FM for Actuaries

Loans and Costs of Borrowing 185
let S m =1+2+···+(m − 1) + m. Thus, m is the term of the loan in
months. In the Rule of 78, the borrower is assumed to pay m S m
of the total
interest in the 1st month, m−1
1
S m
of the total interest in the 2nd month, ···,
S m
of the total interest in the last month. Thus, the interest payment is linearly
decreasing, and the amount of interest paid in the kth installment, denoted
by I k ,is
I k = Total amount of interest paid × m +1− k
S m
= Lnr × m +1− k
S m
.
(a) Show that
m∑
I k = Lnr.
k=1
(b) Show that if you decide to pay off the loan at the kth payment, the
interest rebate, which is defined as the unpaid interest saved, is
R =
(12n − k)(12n − k +1)Lrn
.
12n(12n +1)
Note that if we apply the above rule to a 1-year loan, m =12and S 12 =78.
That is why this method is called the “Rule of 78”.