Is simple interest related to the actuarial method?

Question

I'm trying to understand different methods of calculating interest on installment loans. I keep coming across the terms "simple interest" and "actuarial method". They seem to be related but I haven't found a clear explanation. Here's an example of something I read on another website that has me confused:

Simple interest is computed on the actual balance outstanding on the payment due date. Precomputed interest is calculated on the original principal balance. The interest is added to the original principal balance and divided by the number of payments to determine the payment amount. ... the Rule of 78s is no longer an acceptable method of accounting for loan income. The acceptable method for accounting for loan income is the actuarial method.

Answer 1

Definitions of simple interest differ, e.g. the naïve version here: What is 'Simple Interest'.

Simple Interest = P x I x N

where
      P is the principal
      I is the periodic interest rate
      N is the number of periods

(an example is included below)

However, by contrast, insofar as "Simple interest is computed on the actual balance outstanding on the payment due date.", this is what is used by standard (actuarial) methods.

The simple interest (actuarial) method also differs from the Rule of 78s.

As described here: Rule of 78s - Precomputed Loan

Finance charge, carrying charges, interest costs, or whatever the cost of the loan may be called, can be calculated with simple interest equations, add-on interest, an agreed upon fee, or any disclosed method. Once the finance charge has been identified, the Rule of 78s is used to calculate the amount of the finance charge to be rebated (forgiven) in the event that the loan is repaid early, prior to the agreed upon number of payments.

So taking a simple example: a 12 month loan repaid early, after 9 months.

principal  s = 995.40
no. months n = 12
int. rate  r = 0.03 per month

By simple interest (actuarial) methods, using formula 1 (derived below).

repayments d = r (1 + 1/((1 + r)^n - 1)) s = 100

total int. t = d n - s = 204.60

However if the loan is repaid early, after 9 months, using formula 2.

x = 9
total int. t = ((1 + r)^x - 1) s + (d (1 - (1 + r)^x + r x))/r = 187.46

So the interest saved by repaying early is

204.60 - 187.46 = 17.14

If this was calculated by the Rule of 78s, with the finance charge taken as the total interest due for the 12 month loan.

precomputed interest f = 204.60

precomuputed loan = s + f = 955.40 + 204.60 = 1160

interest forgiven = f (3/78 + 2/78 + 1/78) = 15.74

So in this case it disadvantages the borrower to use the Rule of 78s.

Note the finance charge calculated by the naïve simple interest method in the aforementioned link: What is 'Simple Interest'

Simple Interest = P x I x N = 955.40 x 0.03 x 12 = 343.94

This is a long way from 204.60, but then the demo interest rate is quite high, accentuating the disparity. The naïve simple interest method is otherwise disregarded in this answer.

Demonstrating the interest calculations graphically, it can be observed that the interest payments calculated for months 10, 11 & 12 by the Rule of 78s are less than the simple interest/actuarial calculations.

Formulae derivations

Answer 2

The actuarial method, in relation to a loan (as you mentioned) and its repayment is generally associated with repayment of loan at reducing balance (the method though can be used for many other purpose, on some cases on appreciating balance).

For example, let's say you have borrowed $1000 at an interest rate of 10% to be re-payed in 20 years with one yearly constant payment (which works out to about about $116 per year, put the above values in the calculator, this also gives detailed breakdown).

The idea is at the end of 1st year, when you pay $116, $100 goes towards settling the simple interest of 10% on $1000 for 1 year, the balance $16 is reduced from $1000. So your next years principal is $1000-$16=$984.

In the 2nd year end, you will again pay $116, where now your simple interest on capital outstanding would be 10% of $984, which is $98.4, your principal repayment would be $116-$98.4=$17.6, your principal outstanding at the end of 2nd year would be $984-$17.6=$966.4

Observe the reducing balance, if you keep paying the same amount every year it would finally result in full amortization of the loan in 20 years.

The concept of simple interest is used to calculate the interest on remainder balance.