tvm

The Time Value of Money (rev 1/24/02)

Introduction

People make their decisions in fundamentally the same way. The first step is to identify the available alternatives. There must be two or more from which to choose. If there is only one alternative, there is no choice and, therefore, no decision to be made. Second, all benefits and costs of each alternative are identified. Third, a value is placed on these benefits and costs by the decision-maker so that they can be combined. Fourth, those alternatives for which the benefits are greater than the costs are directly compared, and the alternative whose net benefit is greatest is selected. Once the decision is made, it may turn out that the best alternative was not selected. A cost may have been underestimated or a benefit overvalued, which could have us wishing we had made another choice. It is important to note that every decision-making technique tries to identify which alternative is best before a choice is made.

Economic decision-making tries to improve the quality of the decision making process by objectively measuring the dollar value associated with the benefits and costs of each alternative. This permits the different alternatives to be directly compared. Unfortunately, the economic value of a dollar decreases over time. If someone gave us a choice between receiving $100 today and $100 next year, we would choose the $100 today. Why is it that we would we rather have that money in our hand today than to have the same amount in our hand in one year? The answer is simple. There is an opportunity cost associated money. If we had that $100 today, we could save it or spend it or do nothing with it. Just possessing it has value. This economic concept is called the time preference for money.

The time preference for money means a single dollar today is valued more highly than a single dollar to be received in one year. Therefore, in order to compare an alternative's benefits and costs whose dollars are either received or paid out at different points in time, it is necessary to use dollars that have the same economic value. This only occurs when the dollars are to be received or paid out at the same point in time. The decision-maker's opportunity cost is used to convert the economic value of all dollars into an equivalent amount of dollars at the same point in time. Only dollars at the same point in time can be directly added (benefits) or subtracted (costs). The time value of money (TVM) is a series of relationships that makes this conversion possible.

There are two important concepts underlying TVM. The first is that a dollar you receive today is worth more than a dollar you would receive next period because of the opportunity cost associated with holding money. The second is that a dollar you are more certain of receiving one year from now is worth more today (has a higher present value) than a dollar you are less certain of receiving one year from now.

There are four major TVM relationships. The first solves for the future value given a present sum, opportunity cost and number of periods. The second solves for the present value given a future sum, opportunity cost, and number of periods. The third solves for the future value of an annuity given a series of equal future sums, an opportunity cost, and the number of periods. The fourth solves for the present value of an annuity given a series of equal future sums, an opportunity cost, and the number of periods. Each of these four relationships incorporates four variables. In the first two there is a future sum, a present sum, an opportunity cost (interest rate), and the number of periods. In the last two there is an annuity, a future sum or a present sum, an opportunity cost, and the number of periods. Three of the four variables are always known so that the fourth may be solved for uniquely.

Four TVM variables

A future sum.

"How much will you have in one year if today you put $100 in a savings account earning 6%?"

A present sum.

"How much would you have to put in a savings account earning 6% today to have $1,000 five years from now?"

An interest rate or rate of return.

"What is the interest rate if you borrow $1,000 today and repay $1,060 one year from now?"

The number of periods

"How long will it take for you to double your money if it earns 8% compounded annually?"

NOTE: The interest rate and the time period must match. If interest is compounded twice a year, each period is six months instead of one year; and the interest rate is half the annual rate.

[Listen to a general explanation of the time value of money]

The Future Value Concept (compounding)

Example: You deposit $100 in an account earning 8% compounded annually. How much would you have after 1, 2 and 3 years?

S₁= $100 + $100 (.08) = $108

S₂ = $108 + $108 (.08) = $116.64

S₃ = $116.64 + $116.64 (.08) = $125.97

Algebraic formulation

S₁ = S₀ + (S₀ x i) = S₀(1 + i)¹

S₂ = S₁ + (S₁ x i) = S₁(1 + i)¹ = S₀(1 + i)²

S₃ = S₂ + (S₂x i) = S₂(1 + i)¹ = S₀(1 + i)³

S_n = S₀(1 + i)ⁿ = S₀(FVIF,i,n)

NOTE: (FVIF,i,n) represents the future value interest factor calculated for a specific interest rate (i) and for a specific number of periods (n). This interest factor provides the value of $1 compounded at an interest rate "i" at "n" periods in the future. The value of this interest factor for many different interest rates and for many different periods is often calculated in a table in many finance textbooks. You can view an example of this table.

[Go to FVIF table]

Example - You deposit $100 in a savings account earning 8% compounded semiannually. How much would you have at the end of 1 year, 2 years and 3 years?

S₁ = $100 + $100 (.04) = $104

S₂ = $104 + $104 (.04) = $108.16

S₃ = $108.16 + $108.16 (.04) = $112.49

S₄ = $112.49 + $112.49 (.04) = $116.99

S₅ = $116.99 + $116.99 (.04) = $121.67

S₆ = $121.67 + $121.67 (.04) = $126.54

General formulation

S_n = S₀(1 + i/r)^nr, where r equals the number of times interest is compounded in one year.

