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All_Is_Number

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Registered: 2006-07-10

Posts: 258

How much would you need to invest?

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 The payments continue being paid annually until the balance is withdrawn.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?

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All_Is_Number

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Registered: 2006-07-10

Posts: 258

Re: How much would you need to invest?

Nobody has solved this yet? Is a hint in order?

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Ms. Bitters

Member

Registered: 2008-07-31

Posts: 19

Re: How much would you need to invest?

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 The payments continue being paid annually until the balance is withdrawn.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the blackened portion of your problem statement is defective in the sense that the balance or fund will never be fully withdrawn or totally depleted.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.

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All_Is_Number

Member

Registered: 2006-07-10

Posts: 258

Re: How much would you need to invest?

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.

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.).

Your answer is correct. 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….

(In hindsight, I probably should have made the annual payments start at $150,000, in order to get a nice round number for the initial investment.)

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Ms. Bitters

Member

Registered: 2008-07-31

Posts: 19

Re: How much would you need to invest?

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.

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?

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All_Is_Number

Member

Registered: 2006-07-10

Posts: 258

Re: How much would you need to invest?

What is the balance at the end of the nth year?

If no one else solves this within a week or so, I will.

Perhaps you could crosspost (with mods' permission, of course) your formulas from this thread in the Formulas section, since you took the time to put them in LaTeX and explain them.

Last edited by All_Is_Number (2008-10-23 15:41:25)

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All_Is_Number

Member

Registered: 2006-07-10

Posts: 258

Re: How much would you need to invest?

What is the balance at the end of the nth year?

The balance at the end of the nth year is

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Ms. Bitters

Member

Registered: 2008-07-31

Posts: 19

Re: How much would you need to invest?

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.

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
http://books.google.com/books?id=IKWKPRAu8CAC
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.

Last edited by Ms. Bitters (2008-10-31 04:33:20)

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mathsyperson

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Registered: 2005-06-22

Posts: 4,900

Re: How much would you need to invest?

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.

ie.
www.mathisfunforum.com becomes www.mathisfunforum.com

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

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George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: How much would you need to invest?

Post 5 is correct. Both the formula and the justification.

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Ms. Bitters

Member

Registered: 2008-07-31

Posts: 19

Re: How much would you need to invest?

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.

ie.
www.mathisfunforum.com becomes www.mathisfunforum.com

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

Thanks.

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All_Is_Number

Member

Registered: 2006-07-10

Posts: 258

Re: How much would you need to invest?

Hmmm … I entered the data into a spreadsheet, and calculated the account balance at the end of each year, both before and after payment is made. To simplify the initial amounts, I multiplied the initial balance by 3 to change $833,333.33 to $2,500,000, and did the same with the initial annual payment, which changed from $50,000 to $150,000. (This change will not affect the rate of growth.) I also calculated the annual growth of a $2,500,000 deposit, at 4% effective interest.

The balance of the perpetuity after each annual payment was identical to the value of the $2,500,000 deposit at the same time t. So, I have to stick with my original answer:

(I'll try to attach a pdf of the spreadsheet, but I've never been able to upload attachments on this site.)

Edit to add: the upload did not work.

Last edited by All_Is_Number (2008-10-31 09:49:43)

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You can shear a sheep many times but skin him only once.

All_Is_Number

Member

Registered: 2006-07-10

Posts: 258

Re: How much would you need to invest?

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.

Aha! There's the misunderstanding.

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.

This is because every year, 60% of the interest (i=0.1) earned during the previous year is used to make the annual payment, giving the perpetuity account a net growth rate of 4%.

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

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Ms. Bitters

Member

Registered: 2008-07-31

Posts: 19

Re: How much would you need to invest?

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.

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.

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All_Is_Number

Member

Registered: 2006-07-10

Posts: 258

Re: How much would you need to invest?

My apologies.

No need to apologize.

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