## Annuity

## Sell Your Annuity

Do you have annuity you dont want? Discover When is it Time to Sell Your Annuity? What can I do? Where can I get the money I need? I have an annuity, but I dont know that I can sell it. Is there a good time to sell my annuity? I already have a home improvement loan, but it was used before the roof needed replacing.

## How To Value Annuities

There are two ways to value an annuity, that is, a limited number of cash flows. The slow way is to value each cash flow separately and add up the present values. The quick way is to take advantage of the following simplification. Figure 1.10 shows the cash payments and values of three investments. Row 3. Finally, look at the investment shown in the third row of Figure 1.10. This provides a level payment of 1 a year for each of three years. In other words, it is a 3-year annuity. You can also see that, taken together, the investments in rows 2 and 3 provide Valuing an annuity 3. Three-year annuity exactly the same cash payments as the investment in row 1. Thus the value of our annuity (row 3) must be equal to the value of the row 1 perpetuity less the value of the delayed row 2 perpetuity Present value of a 3-year 1 annuity 1 - 1 The general formula for the value of an annuity that pays C dollars a year for each of ' years is Present value of t-year annuity C 1 - + ANNUITY FACTOR The...

## Future Value of an Annuity

An annuity is a series of equal payments made at fixed intervals for a specified number of periods. For example, 100 at the end of each of the next three years is a three-year annuity. The payments are given the symbol PMT, and they can occur at either the beginning or the end of each period. If the payments occur at the end of each period, as they typically do, the annuity is called an ordinary, or deferred, annuity. Payments on mortgages, car loans, and student loans are typically set up as ordinary annuities. If payments are made at the beginning of each period, the annuity is an annuity due. Rental payments for an apartment, life insurance premiums, and lottery payoffs are typically set up as annuities due. Since ordinary annuities are more common in finance, when the term annuity is used in this book, you should assume that the payments occur at the end of each period unless otherwise noted.

## Annuities Solving for Interest Rate Number of Periods or Payment

Sometimes it is useful to calculate the interest rate, payment, or number of periods for a given annuity. For example, suppose you can lease a computer from its manufacturer for 78 per month. The lease runs for 36 months, with payments due at the end of the month. As an alternative, you can buy it for 1,988.13. In either case, at the end of the 36 months the computer will be worth zero. You would like to know the interest rate the manufacturer has built into the lease if that rate is too high, you should buy the computer rather than lease it. 6This is the equation for an ordinary annuity. Calculators and spreadsheets have a slightly different equation for an annuity due.

## Annuity Present Values

To find annuity present values with a financial calculator, we need to use the PMT key (you were probably wondering what it was for). Compared to finding the present value of a single amount, there are two important differences. First, we enter the annuity cash flow using the PMT key,and,second, we don't enter anything for the future value, FV .So,for example, the problem we have been examining is a three-year, 500 annuity. If the discount rate is 10 percent, we need to do the following (after clearing out the calculator )

## Note on Annuities

So far, we have only discussed ordinary annuities. These are the most important, but there is a variation that is fairly common. Remember that with an ordinary annuity, the cash flows occur at the end of each period. When you take out a loan with monthly payments, for example, the first loan payment normally occurs one month after you get the loan. However, when you lease an apartment, the first lease payment is usually due immediately. The second payment is due at the beginning of the second month, and so on. A lease is an example of an annuity due. An annuity due is an annuity for which the cash flows occur at the beginning of each period. Almost any type of arrangement in which we have to prepay the same amount each period is an annuity due. There are several different ways to calculate the value of an annuity due. With a financial calculator, you simply switch it into due or beginning mode. It is very important to remember to switch it back when you are done Another way to...

## Valuing Level Cash Flows Annuities And Perpetuities

More generally, a series of constant or level cash flows that occur at the end of each period for some fixed number of periods is called an ordinary annuity or, more correctly, the cash flows are said to be in ordinary annuity form. Annuities appear very frequently in financial arrangements, and there are some useful shortcuts for determining their values. We consider these next.

## Present Value for Annuity Cash Flows

Suppose we were examining an asset that promised to pay 500 at the end of each of the next three years. The cash flows from this asset are in the form of a three-year, 500 annuity. If we wanted to earn 10 percent on our money, how much would we offer for this annuity Because the cash flows of an annuity are all the same, we can come up with a very useful variation on the basic present value equation. It turns out that the present value of an annuity of C dollars per period for t periods when the rate of return or interest rate is r is given by Annuity present value C X The term in parentheses on the first line is sometimes called the present value interest factor for annuities and abbreviated PVIFAfr t). The expression for the annuity present value may look a little complicated, but it isn't difficult to use. Notice that the term in square brackets on the second line, 1 (1 + r)t, is the same present value factor we've been calculating. In our example from the beginning of this...

## Future Value for Annuities

On occasion, it's also handy to know a shortcut for calculating the future value of an annuity. As you might guess, there are future value factors for annuities as well as present value factors. In general, the future value factor for an annuity is given by Annuity FV factor (Future value factor - 1) r To see how we use annuity future value factors, suppose you plan to contribute 2,000 every year to a retirement account paying 8 percent. If you retire in 30 years, how much will you have The number of years here, t, is 30, and the interest rate, r,is 8 percent, so we can calculate the annuity future value factor as Annuity FV factor (Future value factor - 1) r (1.0830 - 1) .08 (10.0627 - 1) .08 113.2832 The future value of this 30-year, 2,000 annuity is thus Annuity future value 2,000 X 113.28 226,566.40 Sometimes we need to find the unknown rate, r,in the context of an annuity future value. For example, if you had invested 100 per month in stocks over the 25-year period ended December...

## System of Four Annuity Variables

There is a tight connection between all of the inputs and output to annuity valuation. Indeed, they form a system of four annuity variables (1) Payment, (2) Discount Rate Period, (3) Number of Periods, and (4) Present Value. Given any three of these variables, find the fourth variable. FIGURE 2.3 Spreadsheet for Annuity - System of Four Annuity Variables. Annuity System of Four Annuity Variables 1. Start with the Present Value Spreadsheet, Then Insert and Delete Rows. Open the spreadsheet that you created for Annuity - Present Value and immediately save the spreadsheet under a new name using the File Save As command. Select the range A7 A17 and click on Insert Rows. Select the cell A25, click on Edit Delete, select the Entire Row radio button on the Delete dialog box, and click on OK. Select the range A26 A27, click on Edit Delete, select the Entire Row radio button on the Delete dialog box, and click on OK. 3. Payment. The formula for the Payment (Present Value) ((1 - ((1 +...

## Valuing an Annuity

The ordinary annuity cash flow analysis assumes that cash flows occur at the end of each period. However, there is another fairly common cash flow pattern in which level cash flows occur at regular intervals, but the first cash flow occurs immediately. This pattern of cash flows is called an annuity due. For example, if you win the Florida Lottery Lotto grand prize, you will receive your winnings in 20 installments (after taxes, of course). The 20 installments are paid out annually, beginning immediately. The lottery winnings are therefore an annuity due. Like the cash flows we have considered thus far, the future value of an annuity due can be determined by calculating the future value of each cash flow and summing them. And, the present value of an annuity due is determined in the same way as a present value of any stream of cash flows. Let's consider first an example of the future value of an annuity due, comparing the values of an ordinary annuity and an annuity due, each...

## Valuing a Deferred Annuity

A deferred annuity has a stream of cash flows of equal amounts at regular periods starting at some time after the end of the first period. When we calculated the present value of an annuity, we brought a series of cash flows back to the beginning of the first period or, equivalently the end of the period 0. With a deferred annuity, we determine the present value of the ordinary annuity and then discount this present value to an earlier period. To illustrate the calculation of the present value of an annuity due, suppose you deposit 20,000 per year in an account for 10 years, starting today, for a total of 10 deposits. What will be the balance in the account at the end of 10 years if the balance in the account earns 5 per year The future value of this annuity due is The first step requires determining the present value of a four-cash flow ordinary annuity of 5,000. This calculation provides the present value as of the end of the fourth year (one period prior to the first withdrawal) t...

