# Bond Basics

In the section on tax revenue we explained how property taxes are derived by the municipality in order to fund essential services for the community served. We also saw how we might calculate the amount of tax revenue necessary to fund a project that would only require one taxing period of revenue. As you recall, we wanted to make a $100,000 renovation to the library. Because of the estimated amount of uncollectible taxes (3% in our example), we realized that we needed 100,000/.97 in order to raise the $100,000. In this problem we were not talking about taking out a loan to do the renovation, only that we needed funds to pay for it.

Now we will consider the case where the capital project involved is much too large to be paid back or funded in one year. Instead, in order to carry out the capital project we will have to borrow funds and these borrowed funds will be used to complete the capital project in the near term. But in the long term we are faced with the necessity of both paying back the loan and also for paying interest on it. Such an arrangement is usually financed by a bond.

Bonds are used for several purposes related to libraries:

- to fund construction of new facilities;
- to fund renovation of existing facilities;
- to provide funds to continue operations when the major funding authority is unable to provide operating funds (these latter are called tax anticipation bonds)

To understand bonds you need to understand the property tax revenue structure, time value of money, and annuities (which you do). These three elements are the basic building blocks of a bond.

There are several types of bonds but we will focus on a basic type whereby a municipality issues a bond, pays interest over the life of the bond, and then repays the face amount of the bond at maturity.

# Bond basics

Bonds are created through a contract called a bond indenture or note agreement. The terms of this contract are stated in the bond indenture or note agreement. The bond indenture is a contract that specifies a promise to pay: a) a sum of money at a designated maturity date, plus b) periodic interest at a specified rate on the maturity amount (face value). It may include other aspects of the bond that are not central to this discussion. A bond covenant is an agreement usually part of the bond indenture that specifies additional requirements that the issuer of the bond must observe, such as a specific debt/equity ratio, which is a financial requirement, or non-financial requirements such as providing financial information to bondholders, protecting against the selling of assets by the issuer, or changes of control of the issuer’s assets, or making sure the assets of the company have adequate insurance. These covenants help ensure that the risk involved in lending the money to the issuer is low and that the issuer will have the means to retire the bond (repay it) at maturity.

Bonds may be sold in various ways. For example, they may be sold to an investment banker who acts as a selling agent. The investment banker might underwrite the entire issue by guaranteeing a certain sum to the bond issuer, thus taking the risk of selling the bonds at whatever price they can get (firm underwriting) or they may sell the bond issue for a commission to be deducted from the proceeds of the sale (best efforts underwriting).

There are several types of bonds:

- Secured and unsecured bonds. Secured bonds are backed by a pledge of some sort of collateral. Bonds not backed by collateral are unsecured. A debenture bond is unsecured. Most municipal bonds are unsecured, but safe because of the taxing power of the municipality.
- Term bonds are issues that mature on a single date.
- Serial bonds mature in installments (often used by school or sanitary districts, municipalities, or other local taxing bodies that borrow money through a special maturity).
- Convertible bonds are convertible into another security of a corporation after a specified time.
- Commodity backed bonds are redeemable in measures of a commodity (barrels of oil, tons of coal).
- Deep discount bonds a.k.a. zero interest debenture bonds are sold at a discount that provides the buyer’s total interest payoff at maturity.
- Registered bonds are issued in the name of the owner and require surrender of the certificate and issuance of a new certificate to complete a sale.
- A bearer or coupon bond is not recorded in name of owner and can be transferred from one owner to another.
- Revenue bonds so called because interest on them is paid from specified revenue sources, most frequently issued by governmental bodies.
- Municipal bonds will have a lower interest rate than corporate bonds because their interest is not subject to federal taxes, but do have a default risk incorporated into their interest rate.

# Bond mechanics

In our context only the government body can issue a bond. The library does not issue it, even though the funds from the bond will go towards a library project. The reason for this is that the library does not have taxing power, only the municipality or other government body does. Once it is issued, the municipality receives the money from the bond and uses it to pay for whatever project is targeted for receipt of the funds. Accounting for the bond in government accounting is usually controlled and recorded in a Debt Service Fund. By receiving money from a bond, the municipality/library now has two related but separate financial obligations. First it must pay periodic interest on the bond, typically every 6 months. Second it must repay the bond at maturity. We know from our study of tax revenue that the taxpayers will be footing the bill for both of these payments. The first, interest, is like the problem briefly described at the beginning of this section. A specific amount must be paid each six months to the bond holders for interest. Therefore by using our tax revenue calculations we can determine, based on the tax base, what that constant annual amount of tax revenue should be to meet this annual interest obligation. Let’s say that the municipality, on the library’s behalf, issued a $1,000,000 bond at 6% interest per year. The life of the bond is 20 years.