Here's a calculator which can find the future value given a present sum, an opportunity cost (interest rate), and the number of periods. The opportunity cost (interest rate) must be expressed as a decimal. For example, 7 3/4% should be written as .0775.

[Future Value Calculator]

[Listen to an explanation of future value]

The Present Value Concept (discounting)

Example - You want to have $15,000 in 15 years to use for your child's college education. How much would you have to put in a savings account earning 8% today to achieve this goal?

S₁₅ = S₀(FVIF, i= 8%, n=15)

$15,000 = S₀(3.172)

S₀ = $4728.87

Algebraic formulation for one period

Start with the future value equation, S₁ = S₀(1 + i)¹, and solve for S₀.

S₀ = S₁/(1+ i) = S₁ (1/1+i) = S₁(PVIF, i, 1)

NOTE: (PVIF,i,n) represents the present value interest factor calculated for a specific interest rate (i) and for a specific number of periods (n). This interest factor provides the value of $1 to be received "n" periods in the future discounted at an interest rate "i". This present value is the amount that, if you had it today and put it into an accounted earning an interest rate "i", you would have $1 after "n" periods in the future. The value of this interest factor for many different interest rates and for many different periods is often calculated in a table in many finance textbooks. You can view an example of this table.

[Go to PVIF table]

Example: If S₁ = $1 and i = 5%, then the present value, S₀, equals $.9524 (95.24 cents).

This means that, if your best investment opportunity had an interest rate of 5%, you should be willing to "pay" $.9524 now to receive $1 next period OR $1 to be received next period has the same value as $.9524 received today. Another way to state this is that given your opportunity cost (the return on your best investment opportunity) you would indifferent between the 95.24 cents today or $1 in one year.

Algebraic formulation for two periods

Start with the future value equation, S₂ = S₀(1+i)², and solve for S₀.

S₀ = S₂/(1+i)² = S₂[1/(1+i)²]= S₂(PVIF, i, 2)

Example: If S₂ = $1 and i = 5%, then S₀ = $.9071

This means that, if your best investment opportunity had an interest rate of 5%, you should be willing to "pay" $.9071 now to receive $1 in two periods OR $1 to be received in two periods has the same value as $.9524 received today. Another way to state this is that given your opportunity cost (the return on your best investment opportunity) you would indifferent between the 90.71 cents today or $1 in two years.

General formulation

For one future sum, the present value is

S₀ = S_n/(1+i)ⁿ = S_n[1/(1+i)ⁿ]= S_n(PVIF,i,n) = S_n/(FVIF,i,n)

For a series of future sums, the present value is

S₀ = S₁/(1+i)¹ + S₂/(1+i)² + S₃/(1+i)³ + ... + S_n/(1+i)ⁿ

S₀ = S₁(PVIF,i,n=1) + S₂(PVIF,i,n=2) + S₃(PVIF,i,n=3) + ... + S_n(PVIF,i,n)

Here's a calculator which can find the present value given a future sum, an opportunity cost (interest rate), and the number of periods. The opportunity cost (interest rate) must be expressed as a decimal. For example, 7 3/4% should be written as .0775.

[Present Value Calculator]

[Listen to an explanation of present value]

Annuities

Definition of an annuity: equal sums over equal periods of time.

Types of annuities

Ordinary annuity - each sum occurs at the end of every period.
Annuity due - each sum occurs at the beginning of every period.

Future value of an annuity

How much money would you have saved if you deposited $10 in a bank each year for 5 years and earned 5%?

Assume this is an ordinary annuity. Each of the five $10 sums is deposited at the end of the year.

S₅ = S₁(FVIF,5%,4) + S₂(FVIF,5%,3) + S₃(FVIF,5%,2) + S₄(FVIF,5%,1) + S₅

S₅ = 10(FVIF,5%,4) + 10(FVIF,5%,3) + 10(FVIF,5%,2) + 10(FVIF,5%,1) + 10

S₅ = 10[(FVIF,5%,4) + (FVIF,5%,3) + (FVIF,5%,2) + (FVIF,5%,1) + 1]

S₅ = 10 (FVIFA,5%,5)

S₅ = $55.26

NOTE: (FVIFA,i,n) represents the future value interest factor for an annuity of $1 calculated for a specific interest rate (i) and for a specific number of periods (n). This interest factor provides the value of $1 received each of "n" periods compounded at an interest rate "i" at "n" periods in the future. The value of this interest factor for many different interest rates and for many different periods is often calculated in a table in many finance textbooks. You can view an example of this table.

