Perpetuity

I owe both Morgan Tud and reinout-g and the whole forum a big apology. I am very sorry, I did not read what reinout-g wrote properly.

I did not read and comprehend this line;

"The holy book of solutions has spoken of the mighty €39.84!"

If I had I never would have looked for someone else to offer yet another solution.  I would have taken the defeat graciously and looked harder to see where i had erred.  I must have looked very blinkered in my arrogance.  I am quite embarrased by the way that I behaved.

Thank you Morgan Tud for being so gracious.

Let me see.

Let the effective 4 monthly interest rate be r

Now I need to find the effective 3 yearly interest rate equivalent.  

There are 9 four monthly periods in 3 years so  the effective 3 yearly rate is (1+r)⁹ -1

Consider any perpetuity.

Let the amount at the beginning of a period be A and the period interest rate be i  then the interest earned in any period is Ai.  Now if this is to pay out into perpetuity then the regular period payment must equal the period interest. 

That is  A+Ai-R=A

Ai=R  (I have used A instead of PV)

now I want to look at the 3 year perpetuity.  

Initial investment = €32

after 3 years = €32[ (1+r)⁹ -1+1] = €32[ (1+r)⁹ ]  this is where the perpetuity actually begins. so this is A

A= €32 (1+r)⁹

i = (1+r)⁹ -1

$$\begin{array}{rll}
Ai&=&R\\\\
\left[32(1+r)^{9}\right]\left[(1+r)^{9}-1\right]&=&10 \\\\
32(1+r)^{18}-32(1+r)^{9}-10&=&0\\\\
16(1+r)^{18}-16(1+r)^{9}-5&=&0\\\\
\mbox{let } x=(1+r)^{9}&&\\\\
16x^2-16x-5&=&0 \mbox{ where } x > 0\\\\
\mbox{the quadratic formula gives me }x&=&1.25\\\\
(1+r)^{9}&=&1.25\\\\
1+r&=&1.25^{1/9}\\\\
r&=&1.25^{1/9}-1\approx 0.025103648\\\\
\end{array}$$

Okay now lets look at the 4 monthly perpetuity

Ai=R, A=unknown, i=r=etc, R=€1 (is that what you mean by 1 unit?)

$$\begin{array}{rll}
A&=&\frac{R}{i}\\\\
A&=&\frac{1}{1.25^{1/9}-1}\\\\
A&=&39.83484719\\\\
A&=& 39.83
\end{array}$$

 PV=€39.83

I think that is right but I haven't done any checking.

Can you do that reinout?  I might but it is 1:15am here.  You might be a little less tired than I.

I don't have the time to check it today, but i will tomorrow 

Thank you Melody 

The present value of a perpetuity-immediate (not perpetuity due) be C/i

The present value of a perpetuity-immediate of 10 is 10/i where i be the interest of land's userers. This be the interest for 1 annum, not 3. For 3, the effective  interest  be (i+1)^3 – 1.

Here the value of v is set to the inverse of

$$\ ( i+1): v=\frac{1}{(i+1)}.$$

The units of the Kingdom be Euros and marked as thus: €

$$v^{3}.\frac{10}{(i+1)^3-1}=32 \ Be\ the\ present\ value\ of\ the\ first\ perpetuity$$

$$\ Multiply \ by \ (i+1)^3 \ yieldeth \rightarrow \\
\hspace{150pt} \ 10v^3.(i+1)^3 = 32(i+1)^6 -32(i+1)^3$$

$$\\ In\ that\ v^3*\ (i+1)^3\ = \ 1; \rightarrow 10=32(i+1)^6-32(i+1)^3 \\
\ and \ becometh\ \rightarrow \ 0=32(i+1)^6-32(i+1)^3-10$$

With the magick of the quadratic there be a number that cometh forth. Aye! Two numbers, one be less than nothing. Sir CPhill was hither sent on a quest for nothing, and he findeth nothing, too, or not. This be less than nothing; I shall leave it for the gods, and taketh the weightier part.

$$\ (i+1)^3 = 32 \pm\frac {\sqrt{2304}}{64}=1.25$$

$$\ (i+1)^3=1.25 \
\\ and\ the\ user's\ rate\ be\
\\\sqrt[3]{1.25}-1= 0.0772 \ per\ \ annum\
\\ And\ for\ four\ moons\ be\ \sqrt[3]{1.0772} -1
\\The\ present\ value\ of\ the\ perpetuity-immediate \ be
\\\frac{1}{\sqrt[3]{1.0772} -1} \ =\ 39.84.\ In\ the\ units\ that\ be\ reinout-g's\ Euro$$

                                                                                                                    €

Presented to and for her Royal Highness

by sevice of

Morgan Tud M^-1

You do realise though that you are a medievil Doctor of Medicine.  Not a Dr of Mathematics. ?!?!

I think that thee has made an error Dr Tud,

The effective annual interest rate be of no consequence. 

