From Time Value of Money:

Librarians don’t often consider themselves to be financial experts, but nonetheless, a basic understanding of some financial concepts is mandatory for the library manager. The most basic of these concepts is the time value of money (aka TVM). The time value of money not only has implications for funds in a library’s bank account, but also for tax revenue, bonds for capital projects, and grants. We will approach this in incremental stages, beginning with interest, then to calculation of future value and present value, then to annuities, and finally to bonds, where all these concepts are employed. Remember that our purpose is not to become finance experts, but to understand the concepts underlying financial instruments that affect library operations.

In this section we will explain interest, both simple and compound, as well as present value, future value and discount rates.

Interest

Interest is the fee paid to a lender to borrow money. The fundamental concepts of interest are principal, or the amount borrowed, the interest rate, which is a percentage of the outstanding principal amount borrowed, and the time during which that principal is outstanding. Time is only calculated in years, but in smaller increments, generically called periods, which can be as long as a year or short as a day.

There are three components to interest. A “pure rate of interest” is the amount charged if there is no expectation of default by the borrower or inflation in the economy. A “credit rate of interest” is an additional increment based on the credit worthiness of the borrower. Finally, there is an “expected inflation rate of interest” which is used to compensate for the loss of purchasing power of the amount borrowed due to inflation. Adding these three together gives an effective rate of interest.

Simple interest

Simple interest is computed on the amount of principal for one time period. In such a case a lender lends money, say $100, to a borrower for one period at an interest rate of 6% simple interest. To calculate the interest earned in this transaction we would use the formula

interest = p * i * n where p = principal, i=interest, n=number of periods

In this case the interest would be $6.

Simple interest is often used in commercial transactions in order to get a customer to pay on time. For example, the library might purchase books or other library materials from Alliance Book Services that cost $1,000. The invoice for the shipment states that payment must be made within 30 days, otherwise a 4% penalty will apply. If the library pays the invoice within the time allotted, there would be no penalty. If the payment arrives after the 30 days has elapsed, then the library would owe an additional $40 (4% of $1,000). In essence, if the library has not paid on time, the vender would have effectively “lent” the library $1,000 and therefore is charging interest on this “loan.”

A similar situation might apply with a personal property tax bill, where the tax payer is required to pay the tax by a specific date and if this payment is not made by that date, then the tax due is increased by a set percentage. Such “interest” charges then act as an incentive to pay bills on time.

Compound interest

Compound interest is slightly more complex and is computed on both the outstanding principal and any interest earned that has not been paid or withdrawn.

With compound interest the fundamental variables are:

a. rate of interest (annual rate that must be adjusted to reflect the length of the compounding period if less than a year

b. number of time periods (compounding periods less than or equal to a year)

c. future value (value at a future date of a given sum or sums invested assuming compound interest)

d. present value (value now (present time) of a future sum or sums discounted assuming compound interest

This relationship is depicted in a basic time diagram

Present value | interest | Future value | |||

0 | 1 | 2 | 3 | 4 | 5 |

Number of periods

The calculations relating to the time value of money are usually classified into two areas, single sum problems and more complex situations. We will focus on single sum problems and relevant calculations, then describe the character of more complex situations, leaving the calculation of these problems to accountants and other financial professionals.

Single-sum problems

Single sum problems are concerned either with finding the future value of a present sum at a given interest rate over a specific number of periods or with finding the present value of a future sum that is discounted with a given interest rate over a specific number of periods.

Finding the future value of a present sum

To compute the unknown future value of a present sum, we must know several variables:

1. the present sum

2. the interest rate

3. the number of compounding periods.