[Go to FVIFA table]

Assume this is an annuity due. Each of the five $10 sums is received at the beginning of the year. It is important to note that all interest factors (PVIF, FVIF, FVIFA, PVIFA) are calculated based on the assumption that the monetary sums are received at the end of the period. For an annuity due, the first $10 sum is deposited at the beginning of the first period which is the present (n=0). The second $10 sum is deposited at the beginning of the second year which is the same as the end of the first period (n=1).

S₅ = S₀(FVIF,5%,5) + S₁(FVIF,5%,4) + S₂(FVIF,5%,3) + S₃(FVIF,5%,2) + S₄(FVIF,5%,1)

S₅ = 10(FVIF,5%,5) + 10(FVIF,5%,4) + 10(FVIF,5%,3) + 10(FVIF,5%,2) + 10(FVIF,5%,1)

S₅ = $58.02

Using the FVIFA table,

S₅ = [10(FVIFA,5%,5)](FVIF,5%,1) = $58.02

NOTE: Future value and present value interest factor tables for annuities and all financial calculators assume the annuity is an ordinary annuity with each sum occurring at the end of each period.

Here's a calculator which can find the future value given an annuity, an opportunity cost (interest rate), and the number of periods. The opportunity cost (interest rate) must be expressed as a decimal. For example, 7 3/4% should be written as .0775.

[Future Value of an Annuity Calculator]

Here's a calculator which can find the annuity value given a future sum, an opportunity cost (interest rate), and the number of periods. The opportunity cost (interest rate) must be expressed as a decimal. For example, 7 3/4% should be written as .0775.

[Find the Annuity given the Future Value Calculator]

[Listen to an explanation of the future value of an annuity]

Present value of an annuity

How much would you need to have in an account today earning 5% if you wanted to withdraw $10 each year for the next five years?

Assume this is an ordinary annuity.

S₀ = S₁/(1+i)¹ + S₂/(1+i)² + S₃/(1+i)³ + S₄/(1+i)⁴ + S₅/(1+i)⁵

S₀ = 10/(1+.05)¹ + 10/(1+.05)² + 10/(1+.05)³ + 10/(1+.05)⁴ + 10/(1+.05)⁵

S₀ = 10(PVIF,5%,1) + 10(PVIF,5%,2) + 10(PVIF,5%,3) + 10(PVIF,5%,4) + 10(PVIF,5%,5)

S₀ = 10[(PVIF,5%,1) + (PVIF,5%,2) + (PVIF,5%,3) + (PVIF,5%,4) + (PVIF,5%,5)]

S₀ = 10 (PVIFA,5%,5)

S₀ = $43.29

NOTE: (PVIFA,i,n) represents the present value interest factor of an annuity of $1 received for "n" periods calculated for a specific interest rate (i). This interest factor provides the value today of $1 to be received each of "n" periods in the future discounted at an interest rate "i". This present value is the amount that, if you had it today and put it into an accounted earning an interest rate "i", you could receive $1 each of "n" periods in the future. At the end of the "n" periods, there would be nothing left. The value of this interest factor for many different interest rates and for many different periods is often calculated in a table in many finance textbooks. You can view an example of this table.

[Go to PVIFA table]

Assume this is an annuity due.

S₀ = 10 + 10(PVIF,5%,1) + 10(PVIF,5%,2) + 10(PVIF,5%,3) + 10(PVIF,5%,4)

S₀ = 10[1 + (PVIF,5%,1) + (PVIF,5%,2) + (PVIF,5%,3) + (PVIF,5%,4)]

S₀ = 10[1 + (PVIFA,5%,4)]

S₀ = $45.46

Here's a calculator which can find the present value given an annuity, an opportunity cost (interest rate), and the number of periods. The opportunity cost (interest rate) must be expressed as a decimal. For example, 7 3/4% should be written as .0775.

[Present Value of an Annuity Calculator]

Here's a calculator which can find the annuity given a present sum, an opportunity cost (interest rate), and the number of periods. The opportunity cost (interest rate) must be expressed as a decimal. For example, 7 3/4% should be written as .0775.

[Find the Annuity given the Present Value Calculator]

[Listen to an explanation of the present value of annuity]

Example

You are buying a new home and want to know how big a mortgage you can afford. Assume you can pay $1500 each month for interest and principal and the interest rate is 9%.

Mortgage = Monthly payment (PVIFA, 9%/12, 360)

Mortgage = $1500 (PVIFA, 0.75%, 360)

Mortgage = $186,422.80

NOTE: To use the annuity interest factor tables (either FVIFA or PVIFA), the time period of the annuity must be the same as the time period of the interest rate and "n" equals the total number of these periods. In the above example, because the annuity is a monthly annuity, the interest rate must be a monthly interest rate and "n" must be the total number of months. The only way to solve this problem is to use a financial calculator. Remember, all financial calculators, including those included in these notes, assume that all annuity sums occur at the end of the period, that is, they assume every annuity is an ordinary annuity.