Future Value/Present Value Concepts

Calculating Future Value--A Basic Mathematical Concept

Present Value concepts are vital to the entire field of accounting and finance. In this webpage, I'm hoping that you will get a feel for how present value and future value are related. In the practice of accounting, there are many applications of present value, including bond problems, leases, mortgages, calculation of goodwill, and retirement planning. We start by thinking in mathematical terms, but the concept has broader implications, as you will discover.

To begin with, there are two types of problems that we need to consider--future value problems and present value problems. I am organizing this material into four sections, and will address each section with content and examples:

1. Future Value of a Lump Sum

2. Future Value of an Annuity

3. Present Value of a Lump Sum

4. Present Value of an Annuity

A good place to start is by saying that future value and present value problems involve the receipt or payment of cash. We are not talking about revenues or expenses, unless they involve receipt or payment of cash. Furthermore, we evaluate both the amount of cash as well as the timing of the cash receipt or payment. For example, suppose someone offers you a choice of receiving $1,000 cash today, OR receiving $1,000 cash in one year. Which choice would you make? Obviously, getting the $1,000 today makes more sense, because you could invest the money, and have more than $1,000 a year from now. We summarize this thinking as "the time value of money." The sooner we get the money, the more it is worth to us. Present value calculations are based on the time value of money, and allow us to compare money we spend today with money we will get back in the future. But, before we delve into present value, let's take a look at something you probably already know about, future value.

1. Let's Start with Future Value of a Lump Sum

A future value problem is one where you know how much you will invest today, but you want to figure out how much your investment will grow to in n years. We will use n to denote the periods over which the investment project will take place. Also, note that I've underlined the words "grow to", because we can use a formula to automate the interest calculation. We put in a certain amount, and we can calculate the amount we can take out after n years.

a. FV Problem 1--Mary Invests $100 for 1 year at 4%

Here's the problem I like to start with. Mary starts out with $100. She invests the $100 in a bank account that earns 4% per year. How much will the $100 grow to in one year? You can probably figure out the answer without much difficulty.

The $100 is referred to as the present value of the investment. We are trying to calculate the future value of the investment after one year.

Put in $100; figure out one year's interest = .04 * 100 = $4; add the $4 to $100 and the amount $100 will grow to in one year is $104. So, the future value of $100 in one year is $104.

Here's an alternative, more efficient, way to calculate the answer. Multiply the $100 times (1 + the interest rate) to arrive at the future value. $100(1+.04) = $104.

Using mathematical symbols, we are saying FV =PV(1+r) ; in words, take the present value times (1+the interest rate) to get the future value.

b. FV Problem 2--Mary Invests $100 for 2 years at 4%

If Mary leaves her $100 in the 4% account for two years, how much will it grow to in two years? Let's solve the problem two ways:

Put $100 in the bank at 4%; figure out one year's interest = .04 * 100 = $4; add the $4 to $100 and the amount $100 will grow to in one year is $104. Leave the $104 in the bank account for another year. So, .04*104 = $4.16. Add the $4.16 to $104 and you arrive at $108.16. The future value of $100 for two years at 4% is $108.16.

Note that Mary is getting interest on the original $100 in the first year; but in the second year, she is getting interest on the principal, plus interest on the interest. This is the concept of compound interest, and is a key concept in building wealth. As we did in problem one, we can automate the calculation using a formula.

FV = $100(1+.04)^2

...which says, multiply the present value times (1+the interest rate) raised to the nth power, where n is the number of periods. Here's how it would look with the numbers:

FV = 100(1.04)^2 = 100(1.04)(1.04) = 108.16

In general form, the formula looks like this: FV=PV(1+r)^n, where r is the yearly interest rate, and n is the number of years.

c. FV Problem 3--Mary Invests $100 for 3 Years at 4%

I'm hoping you can grab your calculator and figure this one out for yourself. The formula should be: FV=100(1.04)^3 = $112.48.

d. Creating a Future Value Table

You could create a future value table quite easily. In a future value table, we assume that $1.00 is invested today, and make the calculation to show how much it would grow to in n years at various interest rates. Here's how it would look for a few selected values:

Future Value of $1.00

Per.
4%
6%
8%
10%
1 1.0400      
2 1.0816      
3 1.1248      
4        

The table is saying that if you invest $1.00 at 4% for 3 years, your investment will grow to $1.1248. If the amount invested is $1,000, multiply 1.1248 times $1,000=$1124.80.

2. Future Value of an Annuity

An annuity is a stream of equal payments. Suppose Mary gets a good job and vows to invest $100 at the end of each year in an account that pays 4% interest annually. How much would this investment grow to after three years? After four years?

In this example, we specify that the payments into the account are at the ends of the years, so Mary waits until the end of year 1 to put in the first payment.

The investments and returns can be diagrammed as follows (read down each column, starting at the left):

Period 1 2 3 4
Begin 0 $100.00 $204.00 $312.16
Interest at 4% 0 $4.00 $8.16 $12.48
Deposit $100.00 $100.00 $100.00 $100.00
End $100.00 $204.00 $312.16 $424.64

A Future Value of an Annuity table could be created. This table assumes that all payments are at the ends of the years. Here are selected factors from that table, for a rate of 4%.

