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#1 Re: Help Me ! » Indeterminate forms in Calculus » 2009-01-03 11:44:39

Danbee wrote:

A trust is established in your name which pays t+10 dollars per year for every year in perpetuity, where t is time measured in years ( here the present corresponds to time=0). Assume a constant interest rate of 4%. What is the total value, in today's dollars,of all the money that will be earned by your trust account?

Beacuse of inflation, the value of a dollar decreases as time goes on.Indeed this decrease is directly related to the continuos compounding of interest

Correct me if Im wrong but would you agree that your problem is basically about determining the discounted value (or present value) of a series of payments that start at \$10 at the end of Year One and then increase by \$10 each year, forever; with the further stipulation that interest is i = 4%???

#2 Re: Exercises » Compound Interest » 2008-11-02 06:30:54

OR
Using the formula approach, you first convert your nominal rate of 5% compounded quarterly into its effective rate.  Thus

You then apply the ordinary amortization formula.  Thus

#3 Re: Puzzles and Games » How much would you need to invest? » 2008-11-01 11:22:10

All_Is_Number wrote:

My assertion was that the balance after one year is \$833,333.33 * 1.04 after the payment of \$50,000 is made. Prior to the payment, the balance is \$833,333 * 1.04 + \$50,000.

Beautiful.
A clear case of misinterpretation on my part.
My apologies.
It would seem then that your solution of

is the reduced (and, I must confess, more elegant) version of my long-winded solution of

, the modified retrospective method I was referring to in my last post.
Very handy in calculating the nth balance before the nth payment is

; minimizing, if not eliminating, the need to prepare a spreadsheet schedule, in addition to having a good countercheck of ones spreadsheet schedule.

Somehow,

gives the illusion that its an identity.  After an hour or so of trying to prove it and failing miserably, Im temporarily resigned to the conclusion that it will have to be another puzzle for those who wish to take a crack at it.  Im sure theres an algebra technique out there (that Im presently unaware of) which will resolve this alleged identity.  Or maybe Im just way too sleepy.

All_Is_Number wrote:

Here is an excellent book covering annuities, perpetuities, and much more.

Yes, Im aware of this book courtesy of Don, a.k.a. mathceleb.  Don mentioned it once or twice at mathhelpforum.com.  I have in fact looked for it earlier at that same site before your post.  Its unfortunate that theres no preview available.

#4 Re: Puzzles and Games » How much would you need to invest? » 2008-10-31 04:28:46

mathsyperson wrote:

Most of this stuff is over my head, but I can answer that at least!
You change an URL into a link by enclosing in it [url]tags.

You can also use Click Here! for example, to get a text on the link different to the URL.

Thanks.

#5 Re: Puzzles and Games » How much would you need to invest? » 2008-10-30 18:28:27

All_Is_Number wrote:

Ms. Bitters, are you an actuarial science major (or actuary)? I'm an actuarial science major, though I'm much more interested in the financial aspect than the more traditional actuarial roles.

No, Im not.  My background is in accounting. In a way, I suppose were somewhat alike since Im also much more drawn in finance mathematics than I am with accounting principles and procedures.

All_Is_Number wrote:

The balance at the end of the nth year is

Lets test this conjecture for n = 1.  At the end of 1 year, \$833,333 1/3 would have accumulated to

Since the first years payment is \$50,000, then the perpetuitys balance at the end of the first year would be

If my understanding of your projection is correct, then by your reckoning the perpetuitys balance at the end of the first year would be

This clearly cannot be the case.
Under the presumption that you may have committed a typo, i.e. you must have meant to type in the factor (1.10)^n as the co-factor of \$833,333.33 instead of (1.04)^n, we have

So far so good.  Again, lets test this theory for n = 2.  At the end of the 2nd year, we have

Lets see if this modification will withstand scrutiny.
Since the balance at the end of the 1st year is \$866,666 2/3, it follows that the accumulated amount at the end of the 2nd year before the 2nd years payment would be

Since the 2nd years payment is

it follows that the balance at the end of the 2nd year would be

Thus, my presumed modification failed to deliver.

