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Interest calculations are very important in making financial decisions. In this book you will learn how to do these interest calculations. You may not want to tackle all of them at once. You may prefer to learn one and then wait a while before under taking another, but having them all brought together in this book should make it easy for you to refer to them as you need to. Interest calculations may seem heavy going at first. You may feel like the young knight in olden times whose sword seemed too heavy when he first started training but became not only lighter but a valued friend as he became accustomed to it. Likewise you will find interest calculations valuable tools as you use them enough to become comfortable with them.
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Return to main menuAfter you master the material in this book, you should be able to:
Calculate what a certain amount will accumulate to at a given rate of interest over a specified number of years. FV of1
Find the amount which will result from investing a certain amount each year for a given number of years at a specified rate. FVSERIES (annuity)
Calculate what you could pay now for an amount to be received at a certain time in the future in order to earn the rate of return you desire. PV of1
Determine the rate you will be earning if you pay a given amount now for an amount to be received a certain number of years in the future. RATE1
Decide how much you could pay now for the right to receive a certain amount per year for some number of years and earn the rate of return you desire. PVSERIES (annuity)
Determine the present value of any combination of future amounts, given the pattern of the future amounts and the rate of return desired on them. COMBINATION
Find the price at which a bond would be selling given the inter- est rate it pays, the time until it comes due, and the rate which investors are willing to accept on the bond. BOND
Calculate the rate you would earn on a bond or note if you bought it at a specified price, given its terms. BOND RATE
Find the present value of a set of payments which will begin at some time in the future, given the amount of the payments, the number of years they will continue, and the rate of return desired on them. DEFERRED
Compounding. One question you will be interested in answering is what an amount of money accumulates to by some time in the future if you invest it at a certain rate of interest. Usually you will assume that the amount will be invested at compound interest. Compound interest is interest which is added to the principal and becomes part of the principal for subsequent interest calculations. In addition to the interest rate and length of time necessarily used in interest calculations, to figure with compound interest you also need to know the frequency of the compounding - that is, how often the interest is to be calculated and added to the principal. The total over time amounts to a slightly larger sum if compounded semiannually than if compounded annually. Quarterly or daily compounding increases the results a little more. For example, if you invest $100 at 6 percent interest, it will accumulate as follows:
| compounded: | one year | two years | three years |
|---|---|---|---|
| annually | $106.00 | $112.36 | $119.10 |
| semi-annually | 106.09 | 112.55 | 119.41 |
| quarterly | 106.14 | 112.65 | 119.56 |
| daily | 106.18 | 112.75 | 119.72 |
The interest tables that you will generally have available are based on an assumption of annual compounding. Often, the investment you are making will pay interest compounded more often than annually. As you can see, the more frequent compounding does not change your results substantially, so using the annual compounding tables will not throw your calculations off much.
Formulas. The compound interest calculation process is summarized by the formula:
a = p(1+r)^n
where a is the accumulated amount, p is the principal (beginning amount), r is the rate per period, and n is the number of periods. Actually it is seldom necessary to use the formula as tables are available for many rates of interest.
Using Tables. Let's see how you can use the tables for your compound interest calculations. For example, what will $50 amount to in five years at 8 percent compounded annually. In Table A-1 (the Compound Amount of $1" sometimes called "Amount of 1" Table or "Future Value of 1" Table):
look in the column headed 8%
down to the line labelled 5 in the n column
here you find 1.4693, the amount $1 will accumulate to
Multiply this factor by the $50 in question
1.4693 x 50 = $73.47
so your $50 would amount to $73.47 at the end of five years.
The tables can also be used for other compounding periods if the annual rate divided by the frequency of compounding matches a rate in the table. For example, what will $100 amount to in three years at 8 percent compounded semiannually. Three years is six semi-annual periods with a rate of 4 percent per period. In Table A-1 (the "Compound Amount of $1" Table) look under the 4% column for the figure opposite 6 in the n column. It is 1.2653. Multiplying the factor from the table by $100 gives $126.53, the amount to which $100 will accumulate in three years at 8 percent compounded semiannually.
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An annuity is a series of even payments at intervals. For our purposes here we will deal with annual payments that last a certain number of years. There are also life annuities, which make payments as long as a person lives. We need to distinguish an ordinary annuity and an annuity due when we are dealing with a series of payments.