In practice we will have the following scenario. We have a sum of $1,000 and we want to know how much this will be worth in 3 years at 6% interest if we compound each month. We at least know that our present sum of $1,000 will be worth more than it is today. Again, let us look at the basic time diagram:

Present value | interest | Future value | |||

0 | 1 | 2 | 3 | 4 | 5 |

Number of periods

We invest the $1,000 at period zero. At the end of this period, at “1” on the time diagram, we will earn interest. Since we are compounding each month, we don’t apply the interest rate of 6% because that is the annual rate of interest. Instead, we divide the 6% by 12 to get the “monthly compounding rate” of .5%. and multiply this (.005) by 1,000 to get $5.00

So at point 1 on our time diagram our $1,000 is now worth $1,005. Assuming we don’t withdraw any money from this account and the interest rate remains at 6% we will continue to do this for 36 months. We could do this in a table, the beginning of which would look like this:

Current amount | End of Period | Earned interest | Principal plus interest |

1000 | 1 | 5 | 1005 |

1005 | 2 | 5.03 | 1010.03 |

1010.03 | 3 | 5.05 | 1015.08 |

Instead of calculating this laboriously, we use a formula. We first compute something called a future value factor. This is a factor assuming a present value of one. The only variables are the interest rate for each period and the number of periods. In the table we have only calculated the future value after three periods (one quarter of a year versus our long range plan of 3 years). The formula for the future value factor is:

FVF n,i = (1 +i)n Future value factor

Where FVF n,i is the Future Value Factor for n periods at interest rate i.

For our problem the future value factor of a dollar invested at.5% for 3 periods is:

(1+.005)3 or 1.0151

We then multiply this factor times the present value of the sum to be invested, in our case $1,000, getting $1,000 x 1.0151 or $1,015.08, just like in our table. To compute the future value then of our $1,000 after three years we would compute the Future Value Factor which would be

(1 + .005)36 or 1.1967

We then multiply this factor by our present value sum of $1,000 to get 1000 x 1.1967 = $1,196.68. So at the end of the three year period, if we do not withdraw any principal or earned interest, we will have $1,196.68, our future value of $1,000 at 6% interest for 3 years (36 months) compounded monthly.

The formula for the future value is the present value times the future value factor, or :

PV(FVF n,i)

or in the example above 1,000(1.1967) = $1,196.68

In an analogous manner we can compute the present value of a future sum. That is $1,196.68 three years in the future is worth $1,000 today, given the interest rate of 6%. Why would we want to do this? Let’s say we wanted to make a down payment on a house that we hoped to buy in 5 years. We know that with all our other financial commitments that we won’t be able to save monthly amounts. However, Uncle Gaspar, a rich uncle has left us some money. We know that we can invest this money for five years at an interest rate of 6% that will be compounded monthly. How much of our inheritance from Uncle Gaspar do we need to put in this investment vehicle in order to have $20,000 in five years?

Instead of a future value factor we use a present value factor. We know that over the 60 months (5 years) that any amount invested now will increase. Hence if we remember our time diagram and future amount discounted at a constant rate of interest will decrease each year from the future back to the present. We at least know that we don’t need to invest $20,000 now to get $20,000 in the future, but how much do we have to invest. First we compute the present value factor, given as:

PVF n,i = 1/(1+i)n Present value factor

So at an interest rate of 6% for 60 months (remember .5% per month) our present value factor would be:

1/(1 + .005)60 = 1/1.3489 or .7413

We then use the formula for the present value of a single sum to find our present value:

Present value of a single sum is PV = FV(PVF n,i)

or PV = 20,000(.7413) or $14,826.90

Present value of a single sum and future value of a single sum tables exist where you can determine other variables, such as how many periods it would take at a given interest rate to have a present value equal a desired future value.

While we are talking about single sums, I should mention the “rule of 72” applicable in future value calculations. The rule of 72 says that if we divide 72 by the interest rate we will find out the number of years it will take for a present sum to double. So at 10% interest, it will take 7.2 years to double a present value, compounding monthly. Similarly if we wish to determine what interest rate will double our present value in say, 5 years, we will divide 72 by 5, getting 14.4, i.e. an interest rate of 14.4%.