For one the 4 monthly effective rate is important and for the other the 3 yearly effective interest rate is important.

It is common for the starting point to be the effective yearly rate and work the other 2 out from there but really this only increases the complexity of the question and it is of no importance.

For now at least I am sticking by my answer!  I have checked my logic and it appears to be correct.

In conclusion, I thank thee for your presentation but I think that thou hath erred.

Lady Guinevere.

Hie, Morgan Tud......!!!  M'Lady looketh none too pleased with thy bold reply......!!! She hath already condemned Sisyphus to boulder pushing.......for perpetuity.....

I thank thee for both thou answers, but I believe the answer of Morgan Tud, guardian of the irrealms, rooted of the negative one, is correct.

Okay, lets look at this from my answer backwards

r=1.25^(1/9)-1  This is the answer that I got for the effective 4 monthly interest rate.

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Now, the effective 3 yearly interest rate is (1+r) ^9-1=  This simplifies to 0.25 

We have a perpetuity paying €10 at the end of each 3-year period with the first payement at the end of year 6 with a present value PV1 = €32.

PV=32euros, After 3 years this has grown to 1.25*32= 40euros, at the end of the next 3 years 10 euros will be earned in interest.  This is then paid out as the regular payment.

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Now the 4 monthly interest rate is 1.25^(1/9)-1  and the regular payment is 1 euro

R=PV*interest rate 

PV=1/[(1.25^(1/9)-1] = 39.83 euros

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So far so good.  Is it that you do not accept that 25% every 3 years is equivalent to 1.25^(1/9)-1 every 4 months

Say you invest $P and leave it for 3n years

okay the 4 monthly one will grow to P[1.25^(1/9)-1+1]^(3n*3)=P[1.25^(1/9)]^(9n)=P[1.25^n]

The 3 yearly one will grow to P[1.25^n]

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Reinout, you can accept Morgan Tud's answer if you please, he is a fine Doctor and he also knows many curses, but I do not believe his answer is correct.

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I do owe you an apology Morgan Tud.  Our answers are now the same.  Mine would have be correct from start had there been but an extra 4 months in a year.  A 16month year. umm  Maybe I would have more time to get things done this way.  

My method is still preferable to yours!  Why worry this reinout lad with yearly interest when it has nought to do with the task at hand!

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Melody, I do not wish to offendeth thee, but for some reason I cannot bring my mind to what thou arth doing.

For some reason, what Morgan unravels concurs with my mind and this is validated by one irrefutable truth;

The holy book of solutions has spoken of the mighty €39.84!

I wish I could forclose thee thou fault in thine carefully conceived answer, but I cannot.

May the protector of the irrealms, rooted of the negative one help us define your irregularities.

The reason that you do not like mine is that I have not worked out the annual rate.  And this is upsetting to some people.  What i would like to see is the setting out to show that Morgan Tud's answer is correct (as I have done with mine).  When I am shown that I am wrong I am always happy to accept it.  

I am reasonably  confident that my own answer is correct.  

Morgan Tud has given an annual interest rate I am not sure if he has given any others.  If i get time I will play with his answer as i have done with my own.

Hi Reinout-g and Morgan Tud,

We need more input on this question so I have posted it in a different forum.

This is the address.

http://mathhelpforum.com/business-math/229722-effective-interest-perpetuities.html

Let us see if this can bring a resolution to our impasse.   

I haven't seen you menion that forum in a while Melody .

I see that you really want to get to the bottom of this .

I just realised that in mine i worked on the basis of have 12 lots of 4 months in 3 years.  Even my checking was done on this premise.  I will rework it with 9 lots of 4 months in 3 years and see if it agrees with Morgan Tuds answer.  Little numbers always get me!  Sorry.

Aye, Sir CPhill, one always risks the loss of thy head, if ye challenge the sovereignty of Royalty. Fortunately, Her Majesty preferth honest descent to insincere sucking-up. Though one doeth well to haveth such skill!

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It be a shame there be not a value greater than zero in Sisyphus’ boulder. For even an infinitesimal amount would cause him great wealth after an eternity. Or for the one who seeketh the contents.

By your leave

Morgan Tud M^-1

Forgive the lateness of my return Your Highness. There be a plague in a hamlet that perplexeth me and mine colleagues. Yea, many masses of persons dance for days then fall into sleep, and waketh not to eat nor drink.

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Aye, My Lady an Doctor of Medicine I be …  Still, the magick of math and number doeth fascinate. There be a day when such sorcery leadeth the way to cures for sickness of persons and livestock and pets.

* ---* Why worry this reinout lad with yearly interest*---*

Any known rate of usury may be translated to the annum. This do comforteth many. Especially they that be in the debtor's prison.

My solution be based on the sums with the differences of a geometric ratio. Ye be true, this be more complicated in its observation, but it showeth a uniqueness.

Her Highness hath taken this to the "nines," and her solution be more eloquent. That be why she be the queen and I be a servant.

In your Majesty’s service

Morgan Tud M^-1