If we say "$500 a year", it is ambiguous as to when during the year
each $500 is paid or received. The two cases which we might deal
with in practice are payments at the beginning of each year or payments
at the end of each year. If the payments are at the end of each year, it
is called an ordinary annuity. If the payments are at the beginning of each
year, it is called an annuity due.
Payments that we receive are likely to be at the end of each year, so we
can assume that for calculation purposes. However, if we are saving money,
there is a real question as to whether we are putting it in at the
beginning of the year or at the end. I am used to assuming that we begin
now (the first of the first year). However, text authors often
assume end of year and their table is based on that assumption.
In the next secition we will see how to handle either case.
You will sometimes want to find the accumulated amount which will result from investing a certain amount each year at compound interest. In this case the interest earned each year will be the interest rate times the sum of the previously accumulated amount and the new investment at the beginning of the year. If $100 were invested at the beginning of each year for two years at 6 percent, the accumulated amount would be calculated as follows:
invested at the beginning $100.00
interest at 6% 6.00
amount at the end of first year 106.00
added beginning of second year 100.00
total $206.00
interest on the total at 6% $12.36
amount at the end of two years $218.36
Calculating the amount in this way is seldom necessary since you would usually use interest tables as discussed above.
Using the Tables. In fact, you could use Table A-1 again. In the example we just used, the first $100 would be on deposit for two years and the second $100 for one year. Table A-1 tells you that the first $100 would amount to $112.36 and the second $100 to $106, for a total of $218.36. Even using Table A-1 would require a lot of adding if you planned to accumulate the amounts for many years. Fortunately, Table A-3 summarizes this process so that you do not need to add the results of each year's amount. In Table A-3,
look in the column headed 6%
down to the line labelled 2 in the n column
here you find 2.1836, the amount $1/year will amount to
Multiply this factor by the $100 per year
2.184 x 100 = $218.36
the same figure as using Table A-1. (The figure you get using either Table may a few cents different from the figure which results from the detailed calculations because the tables are accurate only to four digits after the decimal point.)
The calculation illustrated above assumes that you are putting the money in at the beginning of each year (an annuity due as it is called). Sometimes you see an assumption of putting the money in at the end of each year (an ordinary annuity). Table A-3alt can be used for this assumption:
look in the column headed 6%
down to the line labelled 2 in the n column
here you find 2.06, the amount $1/year will amount to
Multiply this factor by the $100 per year
2.06 x 100 = $206
If you put the money in at the beginning of the year instead of the end of the year, you will earn an extra year's interest on the amount. This means that if you have only the table using an end of year assumption, you can get the correct amount for the beginning of year assumption by multiplying the result from that table by 1 plus the rate (this is the same as figuring an extra year's interest and adding it in). In the example above, 206 x 1.06 = 218.36. On the other hand, if you have only the table using a beginning of year assumption, you can get the correct amount for the end of year assumption by dividing the result from that table by 1 plus the rate. In the example, 218.36/1.06 = 206.
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You probably intuitively feel that the right to receive a dollar a year from now is not worth as much to you now as a dollar received right now. This is true in an economic or financial sense because, if you have the amount now, you can invest it in a productive use or lend it at interest. Many times in financial calculations you need to know how much a dollar to be received at some time in the future is worth now. The amount in current dollars which is equivalent to an amount to be received later is called the present value of that future amount. Present dollars and future dollars are equated by using a rate of interest. Thus, the present value of $1.00 can be thought of as the amount which must be invested now to accumulate to $1.00 at the end of a certain number of years at a given rate of interest. It could also be looked at as the price which could be paid now for the right to receive $1.00 a given number of years from now for you to earn a specified rate of compound interest. Thus, finding the present value is, in a sense, the converse of finding the accumulated amount at compound interest. That is, the compound amount is a present sum taken forward into the future (or equated to a future sum) and the present value is a future sum brought back to now (or equated to a present amount.) The process of finding the present value is also spoken of as discounting the future amount.
Formula. The present value of a given sum can be computed by using a formula derived from the compound interest formula given above. It is
a
p = (1+r)^n
where p is the present value, a is the future amount, r is the rate of interest (or discount), and n is the number of years at the end of which the amount will be received.
Using Tables. As with compound interest, it is usually not necessary to use the formula since present value tables have been computed for many interest rates. These tables are called "Present Value of $1" or "Future Worth to Present Worth" tables.