At this point, I must again presume that you would prefer to work out this puzzle on your own rather than having someone hand out the solution to you.  If this is the case, then I would just like to direct your attention to the formula for the future value of the growing (geometric) annuity.  You can find it at
http://en.wikipedia.org/wiki/Time_value_of_money
You can also find it at
http://web.utk.edu/~jwachowi/growing_annuity.pdf
I would also like to draw your attention to
and preview (or review, if youre already familiar with) the so-called retrospective method of determining the outstanding balance of a loan as discussed in chapter 7, page 127 of that excellent book.  A slight modification of this method relative to the future value formula for the growing (geometric) annuity will give you the better solution.  You will also find in that book examples of a growing/increasing (arithmetic) perpetuity.  At any rate, I dare say youll have fun.

P.S. How does one go about attaching/activating a hyperlink on a URL with this forum, like the ones that I've quoted.  I've seen some members do just that and yet I can't seem to find a way.

#6 Re: Puzzles and Games » How much would you need to invest? » 2008-10-20 08:49:51

All_Is_Number wrote:

The balance will never be depleted from the annual payments, but will eventually be withdrawn by the owner, as a transaction separate from the annual payments. Typically, perpetuities don't really provide payments forever, they provide payments until the balance is withdrawn (or the financial institution collapses, etc.).

Of course.  Why didnt think of that?  I guess I was more preoccupied with other things than I realized.

All_Is_Number wrote:

We could use the formula for the present value of a perpetuity with non-level payments, but a much easier way to obtain the answer is by recognizing that the principle of the perpetuity must have a net growth of 4% per year. Since the perpetuity is earning 10%, this implies that 6% interest (i.e. 60% of the total interest earned each year) is used to pay the annual payment. Recognizing that interest compounded annually is the same as simple interest at t = 1 year, we get the equation \$50,000=.06x. Solving for x, we find the perpetuity's value one year before a \$50,000 payment is \$833,333.333.

Your explanation is indeed much easier.  From what I could tell, it is more or less derived from the fact that as

in

from the present value formula of a growing (geometric) annuity as represented by
,
the present value becomes

where
A = present value
R = annual payments of \$50,000
i = 10%
g = 4%

Another perspective consist of an application of infinite geometric series (an application Ive recently learned).  Accordingly, the value of the one time investment one year after the investment is made is

The expression on the right-hand side is the sum of an infinite geometric progression whose first term is
R = 50,000 and whose common ratio, the absolute value of which is less than 1, is

Applying the formula for such a convergent infinite series, we have

The discounted value of B (the perpetuity's value one year before a \$50,000 payment) one year earlier is then determined by

P.S.  Had you not posted as you did suggesting about a hint being in order, I might not have known about your puzzle.  Your first post on 2008-10-13 02:55:19 was a point in time when I had some trouble with my connection (which made me unaware that you posted such a puzzle).  As a counter puzzle:  What is the balance at the end of the nth year?

#7 Re: Puzzles and Games » How much would you need to invest? » 2008-10-19 04:32:16

Suppose you have the opportunity to make a one time investment in an annuity that earns 10% fixed interest per year, compounded annually, from which you will receive annual payments beginning one year after the investment is made. The first year's payment is \$50,000. Each year thereafter, the payment increases by 4% over the previous year's payment. The payments continue being paid annually until the balance is withdrawn. How much money would have to be invested initially?

This is clearly a growing (geometric) perpetuity.
And since this is the case, you will notice that the blackened portion of your problem statement is defective in the sense that the balance or fund will never be fully withdrawn or totally depleted.
I don't have time right now but my initial calculation shows that your initial investment should be
\$833,333.333...

I'll check back in later for a full explanation.

#8 Re: Help Me ! » economy » 2008-09-03 04:36:37

qweiop90 wrote:

i am not a spoon feeder, i just dont have any idea about these messy questions.

From what I've seen of your posts, here and elsewhere, I'd say you've been spoon fed long enough.

qweiop90 wrote:

ASSIGNMENT 5: DUE 4.00PM, 4/09/08

#9 Re: Help Me ! » loan- application of principle. » 2008-08-27 03:31:04

qweiop90 wrote:

suppose that a firm has borrowed \$10,000 in the current year at 8.75% interest rate, with a commitment to repay the loan ( principal and interest ) in equal annual instalments over the following 5 years .

