Editing Interest (section)

===Other formulations===
The outstanding [[balance (accounting)|balance]] ''B<sub>n</sub>'' of a loan after ''n'' regular payments increases each period by a growth factor according to the periodic interest, and then decreases by the amount paid ''p'' at the end of each period:

:<math>B_{n} = \big( 1 + r \big) B_{n - 1} - p,</math>

where

:''i'' = simple annual loan rate in decimal form (for example, 10% = 0.10. The loan rate is the rate used to compute payments and balances.)
:''r'' = period interest rate (for example, ''i''/12 for monthly payments) [http://www.fdic.gov/regulations/laws/rules/6500-1650.html#6500226.14]
:''B''<sub>0</sub> = initial balance, which equals the [[principal sum]]

By repeated substitution, one obtains expressions for ''B''<sub>''n''</sub>, which are linearly proportional to ''B''<sub>0</sub> and ''p'', and use of the formula for the partial sum of a [[geometric series]] results in

:<math>B_n = (1 + r)^n B_0 - \frac{(1+r)^n - 1}{r} p</math>

A solution of this expression for ''p'' in terms of ''B''<sub>0</sub> and ''B''<sub>''n''</sub> reduces to

:<math>p = r \left[ \frac{(1+r)^n B_0 - B_n}{(1+r)^n - 1} \right]</math>

To find the payment if the loan is to be finished in ''n'' payments, one sets&nbsp;''B''<sub>''n''</sub>&nbsp;=&nbsp;0.

The PMT function found in [[spreadsheet]] programs can be used to calculate the monthly payment of a loan:

:<math>p=\mathrm{PMT}(\text{rate},\text{num},\text{PV},\text{FV},) = \mathrm{PMT}(r,n,-B_0,B_n,)</math>

An interest-only payment on the current balance would be

:<math>p_I= r B. </math>

The total interest, ''I''<sub>''T''</sub>, paid on the loan is

:<math>I_{T} = np - B_0. </math>

The formulas for a regular savings program are similar, but the payments are added to the balances instead of being subtracted, and the formula for the payment is the negative of the one above. These formulas are only approximate since actual loan balances are affected by rounding. To avoid an underpayment at the end of the loan, the payment must be rounded up to the next cent.

Consider a similar loan but with a new period equal to ''k'' periods of the problem above. If ''r''<sub>''k''</sub> and ''p''<sub>''k''</sub> are the new rate and payment, we now have

:<math>B_k = B'_0 = (1 + r_k) B_0 - p_k. </math>

Comparing this with the expression for B<sub>k</sub> above, we note that

:<math>r_k = (1 + r)^k - 1</math>

and

:<math>p_k = \frac{p}{r} r_k. </math>

The last equation allows us to define a constant that is the same for both problems:

:<math>B^{*} = \frac{p}{r} = \frac{p_k}{r_k}</math>

and ''B''<sub>''k''</sub> can be written as

:<math>B_k = (1 + r_k) B_0 - r_k B^*.</math>

Solving for ''r''<sub>''k''</sub>, we find a formula for ''r''<sub>''k''</sub> involving known quantities and ''B''<sub>''k''</sub>, the balance after ''k'' periods:

:<math>r_k = \frac{B_0 - B_k}{B^{*} - B_0}</math>.

Since ''B''<sub>0</sub> could be any balance in the loan, the formula works for any two balances separate by ''k'' periods and can be used to compute a value for the annual interest rate.

''B''* is a [[scale invariant]], since it does not change with changes in the length of the period.

Rearranging the equation for ''B''<sup>*</sup>, one obtains a transformation coefficient ([[scale factor]]):

:<math>\lambda_k = \frac{p_k}{p} = \frac{r_k}{r} = \frac{(1 + r)^k - 1}{r} = k\left[1 + \frac{(k - 1)r}{2} + \cdots\right]</math> (see [[binomial theorem]])

and we see that ''r'' and ''p'' transform in the same manner:

:<math>r_k=\lambda_k r</math>
:<math>p_k=\lambda_k p</math>.

The change in the balance transforms likewise:

:<math>\Delta B_k=B'-B=(\lambda_k rB-\lambda_k p)=\lambda_k \, \Delta B </math>,

which gives an insight into the meaning of some of the coefficients found in the formulas above. The annual rate, ''r''<sub>12</sub>, assumes only one payment per year and is not an "effective" rate for monthly payments. With monthly payments, the monthly interest is paid out of each payment and so should not be compounded, and an annual rate of 12·''r'' would make more sense. If one just made interest-only payments, the amount paid for the year would be 12·''r''·''B''<sub>0</sub>.

Substituting ''p''<sub>''k''</sub> = ''r''<sub>''k''</sub> ''B''* into the equation for the ''B''<sub>''k''</sub>, we obtain

:<math>B_k=B_0-r_k(B^*-B_0)</math>.

Since ''B''<sub>''n''</sub> = 0, we can solve for ''B''*: 

:<math>B^{*} = B_0 \left(\frac{1}{r_n} + 1 \right).</math>

Substituting back into the formula for the ''B''<sub>''k''</sub> shows that they are a linear function of the ''r''<sub>''k''</sub> and therefore the ''λ''<sub>''k''</sub>:

:<math>B_k=B_0\left(1-\frac{r_k}{r_n}\right)=B_0\left(1-\frac{\lambda_k}{\lambda_n}\right)</math>.

This is the easiest way of estimating the balances if the ''λ''<sub>''k''</sub> are known. Substituting into the first formula for ''B''<sub>''k''</sub> above and solving for ''λ''<sub>''k''+1</sub>, we obtain

:<math>\lambda_{k+1}=1+(1+r)\lambda_k</math>.

''λ''<sub>0</sub> and ''λ''<sub>''n''</sub> can be found using the formula for ''λ''<sub>''k''</sub> above or computing the ''λ''<sub>''k''</sub> recursively from ''λ''<sub>0</sub> = 0 to ''λ''<sub>''n''</sub>.

Since ''p''&nbsp;=&nbsp;''rB''*, the formula for the payment reduces to

:<math>p=\left(r+\frac{1}{\lambda_n}\right)B_0</math>

and the average interest rate over the period of the loan is

:<math>r_\text{loan} = \frac{I_T}{nB_0} = r + \frac{1}{\lambda_n} - \frac{1}{n}, </math>

which is less than ''r'' if&nbsp;''n''&nbsp;>&nbsp;1.