The concept of present value is an important one in financial decision making because finance often involves the receipt of payment on a debt or a return on an investment at some time in the future. The future receipt may be a single sum receivable at a definite period of time as discussed above or it might be a series of payments to be received over a period of time, or a combination of the two. In either case the return which would be earned or the price which could be paid in order to earn a certain return would be important to you in making decisions.
For example, a $100 non-interest bearing note is due five years from now. How much would you be willing to pay for the note if you wanted to earn 8 percent compound interest on your investment? In the "Present Value of $1" Table (Table A-2):
look in the column headed 8%
down to the line labelled 5 in the n column
here you find .681, the present value of $1
Multiply this factor by the $100 you expect
.681 x $100 =" $68.10
If you paid $68.10 for the note, you would earn 6 percent on your investment.
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Finding a Rate. The present value tables can be
used "backwards" as well as forward. That is, they can be used to determine the
rate of interest which will be earned on an investment purchased at a given price when a
certain specified future payment or payments are to be received from it. For example, a
$10,000 non-interest bearing note due seven years from now is offered for sale at $7,600.
What rate of interest would you earn by buying it at this price? You would pay .76 for
each $1.00 to be received in seven years. In the "Present Value of $1" table (Table A-2) go down the n column to the 7 row. Go
across until you find .76 and see that it is in the 4% column. Therefore, approximately 4
percent would be earned if the note were bought at the offering price. Often this close an
approximation would not be found in the table and interpolation would be necessary.
Interpolation will be explained later in this writeup.
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PV of1 What a dollar in the future is worth now
You could find the present value at a given rate of interest of a series of equal payments to be received for a specified number of years by finding the present value of each payment using the Present Value of $1 table and summing these present values. However, this process has already been summarized for you in a table called the "Present Value of $1 per Year" (or "Annuity to Present Worth" or Present Value of an Annuity, etc. - Table A-4). Thus it is necessary only to look up the present value of a series of $1 payments in the table and multiply this present value by the amount of the annual payment which you expect to receive.
Example. A company is considering purchasing an oil royalty which is expected to pay $10,000 per year for the next seven years and nothing thereafter. If the company desires to earn 12 percent on its investment, how much can it pay for the royalty In this case, the "Present Value of $1 Per Year" table (Table A-4) is used:
look in the column headed 12%
down to the line labelled 7 in the n column
you find 4.5638, the amount $1/year is worth now
Multiply this factor by the $10,000 per year
4.5638 x 10,000 =" $45,638
$45,638 is the maximum which the company could pay and still earn 12 percent on its investment.
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We have discussed four different interest tables. To decide which one to use for a given problem, keep in mind that
The "Compound Amount of $1" (Table A-1) (or Future Value of $1 or Compound Sum of $1) Table tells how much a dollar invested now will amount to in a certain number of years at a given rate of interest.
The "Present Value of $1" Table (Table A-2) gives the amount which must be invested now in order to amount to $1.00 at the end of a certain number of years if a given rate of interest is earned.
The "Compound Amount of $1 per Year" (Table A-3) Table (or Future Value of an Annuity or Sum of an Annuity) tells how much a dollar invested each year will amount to in a certain number of years at a given rate of interest.
The "Present Value of $1 Per Year" Table (Table A-4) (or Present Value of an Annuity) tells how much must be invested now at a given rate of interest in order to receive $1.00 per year for a certain number of years and nothing thereafter.
Ideally, you should become so familiar with the structure of the tables that you could use them even if they had no titles on them.
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Next, let us look at a combination of a series of payments and a lump sum due at some future time, for example, $100 per year for 5 years and $1,000 at the end of five years. This combination is what we sometimes refer to as a bond type problem, because a bond is a basic financial instrument that has this pattern of payments. A bond or note has a series of interest payments that are due over time and then a face value that comes due at some future specified time. We have dealt with series of payments and lump sums separately. All you must do is put them together - to recognize that it is merely a combination of what you have been doing already. Any time you are dealing with sets of payments that might be confusing I would recommend that you draw them out on a time line so that you can see exactly how these payments fall, as illustrated below.