Calculate: ( a ) the amount of the annnual repayment . ( 2 marks )

This is a simple annuity problem.  Take note that
A = \$10,000 (loan amount)
R = equal annual installments (to be determined)
j = .0875 (assumed to be the nominal interest rate)
m = 1 (interest period, as in annually  assumed since the problem statement was rather vague)
t = 5 years (term of loan)
You then solve for R in the following amortization equation:

Solving for R gives us

Calculate: ( b ) the stream of annual principal payments ( 2 marks )

End of Year 1 ≈ \$1,679.27
End of Year 2 ≈ \$1,826.21
End of Year 3 ≈ \$1,986.00
End of Year 4 ≈ \$2,159.77
End of Year 5 ≈ \$2,348.75
Total = \$10,000

Calculate: ( c ) The stream of interest payments which can be entered in the tax calculation of the private benefit-cost analysis.

End of Year 1 = \$2,554.27-\$1,679.27
End of Year 2 = \$2,554.27-\$1,826.21
End of Year 3 = \$2,554.27-\$1,986.00
End of Year 4 = \$2,554.27-\$2,159.77
End of Year 5 = \$2,554.27-\$2,348.75
Total = ???

anyway can anyone show the way of payment and pls list out the principle and interest in annually.

#10 Re: Help Me ! » my family mortgage - » 2008-08-26 07:02:35

Monique wrote:

There are 4 people in my family who contribute to repayments. A, B and C contribute 60 percent of their weekly income and D contributes 25 percent of his weekly income to monthly repayments. A makes \$720 per week. B earns \$690 per week. C earns \$820 per week.
We have taken out a \$560 000 loan at 6.4% over a 30 year period.

a) what amount will I have to repay each month?

b) how much will I have to repay in total on this loan?

c) is there any way of working out how long it will take to pay off the mortgage if we increase the monthly repayments by 10% after 5 years of my 30 year loan period, and how much extra do we individually have to put in?

1) can anyone please explain in words and perhaps even diagrams how I would go about working out this solution, and
2) can anyone please show me the mathematical steps involved in answering this question?

Assumption A. If your interest of 6.4% is compounded monthly and you wish to pay off the loan monthly, then your problem falls under the simple annuity case.  Under this scenario:

a) what amount will I have to repay each month?

This is a simple annuity problem.  Take note that
A = \$560,000 (loan amount)
R = monthly payments (to be determined)
j = .064 (nominal interest rate)
m = 12 (interest period, as in monthly - which also happens to be the payment interval)
t = 30 years (term of loan)
You then solve for R in the following amortization equation:

Solving for R gives us

b) how much will I have to repay in total on this loan?

There are 360 monthly payment intervals (30 years * 12 months).
Thus, your total payments on this loan would be 360*\$3,502.83 or \$1,261,018.80.

c) is there any way of working out how long it will take to pay off the mortgage if we increase the monthly repayments by 10% after 5 years of my 30 year loan period

To work out how sooner you can pay off your mortgage if you increase the monthly repayments by 10% after 5 years of your 30 year loan period, you must first work out how much your outstanding loan balance is at the end of 5 years.  Accordingly, this outstanding loan balance at the end of 5 years is the present value of the payments still to be made.  Since you still have 25 years left in your contract, its reasonable to expect that these (payments) form an annuity of 300 payments [(30 years * 12 months) minus (5 years * 12 months)].  Thus, the outstanding loan balance is:

or \$523,614.40
After this, you can now proceed in working out how long it will take to pay off the mortgage if we increase the monthly repayments by 10% after 5 years of my 30 year loan period.
Note first that

Thus, with
A = \$523,614.40 (loan balance at end of 5 years)
R = \$ 3,502.83*1.10 (new monthly payments after 5 years)
j = .064 (nominal interest rate)
m = 12 (interest period, as in monthly - which also happens to be the payment interval)
t = time it takes to pay off the mortgage if we increase the monthly repayments by 10% after 5 years of my 30 year loan period

Plugging and chugging, t ≈ 20.21208114 or 20.21 years.  This works out to 242 monthly payments of \$3,502.83*1.10 and a smaller payment at the end of the 243rd month (vs. the original remaining 300 monthly payments)

how much extra do we individually have to put in?

This is something that youre gonna have to work out amongst yourselves.

Assumption B.

lets say the interest compounded quartelrly. The loan was issued 5 June 2002. can you help me go through each step from a) to d)...?

If your interest of 6.4% is compounded quarterly and you wish to pay off the loan monthly, then your problem falls under the general annuity case.  Under this scenario:

a) what amount will I have to repay each month?