## What is an annuity certain

Suppose you buy a house with a 100,000 loan and promise to make monthly payments of interest and principal of 733.76 for 30 years. Your monthly payments form a stream of equal payments, at equal time intervals, for 360 months. Such a stream of payments is called an annuity certain. Many readers may have parents who receive a pension, or you may receive a pension yourself. They (or you) could possibly receive a Social Security pension or a retirement pension from a former employer. These pensions are also called annuities, but they are not annuities certain because they have a life contingency feature. When the person receiving the pension dies, the pension stops or perhaps is paid to someone else at a reduced amount. These pensions are called life annuities. Later in the book, we'll study how to compute their values as well. However, an annuity certain has fixed payments at equally spaced time intervals, and the payments will be paid to somebody.

## Examples of annuities certain

You finally lease the car of your dreams. You have signed a 36-month lease, at 1,997.32 per month, on the Phantom Super Imperial Deluxe Year 2003 model. The 36-monthly payments of 1,997.32 form an annuity certain. You have lent your neighbor 100. She promises to repay you at 10 per month for 10 months. The 10 monthly payments of 10 each form an annuity certain. You buy a 1,000 bond issued by the XYZ Corporation. The bond has a 9 coupon rate and matures in 20 years. The coupon interest is paid semiannually. The 40 semiannual coupon payments of 45 form an annuity certain.

## The equation for the present value of an annuity certain

An annuity consists of a series of payments, starting at time t 1 period and ending at time t n periods. This annuity will have n payments. We assume that each of the payments is 1. The annuity will therefore have n equal payments of 1, each one period apart from the previous and from the next payment, and starting in one period. For payments unequal to 1, simply multiply the present value of the annuity by the amount of the payment. Therefore, the present value of the annuity will be (Note an is called a angle n and indicates an annuity certain with n payments.) Substituting v 1 (1 + i), we have

## A look at the tables for an annuity certain

Look at the table values for the present value of an annuity, as shown on pages 86-93. You can see that as the interest rate increases, the value of the annuity goes down. You need to put aside less money to buy the same payments because your money earns at a greater rate. As the number of periods increases, the value of the annuity increases. More money is always worth more money, even if the eventual payment is a long time off. However, the value increases at a decreasing rate, because the additional payments are due at increasingly distant times and have ever-decreasing present values.

## Annuities and How to Calculate Them

An annuity is a series of equal payments or receipts made or received at regular intervals. These payments (receipts) can be made (received) at the beginning or the end of a period. When payments are made at the end of a FIGURE 9.1 Difference between an ordinary annuity and an annuity due Ordinary annuity An annuity of four payments of 1,000 made at the end of each year. Annuity due An annuity due of four payments of 1,000 made at the beginning of the year. period, it is called an ordinary annuity. When payments are made at the beginning of a period, it is called an annuity due. Figure 9.1 illustrates the difference between the two types of annuities. As you might well infer from the time value of money, it is important to recognize the difference between an ordinary annuity and an annuity due. With an annuity due, the payments (receipts) are made (received) earlier, and therefore have a greater present value and future value than an ordinary annuity.

## Future Value of an Ordinary Annuity and Annuity Due Future Value of an Ordinary Annuity

The future value of an ordinary annuity involves payments (receipts) of an equal sum of money at the end of each period for a certain number of periods and allows the interest to accumulate over the total period of time. For example, a deposit of 1,000 made at the end of every year for five years earning 6 percent will grow to 5,637.10, as shown in Table 9.1. TABLE 9.1 Future value of ordinary annuity of 1,000 compounded at 6 for 5 years In Table 9.1 the future value of an ordinary annuity is found by compounding each deposit and summing the interest earned on the deposits. This annuity can also be expressed in equation form as follows This calculation can be cumbersome if you don't have a calculator or if the time period is long. Financial tables can simplify the calculation, as shown in Worksheet 9.1 using the same example. The future value of an ordinary annuity table is shown in Appendix C. WORKSHEET 9.1 How to determine the future value of an ordinary annuity using financial...

## Analysis and calculation of some combination annuities certain

Sometimes you may have a problem that requires you to calculate the present value of part of an annuity certain. You may be able to compute the required value by analyzing the flow of funds and splitting it up into several annuities certain or expressing it in some combination of annuities certain. Here are two examples. EXAMPLE 5.5. You are receiving an annuity of 2 for 10 years, and then an annuity of 1 for 10 additional years. Using i 6 , what is the value of your annuity Split your series of payments into an annuity of 1 for 10 years and an annuity of 1 for 20 years. Adding these two annuities together will give you your annuity. An annuity of 1 for 10 years at 6 is 7.36008705. An annuity of 1 for 20 years at 6 is 11.4699212 (both values from the table). Adding these together gives 18.83000825. This is the value of your annuity. EXAMPLE 5.6. You will receive an annuity of 1 for 10 years starting 11 years from now and ending 20 years from now. Using a 6 interest rate, what is the...

## Using a Financial Calculator to Determine the Future Value of an Annuity Ordinary and Annuity

Worksheet 9.3 outlines the steps to determine the future value of both an ordinary annuity and an annuity due using a financial calculator. The following example illustrates these steps to determine the future value of an ordinary annuity first, and then an annuity due. Example 1 What is the future value of an ordinary annuity of 1,000 to be deposited at the end of every year for five years invested at 6 percent per year Example 2 What is the future value of an annuity due of 1,000 to be deposited at the beginning of each year for five years invested at 6 percent per year WORKSHEET 9.3 How to determine the future value of an annuity (both ordinary and annuity due) using a financial calculator N Total number of discounting periods I Interest rate per period PV Present value FV Future value PMT Annuity payment or receipt semiannual, monthly) into the mode of your calculator. Step 2 Set the mode at the end of the period or beginning of the period. Step 3 Key the number of discounting...

## Future Value of an Annuity Ordinary and Annuity

Microsoft's Excel spreadsheet is used here to illustrate the ease of calculating the future value of an annuity on a computer. There are any number of other software programs that can also be used. Worksheet 9.4 illustrates the determination of the future value of both an ordinary annuity and an annuity due using the same example used in the financial calculator worksheet. Table 9.3 illustrates these two factors using the example of an ordinary annuity and an annuity due of 1,000 deposited each year for a 10-year and a 20-year period invested at 6 percent and 12 percent per annum. TABLE 9.3 Future value of an ordinary annuity and an annuity due invested at 6 and 12 for 10-year and 20-year periods Ordinary annuity of 1,000 invested for 10 years Annuity due of 1,000 invested for 10 years Ordinary annuity of 1,000 invested for 20 years Annuity due of 1,000 invested for 20 years Changing the timing of deposits from the end of the period to the beginning of the period increases the future...

## Using a Financial Calculator to Determine the Present Value of an Annuity Ordinary and Annuity

A financial calculator takes out all the number crunching in the determination of the present value of an annuity. The only difference in determining an annuity due from an ordinary annuity is the setting of the mode of payments (for an annuity due the mode is set to the beginning of the period, and for an ordinary annuity the mode is set to the end of the period). All the other keys are punched in the same way with the same information. Worksheet 9.7 outlines the steps in calculating the present value of an ordinary annuity in example 1 and the present value of an annuity due in example 2.

## Funding Jill Morans Retirement Annuity

Sunrise Industries wishes to accumulate funds to provide a retirement annuity for its vice president of research, Jill Moran. Ms. Moran, by contract, will retire at the end of exactly 12 years. Upon retirement, she is entitled to receive an annual end-of-year payment of 42,000 for exactly 20 years. If she dies prior to the end of the 20-year period, the annual payments will pass to her heirs. During the 12-year accumulation period, Sunrise wishes to fund the annuity by making equal, annual, end-of-year deposits into an account earning 9 interest. Once the 20-year distribution period begins, Sunrise plans to move the accumulated monies into an account earning a guaranteed 12 per year. At the end of the distribution period, the account balance will equal zero. Note that the first deposit will be made at the end of year 1 and that the first distribution payment will be received at the end of year 13. Draw a time line depicting all of the cash flows associated with Sunrise's view of the...

## The System of Four Variables for Constant Annuities

As we can see from the annuity formulas we derived, there are four variables involved here the present value (or equivalently, the future value), the constant annuity amount A, the number of periods n, and the discount rate r. Given any three of these we can calculate the fourth. For example, assume that I have 100,000 today (the present value) and that I invest earning 10 interest per year (the discount rate). I can calculate the maximum amount I can withdraw every year (the annuity amount) if I want to be able to make 10 withdrawals (the number of periods) starting today. We can rearrange the present value formulas to calculate the annuity amount or the number of periods when the other variables are given. If the discount rate is the unknown, however, we have to calculate the answer by trial and error (iteration). In any case, you will almost always use Excel's built-in functions to do this type of calculations. You only need to recognize that these four variables are tied to one...