Valuing a Bond. Take an example of a $1,000 bond or note that is due 5 years from now. $1,000 is the standard unit of trading for bonds. (In fact, it is customary in doing bond calculations not to say anything about face value, but to assume that you are talking about a standard $1,000 bond.) So take the bond or note due in 5 years, and say it pays a 10 percent rate of interest on its face value, so it will be paying $100 a year for 5 years. Now, if you were to be technical with bonds, you would say that bonds pay interest twice a year and so therefore you ought to use semi-annual compounding and take the payment as $50 every 6 months. If you get to dealing in large amounts of money you probably would want to do that, but for your purposes here, it is sufficient to use the annual compounding assumption and treat it as an annual payment (the annual compounding assumption is automatically built into the tables). One thing to keep in mind if you want to use the semi-annual assumption is that you must do it for both the face value and the series of payments. It throws the results off if you use semi-annual payments and then assume annual compounding for the principal amount.
To see the situation clearly, you could lay it out on a time line like this:
100 100 100 100 100
/________/_______/_______/_______/_______/
1000
You want to find the present value of the stream of $100 payments plus the present value of the $1,000 lump sum payment at the end of the period. Let's say you would be willing to accept an 8 percent return on this note - that it is certain enough of payment that you would be satisfied with an 8 percent return on it. If you were to use a semi-annual assumption instead of the annual assumption, you would say this is the same as $50 per period for 10 six month periods with a 4 percent rate per period. You usually think of the tables as being per year, but you can use them for any other time period as well, as long as you are careful to be consistent.
Let's go ahead on the annual basis. You expect a series of $100 payments so you go to Table A-4. You have said you want an 8 percent rate of return, so you need to find the present value of this series of payments discounted at 8 percent. In Table A- 4:
you go to the 8% column
down to the 5 year line
you find there 3.9927, the present value of $1/year
$100 x 3.9927 =" $399.27
Thus, the value of the $100 a year interest on the note is $399.27. That is not all you get from this note, though. You also get the $1,000 at the end of the period, so for that you must go to Table A-2:
look in the column headed 8%
down to the line labelled 5 in the n column
here you find .6806, the present value of $1
multiply this factor by the $1,000 you expect
.6806 x $1,000 =" $680.60
If you discount the $1,000 back to the present, it's worth $681.00. That gives you a total present value for this note of $1,080 ($399 + $681). That is the same thing as saying that you can pay $1,080 for the note and be making an 8 percent return on your money, if it pays off as you expect over the time period. To summarize, your calculations would look like this:
Series of payments - go to table A-4
8% column to 5 year line > 3.993 * 100 = $399.27
Single amount - go to table A-2
8% column to 5 year line > .681 * 1,000= 680.60
Total present value of the note $1,079.87
Another Example. A building is leased to a tenant for a net annual rent of $5,000. The tenant has agreed to buy the building at the end of 10 years for $50,000. How much could you pay for the building if you required a 12 percent return on your money? (The first year's rent has been paid to the present owner. The next payment is due one year from now.) The solution to this problem requires the valuing of both a stream of payments to be received over a period of time and a payment to be received at the end of that time. The value of the payments is found as follows:
Rent - Series of payments - go to table A-4
12% column to 9 year line > 5.3282 x 5,000 =" $26,641
Purchase price - Single amount - go to table A-2
12% column to 10 year line > .3220 x 50,000 =" 16,100
Total present value of the building $42,741
Adding the present value of the rent to the present value of the purchase price gives $42,741 as the price which would give you a 12 percent return on your investment.
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Again you can work this bond type of problem "backward". Say you can buy for $950 a $1,000 note due 5 years from now paying $100 a year in interest. Your question then would be what rate of return will you be making on your money, if you pay $950 for this note. To find the rate in a case like this requires an "iterative" or trial and error process. If you were going to make 10 percent, you would have to pay exactly $1000 for this note, because that is what it pays on its face value, so you would try something over 10 percent. You might try 12 percent
Interest - Series of payments - go to table A-4
12% column to 5 year line > 3.6048 x 100 =" $360.48
Face value - single payment - go to table A-2
12% column to 5 year line > .5674 x 1,000=" 567.40
Total present value of the note $927.88
Thus, if you paid $927.88 for this note you would be making right at 12 percent on your money. You are paying a little more than the $927.00 - you are paying $950 for the note. That means that you are making something less than 12 percent. So you try a lower rate, say 11 percent. You do the same thing that you did at 12 percent:
Interest - Series of payments - go to table A-4
14% column to 5 year line > 3.430 x 100 = $343.00
Face value - single payment - go to table A-2
14% column to 5 year line > .519 x 1,000= 519.00
Total present value of the note $862.00
This shows that if you paid $862.00 for this note, you would be making 14 percent on your money. You are paying more than $862, so you must be making less than 14 percent. Next you would try 13 percent
Interest - Series of payments - go to table A-4
13% column to 5 year line > 3.517 x 100 =" $351.70
Face value - single payment - go to table A-2
13% column to 5 year line > .543 x 1,000=" 543.00
Total present value of the note $894.70
That means that if you paid about $895 for the note, you would be making 13 percent on your money. That is close to what you are paying, but not exactly the same. You are paying $900 which is between the $895 value at 13 percent and the $927 value at 12 percent, so you know your actual rate of return is between 12 percent and 13 percent.