At 6% a year we know that the municipality has to pay $60,000 per year to bondholders in two installments of $30,000 each. This amount will not change for 20 years. It will be $60,000 per year for the life of the bond.

We also know that after 20 years the municipality will have to repay the $1,000,000 in full. This is an annuity problem. From our tax base we will have to collect a constant payment each year, whereby the compounded interest of each payment will eventually add up to $1,000,000. This is the future value of an ordinary annuity problem.

*Calculation of tax revenue for generating $60,000 in bond interest payments*

Assume we have a net assessed tax base, after deducting for non-taxable property and exemptions, of $50,000,000. We assume that 3% of assessed taxes will not be collectible. Therefore we will have to raise 60,000/.97 (or $61,856) in order to be assured we have sufficient funds to pay interest on the bonds. Our tax rate to raise this specific amount will be 61,856/50,000,000 or 0.00123712. So each taxable piece of property will generate tax revenue equal to the assessed value of their property times 0.00123712. This annual tax will pay for the interest on the bond. An owner of a house with an assessed value of $100,000 will pay $123.71 per year to pay this interest on the bond.

*Calculation of tax revenue for generating the future value of an ordinary annuity of $1,000,000.*

We now have to figure out how much tax revenue to collect each year in order to pay back the borrowed amount (face value of the bond) after twenty years. We should think of the amount we collect for this part of our obligation of the annuity payment (rent). We know from our study of annuities that we use the following formula to find the future value of an ordinary annuity:

(R)FVF-OA n,i or (R)(1+i)n – 1/i

Where R is the rent or payment per period and FVF-OA n,I is the future value factor of an ordinary annuity. In those previous problems we were solving for the future value. We knew the periodic payment, the number of periods and the interest rate.

Let’s assume that over twenty years we can find a guaranteed interest rate of 4%. That is, each year when we collect our special tax that will be used to pay off the bond at maturity, that the tax revenue we receive will be invested in an investment vehicle that will earn 4% per year, compounded monthly. With this assumption in hand (and this is essentially what is done when computing such an amount), let us look at our suggested calculation.

In analyzing the variables of this equation, we know the following:

- n = 20 (yearly periods) (We will use annual periods here because the rent is collected annually. For the purposes of this illustration we will also assume that the annual payment is invested in an instrument that is compounded monthly and yields 4% per year with compounding. This is just to make our calculations and the presentation more clear.)
- i = 4% per year or .04
- FV = 1,000,000 because this is the future value we need to pay off the bond.
- But we don’t know what the periodic payment will be, so we will have to solve for it. We know all the other variables.
- (R)(1+i)n – 1/i = 1,000,000
- R (1 + .04)20 -1/.04 = 1,000,000
- R (1.04)20 -1/.04 = 1,000,000
- R (2.191123143)-1/.04 = 1,000,000
- R(1.191123143/.04) = 1,000,000
- R(29.778078576) = 1,000,000
- So R = 1,000,000/29.778078576 = $33,581.75

This means that if we collect $33,581.75 per year in taxes and invest the collected amount each year in an investment vehicle that pays 4% per year after compounding, that we will have $1,000,000 at the end of twenty years.

So back to tax revenue and our assessed tax base. We calculated above that we would need to collect $61,856 from property owners annually in order to pay for the interest on the bond. An owner of a house with an assessed value of $100,000 will pay $123.71 per year to pay this interest fee for the bond.

In order to net $33,581.75 per year to pay for the bond amortization, we will again have to assume that 3% of assessed tax is uncollectible. So we will need to levy a tax of $33,581.75/.97 or $34,620.36.

Our tax rate for this amount will be 34620.36/50,000,000 or .000692. For the owner of a house with an assessed value of $100,000 will pay an additional $69.24 annually towards repayment of the principal. In total the homeowner’s tax bill will be an additional $192.95 ( $123.71 for interest plus $69.24 for principal) for the next 20 years.

