Interest - Formulae

Formulae

The balance of a loan with regular monthly payments is augmented by the monthly interest charge and decreased by the payment so

,

where

i = loan rate/100 = annual rate in decimal form (e.g. 10% = 0.10 The loan rate is the rate used to compute payments and balances.)

r = period rate = i/12 for monthly payments (customary usage for convenience)

B₀ = initial balance (loan principal)

B_k = balance after k payments

k = balance index

p = period (monthly) payment

By repeated substitution one obtains expressions for B_k, which are linearly proportional to B₀ and p and use of the formula for the partial sum of a geometric series results in

A solution of this expression for p in terms of B₀ and B_n reduces to

To find the payment if the loan is to be finished in n payments one sets B_n = 0.

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

An interest-only payment on the current balance would be

.

The total interest, I_T, paid on the loan is

.

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. The final payment would then be (1+r)B_n-1.

Consider a similar loan but with a new period equal to k periods of the problem above. If r_k and p_k are the new rate and payment, we now have

.

Comparing this with the expression for B_k above we note that

and

.

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

and B_k can be written as

.

Solving for r_k we find a formula for r_k involving known quantities and B_k, the balance after k periods,

Since B₀ 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* one gets a transformation coefficient (scale factor),

(see binomial theorem)

and we see that r and p transform in the same manner,

The change in the balance transforms likewise,

which gives an insight into the meaning of some of the coefficients found in the formulas above. The annual rate, r₁₂, 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·B₀.

Substituting p_k = r_k B* into the equation for the B_k we get,

Since B_n = 0 we can solve for B*,

.

Substituting back into the formula for the B_k shows that they are a linear function of the r_k and therefore the λ_k,

This is the easiest way of estimating the balances if the λ_k are known. Substituting into the first formula for B_k above and solving for λ_k+1 we get,

λ₀ and λ_n can be found using the formula for λ_k above or computing the λ_k recursively from λ₀ = 0 to λ_n.

Since p=rB* the formula for the payment reduces to,

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

,

which is less than r if n>1.