The general annuity formula for the present value of an ordinary annuity (i.e. end of interval payment) is given by

where
A = \$560,000 (loan amount)
R = monthly payments (to be determined)
j = .064 (nominal interest rate)
m = 4 (interest period, as in quarterly)
t = 30 years (term of loan)
p = 12 (payment interval, as in monthly)

Solving for R gives us: R ≈ 3,490.436822 or \$3,490.44
Note: 6.40% compounded quarterly is equivalent to approximately 6.3661668752351% compounded monthly since both have an effective nominal rate of approximately 6.55524495360014%
We show this with the following:

To simplify and to duplicate our discussion of assumption A, we convert this complex annuity problem into its equivalent simple case.  Thus with
A = \$560,000 (loan amount)
R = monthly payments (to be determined)
j = .063661668752351 (the equivalent nominal interest rate compounded monthly of 6.40% compounded quarterly)
m = 12 (interest period, as in monthly)
t = 30 years (term of loan)
You then solve for R in the following amortization equation:

Solving for R gives us

b) how much will I have to repay in total on this loan?

There are 360 monthly payment intervals (30 years * 12 months).
Thus, your total payments on this loan would be 360*\$3,490.44 or \$1,256,558.40.

c) is there any way of working out how long it will take to pay off the mortgage if we increase the monthly repayments by 10% after 5 years of my 30 year loan period

To work out how sooner you can pay off your mortgage if you increase the monthly repayments by 10% after 5 years of your 30 year loan period, you must first work out how much your outstanding loan balance is at the end of 5 years.  Accordingly, this outstanding loan balance at the end of 5 years is the present value of the payments still to be made.  Since you still have 25 years left in your contract, its reasonable to expect that these (payments) form an annuity of 300 payments [(30 years * 12 months) minus (5 years * 12 months)].  Thus, the outstanding loan balance is:

or \$523,407.17
After this, you can now proceed in working out how long it will take to pay off the mortgage if we increase the monthly repayments by 10% after 5 years of my 30 year loan period.
Note first again that

Thus, with
A = \$523,407.17 (loan balance at end of 5 years)
R = \$3,490.44*1.10 (new monthly payments after 5 years)
j = .063661668752351 (the equivalent nominal interest rate compounded monthly of 6.40% compounded quarterly)
m = 12 (interest period, as in monthly)
t = time it takes to pay off the mortgage if we increase the monthly repayments by 10% after 5 years of my 30 year loan period

Plugging and chugging, t ≈ 20.23034792 or 20.23 years.  This also works out to 242 monthly payments of \$3,490.44*1.10 and a smaller payment at the end of the 243rd month (vs. the original remaining 300 monthly payments)

how much extra do we individually have to put in?

Again, this is something that youre gonna have to work out amongst yourselves.

and perhaps even diagrams how I would go about working out this solution

is there any way possible to post a graph here showing my repayments over this period?

#11 Re: Help Me ! » Effective Interest » 2008-08-25 07:33:16

Nestea wrote:

JenCo wants to drop the effective rate of interest on its credit card by 2.40%. If it currently charges a nominal rate of 5.75% compounded monthly, at what value should it set the new nominal rate?

I need help finding the nominal rate.

Accordingly, the effective rate of a nominal rate of 5.75% compounded monthly is given by

If we take the statement JenCo wants to drop the effective rate of interest on its credit card by 2.40%. to mean
(100% - 2.40%)*w, then the new effective rate of interest that JenCo wishes to implement is 97.6% of w. The new nominal rate j can then be determined from the following equation:

Solving for j gives us: j ≈ .05615480096

#12 Re: Help Me ! » Help Me!! » 2008-08-25 07:22:20

Nestea wrote:

If you went to the bank for a 15 year, \$150,000 mortgage at an interest rate of 6.75% pa, compounding quarterly and you wanted to give payments semi-annually how much would each of your payments have to be over the 15 years?

The general annuity formula for the present value of an ordinary annuity (i.e. end of interval payment) is given by

where
A = \$150,000 (amount of mortgage loan)
R = semi-annual payments (to be determined)
j = .0675 (nominal interest rate)
m = 4 (interest period, as in quarterly)
t = 15 years (term of loan)
p = 2 (payment interval, as in semi-annually)

Solving for R gives us: R ≈ 8,057.36905 or \$8,057.37.

