Re: solution for the interest rate of an annuity

On Fri, 15 Feb 2008 07:50:29 -0800 (PST), pjmercurio@xxxxxxxxx wrote:

On Feb 14, 9:34 pm, David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:
pjmercu...@xxxxxxxxx wrote:
Hi, I recently developed an equation that explicitly solves for the
interestratein the present-value calculation of an annuity. It is in
the form of an infinite convergent series, and was derived using an
inversion technique. Since practical use prohibits calculating
infinite sums, I bounded the terms of the series, and thus found an
upper bound on the error when calculating only a finite set of terms.
Then theinterestratecan be calculated to any desired accuracy by
just taking the required amount of terms of the sum.

I have submitted my paper to several journals, but I cannot seem to
find one that is very interested in such a formula. In comparison to
Newton's Method, my equation takes a lot more iterations. I think it
would be beneficial in practice, however. I was working at BMO Bank of
Montreal at the time of my discovery, when I was trying to implement a
risk assessment database to calculateinterestrates on car loans. The
"solver" tool found in Excel was not capable of calculating many
records at once, whereas my formula managed to calculate rates for
tens of thousands of records in short time.

I am in the process of doing an efficiency analysis on my formula to
help sell it.
I am wondering if anyone has any recommendations of any journals that
would be interested in this combinatorial approach to risk
management.
Any ideas?

You might consider posting your result in this newsgroup -- indeed, in this
very thread. If we see the formula, I hope we can either diagnose why you
can't seem to get people interested in it or else confirm for you that
people actually _should_ be interested in it.

David- Hide quoted text -

- Show quoted text -

In finance, there arise situations in which it is necessary to solve
for the interest rate of an annuity, given the present-value, the
periodic cash flow, and the number of periods. Although this problem
can be solved using iterative methods, it would be beneficial to have
an explicit formula for the interest rate that could be easily
incorporated into a spread*** or database.

Using a reversion technique on the present-value formula for an
annuity, a solution for the interest rate is reached in the form of a
convergent infinite series. By bounding this series from above, the
interest rate can be calculated using only a finite subset of terms,
and is guaranteed to be within any desired degree of accuracy to the
actual rate.


INTRO

We take the PV formula:

PV = A * (1 - (1+i)^(-n))/i

where A is the periodic cash flow, i is the interest rate, and n is
the number of periods.

Attempting to isolate for i, we reach:

i = R - R*(1+i)^(-n) (Eq.2)

where R = A/PV is the ratio of payment to present-value.


METHOD

Lagrange's Reversion Theorem (a form of Lagrange's Implicit Function
Theorem) generates a formal power series expansion of implicitly
defined functions, and can be applied to (Eq.2).

...

By Lagrange, we get:

i = R + sum_k=1_inf {(-1)^k * R^k/k! * partial{R}^(k-1) * (1+R)^(-kn)}

which simplifies to

i = R - sum_k=1_inf {S_k} (Eq.8)

where S_k = R^k * (kn+k-2)!/(k!*(kn-1)!) * (1+R)^(-(kn+k-1))



So that's the formula. In my paper, I go on to discuss bounding S_k to
allow taking a finite number of terms of the sum, resulting in a
solution with error. But the error can be bounded as well.


DISCUSSION

Let d represent the number of terms of the summation to be computed in
the calculation (i.e., the degree of accuracy). Then (Eq.8) can be
written as:

i = R - sum_k=1_d {S_k} - sum_k=d+1_inf {S_k}

and so the error, E, is precisely:

E = sum_k=d+1_inf {S_k}

This is a difficult calculation as well, but using an upper bound on
S_k could achieve a reasonable upper bound for the error.


UPPER BOUND

Stirling's Inequality gives a good upper bound for the k!'s in S_k.

...

By Stirling, we get:

S_k < c*u^k =: U_k

where c = (e^(5/4) * root{n^2 + n})/(n+1)^2 * (1+R)

and u = ((R * (1+1/n)^n * (n+1))/(1+R)^(n+1))^k


ERROR APPROX

Using the upper bound U_k, E can be bounded:

E = sum_k=d+1_inf {S_k} < sum_k=d+1_inf {U_k}

The summation sum_k=d+1_inf {U_k} forms an infinite geometric series
if |u| < 1 in U_k = c*u^k


Here I go into proving 2 claims,

Claim 1: u=1 if and only iff R = 1/n (precisely when i=0) <-- very
interesting!!
Claim 2: |u|<1 for all R > 1/n (precisely when i>0)

So |u|<1 for all situations as long as the cash flow is actually
compounding, so then the geometric series converges for all values as
long as there is an interest rate above 0%.

So the error can be bounded above.


E < sum_k=d+1_inf {U_k} = c * (u^(d+1))/(1-u)

implies =>

d < (ln(E) + ln(1-u) - lnc)/ln(u) - 1

So there exists some integer d such that E < sum_k=d+1_inf {U_k}.
Since E decreases as d increases, then by taking d to be

d > (ln(E) + ln(1-u) - lnc)/ln(u) - 1

will guarantee an error of less than E.


That's pretty much it.
Some examples will show that for large values of n (eg. 300 months,
like a mortgage) this series converges VERY fast to several digits of
accuracy: within 6 iterations usually.

But for smaller values of n (eg. 60 months, like a car loan) this
series converges quite slower (could require several hundred
iterations) but converges nonetheless.


Let me know what you think. I hope my syntax is okay, I kinda just
made it up on the spot. And I hope all my brackets make sense.
Thanks for reading!

Peter

Two links for you to check out:
<http://oakroadsystems.com/math/loan.htm#LoanInterest>
<http://oakroadsystems.com/math/loan.htm#Cantrell>

Since David C. has already posted in this thread, I expect he will
post a follow-up with his observations on what you've written above.

.

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