## Growing Annuities in Advance

For a growing annuity in advance, as shown in Figure 7.5 we can write As before, with proper substitutions and adjustments, we can use the formulas for constant annuities in advance or Excel's built-in functions for them to do calculations for growing annuities in advance. Here, the first annuity amount is A and not A(1 + g) as was the case with growing annuities in arrears. Here are the compact formulas for growing annuities in advance

## The System of Five Variables for Growing Annuities

We discussed before that constant annuities are governed by a system of four variables. Growing annuities are governed by a system of five variables the old ones plus the growth rate g and given any four of these you can calculate the fifth. You should also recognize that a constant annuity is the same as a growing annuity with a growth rate of zero. Most of the other issues I discussed under the system of four variables for constant annuities apply to this system of variables as well.

## The Future Value or Amount of an Annuity

Suppose you make a periodic payment, of a fixed amount, into an investment fund, and the fund earns a given interest rate for a period of time. How much will it amount to at the end of the given time period Your payments form an annuity certain and the value they will amount to are referred to as the future value of the annuity, or the amount of the annuity. When you finish this chapter, you should understand what the future value of an annuity means, how to compute its value, and where it might be used in practical work. uses of the future value of an annuity Here are some examples of the future value of an annuity. 3. You have bought an annuity from an insurance company and contribute a certain amount annually to the annuity. How much will you have at a given future time the equation for the future value of an annuity From Chapter 5, we have the equation for the present value of an annuity This is the equation for the future value of an annuity of 1, after t periods, at interest...

## Present Value of an Annuity

The calculation of the present value of an annuity is the reversal of the compounding process for the future value. The idea is to discount each individual payment from the time of its expected receipt to the present. We can illustrate by building another table to calculate the present value of the annuity in Figure 3.4 ten payments of 100, made on December 31st of each year, discounted at a rate of 5 . Figure 3.5 Present Value of an Annuity Figure 3.5 Present Value of an Annuity annuity The present value of this annuity is 772.17. This means that an investor with opportunities to invest in other investments of equal risk at 5 would be willing to pay 772.17 for this annuity. Notice that because the payments are received at the end of each period, we discount them for the entire period. Formula for The formula for calculating the present value of an annuity is

## Constant Annuities in Arrears

Figure 7.2 shows the timeline for a regular constant annuity where A is the constant periodic cash flow, n is the number of cash flows, and r is the periodic discount rate. Most often the word constant is dropped and unless otherwise qualified, an annuity is assumed to be a constant annuity. In a regular annuity, the first annuity payment occurs at time 1, that is, in one period from now or at the end of the first period. Such an annuity is called a regular annuity or an annuity in arrears. We can calculate the present value PVC0 and future value FVCn of an annuity in arrears as

## Annuities An odd investment

Annuities are peculiar investment products. They're contracts that are backed by an insurance company. If you, the annuity holder (investor), die during the so-called accumulation phase (that is, prior to receiving payments from the annuity), your designated beneficiary is guaranteed to receive the amount of your contribution. In this sense, annuities look a bit like life insurance. Annuities, like IRAs, allow your capital to grow and compound without taxation. You defer taxes until withdrawal. Annuities carry the same penalties for withdrawal prior to age 591 2 as do other retirement accounts. Unlike an IRA, which has an annual contribution limit, you can deposit as much as you want into an annuity in any year even a million dollars or more if you have it As with a so-called nondeductible IRA, you get no upfront tax deduction for your contributions. Although annuities are retirement vehicles, as noted earlier in this chapter, they have no place inside retirement accounts. Annuities...

## Present Value of an Annuity Ordinary and Annuity

Worksheet 9.8 illustrates the use of Microsoft Excel's spreadsheet software to determine the present value of both an ordinary annuity and an annuity due using the same examples as in the previous section. As seen from the totals in Worksheet 9.8, the present value of an annuity due is always greater than the present value of an ordinary annuity because the payments are received sooner and the first payment is not discounted. The following conclusions can be drawn for the present value of an annuity WORKSHEET 9.5 How to determine the present value of an ordinary annuity using financial tables PMT Annuity or equal payments receipts PVIFA Present value interest factor of an annuity i Interest discount rate per period n Number of periods Step 3 Multiply the annuity by the interest factor to obain the present value. The greater the rate of interest used to discount the annuity payments, the smaller the present value of the annuity. The sooner the annuity payments are received, the greater...

## Growing Annuities in Arrears

For the growing annuity in arrears shown in Figure 7.4 where the first cash flow of A(1 + g) at time 1 grows at the rate of g per period, we can calculate the present value as This looks a little more complex than the corresponding constant annuity formula, but notice that if we substitute (1 + k) for (1 + r) (1 + g) then the formula for PVG0 becomes the same as the formula for PVC0 with k substituted for r. So FIGURE 7.4 A timeline for a regular growing annuity. we can use Excel's built-in formulas for a constant annuity in arrears to do calculations for a growing annuity in arrears by substituting a (growth) adjusted discount rate k for the regular discount rate r. It is important to note that for the equivalence to work, the first cash flow for the growing annuity in arrears has to be A(1 + g) and not A. If in a problem the first cash flow for a growing annuity in arrears is given as 1,000 and the growth rate is 5 , and you want to use the constant annuity formula or Excel's...

## Combining Single Amounts and Annuities

In some cases an investment involves both a single amount and an annuity. For example, suppose you could purchase an investment that offered to pay 100 at the end of each year for 10 years and 1,000 at the end of the 10-year period. If you expect the investment to earn 8 interest, how much would you pay for the investment at the beginning of the 10-year period To answer this question, compute the present value of the annuity and add the present value of the single amount. Any investment problem can be thought of as a single amount, a series of single amounts, an annuity, or a combination of these arrangements.

## These factors come from a Present Value of an Annuity table

By referring to a Present Value of An Annuity table, the corresponding discount rate can be found. In this example the discount rate represented by the cash flows in the example is about 21.8 percent. This is consistent with the NPV calculation above, where NPV at 16 percent equaled a positive 15,559. Obviously, this means that the IRR would be substantially in excess of 16 percent. Determining IRR with uneven cash flows is more complicated than with level cash flows. It involves a trial-and-error process and since the cash flows are not the same every year, the present value must be calculated on a year by year basis (rather than on an annuity basis). The first step is to determine a discount rate that may be close to the actual IRR and use this discount rate to calculate the PV of the cash flows for the project. If the PV of the cash inflows exceeds the PV of the investment, the discount rate selected was too low while if the discounted cash flow is negative, the discount rate...

## Present Value of an Ordinary Annuity

As with the future value of an annuity, the timing of the payments will have an impact on the amount of the present value. For an ordinary annuity, payments (deposits) are made at the end of a period, and for an annuity due, payments (deposits) are made at the beginning of a period. As an example, the present value of an ordinary annuity of 1,000 discounted at 6 percent for five years is shown in Table 9.4. TABLE 9.4 Present value of an ordinary annuity of 1,000 discounted at 6 for 5 years Financial tables can simplify the calculation of the present value of an ordinary annuity. Worksheet 9.5 illustrates the use of financial tables in determining the present value of an ordinary annuity. The present value of an annuity table can be found in Appendix D.

## Discrete Time Markov And Semimarkov Reward Processes And Generalised Annuities

In this section, after a short introduction of DTMRWP and their natural relation with annuities, we will show how it is possible to give a further generalisation to the annuity concept by means of SMRWP. 2.1 Annuities And Markov Reward Processes The annuity concept is very simple and can be easier understood by means of Figure 2.1 for the immediate case, Figure 2.1 constant rate annuity-immediate. Figure 2.2 constant rate annuity-due. Clearly the periodic payments can be variable. The simple problem to face is to compute the value at time 0 (present value) or at time n (capitalisation value) of the annuity. The present value formulas are presented at the beginning of Chapter 4. There exist similar results for capitalisation values (see Kellison (1991). Instead of assuming as usual that S is deterministic, we will now consider S as r.v. with a set of possible values The figure clearly shows that MRWP can be considered a natural generalisation of the annuity...