#13 Re: Help Me ! » Compound Interest » 2008-08-25 07:18:14

Nestea wrote:

Oleg borrowed \$4,500, \$4,000, \$4,500, and \$4,000 from his dad on September 1 of each of four successive years for college expenses. Oleg and his dad agreed to a loan at the rate of 7% compounded quarterly. If it is now 1 year from the last day that he borrowed money, how much would Oleg owe?
For full marks your answer should be rounded to the nearest cent.

#14 Re: Maths Is Fun - Suggestions and Comments » 9 x tables » 2008-08-21 05:03:36

this is a trick i learnt in class hold up both of you hands in front of you and if you wanted to do
9x7 put down you sevent finger and that represents the gap the left is the tens and the right is the ones also the 2 numbers add up to 9

Second time I came across this trick this month.  Check out this URL:
http://forums.xkcd.com/viewtopic.php?f=17&t=24600

#15 Re: Formulas » Serial Bonds » 2008-08-06 05:04:33

MathsIsFun wrote:

Thank you for that wonderful treatment of the subject, Ms Bitters.

You're welcome.  Hopefully, somebody far more enlightened than me finds better use for it.

MathsIsFun wrote:

Are you perhaps missing something between (4.1) and (6) ... ?

Quite true.  It's whole now.  Apparently there's a slight glitch or variation between the Previewed form and the Submitted form.  The Previewed form has a wider display and everything displayed without a hitch.  The Submitted form, on the other hand, has a narrow display and needs a little bit of tweaking to make it just about right.

#16 Formulas » Serial Bonds » 2008-08-05 06:35:59

Ms. Bitters
Replies: 3

Upon revisiting my notes on the subject of bonds, serial bonds in particular, it occurred to me that I could perhaps reclaim several lines of bytes from my calculators memory if I could but derive a single plug and chug formula for the calculation of the purchase price (flat price) of a serial bond on an interest date to replace (hopefully) the byte guzzler BASIC like program that I wrote on my calculator for such a calculation (a program I used primarily for automating purposes and for checking the results I obtained in going through the motion of the traditional methods).

Eventually, I did somehow managed to derive such a formula, or formulas.  Judging by the nature of the formula(s), it turned out that I was dealing with something similar to the concept of the general annuity formulas. Accordingly, general annuity formulas are used to deal with the so-called complex or general cases where the payment interval is not the same as the interest period (including those simple or special cases where the payment interval is the same as the interest period).  In the case of serial bonds, there is apparently such a thing as a general-purpose formula that can deal with different cases.

In deriving these formulas, I was guided by three known (well known I dare say) statements which in turn somehow proved to be special cases.  To illustrate these three statements, lets consider some examples.

Example 1.  On 2/1/77, Lawrence buys a farm worth \$40,000 cash from Watkins, paying \$10,000 cash and signing a contract to pay interest semiannually at the rate 7% and to pay the remaining principal in 6 equal semiannual installments.  What does an investor pay on buying this contract to yield 8% compounded semiannually on 8/1/78?

The contract is a serial bond involving six distinct bonds of \$5,000 each.  At the end of a year and six months (on 8/1/78), three bonds remain alive. The price of the contract is the sum of the prices of three \$5,000 bonds, to be redeemed at the ends of ½, 1, and 1½ years respectively.

The investors yield rate is greater than the bond rate; thus a discount is in order.  By use of the general method, we have

Price (on 8/1/78) = \$4,975.96 + \$4,952.85 + \$4,930.62 = \$14,859.43.

Those acquainted with annuity formulas will recognize that

≈ \$13,875.45
On the other hand

can be summarized by use of a statement which says that

Thus,

This then gives us

= \$13,875.45 + \$983.98 =\$14,859.43.
This is in fact the essence of another statement that is a direct consequence of (1).  It pretty much says that the price \$P to yield the investor/buyer the rate i per period of a serial bond with the bond rate q per interest/coupon period, whose face value will be redeemed in equal installments of \$H at the end of each period for n periods is given by

By use of the discount method, we have

Price (on 8/1/78) = \$4,975.96 + \$4,952.85 + \$4,930.62 = \$14,859.43.

By making use of statement (1), we arrive at the same sum by the following:

The next example serves to illustrate the third statement.
Example 2.  A man borrows \$10,000, contracting to pay \$2,000 of the principal at the end of each year for 5 years, and to pay interest semiannually at the rate 7%.  His contract is sold 2 years later to yield the investor 6% compounded semiannually.  Find the price he pays.

