Understanding the Time Value of Money

Congratulations!!! You have won a cash prize! You have two payment options: A - Receive $10,000 now OR B

- Receive $10,000 in three years. Which option would you choose?

What Is Time Value?

If you're like most people, you would choose to receive the $10,000 now. After all, three 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.

But why is this? A $100 bill has the same value as a $100 bill one year from now, doesn't it? Actually, although the

bill is the same, you can do much more with the money if you have it now because over time you can earn

more interest on your money.

Back to our example: by receiving $10,000 today, you are poised to increase the future value of your money

by investing and gaining interest over a period of time. For Option B, you don't have time on your side, and the

payment received in three years would be your future value. To illustrate, we have provided a timeline:

If you are choosing Option A, your future value will be $10,000 plus any interest acquired over the three years. The

future value for Option B, on the other hand, would only be $10,000. So how can you calculate exactly how

much more Option A is worth, compared to Option B? Let's take a look.

SEE: Internal Rate Of Return: An Inside Look

Future Value Basics

If you choose Option A and invest the total amount at a simple annual rate of 4.5%, the future value of your

investment at the end of the first year is $10,450, which of course is calculated by multiplying the principal amount

of $10,000 by the interest rate of 4.5% and then adding the interest gained to the principal amount:

Future value of investment at end of first year:

= ($10,000 x 0.045) + $10,000

= $10,450

You can also calculate the total amount of a one-year investment with a simple manipulation of the above

equation:

Original equation: ($10,000 x 0.045) + $10,000 = $10,450

Manipulation: $10,000 x [(1 x 0.045) + 1] = $10,450

Final equation: $10,000 x (0.045 + 1) = $10,450

The manipulated equation above is simply a removal of the like-variable $10,000 (the principal amount) by dividing

the entire original equation by $10,000.

If the $10,450 left in your investment account at the end of the first year is left untouched and you invested it at

4.5% for another year, how much would you have? To calculate this, you would take the $10,450 and multiply it

again by 1.045 (0.045 +1). At the end of two years, you would have $10,920:

Future value of investment at end of second year:

= $10,450 x (1+0.045)

= $10,920.25

The above calculation, then, is equivalent to the following equation:

Future Value = $10,000 x (1+0.045) x (1+0.045)

Think back to math class and the rule of exponents, which states that the multiplication of like terms is equivalent

to adding their exponents. In the above equation, the two like terms are (1+0.045), and the exponent on each is

equal to 1. Therefore, the equation can be represented as the following:

We can see that the exponent is equal to the number of years for which the money is earning interest in an

investment. So, the equation for calculating the three-year future value of the investment would look like this:

This calculation shows us that we don't need to calculate the future value after the first year, then the second year,

then the third year, and so on. If you know how many years you would like to hold a present amount of money in

an investment, the future value of that amount is calculated by the following equation:

SEE: Accelerating

Present Value Basics

If you received $10,000 today, the present value would of course be $10,000 because present value is what your

investment gives you now if you were to spend it today. If $10,000 were to be received in a year, the present value

of the amount would not be $10,000 because you do not have it in your hand now, in the present. To find the

present value of the $10,000 you will receive in the future, you need to pretend that the $10,000 is the total future

value of an amount that you invested today. In other words, to find the present value of the future $10,000, we

need to find out how much we would have to invest today in order to receive that $10,000 in the future.

To calculate present value, or the amount that we would have to invest today, you must subtract the

(hypothetical) accumulated interest from the $10,000. To achieve this, we can discount the

future payment amount ($10,000) by the interest rate for the period. In essence, all you are doing is rearranging

the future value equation above so that you may solve for P. The above future value equation can be rewritten by

replacing the P variable with present value (PV) and manipulated as follows:

Let's walk backwards from the $10,000 offered in Option B. Remember, the $10,000 to be received in three years

is really the same as the future value of an investment. If today we were at the two-year mark, we would discount

the payment back one year. At the two-year mark, the present value of the $10,000 to be received in one year is

represented as the following:

Present value of future payment of $10,000 at end of year two:

Note that if today we were at the one-year mark, the above $9,569.38 would be considered the future value of our

investment one year from now.

Continuing on, at the end of the first year we would be expecting to receive the payment of $10,000 in two years.

At an interest rate of 4.5%, the calculation for the present value of a $10,000 payment expected in two years

would be the following:

Present value of $10,000 in one year:

Of course, because of the rule of exponents, we don't have to calculate the future value of the investment every

year counting back from the $10,000 investment at the third year. We could put the equation more concisely and

use the $10,000 as FV. So, here is how you can calculate today's present value of the $10,000 expected from a

three-year investment earning 4.5%:

So the present value of a future payment of $10,000 is worth $8,762.97 today if interest rates are 4.5% per year. In

other words, choosing Option B is like taking $8,762.97 now and then investing it for three years. The equations

above illustrate that Option A is better not only because it offers you money right now but because it offers you

$1,237.03 ($10,000 - $8,762.97) more in cash! Furthermore, if you invest the $10,000 that you receive from Option

A, your choice gives you a future value that is $1,411.66 ($11,411.66 - $10,000) greater than the future value of

Option B.

SEE: Economics And The Time Value Of Money

Present Value of a Future Payment

Let's add a little spice to our investment knowledge. What if the payment in three years is more than the amount

you'd receive today? Say you could receive either $15,000 today or $18,000 in four years. Which would you

choose? The decision is now more difficult. If you choose to receive $15,000 today and invest the entire amount,

you may actually end up with an amount of cash in four years that is less than $18,000. You could find the future

value of $15,000, but since we are always living in the present, let's find the present value of $18,000 if interest

rates are currently 4%. Remember that the equation for present value is the following:

In the equation above, all we are doing is discounting the future value of an investment. Using the numbers above,

the present value of an $18,000 payment in four years would be calculated as the following:

Present Value

From the above calculation we now know our choice is between receiving $15,000 or $15,386.48 today. Of course

we should choose to postpone payment for four years!

