FM for Actuaries

170 CHAPTER 5
where A and B are given in (5.13) and (5.17), respectively. Solving the above
equation for various values of k we obtain the following results:
k 12 24 36 48 60
r ∗ (%) 5.369 5.085 4.972 4.906 4.867
If the loan is redeemed at maturity, the annual effective rate of interest is (1 +
0.04853
12
) 12 − 1 = 4.9624%. However, if the loan is redeemed after 1 year, the
effective rate of interest is (1 + 0.05369
12
) 12 − 1=5.5503%. In the case the loan
involves a further penalty for early redemption, the difference in the borrowing
costs may be even higher.
5.8 Summary
1. The balance of a loan can be computed using the prospective or retrospective
methods. The prospective method computes the present value of the
future installments required to pay back the loan. The retrospective method
computes the accumulated value of the installments that have been paid, and
subtracts this figure from the original loan amount with accumulated interest.
2. The amortization method of redeeming a loan computes the interest incurred
since the last installment. The current installment is used to offset the interest
with the remaining part applied to reduce the principal.
3. The sinking fund method of redeeming a loan involves two part payments.
There is a portion that serves the interest calculated on the original principal.
Another portion is deposited into a sinking fund that accumulates with
interest to redeem the principal. The interest credited to the sinking fund
may differ from the interest charged to the loan. If the interest credited to
the sinking fund is the same as the interest charged on the principal, the total
installments in the two methods are the same.
4. If the interest credited to the sinking fund is less than the interest charged on
the loan, then the total amount of payments in the sinking fund method is
higher than the total amount of payments in the amortization method.
5. To compare the costs of different loans, we may use the repayment installments
or the rates of interest. When loans are repaid by installments of the
same frequency and timing, they can be compared by the required repayment
installments. When the frequencies or timing of the repayments are different,
it is appropriate to compare the loans by their rates of interest.