## Tax Deferred Annuities

There are investment vehicles offered by insurance companies that give an incredible advantage over traditional savings vehicles such as bank savings accounts and certificates of deposits. The life insurance companies in this country offer programs called fixed-rate, tax-deferred annuities. They typically offer slightly higher rates than banks and, best of all, these annuity accounts accumulate interest on a tax-deferred basis. This means that your account earns interest without causing your marginal tax bracket to go up. Money that accumulates free of current income tax will grow much quicker than funds that are taxable each year. These accounts typically accumulate free of current state income tax as well. Many retirees have also figured out that while the interest in these accounts is in the accumulation phase, no 1099 statements are generated. Therefore, the interest from these accounts while accumulating is not taken into account for taxation of your Social Security benefits

## Why annuities certain are important

Paying off loans in installments over the life of the loan is good financial policy, both for the borrower and for the lender. The annuity certain, as a means of repaying the loan, is frequently an easy and convenient way to do this. Annuities certain are also easy to evaluate because the payments are the same size, they are made at regular time intervals, and the payments are all evaluated using the same interest rate. We'll get to evaluations using different interest rates later in the book. Annuities certain occur very often in finance. Bond coupon payments, loan payments, annuity payments, and many other contractual payments all often form annuities certain. You should make sure you understand how annuities certain work and how to compute the mathematical present value of an annuity certain.

## Index Annuities and Variable Annuities

Imagine being able to obtain stock market rates of return without worrying about current income taxes being created. All tax-deferred annuities offer the ability to defer the taxes on your account until some future point in time. While living, you usually control the timing of when you want to pay taxes. This can give you an extreme advantage over more traditional investments such as mutual funds. Individual stocks operate much the same way in that your growth is tax-deferred until you sell the stock. The problem with individual stocks, however, is that it is much harder to obtain true diversification than with a variable annuity. Variable annuities typically offer very diversified subaccounts, which allow you to have adequate diversification. Also, variable annuities will allow you to switch from one type of subaccount to another without any current capital gains or income tax. For example, if you have invested with a variable annuity and are currently invested in the overseas...

## Variable Annuity Death Benefits

Other insurance benefits offered by various variable annuity companies today are death benefits. These typically come in play if the account owner or annuitant passes away before a certain age. In these cases, the insurance company will usually pay out the account holder's original deposit amount or current account value, whichever is greater. Keep in mind that mutual funds and variable annuity subaccounts typically are not protected against normal market volatility while the account holder is living. Therefore, a client can feel more secure investing in a variable annuity knowing that perhaps the spouse or family would be taken care of in the event the market value of the annuity was down drastically at the time of his or her demise. Critics of variable annuities state that these types of death benefit guarantees are nothing more than expensive gimmicks. In other words, these critics are not against the insurance benefits, they are just against the insurance company charging too much...

## Variable Annuity Performance

Variable annuities have subaccounts that are very similar to mutual fund accounts. They both are usually actively managed accounts that offer diversification. Critics point out that variable annuities are more expensive to own because in addition to normal management fees imposed by the fund manager of the subaccount, you have the added insurance expenses. They point out that mutual funds don't have these added expenses and, therefore, are a better choice. What most critics fail to take in account is that the actual performance of variable annuities and mutual funds are, in fact, different Mutual fund managers typically carry a higher cash balance within their portfolios due to frequent redemptions. How the portfolios are usually allocated creates a difference in the performance numbers. The average returns on variable annuity subaccounts in the categories of growth, growth and income, and international funds are equivalent, if not greater, than the average mutual fund's performance,...

## Comparison of a Tax Efficient Mutual Fund and a Tax Deferred Variable Annuity Account

Assuming you could find a tax-efficient mutual fund that could manage to obtain the most favorable long-term capital gains rate available of 18 percent (which became available starting December 31, 2000, for capital gains assets purchased from that date and beyond and held five years), the tax-deferred variable annuity could still be the better investment after taxes The following illustrations in Figures 7.12 and 7.13 assumes the same rate of return for each account.

## Annuities and Perpetuities

There are several special cases of the present value formula, equation (9.5), described earlier. When C1 C2 . . . CT C, the stream of payments is known as a standard annuity. If T is infinite, it is a standard perpetuity. Standard annuities and perpetuities have payments that begin at date 1. The present values of standard annuities and perpetuities lend themselves to particularly simple equations. Less standard perpetuities and annuities can have any payment frequency, although it must be regular (for examples, every half period). Many financial securities have patterns to their cash flows that resemble annuities and perpetuities. For example, standard fixed-rate residential mortgages are annuities straight-coupon bonds are the sum of an annuity and a zero-coupon bond (see Chapter 2). The dividend discount models used to value equity are based on a growing perpetuity formula.

## What About Annuities

Annuities due, in case you've never heard of them, are simply annuities where the payments happen at the beginning of each period instead of the end, as they do with ordinary annuities. There are adjusted versions of PVA and FVA for these types of annuities, as well as new rules concerning where those adjusted formulas will give you values on the timeline. However, they're probably not worth learning you can usually handle almost all problems involving annuities due using the ordinary annuity formulas and a little creativity. For example, consider the annuity shown in Figure 5-10 is this an annuity Yes, and the question of whether it's an ordinary annuity or an annuity due only effects where we want to find the value of this annuity. Or, consider another example suppose we knew that the present value of this five-payment annuity due was 625.48 when the interest rate is 10 percent how would we go about solving for the payment amount Well, once again we'd manipulate what we had to make...

## Annuities

Consider a series of T payments, each of amount 1 at times 1 , ,T. Such a stream of payments is called annuity immediate (with payments at the end of the period). The present value of this cash flow for i 0 is and often it is also called the Present Value Interest Factor of an Annuity abbreviated as PVIFAi-T For i 0 we have T. Sometimes the interest rate is attached to symbol a a ii or o Y .i - Consider again a series of T payments, cach of amount 1 but now at times 0, ,T 1. Such a stream of payments is called annuity due (with payments at the beginning of the period). The present value of this cash flow for i 0 is For an infinite stream of constant payments of amount 1, the annuity is called perpetuity and if it is immediate or due, its present value is

## Ordinary Annuities

An ordinary, or deferred, annuity consists of a series of equal payments made at the end of each period. If you deposit 100 at the end of each year for three years in a savings account that pays 5 percent interest per year, how much will you have at the end of three years To answer this question, we must find the future value of the annuity, FVAn. Each payment is compounded out to the end of Period n, and the sum of the compounded payments is the future value of the annuity, FVAn. The first line of Equation 2-4 represents the application of Equation 2-1 to each individual payment of the annuity. In other words, each term is the compounded amount of a single payment, with the superscript in each term indicating the number of periods during which the payment earns interest. For example, because the first annuity payment was made at the end of Period 1, interest would be earned in Periods 2 through n only, so compounding would be for n 1 periods rather than n periods. Compounding for the...

## Growing Annuities

Normally, an annuity is defined as a series of constant payments to be received over a specified number of periods. However, the term growing annuity is used to describe a series of payments that is growing at a constant rate for a specified number of periods. The most common application of growing annuities is in the area of financial planning, where someone wants to maintain a constant real, or inflation-adjusted, income over some specified number of years. For example, suppose a 65-year-old person is contemplating retirement, expects to live for another 20 years, Differentiate between a regular and a growing annuity. What three methods can be used to deal with growing annuities

## Annuity

An annuity is a level stream of regular payments that lasts for a fixed number of periods. Not surprisingly, annuities are among the most common kinds of financial instruments. The pensions that people receive when they retire are often in the form of an annuity. Leases and mortgages are also often annuities. To figure out the present value of an annuity we need to evaluate the following equation Annuity The present value of having cash flows for T years is the present value of a consol with its first payment at date 1 minus the present value of a consol with its first payment at date T + 1. Thus, the present value of an annuity is formula (4.11) minus formula (4.12). This can be written as Formula for Present Value of Annuity

## The Annuity Formula

An Annuity pays the The second type of cash flow stream that lends itself to a quick formula is an annuity, which is a stream of cash flows for a given number of periods. Unlike a perpetuity, payments stop after T periods. For example, if the interest rate is 10 per period, what is the value of an annuity that pays 5 per period for 3 periods What is the shortcut to compute the net present value of an annuity It is the annuity formula, which is Is this really a short-cut Maybe not for 3 periods, but try a 360-period annuity, and let me know which method you prefer. Either works.