Interpolating. Your question then is where between 12 and 13 and you "interpolate" to find out. We can say that the total distance between the present values here is $32.00. $900 is $27.00 from the value at the 12 percent rate, so you would take (27/32) x 1 (because there is one percentage point difference between your trial rates) and that works out to approximately .8. You would be making approximately 12.8 percent on this note if you buy it at $900. You interpolate to the nearest tenth of a percent - to 12.8 percent not 12.84 percent. In interpolating you think of it as relative distances
1%
/ \
12% 13%
/------------------------/--------/
$927 $900 $895
\ /
$27
\ /
$32
The distance in percentage points is 1. The distance in dollars is $32.00 and we are saying then that the $900 is 27/32 of the way from the 12 percent amount to the 13 percent amount and so the rate at $900 would be 27/32 of the way from 12 percent to 13 percent. If you are using a financial calculator, interpolation is not necessary because it's calculations are automatically carried to several decimal places.
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You may also need to deal with a series of payments that starts sometime in the future. Let's imagine that you have a contract that will pay you $100 a year for three years, but the payments don't start until three years from now.
$100 $100 $100
/_______/_______/_______/_______/_______/
There are five years involved but you will not get any payments in the first two - at the end of the third year you get a payment, at the end of the fourth year you get a payment, and at the end of the fifth year you get a payment.
Now you could approach that series as simply a set of payments and say the present value of $100 at the end of three years plus the present value of $100 at the end of four years plus the present value of $100 at the end of five years. It wouldn't be too bad with three payments of $100 each, but if you had 20 payments of $67.56 each, that is a lot of figuring to do. You want a compact way of doing this and again you can use the idea of equivalents. You can find the equivalent of this set of payments as of the end of the second year and find the equivalent of that amount now. To bring deferred payments back to the present you use the fact that you can find the present value of any set of payments at any time by using the present value tables.
You can find the present value of this series of payments as of the beginning of the year in which those payments begin by using Table A-4. Of course, you must specify a rate. Let's say you want an 8 percent return. You can go to Table A-4 and find that at 8 percent $1 a year for three years is worth 2.577. Multiplying 2.577 by the $100 a year you expect gives $257.70. But that is not at the present, that is at the beginning of the third year. These payments are worth the equivalent of $257.70 at the beginning of the third year, but you want the value at the beginning of the first year.
If you had a payment of $257.70 coming in at the beginning of the third year, that wouldn't cause you any problem because that is the same as an amount received at the end of the second year. Think about 12:01 a.m. on January 1 vs. 11:59 p.m. on December 31; there is no real difference between the end of one year and the beginning of the next year. If you had the amount of $257.70 at the end of the second year, you could bring that back to the present using Table A-2. The $257.70 is equivalent to $220.85 now. Thus, the $100 a year also has a present value of $220.85. If you paid $220.85 for this set of payments, you would be making 8 percent on your money. In summary, the calculations would be:
Series of three payments (go to Table A2-4)
8% column to 3-year line > 2.577 x $100 = $257.70
Bring value beginning of third year to present
(go to Table A2-2) 8% column to 2-year line x .857
Present value now $220.85
You can test this result with a savings account analogy. Imagine putting $220.85 in a savings account that pays 8 percent compounded annually, then starting to draw out the $100 a year at the end of the third year. Try it and see that you come out just about even.
Future payments received are generally assumed (unless stated otherwise) to be at the end of the year that you are talking about. If you say a series of payments over five years, you mean that at the end of each of those five years you expect to get one of those payments. Then you must remember that when you bring a series of payments back to the present value using Table A-4 you are bringing it back to the beginning of the year at the end of which the first payment is to be received. That is easy to confuse. You can either bring it back too far or not bring it back far enough. Again it is wise to lay the payments out on a time line to see them clearly.
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