 percent opportunity cost of capital project

B has an NPV of –$58:

The third project, C, involves the same initial outlay as the other two projects but

its first-period cash flow is larger. It has an NPV of +$50

The net present value rule tells us to accept projects A and C but to reject project

B.

Some companies discount the cash flows before they compute the payback period.

The discounted-payback rule asks, How many periods does the project have to last in

order to make sense in terms of net present value? This modification to the payback rule

surmounts the objection that equal weight is given to all flows before the cutoff date.

However, the discounted-payback rule still takes no account of any cash flows after the

cutoff date.

1.3 The internal rate of return

The internal rate of return is defined as the rate of discount which makes NPV=0.

This means that to find the IRR for an investment project lasting T years, we must solve

for IRR in the following expression:

For example, consider a project that produces the following flows:

The internal rate of return is IRR in the equation:

Let us arbitrarily try a zero discount rate. In this case NPV is not zero but +$2,000:

The NPV is positive; therefore, the IRR must be greater than zero. The next step

might be to try a discount rate of 50 percent. In this case net present value is –$889:

The NPV is negative; therefore, the IRR must be less than 50 percent. From this,

we can see that a discount rate of 28 percent gives the desired net present value of zero.

Therefore, IRR is 28 percent.

Now the internal rate of return rule is to accept an investment project if the

opportunity cost of capital is less than the internal rate of return. If it is equal to the IRR,

the project has a zero NPV. And if it is greater than the IRR, the project has a negative

NPV. Therefore, when we compare the opportunity cost of capital with the IRR on our

project, we are effectively asking whether our project has a positive NPV. This is true not

only for our example. The rule will give the same answer as the net present value rule

whenever the NPV of a project is a smoothly declining function of the discount rate. And

any time you find and launch a positive-NPV project you have made your company’s

stockholders better off.

2. Implications of TVM in financing decisions:

When an investment opportunity or “project” is identified, if the financial manager

admits that the project is worth more than the capital required to undertake it, he or she

then considers how the project should be financed. Part of the money for new

investments comes from profits that companies retain and reinvest. The other can come

from selling new debt or equity securities. These financing patterns raise several

interesting questions. In this part what we focus on is how firms can apply the TVM in

bonds and stocks pricing to decide which capital structure policy is worth undertaking.

2.1. Stock pricing

Cash flows from stock include cash flows from dividends and cash flows from

stock price. The current price of the stock can be written as the present value of the

dividends beginning in one period and extending out forever:

We have illustrated here that the price of the stock today is equal to the present

value of all of the future dividends.

An example is that you are considering buying a share of stock today. You plan to

sell the stock in one year. You somehow know that the stock will be worth $70 at that

time. You predict that the stock will also pay a $10 per share dividend at the end of the

year. If you require a 25 percent return on your investment, what is the most you would

pay for the stock? In other words, what is the present value of the $10 dividend along

with the $70 ending value at 25 percent?

If you buy the stock today and sell it at the end of the year, you will have a total of

$80 in cash. At 25 percent:

Therefore, $64 is the value you would assign to the stock today.

In the case of preferred stock with the dividend on a share of zero growth, the

stock can be viewed as an ordinary perpetuity with a cash flow equal to D every period.

Suppose we know that the dividend for some company always grows at a steady rate,

which is called a growing perpetuity.

2.2 Bond pricing:

Cash flows from bonds include: Coupon and Maturity value. Value of bonds is the

present value of the stream of cash flows it is expected to generate. Hence, the value of a

bond is obtained by discounting the bond's expected cash flows to the present using the

appropriate discount rate.

Summary

Time Value of Money (TVM) is an important concept in financial management. It

can be used to compare investment alternatives and to solve problems involving loans,

mortgages, leases, savings, and annuities. Time value of money results from the concept

of interest. We have explained the distinction between compound interest and simple

interest. Then we analyzed the difference between the nominal interest rate and the real

interest rate. This difference arises because the purchasing power of interest income is

reduced by inflation.

As a whole, this study covers an introduction to calculate the present value and

future value of money, illustrates the use of time value of money in making decisions

process, shows a matrix approach to solving time value of money problems and

introduces the concepts of annuities due, and perpetuities. A simple introduction to

working time value of money problems on a financial calculator is included as well as

additional figures to help understand time value of money.

The studied issue serves as the foundation for all other notions in finance. It

impacts business finance, consumer finance and government finance. Because of limited

time and knowledge, we can only present one among many approaches to the problem.

We would like to receive comments and addition from lecturers to improve this study.

References:

1. Brigham, E. F., & Ehrhardt, M. C. (2008). Financial Management (12 ed.).

Mason, OH: South-Western Cengage Learning.

2. Fundamental of Corporate Finance, third edition, Brealey Myers Marcus, Chapter

two.

3. http://www.frickcpa.com/tvom/TVOM_PV_SS.asp