As with anything in real life, bonds are more complicated than this simple example shows. In the real world a bond with a face value of $1,000,000 paying 9% interest every six months will have some additional complexities involved. For example if the prevailing interest rate for similar investments is 10%, then you might imagine that this bond investment may not be as desirable because the interest that can be earned on the bond is less than other available investment vehicles. Therefore what happens when a prevailing interest rate is higher than the stated or coupon rate on the bond is that the proceeds from the sale of the bond will be less than the face value of the bond! So instead of receiving $1,000,000 for the bond, the bond issuer will receive less. How much less? The value of the bond for investors is equal to two elements: the present value of the $1,000,000 discounted over 20 years at 10% plus the present value of the interest payments of $90,000. The first element’s computation is the present value of a single sum problem. The second element’s computation is the present value of an ordinary annuity problem. (In our discussion about annuities we only illustrated the future value of an annuity. Conceptually we learned that this was similar to the future value of a single sum. A similar analogy exists with finding the present value of a single sum and the present value of an annuity. The formula is just more complex.) We use 10% and not 9% for discounting the face value of the bond because investors can earn 10% elsewhere, but only 9% here.

We also discount by 10% the interest payments or coupon payment of the bond of $90,000 (according to what the bond actually pays, computed at 9% of face value). In computation terms we have for the principal, the present value of a single sum:

FV(PVF20,10) = ($1,000,000) 1/ (1 + .10)20

1,000,000 * .1486 or $148,643

For the interest payments we have in computation terms the present value of an ordinary annuity:

R [(1 - (1 / (1 + i)n)) / i] or $90,000(8.5136) = $766,224.

So the investor will pay the sum of these two present values:

- Present value of $1,000,000 (principal) discounted over 20 years at 10% = $148,643
- Plus Present value of interest payments ($90,000) discounted over 20 years at 10% = $766,224
- Total to be paid for the bond at sale (realizable amount) = $914,867

The difference between $1,000,000 and $914,867 of $85,133 under acceptable accrual accounting is amortized (written off) over the life of the bond issue to interest expense. The $85,133 is called a discount on the bond.

In a similar manner, if the bond would pay interest of 10% but the going rate of a comparable investment were 9% then this bond investment would be more attractive and an investor would be willing to pay more for it. So we would receive more than $1,000,000 for our bond under these conditions. The difference between the face value of the bond of $1,000,000 and the price paid (which would be more than this) is called a bond premium. Under acceptable accrual accounting this premium is also amortized over the life of the bond and CREDITED to interest expense each period. So bond interest expense is increased by amortization of a discount and decreased by amortization of a premium.

One additional complexity and that is that a fee is usually paid to the investment bank or underwriter and this expense is also borne by the institution issuing the bond.

There is no practical need for the librarian to know any more details than what is presented above. The main points to take away from this discussion are:

- When a municipality issues a bond it must consider two revenue/expense streams, the interest expense on the bond and the repayment of principal.
- The payment of the interest to bond holders is usually collected from tax revenues and paid annually so there is little need for computation of compound interest (exceptions, of course, exist).
- The repayment of the principal on the bond is treated in the same manner as an ordinary annuity (or in some cases an annuity due). Tax revenue is collected each period and invested in a vehicle that uses compound interest. At the end of the bond (or annuity period) with a predictable interest rate, the principal will be available for retiring the bond (paying back the bond holders).

Other complexities exist but understanding these main points is sufficient for our purposes.

# Annuities

You might wonder why we are talking about annuities in a library financial management course. After all, annuities are for people and not for libraries, right? Well, not entirely. We’ll look at annuities because they are important in two areas, bonds and saving internally in the library for any kind of future project. The concept of an annuity builds on our previous study of compound interest and the computation of a future value and a present value of a single sum.

At the core of an annuity is the concept that a series of dollar amounts are either invested or received periodically, such as a loan that is to be repaid in one lump sum at the end of a specified period of time.

By definition an annuity requires three elements:

1) periodic payments (rents) always be the same;

2) intervals between the payments is always the same; and

3) interest be compounded once each interval.

There are two types of annuities (i.e. two types of simple annuities) based on when the rents or payments occur. For an ordinary annuity payments occur at the end of each period. For an annuity due the rents occur at the beginning of each period.

The future value of an annuity is the sum of all the payments plus the accumulated compound interest on them. In this way it is very similar to the future value of a single sum. The only difference is that instead of one single sum being compounded over several periods to obtain the future value of this sum, the future value of an annuity results from periodic payments made during the life of the annuity and the resulting accumulated compound interest on each of these payments. The total of all the payments made plus the accumulated interest on them is the future value of the annuity.