## Annuities The Census

A census was an obligation to pay an annual return from fruitful property. (157) These contracts somewhat resembled the modern annuity, by which term the census is usually rendered, and also resembled the modern mortgage. The buyer of a census often paid cash the seller was a debtor and must make annual payments. This form of contract developed in early feudal times. At first, the returns were paid in real fruits, and hence they were considered as sales of future goods. They became the sale of a right to future money, sometimes in variable, and more often in fixed, amounts. A life census ran for the life of buyer or seller and resembled a life annuity. A perpetual census ran forever. A temporary census ran for a fixed number of years and resembled a mortgage. A census might be nonredeemable or redeemable at the option of the buyer, or of the seller, or of both. Some of these forms were the equivalent of personal loans and were challenged as

## The annuity due

Occasionally you may come across the term annuity due. This means an annuity with the first payment due right away, rather than one period in the future as is the case with the annuity certain. Clearly, an annuity due with n + 1 payments at a given interest rate is worth the value of an annuity (at the same rate) with n payments, plus 1, because of the payment of 1 made right away. The annuity due symbol is sometimes a with two dots over it (a) and called a double-dot. We won't use the annuity due concept in this book, but you should know what it is.

## Constant Annuities

A constant annuity is a series of equal cash flows that occur at regular intervals. Learning to do calculations with constant annuities efficiently is important because they occur in many practical situations. In dealing with annuities, we almost always assume that the applicable discount rate is the same for the entire period of the annuity. This happens to be the case in many practical situations and makes it possible to derive compact formulas to do various types of annuity calculations. Keep in mind, however, that this is an assumption, and if it does not apply to a problem you are working on, then you should not use the compact formulas in your problem. Excel has several built-in functions to do various types of annuity calculations, which use the same formulas we will discuss. So you will rarely have to write your own formulas to do annuity calculations. Nonetheless, it is always important to understand how a built-in function does its calculations because that will reduce your...

## Annuity method

With the annuity method, the asset, or rather the amount of capital representing the asset, is regarded as being capable of earning a fixed rate of interest. The sacrifice incurred in using the asset within the business is therefore two-fold the loss arising from the exhaustion of the service potential of the asset and the interest forgone by using the funds invested in the business to purchase the fixed asset. With the help of annuity tables, a calculation shows what equal amounts of depreciation, written off over the estimated life of the asset, will reduce the book value to nil, after debiting interest to the asset account on the diminishing amount of funds that are assumed to be invested in the business at that time, as represented by the value of the asset. Figure 15.4 Annuity method Figure 15.4 Annuity method An extract from the annuity tables to obtain the annual equivalent factor for year 5 and assuming a rate of interest of 10 would show The annuity method, with its...

## Annuity Functions

Excel provides several functions to do different kinds of calculations for constant annuities, which you can use for both loans and investments. Constant annuities are equal cash flows (either all inflows or all outflows) that take place at equal time intervals. Constant annuities are governed by a set of 4 variables if you specify 3, then the fourth can be calculated. The Excel functions for calculating these variables are PV, FV, PMT, RATE, and NPER. These functions actually go one step beyond handling just constant annuities in that in addition to the annuity payment you can also specify a cash flow pv at time 0 and another cash flow fv at the end of the annuity period, that is, nper periods from time 0. I have also included a few other functions here that Excel provides for annuity-related calculations. However, you will use the five functions I mentioned before much more frequently than these others. In the annuity functions, the commonly used arguments have the following...

## Annuities due

Annuity due Level stream of cash flows starting immediately. The perpetuity and annuity formulas assume that the first payment occurs at the end of the period. They tell you the value of a stream of cash payments starting one period hence. However, streams of cash payments often start immediately. For example, Kangaroo Autos in Example 3.8 might have required three annual payments of 4,000 starting immediately. A level stream of payments starting immediately is known as an annuity due. If Kangaroo's loan were paid as an annuity due, you could think of the three payments as equivalent to an immediate payment of 4,000 plus an ordinary annuity of 4,000 for the remaining 2 years. This is made clear in Figure 3.10, which compares the cash-flow stream of the Kangaroo Autos loan treating the three payments as an annuity (panel a) and as an annuity due (panel b). In general, the present value of an annuity due of t payments of 1 a year is the same as 1 plus the present value of an ordinary...

## Fixed Annuities

This gives the fixed annuity distinct advantages over a regular taxable account. The only downside is that you will have to pay taxes later down the road. Taxes are due when you pull your interest out and, in most cases, upon your demise. If married, most insurance companies will allow the spouse to continue the annuity certificate without tax consequences if the surviving spouse is named as primary beneficiary. Fixed annuities offer an excellent bond substitute. Purchasing individual bonds requires knowledge and effort. Investing in bond mutual funds can expose you to market volatility. These fixed annuities actually are a proxy for bonds, because an insurance company typically will have the majority of the funds backing your fixed annuity account invested in a portfolio of bonds. A competitive annuity company will offer high yields with security of principal. Many will offer guaranteed minimum interest rates while providing high current yields to the consumer. Look for a company...

## Growing Annuity

Cash flows in business are very likely to grow over time, due either to real growth or to inflation. The growing perpetuity, which assumes an infinite number of cash flows, provides one formula to handle this growth. We now consider a growing annuity, which is a finite number of growing cash flows. Because perpetuities of any kind are rare, a formula for a growing annuity would be useful indeed. The formula is13 Formula for Present Value of Growing Annuity r - g r - g V1 + rj _ where, as before, C is the payment to occur at the end of the first period, r is the interest rate, g is the rate of growth per period, expressed as a percentage, and T is the number of periods for the annuity.

## Swiss Annuity

The Swiss annuity offers many benefits, and it is a great way to create retirement income and receive tax-free withdrawals at any time. Unlike the life insurance policy, designed to cover you in the event you should die early, the annuity anticipates that you might live longer and that you will need to preserve your assets and have income for the duration. The annuity can either be single or joint. The word annuity actually means annual payment. Annuities can be for an individual (single) or for a couple (joint). There are different types of annuities. The immediate annuity starts as soon as it has been created or up to anytime in the first year, and begins paying you a guaranteed income on a schedule to meet your needs. This time frame could be over your lifetime or for a specific number of years. It also may be based on someone else 's lifetime. Based on the amount you purchase, the insurance company will compute the amount you will receive over that period. The annuity is an...

## Variable Annuities

Policyholders of annuities will select these vehicles in case they need a secure source of income in their future. You are under no obligation to annuitize your annuity with an insurance company. This is one of the biggest misconceptions circulating today. Many investors only half read their annuity policies or, worse, they listen to others who are confusing a tax-deferred annuity with another annuity product called an immediate annuity. A tax-deferred annuity gives you the option to annuitize, not the obligation. Most tax-deferred annuities will never be annuitized. Typically, an insurance company states that if you still have your funds with the company at age 90 and up, it will want you to annuitize. If you let the insurance company know in writing that you don't want to, it will usually give you another five years to make a decision. If an insurance company puts pressure on you to annuitize, you always have the option of cashing in your account or moving your annuity account to...

## The Plan Of This Book

This history attempts to report the rates on loans of all maturities, from very short-term personal or trade loans and government bills to perpetual annuities with no maturity date at all. For modern times, maturity is accurately defined, and no reporting problem arises except for the ambiguities inherent in bond averages or optional redemption features. Later chapters present tables showing gradual and unbroken series of interest rates, from the shortest to the longest maturity. maturity categories, long and short, because the bulk of the data is not more precise. Long-term loans will include the perpetual debt of Italian cities, the French rentes, the perpetual annuities issued by many European towns, and other loans that were clearly intended to run for many years. Short-term loans will include bills of exchange, bank deposits, pawnshop and other collateral loans, and the floating debt of princes. Much of this short-term debt, in fact, probably ran for years, but the form...

