1 the time value of money timothy r. mayes, ph.d. fin 3300: chapter 5

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The Time Value of MoneyThe Time Value of Money

Timothy R. Mayes, Ph.D.

FIN 3300: Chapter 5

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What is Time Value?What is Time Value?

We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return

In other words, “a dollar received today is worth more than a dollar to be received tomorrow”

That is because today’s dollar can be invested so that we have more than one dollar tomorrow

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The Terminology of Time The Terminology of Time ValueValue

Present Value - An amount of money today, or the current value of a future cash flow

Future Value - An amount of money at some future time period

Period - A length of time (often a year, but can be a month, week, day, hour, etc.)

Interest Rate - The compensation paid to a lender (or saver) for the use of funds expressed as a percentage for a period (normally expressed as an annual rate)

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AbbreviationsAbbreviations

PV - Present value FV - Future value Pmt - Per period payment amount N - Either the total number of cash flows

or the number of a specific period

i - The interest rate per period

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TimelinesTimelines

0 1 2 3 4 5

PV FV

Today

A timeline is a graphical device used to clarify the timing of the cash flows for an investment

Each tick represents one time period

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Calculating the Future ValueCalculating the Future Value

Suppose that you have an extra $100 today that you wish to invest for one year. If you can earn 10% per year on your investment, how much will you have in one year?

0 1 2 3 4 5

-100 ?

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Calculating the Future Value Calculating the Future Value (cont.)(cont.)

Suppose that at the end of year 1 you decide to extend the investment for a second year. How much will you have accumulated at the end of year 2?

0 1 2 3 4 5

-110 ?

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Generalizing the Future Generalizing the Future ValueValue

Recognizing the pattern that is developing, we can generalize the future value calculations as follows:

If you extended the investment for a third year, you would have:

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Compound InterestCompound Interest

Note from the example that the future value is increasing at an increasing rate

In other words, the amount of interest earned each year is increasing• Year 1: $10• Year 2: $11• Year 3: $12.10

The reason for the increase is that each year you are earning interest on the interest that was earned in previous years in addition to the interest on the original principle amount

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Compound Interest Compound Interest GraphicallyGraphically

0

500

1000

1500

2000

2500

3000

3500

4000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Years

Fu

ture

Val

ue

5%

10%

15%

20%

3833.76

1636.65

672.75

265.33

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The Magic of CompoundingThe Magic of Compounding

On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly purchased Manhattan from the Indians for $24 worth of beads and other trinkets. Was this a good deal for the Indians?

This happened about 371 years ago, so if they could earn 5% per year they would now (in 1997) have:

If they could have earned 10% per year, they would now have:

$54,562,898,811,973,500.00 = 24(1.10)371

$1,743,577,261.65 = 24(1.05)371

That’s about 54,563 Trillion dollars!

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The Magic of Compounding The Magic of Compounding (cont.)(cont.) The Wall Street Journal (17 Jan. 92) says that all of New

York city real estate is worth about $324 billion. Of this amount, Manhattan is about 30%, which is $97.2 billion

At 10%, this is $54,562 trillion! Our U.S. GNP is only around $6 trillion per year. So this amount represents about 9,094 years worth of the total economic output of the USA!

At 5% it seems the Indians got a bad deal, but if they earned 10% per year, it was the Dutch that got the raw deal

Not only that, but it turns out that the Indians really had no claim on Manhattan (then called Manahatta). They lived on Long Island!

As a final insult, the British arrived in the 1660’s and unceremoniously tossed out the Dutch settlers.

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Calculating the Present ValueCalculating the Present Value

So far, we have seen how to calculate the future value of an investment

But we can turn this around to find the amount that needs to be invested to achieve some desired future value:

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Present Value: An ExamplePresent Value: An Example

Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about $100,000 on her 18th birthday to pay for four years of college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal?

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AnnuitiesAnnuities

An annuity is a series of nominally equal payments equally spaced in time

Annuities are very common:• Rent• Mortgage payments• Car payment• Pension income

The timeline shows an example of a 5-year, $100 annuity

0 1 2 3 4 5

100 100 100 100 100

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The Principle of Value The Principle of Value AdditivityAdditivity

How do we find the value (PV or FV) of an annuity?

First, you must understand the principle of value additivity: • The value of any stream of cash flows is equal

to the sum of the values of the components In other words, if we can move the cash

flows to the same time period we can simply add them all together to get the total value

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Present Value of an AnnuityPresent Value of an Annuity

We can use the principle of value additivity to find the present value of an annuity, by simply summing the present values of each of the components:

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Present Value of an Annuity Present Value of an Annuity (cont.)(cont.)

Using the example, and assuming a discount rate of 10% per year, we find that the present value is:

0 1 2 3 4 5

100 100 100 100 100

62.0968.3075.1382.6490.91379.0

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Actually, there is no need to take the present value of each cash flow separately

We can use a closed-form of the PVA equation instead:

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We can use this equation to find the present value of our example annuity as follows:

This equation works for all regular annuities, regardless of the number of payments

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The Future Value of an The Future Value of an AnnuityAnnuity

We can also use the principle of value additivity to find the future value of an annuity, by simply summing the future values of each of the components:

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The Future Value of an Annuity The Future Value of an Annuity (cont.)(cont.)