Remember that there are two types of simple annuities, the ordinary annuity and the annuity due. They differ only by the time at which the rent is deposited. For the ordinary annuity, the rent is deposited at the end of a period and for the annuity due the rent is deposited at the beginning of the period.

So to illustrate with a diagram:

Future value of an ordinary annuity (payments made at the end of each period)

Imagine $10 invested at the end of each period (month) for 5 periods at 5% interest per year (.42% per month).

Present 1 2 3 4 5 Yr end value

|---$10||_|_| 10.1677

|------—$10_|_|_| 10.1255

|-----------$10|_| 10.0835

|----------------$10__| 10.0417

|--------------------$10 10.0000

Total Future Value of an ordinary annuity of $10 for 5 months at .42% $50.4184

interest per month (5% annual interest)

Since this is an ordinary annuity, the payments or rents are paid at the end of each period. For the first payment, paid at the end of period one, the sum earns interest for four periods, the second payment for three periods, the third for two periods and the fourth for one period. The last payment has no chance to earn interest and so is recorded at $10 since no compounding took place.

The resulting sums total $50.4184, the future value of an ordinary annuity.

While this is a visually instructive way to calculate the future value of an ordinary annuity, the calculations could get cumbersome if there are many periods involved. What we need is the same type of tool used in determining the future value of a single sum. But instead of just a future value factor that we used in future value problems using compounding, we need a future value factor of an ordinary annuity. This takes into account the periodic payments, unlike finding the future value of a single sum.

Factor tables for various periods and interest rates exist, but we can also calculate this factor by the formula:

FVF-OA n,i = (1+i)n – 1/I

where FVF-OA n,i is the future value factor of an ordinary annuity

i is the interest rate and

n is the number of periods

Once we have this factor we multiply it by the periodic payment (rent) to get the future value of an ordinary annuity, or

Future value of an ordinary annuity = R (FVF-OA n,i )

where R = the rent due

For the problem above the future value factor of an ordinary annuity is

(1+.0042)5 – 1/.0042 = 5.0422

When we multiply this by the rent we get the future value of an ordinary annuity or

10 * 5.0422 = 50.422 (the difference in thousandths of cents from the above figure attributed to rounding)

Future value of an annuity due

The future value of an annuity due differs from an ordinary annuity because the rent is deposited at the beginning of the first period. So we use same formula for future value of an ordinary annuity factor and multiply it by 1 plus the interest rate and then use this factor to multiply by periodic rent.

Again to illustrate by diagram:

Present 1 2 3 4 5 Yr end value

$10||_|_|_| 10.2188

|----$10_|_|_|_| 10.1677

|--------$10|_|_| 10.1255

|-------------$10|_| 10.0835

|-----------------—$10__| 10.0417

Total Future Value of an ordinary annuity of $10 for 5 months at .42% $50.6372

interest per month (5% annual interest)

Again we can use a formula to find the future value factor of an annuity due. It is similar to the future value factor of an ordinary annuity, but it differs because the cash flows become due one period earlier than an ordinary annuity. The future value factor of an annuity due is therefore greater than the future value factor of an ordinary annuity. The formula is:

FVF-AD n,i = ((1+i)n -1/i) * (1+i)

Above for the ordinary annuity we had the calculation for the future value factor of an ordinary annuity being

(1+.0042)5 – 1/.0042 = 5.0422

so to determine the future value factor of an annuity due we multiply this result by 1 plus the periodic interest rate of .0042 which gives us 5.0422 * 1.0422 = 5.0634

We then multiply this factor by the periodic payment (rent) to get the future value of our annuity due which would be 5.0634 * 10 = 50.6338 (again the few thousandths of a cent difference attributable to rounding)

So let’s say we have $100 a month that is burning a hole in our pocket. Rather than spend it on I-tunes or beer we decide to invest it so that we can retire early and then spend money on I-tunes and beer. If we invest $100 a month at an annual rate of 6% a year (.5% monthly) compounded monthly, how much would we have after 10 years? We know that if we just saved this in our bottom drawer we would have $12,000 ($100 times 120 months). How much would we have if we invested in an ordinary annuity? How much more would we save if we invested in an annuity due?