## What to Use for Your Fixed Investments

When investment professionals talk of fixed investments, they generally are talking about bonds, notes, and bills. These are known as debt instruments. These fixed investments generally denote that you are not an equity owner of any issuer but, rather, lending these issuers funds. You are also in a lending position when you place your money into a money market fund or certificate of deposit. An annuity cer-

## More Time Value of Money

Quick Formulas for Perpetuities and Annuities The answer to the second question will be that there are indeed valuation formulas for projects that have peculiar cash flow patterns perpetuities and annuities. Their values are easy to compute when interest rates are constant. This often makes them useful quick-and-dirty tools for approximations. They are not only in wide use, but also are necessary to compute cash flows for common bonds (like mortgages) and to help you understand the economics of corporate growth.

## Formulation Of The Basic Model

We begin by defining the tangible value (TV) to be the economic book value associated with the earnings stream that can be derived from the current business without the addition of further capital. A valuable simplification is obtained by using a fixed annual payment E to create a figurative annuity that has the same present value (PV) as the literal earnings stream. If k is the appropriated risk-adjusted discount rate (usually taken as equivalent to the COC the cost of equity capital), then ied by an appropriately chosen fixed return R per dollar invested that is received year after year in perpetuity. The project's net return pattern can thus be rendered equivalent (in PV terms) to a level annuity based on the dollar amount invested at a fixed spread (R - k) over the COC k. Piling these heroic assumptions one upon another, all of a firm's franchise projects can be compressed into an average annual spread per dollar of funding that reaches from the current time into perpetuity....

## Application Fixed Rate Mortgage Payments

Mortgages are annuities, so the annuity formula is quite useful. Most mortgages are fixed rate mortgage loans, and they are basically annuities. They promise a specified stream of equal cash payments each month to a lender. A 30-year mortgage with monthly payments is really a 360-payments annuity. (The annu-ity formula should really be called a month-ity formula in this case.) What would be your monthly payment if you took out a 30-year mortgage loan for 500,000 at an interest rate of 7.5 per annum A 30-year mortgage is an annuity with 360 equal payments with a discount rate of 0.625 per month. Its PV of 500,000 is the amount that you are borrowing. You want to determine the fixed monthly cash flow that gives the annuity this value The mortgage payment can be determined by solving the Annuity formula.

## Application Projects With Different Lives and Rental Equivalents

Comparing contracts Annuities have many applications, not only in a fixed income context, but also in the corporate with unequal lives. realm. For example, let me offer you a choice between one of the following two contracts, e.g., for a building or equipment Rental Value Many capital budgeting problems consist of such an upfront payment plus regular fixed pay-Equivalents. ments thereafter. It is often useful to think of these contracts in terms of a rental equivalent cost. This is sometimes called EAC for equivalent annual cost in this context, but it is really just another name for the C in the annuity formula. It is the dollar figure that answers What kind of annuity payment would be equal to the contract's present value Turn the annuity for

## Summary of Special Cash Flow Stream Formulas

The growing annuity I am not a fan of memorization, but you must remember the growing perpetuity formula. (It would likely be useful if you could also remember the annuity formula.) These formulas are used in many different contexts. There is also a growing annuity formula, which nobody remembers, but which you should know to look up if you need it. It is A summary Figure 3.1 summarizes the four special cash flow formulas. The present value of a growing perpetuity must decline (r g), but if g 0 declines at a rate that is slower than that of the simple perpetuity. The annuity stops after a fixed number of periods, here T 7, which truncates both the cash flow stream and its present values. Simple Annuity (T 7) Growing Annuity (T 7)

## Bond Prices Yields to Maturity and Bond Market Conventions

The annuity on the right side of Exhibit 2.13 has a yield-to-maturity, r, that if P is the price of the annuity. If both these bonds have prices equal to 100,000, their yields-to-maturity are both 10 percent per annum.26 For straight-coupon, deferred-coupon, zero-coupon, perpetuity, and annuity bonds, the bond price is a downward sloping convex function of the bond's yield to maturity.

## Business partner or economic web capital

Skandia's Assurance and Financial Services (AFS) division defines itself as a virtual corporation. Its focus is not on traditional life insurance, but on a combined product consisting of savings and insurance such as variable annuities, where a part of the payment is invested in mutual funds, the value of which forms the ultimate policy value. AFS puts its money and brainpower into developing insurance products, operating an internal and external network, and opening global markets. But AFS neither manages mutual funds nor deals directly with the public. That is done by partners. Downstream there are local sellers - banks, brokerages and financial advisers - who want to sell insurance and value AFS's product-development expertise. Upstream are well-known fund managers such as J.P. Morgan, and others, who value AFS

## Using Present Value Formulas To Value Bonds

You may have noticed a shortcut way to value the Treasury bond. The bond is like a package of two investments The first investment consists of five annual coupon payments of 70 each, and the second investment is the payment of the 1,000 face value at maturity. Therefore, you can use the annuity formula to value the coupon payments and add on the present value of the final payment PV(bond) PV(coupon payments) + PV(final payment) (coupon X five-year annuity factor) + (final payment X discount factor) Any Treasury bond can be valued as a package of an annuity (the coupon payments) and a single payment (the repayment of the face value).

## How To Build Your Own Spreadsheet Model

The Excel PV function can be used to calculate the present value of a single cash flow, the present value of an annuity, or the present value of a bond. For a single cash flow, the format is -PV(Discount Rate Period, Number of Periods, 0, Single Cash Flow). Enter -PV(B5,B6,0,B4) in cell B18.

## Ppendix A Calculating and Recording Goodwill

There are several methods that use the expected annual earnings in excess of normal to estimate goodwill. A common approach is to pay for a given number of years' excess earnings. For instance, Acquisitions Inc. might offer to pay for four years of excess earnings, which would total 22,000. Alternatively, the excess earnings could be viewed as an annuity. The most optimistic purchaser might expect the excess earnings to continue forever. If so, the buyer might capitalize the excess earnings as a perpetuity at the normal industry rate of return according to the following formula Goodwill Discounted present value of a 5,500-per-year annuity for 10 years at 10 5,500 X 10-year, 10 present value of annuity factor 5,500 X 6.145 33,798

## Twelfth Century Interest Rates

Annuities and Mortgages (Long Term) * As explained more fully elsewhere, the subtitles short term and long term in these summaries are in part arbitrary. Thus, princely and state loans not specifically designated as annuities were usually floating debt and so are tabulated-as short term, even though in fact I hey may have been outstanding for a long period. Similarly, mortgages and annuities are assumed to be long term, although some may have had near-term maturity dates. It is probable that in the Netherlands, as early as 1200, rich merchants bought land rentes and house rentes, that is, loaned money at long term against real security, at 8-10 per annum. Such long-term annuities or mortgages, the census described in an earlier chapter, were generally at rates lower than the rates on short-term loans either princely, personal or commercial. (219)

## Thirteenth Century Interest Rates

In the Netherlands long-term real estate loans (the census) were reported at 8-10 in the thirteenth and early fourteenth centuries. At the end of the thirteenth century, however, the rentiers of Arras, who had accumulated fortunes in the cloth trade with Genoa and at the Champagne fairs, invested in land and loans at rates as high as 14 . (237) No certificates were given to the owners of these prestiti, which could be called Venetian government bonds, but their claims were recorded in the Loan Office. These claims could be sold and transferred to others. Thus, although the nominal rate of 5 did not represent a going or acceptable rate of interest, this rate, divided by the price paid freely in the open market and modified by other advantages or disadvantages of ownership, should reflect the value of money on a long-term annuity basis. The purchase of prestiti in the open market became a popular form of voluntary investment by Venetian nobles. They were used...

## Example 315 How Inflation Might Affect Bill Gates

We showed earlier (Example 3.11) that at an interest rate of 9 percent Bill Gates could, if he wished, turn his 96 billion wealth into a 40-year annuity of 8.9 billion per year of luxury and excitement (L&E). Unfortunately L&E expenses inflate just like gasoline and groceries. Thus Mr. Gates would find the purchasing power of that 8.9 billion steadily declining. If he wants the same luxuries in 2040 as in 2000, he'll have to spend less in 2000, and then increase expenditures in line with inflation. How much should he spend in 2000 Assume the long-run inflation rate is 5 percent. Mr. Gates needs to calculate a 40-year real annuity. The real interest rate is a little less than 4 percent so the real rate is 3.8 percent. The 40-year annuity factor at 3.8 percent is 20.4. Therefore, annual spending (in 2000 dollars) should be chosen so that

## Example 8 Back to Kangaroo Autos

Let us return to Kangaroo Autos for (almost) the last time. Most installment plans call for level streams of payments. So let us suppose that this time Kangaroo offers an easy payment scheme of 4,000 a year at the end of each of the next 3 years. First let's do the calculations the slow way, to show that if the interest rate is 10 , the present value of the three payments is 9,947.41. The time line in Figure 1.11 shows these calculations. The present value of each cash flow is calculated and then the three present values are summed. The annuity formula, however, is much quicker

## Fourteenth Century Interest Rates

Rates of 8-10 on real estate loans are cited in this century for the Netherlands this range was unchanged from the thirteenth century. At this time the city of Bruges did most of its financing by selling life annuities and rentes to private individuals the rates it paid are not stated, but are said to have been far below commercial rates.