Future Value of Annuity of $1.00

Period
4%
6%
1 1.000  
2 2.040  
3 3.122  
4 4.246  

Can you figure out the table factors for the 6% column?

The formula for calculating each interest factor can be computed directly using the following formula:

FV = PMT [((1 + i) ^n - 1) / i]

The PMT for table purposes would be $1.00. For the 4% factors, the calculations would go like this:

Future Value of Annuity of $1.00

Period 4%
1 FV factor=1(1.04-1)/.04 = 1(.04/.04) = 1.000
2 FV factor =1(1.0816-1)/.04=1(2.04) = 2.040
3 FV factor = 1(1.124864-1)/.04=1(3.1216)=3.1216
4 FV factor=1(1.16986-1)/.04=1(4.2465)=4.2465

We don't have any particular applications of the FV Annuity calculation, but it is useful to know that the idea exists.

3. Present Value of a Lump Sum

A present value problem is one in which we know (or can estimate) the amount of money that will come to us in the future, but wish to know how much we should spend to get it--at some desired interest rate. This may sound strange at first, because, how can we know how much money is coming to us in the future? But, there are situations in which the future cash flow is contractual, so in fact, it is known. Here's an example, that you are already familiar with.

a. PV of a Lump Sum--an Investment Proposal

An investment consultant tells you that if you invest in a certain contract, you will receive exactly $104 in one year. How much should you invest in this contract, assuming you demand a 4% return (yield) on your money? In this problem (diagrammed below), we know how much money is coming to us at the end of one year. The thing we don't know is, how much should we pay for it to earn a yield of 4%?

Time Period 0

Time Period 1

Amount to invest to earn 4% yield ??? 104

If you refer to the Future Value Problem 1, shown earlier, you already know the answer to this problem. We know that $100 invested at 4% will amount to $104 in one year. In this present value problem, the given figure is in the future. We must "discount" this future value back to present to figure out the amount we should invest to earn our desired yield. In order to earn an exactly 4% yield, we must invest exactly $100. If we invest $101, our yield will be less than 4%.

Formula for the Present Value of a $1 Lump Sum

Here is the formula we use to determine the present value of $104, discounted at 4%. Here's the formula:

PV = FV(1/1+interest rate)^n

This formula says, the present value equals the future value we receive times 1 divided by (1 + the interest rate) raised to the nth power. To solve the investment problem, the calculation would be:

PV = 104(1/1.04^1) = 104(1/1.04)=$100

b. PV of a Lump Sum--Another Investment Proposal

An investment consultant tells you that if you invest in a certain contract, you will receive exactly $108.16 in two years. How much should you invest in this contract, assuming you demand a 4% return (yield) on your money?

The cash flow situation can be diagrammed as follows:

Time Period 0

Time Period 1

Time Period 2

Amount to invest to earn 4% yield ???   108.16

Again, we have already seen this problem as a future value problem. Here, in this present value problem, we are trying to determine the amount we should invest today in order to earn a yield of 4%.

Solution: PV = 108.16(1/1.04^2) = 108.16(1/1.0816)=$100

A Present Value of $1 Table

As we saw with future value calculations, a table can be constructed that gives us the present value factor for a certain interest rate for a certain number of periods. Shown below is a demonstration of how such a table can be calculated.

Present Value $1.00

Per. 4% Calc. 6% Calc.
1 .9615 1/1.04 ________ ________
2 .9245 1/(1.04)^2 ________ ________
3 .8889 1/(1.04)^3 ________ ________
4 .8548 1/(1.04)^4 ________ ________

I have left the 6% columns open so you can fill them in. Change the formulas under "Calc" to reflect a 6% rate.

Below is a 5-period table with interest rates of 6, 8 and 10%.

PV of $1.00 ______________________PV of a series of $1.00 annual payments

  6% 8% 10%       6% 8% 10%
1 0.9434 0.9259 0.9091   1 0.9434 0.9259 0.9091
2 0.8900 0.8573 0.8264   2 1.8334 1.7833 1.7355
3 0.8396 0.7938 0.7513   3 2.6730 2.5771 2.4869
4 0.7921 0.7350 0.6830   4 3.4651 3.3121 3.1699
5 0.7473 0.6806 0.6209   5 4.2124 3.9927 3.7908

I have a link to a full PV table on your website.

Application of Present Value of $1

Presented below are several examples of how the PV of $1 table can be used. Not e that there are several different phrases that imply "compute the present value", also illustrated in these examples.

PV Problem 1

What is the present value of $1,000 to be received in three years, discounted at 4%?

Solution: 1000*.8889=888.90.

PV Problem 2

How much would you pay today to receive $5,000 in four years, if you require a 4% yield on your investment?

Solution: 5000*.8548 = $4274

PV Problem 3

An investment counselor presents the following proposal to you. If you pay $1886 today, you will receive $1,000 one year from now, and $1,000 two years from now. You require at least a 4% return on your investment. Does this investment qualify?