The contract is a serial bond involving five distinct bonds of \$2,000 each.  At the end of 2 years, three bonds remain alive.  The price of the contract is the sum of the prices of three \$2,000 bonds, to be redeemed at the ends of 1, 2, and 3 years respectively.

The bond rate is greater than the investors yield rate; thus a premium is in order.  By use of the general method, we have

Price (At the end of 2 years) = \$2,019.135 + \$2,037.171 + \$2,054.172 = \$6,110.478 or \$6,110.48.

A compact expression for the sum of

is

On the other hand

can be summarized by use of a statement which says that

Thus,

≈ \$773.343.  This then gives us

= \$5,337.134 + \$773.343 = \$6,110.478 or \$6,110.48.

By use of the premium method, we have

Price (At the end of 2 years) = \$2,019.135 + \$2,037.171 + \$2,054.172 = \$6,110.478 or \$6,110.48.

By making use of statement (2), we duplicate the same sum by

≈ \$6,110.4775954 or \$6,110.48.

With these special cases aside, we now consider examples that dont seem to exhibit the same format as those represented by (1) and (2).

Example 3.  A \$1,000,000 serial issue of 7% bonds pays coupons on 2/1 and 8/1 and will be redeemed in 5 equal annual installments.  The bonds were issued on 2/1/76.  An insurance company buys all bonds outstanding on 8/1/78, to yield 8%.  Find the price paid.

This serial bond involves five distinct bonds of \$200,000 each.  At the end of 2.5 years (on 8/1/78), three bonds remain alive. The price of the contract is the sum of the prices of three \$200,000 bonds, to be redeemed at the ends of ½, 1½, and 2½ years respectively.

The investors yield rate is greater than the bond rate; thus a discount is in order.  By use of the general method, we have

Price (on 8/1/78) = \$199,038.46 + \$197,224.91 + \$195,548.18 = \$591,811.55.

A compact expression for the sum of

is

≈ \$534,492.39
On the other hand, a compact expression for

is

≈ \$57,319.16.  Thus, we duplicate the result obtained by the general method with

= \$534,492.39 + \$57,319.16 = \$591,811.55.

By use of the discount method, we have

Price (on 8/1/78) = \$199,038.46 + \$197,224.91 + \$195,548.18 = \$591,811.55.
A compact expression for this method is of course given by

≈ \$591,811.5482 or \$591,811.55.

Example 4.  On 3/1/77, a corporation sells a \$2,000,000 issue of 8% bonds paying coupons on 3/1 and 9/1.  The issue is redeemable in equal installments at the end of each 2 years for 10 years.  On 9/1/79, find the value of all outstanding bonds to yield 7%.

This serial bond involves five distinct bonds of \$400,000 each.  At the end of 2.5 years (on 9/1/79), four bonds remain alive.  The value of all outstanding bonds is the sum of the prices of four \$400,000 bonds, to be redeemed at the ends of 1½, 3½, 5½, and 7½ years respectively.

The bond rate is greater than the yield rate; thus a premium is in order.  By use of the general method, we have

Price (on 8/1/78) = \$405,603.274 + \$412,229.088 + \$418,003.1021 + \$423,034.8218
= \$1,658,870.2859 or \$1,658,870.29.

A compact expression for the sum of

is

On the other hand, a compact expression for the sum of

is

Thus, we duplicate the result obtained by the general method with

= \$1,187,907.999 + \$470,962.2863 = \$1,658,870.2853 or \$1,658,870.29.

By use of the premium method, we have

Price (on 8/1/78) = \$405,603.274 + \$412,229.088 + \$418,003.1021 + \$423,034.8218
= \$1,658,870.2859 or \$1,658,870.29.

A compact expression for this method is given by

≈ \$1,658,870.286 or \$1,658,870.29

Attempts on my part to discern empirically a pattern for a general formula from the preceding four examples and several others (notwithstanding their tantalizing germ of an idea/trend) have initially met with frustration after frustration.  Fortunately, a common denominator of these examples is the arithmetic progression format of the addends exponent when added vertically.  From this observation and after going through an accelerated subconscious assimilation, it becomes apparent that the key to linking these examples together lies in decomposing the elements of n (n being the number of interest periods) in the compound interest formula.