## Fifteenth Century Interest Rates

Census loans, sometimes called rentes, on land in Lombardy were now usually at 6 , although it was not unusual to find lower and higher rates. In Southern France the prevailing rate for such loans was 10 . (288) By 1500 a census loan on land in Italy and Germany was usually at 5 . (289) In 1489, when King Ferdinand and Queen Isabella of Spain were in pressing need to finance their war against the Moors, they invited cities and individuals to make loans to them and granted 10 annuities for the sums so obtained. (291) Castile had long financed itself by such perpetual annuities. These annuities became such a burden that in her last will Isabella advised her successors never to sell perpetual annuities she ordered that all free revenues of Granada should be applied to repayment of these loans. Her injunctions were not obeyed, and in subsequent centuries the record of the Spanish Crown became a dreary succession of defaults in spite of the wealth of Spanish...

## Compounding Intervals and Bond Prices

The third is an annuity that pays C dollars a year for t years. To find its present value we take the difference between the values of two perpetuities Use the annuity factors shown in Appendix Table 3 to calculate the PV of 100 in each of c. Given these one- and two-year discount factors, calculate the two-year annuity factor. d. If the PV of 10 a year for three years is 24.49, what is the three-year annuity factor Siegfried Basset is 65 years of age and has a life expectancy of 12 more years. He wishes to invest 20,000 in an annuity that will make a level payment at the end of each year until his death. If the interest rate is 8 percent, what income can Mr. Basset expect to receive each year Use a spreadsheet program to construct your own set of annuity tables. Derive the formula for a growing (or declining) annuity.

## Do Worry about these but not too much

V Volatility of your investment balance. When you invest in mutual funds that hold stocks and or bonds, the value of your funds fluctuates with the general fluctuations in those securities markets. These fluctuations don't happen if you invest in a bank certificate of deposit (CD) or a fixed insurance annuity that pays a set rate of interest yearly. With CDs or annuities, you get a statement every so often that shows steady but slow growth in your account value. You never get any great news, but you never get any bad news either (unless your insurer or bank fails, which could happen).

## Bond Prices and Yields

Did you notice that the coupon payments on the bond are an annuity In other words, the holder of our 6 percent Treasury bond receives a level stream of coupon payments of 60 a year for each of 3 years. At maturity the bondholder gets an additional payment of 1,000. Therefore, you can use the annuity formula to value the coupon payments and then add on the present value of the final payment of face value (coupon X annuity factor) + (face value X discount factor)

## Fv5 for individual cash flows received attotal

TIP Convert 1,000 every 2 years into an equivalent annual annuity (i.e., an annuity that would provide an equivalent present or future value to the actual cash flows) pattern. Solving for a 2-year annuity that is equivalent to a future 1,000 to be received at the end of year 2, we get Therefore, R 1,000 2.100 476.19. Replacing every 1,000 with an equivalent two-year annuity gives us 476.19 for 20 years.

## Solving for i with Uneven Cash Flow Streams

It is relatively easy to solve for i numerically when the cash flows are lump sums or annuities. However, it is extremely difficult to solve for i if the cash flows are uneven, because then you would have to go through many tedious trial-and-error calculations. With a spreadsheet program or a financial calculator, though, it is easy to find the value of i. Simply input the CF values into the cash flow register and then press the IRR key. IRR stands for internal rate of return, which is the percentage return on an investment. We will defer further discussion of this calculation for now, but we will take it up later, in our discussion of capital budgeting methods in Chapter 7.8

## Analysis of Fixed Income Securities 731 The NPV Approach

Figure 7-1 shows the time profile of the cash flows Ct for three bonds with initial market value of 100, 10 year maturity and 6 annual interest. The figure describes a straight coupon-paying bond, an annuity, and a zero-coupon bond. As long as the cash flows are predetermined, the valuation is straightforward.

## Example 15 How Inflation Might Affect Bill Gates

We showed earlier (Example 11) that at an interest rate of 9 percent Bill Gates could, if he wished, turn his 96 billion wealth into a 40-year annuity of 8.9 billion per year of luxury and excitement (L&E). Unfortunately L&E expenses inflate just like gasoline and groceries. Thus Mr. Gates would find the purchasing power of that 8.9 billion steadily declining. If he wants the same luxuries in 2040 as in 2000, he'll have to spend less in 2000, and then increase expenditures in line with inflation. How much should he spend in 2000 Assume the long-run inflation rate is 5 percent. Mr. Gates needs to calculate a 40-year real annuity. The real interest rate is a little less than 4 percent so the real rate is 3.8 percent. The 40-year annuity factor at 3.8 percent is 20.4. Therefore, annual spending (in 2000 dollars) should be chosen so that

## Interest As Distinct From Usury

But soon interest as a penalty was by some, and in certain circumstances, permitted from the beginning of a loan. One such case was as compensation for time and effort in making loans, a sort of wage. Another was when the money could clearly have been used profitably if not loaned, and still another was when loss was incurred by the lender, say in selling property, in order to raise the money to lend. Again, in the case of the census (annuity secured by property) discussed below, interest from the start was allowed if it was no greater than the gain from investing the money in the outright purchase of the property. Finally, interest on a forced loan by a state was by some considered licit as damages as long as the loanholder would rather have his principal back than receive the interest. (151) Risk was generally not considered a legal ground for accepting interest or profit on a loan. Perhaps this was because loans were usually secured by property worth many...

## Longterm Interest Rates

When a new war with France began in 1702, it met with early success at Blenheim and Gibraltar. It was financed relatively smoothly by loans at the Bank of England for short-term requirements and by the sale to the public of annuities for long-term requirements. These annuities were negotiated at a wide variety of terms and at rates between 6 and 8.7 or higher (see Table 12). They were sometimes for one, two, or three lives, or ran to 96-99 years, or were perpetual. They were often accompanied by prizes and lotteries. Therefore, their nominal rates cannot always be translated accurately into a rate of interest. Interest rates fell in the second decade. England was on the verge of its first easy-money period. In 1714 the usury laws were amended, reducing the maximum rate from 6 to 5 . In 1715 the government sold a 5 perpetual annuity, and in 1717 it sold a 4 funding loan. This was a period of moderate debt reduction and close cooperation between the Bank and the Exchequer. Dealings with...

## Partial Amortization or Bite the Bullet

The monthly payment can be calculated based on an ordinary annuity with a present value of 100,000. There are 240 payments, and the interest rate is 1 percent per month. The payment is In this case, we have an ordinary annuity of 1,750 per month for some unknown number of months. The interest rate is 7 12 .5833 percent per month, and the present value is 234,000. See if you agree that it will take about 260 months, or just under 22 years, to pay off the loan. Maybe MD really stands for mucho debt

## Summary And Conclusions

A series of constant cash flows that arrive or are paid at the end of each period is called an ordinary annuity, and we described some useful shortcuts for determining the present and future values of annuities. 4. Many loans are annuities. The process of providing for a loan to be paid off gradually is called amortizing the loan, and we discussed how amortization schedules are prepared and interpreted.

## RJ2 1 fj2 1 023 1 fj2

To understand more fully the relationships among the yield of a security, its cash flow structure, and spot rates, consider three types of securities zero coupon bonds, coupon bonds selling at par (par coupon bonds), and par nonprepayable mortgages. Mortgages will be discussed in Chapter 21. For now, suffice it to say that the cash flows of a traditional, nonprepayable mortgage are level that is, the cash flow on each date is the same. Put another way, a traditional, nonprepayable mortgage is just an annuity.

