All I am looking for is some formulas or good websites to figure the following:

If I have a 4 year amortizing lown and I borrow 1,oooo initally and
repay in four equal annual year-end payments and interest rate is 8%
how do I show the annual payment is 301.92?

Also, how do I show that the loan balance of of one year (778.08) is
equal to the year-end payment of 301.92 times the 3 year annuity
factor.

Again, not looking for answers....just formulas or tools to assist me. Thanks

Hi!!

Let me start defining the topic.
You want to find four equal annual payments that equals a present
value of $1,000 at a rate of 8%. This is clearly a present value
problem. Suppose that you know the amount of the future payments or
cash flows, call them CF (they are all equal). These future cash flows
must have a present value of $1,000.
These type of cash flows are known as annuities, and the formula for
the present value of the annuities is:

PV = CF/r * [1 - 1/(1 + r)^n]

where:
PV = present value (principal)
CF = cash flows (annual payment or annuity)
r = interest rate
n = number of years

The above formula derives from the following definition of present
value of future cash flows:
"The present value of a sum of future cash flows is equal to the
summation of  the present value of each cash flow":
PV = Sum(i=1 to n)[CFi/(1+r)^i]

In your case you know the PV (it is the amount borrowed), you only
need to find the value of the cash flows, from the first PV formula we
can isolate CF to find:
CF = PV * r / [1 - 1/(1 + r)^n]} =
   = $1,000 * 0.08 / [1-1/(1.08)^4]} =
   = $80 / 0.26497 =
   = $301.92


For your second question consider that the loan balance of one year is:
1 year Balance = PV*(1+r) - CF/r * [(1+r)^1 - 1] =
               = PV*(1+r) - CF/r * r =
               = PV*(1+r) - CF =
               = CF/r * [1 - 1/(1 + r)^4] * (1+r) - CF =
               = CF/r * [(1+r) - 1/(1 + r)^3] - CF/r * r =
               = CF/r * [1 + r - 1/(1 + r)^3 - r] =
               = CF * 1/r * [1 - 1/(1 + r)^3] =
               = $301.92 * [3 year annuity factor]

An interpretation of the above result is that in one year (after the
first payment) you still owe the next three payments of $301.92, the
PV of these three future payments is:
PV_1 = CF/r * [1 - 1/(1 + r)^3] = $301.92 * [3 year annuity factor],
and the amount that you still owe in one year (just after the firtst
payment) is the 1-year balance.

For more formulas and their derivation see the following great page, I
strongly recommend you to read it:
"Loan or Investment Formulas":
http://oakroadsystems.com/math/loan.htm


Note that most formulas for annuities can be easily adapted to shorter
period payments by replacing the terms (1 + r)^n by (1 + r/q)^(n*q),
where q is the number of periods per year (for example if there are
monthly payments q=12).

Other useful pages are:
"Present Value of an Annuity":
http://www.getobjects.com/Components/Finance/TVM/pva.html

"Annuity Payments":
http://www.getobjects.com/Components/Finance/TVM/pmt.html

"Future Value of an Annuity":
http://www.getobjects.com/Components/Finance/TVM/fva.html


"Annuities":
http://www.netmba.com/finance/time-value/annuity/



I hope that this helps you, feel free to request for a clarification if you need it.

---

Request for Answer Clarification by baseball2-ga on 17 Jul 2005 15:02 PDT

You certainly put in much effort into this! I really appreciate it and
especailly like
the additional web-sites you provided me. I do have on question in
regards to my second question. If I am reading the formula right this
is what I should be doing.

778.08 (balance after one year) / .o8 * 1-1 % ( 1 + .08) ^ 3 = 301.92.
I don't seem to be coming up with this. Maybe I have the wrong balance
after one year...but that is not your problem! Just wanted to clarify
my interpetation of the formula. Thank you

---

Clarification of Answer by livioflores-ga on 17 Jul 2005 15:41 PDT

Hi!!

1 year Balance = PV*(1+r) - CF/r * [(1+r)^1 - 1] =
               = $1000*(1.08) - $301.92/0.08 * [1.08-1] =
               = $1080 - $301.92 =
               = $778.08

On the other hand we had found that:
1 year Balance = CF * 1/r * [1 - 1/(1 + r)^3] = $778.08
where 1/r * [1 - 1/(1 + r)^3] is the 3-year annuity factor.

Then, if we isolate CF (the payment) from the above:
CF = $778.08 / {1/r * [1 - 1/(1 + r)^3]} =
   = $778.08 / {1/0.08 * [1 - 1/(1.08)^3]} =
   = $778.08 / 2.5771 =
   = $301.92

The only difference with your proposed solution is the 1/r factor (you
use mistakely r).

Hope that this clarify the answer, if not do not hesitate to request
for further assistance on this.

Regards,
livoflores-ga