Consider an \$M serial issue of j% bonds that pays interest/coupons every m periods and will be redeemed in equal installments.  Suppose that this serial issue is to be sold on an interest/coupon date.  We introduce the following notations:

H = equal installment of the serial bond
r = bond rate
j = yield rate or buyers/investors rate
m = coupon/interest period
b = time from purchase date (which is a coupon/interest date) to 1st or next redemption date (whichever
applies)
d = common difference between redemption dates, i.e., time from 1st redemption date to 2nd redemption
date, time from 2nd redemption date to 3rd redemption date, and so on and so forth.
h = number of terms (i.e., number of serial bond equal installment to be retired) as defined by

where
is the time from purchase date (which is a coupon/interest date) to last redemption date.
= flat price to yield j on an interest date using the general method
= flat price to yield j on an interest date using the premium method
= flat price to yield j on an interest date using the discount method

We then obtain the following results.

where X is

and where Y is

Using (3) and (4),

For the premium method version, we have:

For the discount method version we have:

From the looks of it, the general method form of (5.1) should be sufficient for counterchecking the results one obtains from the traditional methods.  For additional countercheck, depending upon the case on hand, I suppose (6) or (7) might be useful (it does require more steps though).

Revisiting Example 1, we get the following:
Example 1.
(Bond due in ½ year)      = {Bond due in [½ + (1-1)*½] year}
(Bond due in 1 year)       = {Bond due in [½ + (2-1)*½] year}
(Bond due in 1½ year)     = {Bond due in [½ + (3-1)*½] year}

With H = \$5,000, r = .07, j = .08, m = 2, b = ½, d = ½, and h = 3, and using (5) to emphasize and highlight the link that example 1 has with the trend set forth by examples 3 and 4, we get

Notice that (1.05) and (8) are practically the same.

Similarly, using (5.1), we get

The breakdowns of the remaining examples are as follows:

Example 2.
(Bond due in 1 year)     = {Bond due in [1 + (1-1)*1] year}
(Bond due in 2 years)    = {Bond due in [1 + (2-1)*1] years}
(Bond due in 3 years)    = {Bond due in [1 + (3-1)*1] years}

H = \$2,000, r = .07, j = .06, m = 2, b = 1, d = 1, and h = 3

Example 3.
(Bond due in ½ year)        = {Bond due in [½ + (1-1)* 1] year}
(Bond due in 1½ year)      = {Bond due in [½ + (2-1)* 1] years}
(Bond due in 2½ years)     = {Bond due in [½ + (3-1)* 1] years}

H = \$200,000, r = .07, j = .08, m = 2, b = ½, d = 1, and h = 3

Example 4.
(Bond due in 1½ year)    = {Bond due in [1½ + (1-1)*2] years}
(Bond due in 3½ years)   = {Bond due in [1½ + (2-1)*2] years}
(Bond due in 5½ years)   = {Bond due in [1½ + (3-1)*2] years}
(Bond due in 7½ years)   = {Bond due in [1½ + (4-1)*2] years}

H = \$400,000, r = .08, j = .07, m = 2, b = 1½, d = 2, and h = 4

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On a more personal note, Im inclined to call formula (5.1) as the Campos-Hart general formula for the calculation of the purchase price of a serial bond on an interest date in honor of the late Mr. Jose Y. Campos, Founder and Chairman Emeritus of United Laboratories, Inc. (whose genius, vision, and generosity enabled my family and countless others to live prosperous lives, and whose charitable foundation to this day through his successors, namely his children and grandchildren, continues to help those who are in need of assistance) and the late Dr. William L. Hart, mathematician, book author and professor (whose book introduced me to the mathematics of finance).

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While searching the net for existing similar work on serial bonds, I came across a preview of two JSTOR webpages that pursued a different line (so it seemed to me), namely that of approximating the yield rate of serial bonds.  The notation used was similar to those used by an actuary. Unfortunately for me, I wasnt granted access to these two fascinating documents (Im not a member of JSTORs affiliated organizations) to see if the work of the author, a certain Mr. Ralph W. Snyder, has already anticipated the formulas that Ive just derived.  The documents in question are as follows:

Direct Yield Formulas for Serial Bonds
Ralph W. Snyder
Geo. S. Olive & Co.
The Accounting Review, Vol. 30, No. 2 (Apr., 1955), pp. 257-267 (article consists of 11 pages)

Some More Notes on the Bond Yield Problem: Serial Bonds
Ralph W. Snyder
The Accounting Review, Vol. 28, No. 3 (Jul., 1953), pp. 412-421 (article consists of 10 pages)