## Concepts Review and Critical Thinking Questions

Annuity Factors There are four pieces to an annuity present value. What are they 2. Annuity Period As you increase the length of time involved, what happens to the present value of an annuity What happens to the future value 3. Interest Rates What happens to the future value of an annuity if you increase the rate r What happens to the present value

## Shortterm Interest Rates

Table 14 is based largely on the established rates of discount of the Bank of England. The table is not complete. There are indications that market rates were at times below bank rate. Chart 4 (page 152) compares bank rate with the effective yields of new long-term government securities until 1728, and thereafter with decennial averages of the market yield of 3 annuities and consols, derived from Table 13. It is evident that bank

## How Large Will This Balloon Payment Have To Be For You To Keep Your Monthly Payments At

Comparing Cash Flow Streams You've just joined the investment banking firm of Dewey, Cheatum, and Howe. They've offered you two different salary arrangements. You can have 75,000 per year for the next two years, or you can have 55,000 per year for the next two years, along with a 30,000 signing bonus today. If the interest rate is 10 percent compounded monthly, which do you prefer Calculating Present Value of Annuities Peter Piper wants to sell you an investment contract that pays equal 10,000 amounts at the end of each of the next 20 years. If you require an effective annual return of 9.5 percent on this investment, how much will you pay for the contract today 39. Present Value and Interest Rates What is the relationship between the value of an annuity and the level of interest rates Suppose you just bought a 10-year annuity of 2,000 per year at the current interest rate of 10 percent per year. What happens to the value of your investment if interest rates suddenly drop to 5 percent...

## Questions and Problems14

Annuities and Growing Annuities 4.30 What is the value of a 15-year annuity that pays 500 a year The annuity's first payment is at the end of year 6 and the annual interest rate is 12 percent for years 1 through 5 and 15 percent thereafter. b. You receive 1,750,000 now, but you do not have access to the full amount immediately. The 1,750,000 would be taxed at 28 percent. You are able to take 446,000 of the after-tax amount now. The remaining 814,000 will be placed in a 30-year annuity account that pays 101,055 on a before-tax basis at the end of each year. 4.50 A 10-year annuity pays 900 per year, with payments made at the end of each year. The first 900 will be paid 5 years from now. If the APR is 8 and interest is compounded quarterly, what is the present value of this annuity

## Putting It All Together Savant Concepts Illustrated

Will there be an increase in value-added The acquisition would be financed with 8 debt after tax-deductable interest expense, the cost of this capital is 8 x (1 - 34 ) x 10 million, or 528,000. After-tax operating profit for the acquisition is 800,000 per year. Thus, value-added is 800,000 - 528,000 272,000 this is also the net increase in financial-accounting earnings and the annual cash flow. What transaction costs would be involved You ascertain that 400,000 of legal, accounting, and loan fee costs would be incurred. Half of them are tax deductible, so the after-tax cost is 400,000 - (.34)(1 2)( 400,000) 332,000. You note that this is 60,000 in excess of the first year value-added. However, assuming you hold the company for 10 years, there is still a positive net present value of - 60,000(year 1) + 4,625,360(sum of years 2 through 10), or 4,565,360. The latter figure uses an 8 cost of capital, for 10 years of annuity, with a resulting factor of 6.7101 less .9259(year 1 factor), or...

## Imagine You Are Discussing A Loan With A Somewhat Unscrupulous

Calculating Present Values A 5-year annuity of ten 8,000 semiannual payments will begin 9 years from now, with the first payment coming 9.5 years from now. If the discount rate is 14 percent compounded monthly, what is the value of this annuity five years from now What is the value three years from now What is the current value of the annuity 56. Calculating Annuities Due As discussed in the text, an ordinary annuity assumes equal payments at the end of each period over the life of the annuity. An annuity due is the same thing except the payments occur at the beginning of each period instead. Thus, a three-year annual annuity due would have periodic payment cash flows occurring at Years 0, 1, and 2, whereas a three-year annual ordinary annuity would have periodic payment cash flows occurring at Years 1, 2, and 3. a. At a 10.5 percent annual discount rate, find the present value of a six-year ordinary annuity contract of 475 payments. b. Find the present value of the same contract...

## Calculating Present Value Of A Perpetuity Given An Interest Rate Of 6.5 Percent

Calculating Present Value of Annuities Congratulations You've just won the 15 million first prize in the Subscriptions R Us Sweepstakes. Unfortunately, the sweepstakes will actually give you the 15 million in 375,000 annual installments over the next 40 years, beginning next year. If your appropriate discount rate is 11 percent per year, how much money did you really win 50. Variable Interest Rates A 10-year annuity pays 1,500 per month, and payments are made at the end of each month. If the interest rate is 15 percent compounded monthly for the first four years, and 12 percent compounded monthly thereafter, what is the present value of the annuity 51. Comparing Cash Flow Streams You have your choice of two investment accounts. Investment A is a 10-year annuity that features end-of-month 1,000 payments and has an interest rate of 11.5 percent compounded monthly. Investment B is an 8 percent continuously compounded lump-sum investment, also good for 10 years. How much money would...

## Replacing an old machine

We can calculate the NPV of the new machine and its equivalent annual cost, that is, the 5-year annuity that has the same present value. 5-year annuity_13.93 13.93 13.93 13.93 13.93 58.70 The cash flows of the new machine are equivalent to an annuity of 13,930 per year. So we can equally well ask at what point we would want to replace our old machine, which costs 12,000 a year to run, with a new one costing 13,930 a year. When the question is posed this way, the answer is obvious. As long as your old machine costs only 12,000 a year, why replace it with a new machine that costs 1,930 more

## Generalization To Other Financial Applications See Chapter

Chapter 11 represents an attempt to show that the same decomposition used in the franchise value can be generalized to model a number of financial situations. The subject of this chapter is a defined contribution (DC) retirement plan. The investor specifies his or her objective as achieving a retirement annuity that pays a prescribed percentage of his or her final salary. The objective is to find the current asset salary ratio required to keep the plan on track, given values for salary growth, investment returns, and the savings rate. The asset salary ratio has a certain similarity to a P E ratio. However, the real analogy lies in the problem's structure of having (1) a current pool of earning assets (the current savings), (2) a source of future growth (in this case, the prospective salary growth), (3) a procedure that specifies the investment opportunities derived from this growth (the fraction of future salary that can be saved), and (4) an investment return that can be applied to...

## Example 61 Valuing a New Computer System

Don't be put off by the fact that the computer system does not generate any sales. If the expected cost savings are realized, the company's cash flows will be 22,000 a year higher as a result of buying the computer. Thus we can say that the computer increases cash flows by 22,000 a year for each of 4 years. To calculate present value, you can discount each of these cash flows by 10 percent. However, it is smarter to recognize that the cash flows are level and therefore you can use the annuity formula to calculate the present value PV cash flow x annuity factor 22,000 x

## But It Unfortunately Ran A Bit Over Budget. All Is Not Lost. You Just Received An Offer In The Mail To Transfer Your 5

Calculating Annuity Payments This is a classic retirement problem. A time line will help in solving it. Your friend is celebrating her 35th birthday today and wants to start saving for her anticipated retirement at age 65. She wants to be able to withdraw 80,000 from her savings account on each birthday for 15 years following her retirement the first withdrawal will be on her 66th birthday. Your friend intends to invest her money in the local credit union, which offers 9 percent interest per year. She wants to make equal annual payments on each birthday into the account established at the credit union for her retirement fund.

## Exhibit 211 Bond Types Based on Coupon or Cash Flow Pattern

(also called a consol) Annuity bond Amortization of Annual Pay Annuity Bonds. The promised cash flows of some bonds resemble an annuity rather than a straight-coupon bond. For instance, residential mortgages and many commercial mortgages pay a level monthly payment, a portion of which is interest and another portion principal. The right-hand side of Exhibit 2.13 illustrates the pattern of principal paydown for a 30-year annual pay mortgage over time (that is, its amortization). It shows that early in the life of this annuity bond the largest portion of the payment is interest. With each subsequent payment, a greater portion of the payment applies to principal and a smaller portion applies to interest. For example, 10,000 of the 10,607.925 first year payment is interest, but this interest drops to 9,939.208 in the second year, even though the first and second year payments are identical. The Amortization of an Annual-Pay Straight Coupon Bond. The left-hand side of Exhibit 2.13...