The Bottom Line

These calculations demonstrate that time literally is money - the value of the money you have now is not the same

as it will be in the future and vice versa. So, it is important to know how to calculate the time value of money so

that you can distinguish between the worth of investments that offer you returns at different times.

Calculating the Present and Future Value Of Annuities

At some point in your life, you may have had to make a series of fixed payments over a period of time - such as

rent or car payments - or have received a series of payments over a period of time, such as bond coupons. These

are called annuities. If you understand the time value of money, you're ready to learn about annuities and how

their present and future values are calculated.

What Are Annuities?

Annuities are essentially a series of fixed payments required from you or paid to you at a specified frequency over

the course of a fixed time period. The most common payment frequencies are yearly, semi-annually (twice a year),

quarterly and monthly. There are two basic types of annuities: ordinary annuities and annuities due.

Ordinary Annuity: Payments are required at the end of each period. For example, straight bonds usually

pay coupon payments at the end of every six months until the bond's maturity date.

Annuity Due: Payments are required at the beginning of each period. Rentis an example of annuity due.

You are usually required to pay rent when you first move in at the beginning of the month, and then on

the first of each month thereafter.

Since the present and future value calculations for ordinary annuities and annuities due are slightly different, we

will first discuss the present and future value calculation for ordinary annuities.

Calculating the Future Value of an Ordinary Annuity

If you know how much you can invest per period for a certain time period, the future value of an ordinary

annuity formula is useful for finding out how much you would have in the future by investing at your given interest

rate. If you are making payments on a loan, the future value is useful in determining the total cost of the loan.

Let's now run through Example 1. Consider the following annuity cash flow schedule:

To calculate the future value of the annuity, we have to calculate the future value of each cash flow. Let's assume

that you are receiving $1,000 every year for the next five years, and you invested each payment at 5%. The

following diagram shows how much you would have at the end of the five-year period:

Since we have to add the future value of each payment, you may have noticed that if you have an ordinary annuity

with many cash flows, it would take a long time to calculate all the future values and then add them together.

Fortunately, mathematics provides a formula that serves as a shortcut for finding the accumulated value of all cash

flows received from an ordinary annuity:

C = Cash flow per periodi = interest raten = number of payments

Using the above formula for Example 1 above, this is the result:

= $1000*[5.53]= $5525.63

Note that the 1 cent difference between $5,525.64 and $5,525.63 is due to a rounding error in the first calculation.

Each value of the first calculation must be rounded to the nearest penny - the more you have to round numbers in

a calculation, the more likely rounding errors will occur. So, the above formula not only provides a shortcut to

finding FV of an ordinary annuity but also gives a more accurate result.

Calculating the Present Value of an Ordinary Annuity

If you would like to determine today's value of a future payment series, you need to use the formula that

calculates the present value of an ordinary annuity. This is the formula you would use as part of abond pricing

calculation. The PV of an ordinary annuity calculates the present value of the couponpayments that you will

receive in the future.

For Example 2, we'll use the same annuity cash flow schedule as we did in Example 1. To obtain the total

discounted value, we need to take the present value of each future payment and, as we did in Example 1, add the

cash flows together.

Again, calculating and adding all these values will take a considerable amount of time, especially if we expect many

future payments. As such, we can use a mathematical shortcut for PV of an ordinary annuity.

C = Cash flow per periodic = interest rate = number of payments

The formula provides us with the PV in a few easy steps. Here is the calculation of the annuity represented in the

diagram for Example 2:

= $1000*[4.33]= $4329.48

Calculating the Future Value of an Annuity Due

When you are receiving or paying cash flows for an annuity due, your cash flow schedule would appear as follows:

Since each payment in the series is made one period sooner, we need to discount the formula one period back. A

slight modification to the FV-of-an-ordinary-annuity formula accounts for payments occurring at the beginning of

each period. In Example 3, let's illustrate why this modification is needed when each $1,000 payment is made at

the beginning of the period rather than at the end (interest rate is still 5%):

Notice that when payments are made at the beginning of the period, each amount is held longer at the end of the

period. For example, if the $1,000 was invested on January 1 rather than December 31 each year, the last payment

before we value our investment at the end of five years (on December 31) would have been made a year prior

(January 1) rather than the same day on which it is valued. The future value of annuity formula would then read:

Therefore,

= $1000*5.53*1.05= $5801.91

Calculating the Present Value of an Annuity Due

For the present value of an annuity due formula, we need to discount the formula one period forward as the

payments are held for a lesser amount of time. When calculating the present value, we assume that the first

payment was made today.

We could use this formula for calculating the present value of your future rent payments as specified in a lease you

sign with your landlord. Let's say for Example 4 that you make your first rent payment at the beginning of the

month and are evaluating the present value of your five-month lease on that same day. Your present value

calculation would work as follows:

Of course, we can use a formula shortcut to calculate the present value of an annuity due:

Therefore,

= $1000*4.33*1.05= $4545.95

Recall that the present value of an ordinary annuity returned a value of $4,329.48. The present value of an

ordinary annuity is less than that of an annuity due because the further back we discount a future payment, the

lower its present value: each payment or cash flow in an ordinary annuity occurs one period further into the

future.

Conclusion

Now you can see how annuity affects how you calculate the present and future value of any amount of money.

Remember that the payment frequencies, or number of payments, and the time at which these payments are

made (whether at the beginning or end of each payment period) are all variables you need to account for in your

calculations.