The cash flow picture for this one looks like this:

Time Period 0

Time Period 1

Time Period 2

Pay $1886
Receive $1,000
Receive $1,000

You will need to discount each cash flow at 4% and then add the present values together. For the first year, 1000*.9615=$961.50. For the second year, 1000*.9245=$924.50. Add the two present values together: 961.50 + 924.50=$1886. The investment yield is exactly 4%.

Proving that Your PV Calculation is Correct

A useful strategy to prove that your PV calculation is correct, is to turn the problem around, and make it a future value problem. For example, Donna calculates the present value of $1,000 to be received in four years, discounted at 4%. She determines that the PV = $854.80. Is this correct?

Let's turn it around. If you invest 854.80 for four years at 4%, how much does it grow to?

FV = 854.80 * 1.04 * 1.04 * 1.04 *1.04 = $1,000. The PV of 854.80 is correct. This strategy reminds us that present value and future value are related to one another.

4. The Present Value of an Annuity

Many investment contracts are set up as annuities. An annuity is a stream of equal cash flows. A house mortgage, a car loan, a lease, and many retirement plans are set up as annuities.

Time Period 0

Time Period 1

Time Period 2

Pay $1886
Receive $1,000
Receive $1,000

The problem you looked at a few moments ago was an annuity. There were two payments, and they were equal--$1,000 each. You can compute the present value of an annuity using the PV of $1 table, as was demonstrated for this annuity. However, annuities can go for many years, so there is a separate table that can be used if the cash flows each year are equal, making the calculation more convenient. Let's look at the PV of $1 and PV of Annuity of $1 side by side.

Present Value $1.00 ................................................Present Value of Annuity of $1.00

Per. 4% Calc. 4% Calc.
1 .9615 1/1.04 .9615 .9615
2 .9245 1/(1.04)^2 1.8860 .9615+.9245
3 .8889 1/(1.04)^3 2.7749 .9615+.9245+.8889
4 .8548 1/(1.04)^4 3.6297 .9615+.9245+.8889+.8548

Note the pattern of the calculation of the annuity factors. For a two-year annuity at 4%, the annuity factor is 1.886, which is equal to .9615 + .9245, the sum of the first two factors in the PV of $1 table. This pattern continues. Conclusion? You could construct the PV of Annuity of $1 entirely, by using the factors from the PV of $1 table.

A Formula to Calculate the PV of An Annuity

The annuity factor for the present value of an annuity can be calculated directly using the following formula:

PV = PMT [(1 - (1 / (1 + i)^n )) / i]

Present Value of Annuity of $1.00

Period 4% Calculation by Formula
1 .9615 PV=1[(1-(1/1.04)^1)) / .04] = .9615
2 1.886 PV=1[(1-(1/1.04)^2)) /.04 ] = 1.886
3 2.775 *PV=1[(1-(1/1.04)^3))/.04] = 2.77509
4 3.630 Try it; see if you get something like 3.6297

* 1.04^3 is 1.124864. Then, 1/1.124864 is .888996. Subtract .888996 from 1, giving .1110036. Divide .1110036 by .04 = 2.77509

Proving that Your PV of Annuity is Correct

Bob is trying to calculate how much he should pay to receive a $1,000 annual payment for four years. For this investment, Bob demands a 4% yield. Bob's cash flow picture is as follows:

Time Period 0

Time Period 1

Time Period 2

Time Period 3

Time Period 4

Pay How Much?
Receive $1,000
Receive $1,000
Receive $1,000
Receive $1,000

Bob looks up the annuity factor for 4 years at 4%, which is 3.630. The present value of the $1,000 annuity is 3.630*1,000=$3,630. Is this correct? Is there some way to prove that it is correct. Yes. Turn the problem into a future value problem: If Bob invests $3,630 today at 4% interest, will he be able to withdraw $1,000 per year for 4 years? Let's see.

Year 1: Invest $3630 today at 4%; It will grow to 1.04 * 3630 in one year = $3775.20. However, we are going to take out the first payment of $1,000, leaving $2775.20.

Year 2: Invest the remainder, $2775.20 for the second year and it will grow to 2775.20 * 1.04 = $2886.21. But, we must withdraw the second annuity payment, leaving 1886.21.

Year 3: Invest the $1886.21 for the third year at 4%; it will grow to 1886.21 * 1.04=$1961.65. Take out the third annuity payment, leaving $961.65.

Year 4: Invest the remaining $961.65 at 4% and it will grow to 961.65 * 1.04 = $1,000. Take out the last payment of $1,000, leaving a balance of 0 in the investment.

After making this calculation a few times, it may strike you that there is an economic equivalence between the amount you pay today for the investment ($3630) and the four-year annuity of $1,000, if the relevant interest rate is 4%.

Precision with Present Value Problems

You don't have to be correct to the penny with present value problems--unless you are making bank loans. In particular, if you are getting your present value factors from a table, there will be some inevitable rounding error. Don't worry about it. For our purposes, if your calculations are correct within ten dollars, that should be sufficient precision in most cases.