Using the example, and assuming a discount rate of 10% per year, we find that the future value is:

100 100 100 100 100

0 1 2 3 4 5

146.41133.10121.00110.00}=

610.51at year 5

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Just as we did for the PVA equation, we could instead use a closed-form of the FVA equation:

This equation works for all regular annuities, regardless of the number of payments

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We can use this equation to find the future value of the example annuity:

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Annuities DueAnnuities Due

Thus far, the annuities that we have looked at begin their payments at the end of period 1; these are referred to as regular annuities

A annuity due is the same as a regular annuity, except that its cash flows occur at the beginning of the period rather than at the end

0 1 2 3 4 5

100 100 100 100 100100 100 100 100 1005-period Annuity Due

5-period Regular Annuity

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Present Value of an Annuity Present Value of an Annuity DueDue

We can find the present value of an annuity due in the same way as we did for a regular annuity, with one exception

Note from the timeline that, if we ignore the first cash flow, the annuity due looks just like a four-period regular annuity

Therefore, we can value an annuity due with:

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Present Value of an Annuity Due Present Value of an Annuity Due (cont.)(cont.)

Therefore, the present value of our example annuity due is:

Note that this is higher than the PV of the, otherwise equivalent, regular annuity

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Future Value of an Annuity Future Value of an Annuity DueDue

To calculate the FV of an annuity due, we can treat it as regular annuity, and then take it one more period forward:

0 1 2 3 4 5

Pmt Pmt Pmt Pmt Pmt

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Future Value of an Annuity Due Future Value of an Annuity Due (cont.)(cont.)

The future value of our example annuity is:

Note that this is higher than the future value of the, otherwise equivalent, regular annuity

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Deferred AnnuitiesDeferred Annuities

A deferred annuity is the same as any other annuity, except that its payments do not begin until some later period

The timeline shows a five-period deferred annuity

0 1 2 3 4 5

100 100 100 100 100

6 7

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PV of a Deferred AnnuityPV of a Deferred Annuity

We can find the present value of a deferred annuity in the same way as any other annuity, with an extra step required

Before we can do this however, there is an important rule to understand:

When using the PVA equation, the resulting PV is always one period before the first payment occurs

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PV of a Deferred Annuity PV of a Deferred Annuity (cont.)(cont.)

To find the PV of a deferred annuity, we first find use the PVA equation, and then discount that result back to period 0

Here we are using a 10% discount rate

0 1 2 3 4 5

0 0 100 100 100 100 100

6 7

PV2 = 379.08

PV0 = 313.29

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PV of a Deferred Annuity PV of a Deferred Annuity (cont.)(cont.)

Step 1:

Step 2:

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FV of a Deferred AnnuityFV of a Deferred Annuity

The future value of a deferred annuity is calculated in exactly the same way as any other annuity

There are no extra steps at all

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Uneven Cash FlowsUneven Cash Flows

Very often an investment offers a stream of cash flows which are not either a lump sum or an annuity

We can find the present or future value of such a stream by using the principle of value additivity

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Uneven Cash Flows: An Uneven Cash Flows: An Example (1)Example (1)

Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment?

0 1 2 3 4 5

100 200 300

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Uneven Cash Flows: An Uneven Cash Flows: An Example (2)Example (2)

Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year?

0 1 2 3 4 5

300 500 700

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Non-annual CompoundingNon-annual Compounding

So far we have assumed that the time period is equal to a year

However, there is no reason that a time period can’t be any other length of time

We could assume that interest is earned semi-annually, quarterly, monthly, daily, or any other length of time

The only change that must be made is to make sure that the rate of interest is adjusted to the period length

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Non-annual Compounding Non-annual Compounding (cont.)(cont.)

Suppose that you have $1,000 available for investment. After investigating the local banks, you have compiled the following table for comparison. In which bank should you deposit your funds?

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To solve this problem, you need to determine which bank will pay you the most interest

In other words, at which bank will you have the highest future value?

To find out, let’s change our basic FV equation slightly:

In this version of the equation ‘m’ is the number of compounding periods per year

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We can find the FV for each bank as follows:

First National Bank:

Second National Bank:

Third National Bank:

Obviously, you should choose the Third National Bank

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Continuous CompoundingContinuous Compounding

There is no reason why we need to stop increasing the compounding frequency at daily

We could compound every hour, minute, or second We can also compound every instant (i.e.,

continuously):

Here, F is the future value, P is the present value, r is the annual rate of interest, t is the total number of years, and e is a constant equal to about 2.718

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Continuous Compounding Continuous Compounding (cont.)(cont.)

Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. What is the future value of your $1,000 investment?

This is even better than daily compounding The basic rule of compounding is: The more

frequently interest is compounded, the higher the future value

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Continuous Compounding Continuous Compounding (cont.)(cont.)

Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. If you plan to leave the money in the account for 5 years, what is the future value of your $1,000 investment?

1 the time value of money timothy r. mayes, ph.d. fin 3300: chapter 5

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