Future Value of an Ordinary annuity

R = 100

i = .005

n = 120

FVF-OA n,i = (1+i)n – 1/i

FVF-OA n,i = (1 + .005)120 -1/.005 = 1.8194 -1/.005 = .8194/.005 = 163.8793

R (FVF-OA n,i ) = 100 * 163.8793 =$16,387.93 (you earned $4,387.93 in interest)

If we instead used an annuity due we would have:

Future value of an annuity due

R = 100

i = .005

n = 120

FVF-AD n,i = ((1+i)n -1/i) * (1+i)

FVF-AD n,i = ((1 + .005)120 -1/.005) * (1.005) = (1.8194 -1/.005) * (1.005) = (.8194/.005) * 1.005 = 163.8793 *1.005 = 164.6987

R (FVF-AD n,i ) = 100 * 164.6987 = 16,469.87 (you earned $4,469.87 in interest, or $81.94 more in interest than the ordinary annuity.

Computation of the rent or number of periodic rents

If you wanted to compute the rent or number of period rents, this can easily be done by algebraic manipulation of the formulas.

Computation of the rent when the number of rents and interest rate and future value are known.

You know that the future value of an ordinary annuity is R (FVF-OA n,i )

Let’s say you want to accumulate $20,000 on a down payment on a house and that your Uncle Gaspar did not die, but still wants to help you out but he can only contribute a specific amount per month. How much will you ask him to contribute? You need the money in 5 years (60 periods) and you can get a 6% interest rate for an ordinary annuity.

$20,000 = R (FVF-OA n,i ) = R ((1+i)n – 1/I) = R (1 + .005)60 -1/.005 or R (1.3489)-1/.005 = .3489/.005=69.77

So by algebraic manipulation R = 20,000/69.77 = $286.65 a month.

If you saved this amount in your mattress you would only accumulate $17,199.37 in 5 years. The monthly compounding allow you accumulate $20,000 in five years with this payment.

To compute the number of rents you would use a future value of ordinary annuity table, which would give the number of periods required. This is beyond the scope of this explanation. The important thing to know is that this can be computed with relative ease.

As you might infer, we can also find the present value of both an ordinary annuity and an annuity due. That is if we have a sum and want to receive periodic rents from it until its future value is zero, we can use analogous procedures using a formula for a present value factor of an ordinary annuity or a present value factor of an annuity due. Such calculations, while interesting and potentially useful for some library applications are beyond the scope of this section.

Applications

Our discussions of the time value of money, beginning with simple and compound interest, determining the future value and the present value of a single sum, and finally here learning about the mechanisms by which annuities work, all lead up to a discussion of bonds. A basic knowledge of bonds is essential for any library manager who will ever be involved in securing funding for a capital building or renovation project. In addition, a knowledge of bonds is also important for anyone wanting to understand personal finance and investing. The explanation of bonds, with applications to library projects, will be covered in the next section on bonds. In that section, for example, we will see how to set up a debt service fund that will be funded by a special tax referendum. The proceeds from the referendum will not only pay the periodic interest expense (from the face value of the bond), but also be structured like the future value of an annuity due, where each period a tax is collected. Part of the tax revenue goes to pay the interest and part is deposited in an investment vehicle that earns a given rate of interest itself so that by the end of the term of the bond, when the bond principal is due, the funds are available to pay off the bond principal.

Annuity problems:

- Find the future value of an ordinary annuity with a periodic rent of $200 invested at an annual interest rate of 12% for 4 years.
- Find the future value of an annuity due with a periodic rent of $75 invested at an annual interest rate of 9% for 12 years.

Annuity problem solutions

- Find the future value of an ordinary annuity with a periodic rent of $200 invested at an annual interest rate of 12% for 4 years.

FVF-OA n,i = (1+i)n – 1/I

FVF-OA n, = (1 +.01)48 -1/.01 = 1.6122-1/.01 =.6122/.01 = 61.2226

Future value of ordinary annuity = R (FVF-OA n,i ) = 200(61.2226) = 12,244.52 - Find the future value of an annuity due with a periodic rent of $75 invested at an annual interest rate of 9% for 12 years.

FVF-AD n,i = ((1+i)n -1/i) * (1+i) = ((1+.0075)144 -1/.0075) * 1+ .0075) = (2.9328-1)/.0075 * 1.0075

. = 1.9328/.0075 * 1.0075 = 257.7116 * 1.0075 = 259.6444

Future value of an annuity due is R(FVF-AD n,i) = 75 * 259.6444 = 19,473.33