Commissions and Costs of Amortising Loans

Many sources of revenue and costs in finance are proportional to the balance of amortising loans over time. This includes:

commissions and servicing fees incurred for originating and managing loans;
funding costs for loans.

For example the nominal value of the commissions payable on an amortising loan might be 1% of the balance for the life of the loan. As the loan amortises, as the balance reduces, the trail commission reduces proportionately. The commission might be payable base don the monthly balance over the first 3 years.

where:

represents the commission rate per period, for example 1 per cent per annum;
represents the discount factor for period t: that is ;
represents the expected balance of the loan in period t (as a percentage of its opening balance ()), it is calculated based on the amortising loan balance projection as: , where the payment amount is and is the loan interest rate;
from 0 to T-1, as it is often customary to pay commmissions based on balances at the start of each reference period.

The balance of an amortising loan over an horizon is the net effect of the interest accumulated per month and the amount repaid, a loan making initial payments of 8% per annum, and accumulating interest at 5% per annum, has a balance that reduces at 3% in the first year and progressively more each period thereafter.

The sum of the balances of an amortising is the sum of a geometric series, the discounted sum of the balances is the same underlying amortisation but with the value of accumulated interest reduced by the discount rate. This is evident from the following derivation:

Then the sum of the balance over the term is:

Applying the sum of a geometric series this becomes:

A commission or cost over the term is then simply:

An easy to calculate closed form solution.

Sum of a Geometric Series

A closed from solution to the sum of geometric series ( ) of a fixed length T can be derived as follows:

Or over the interval 0 to T-1:

Example calculation

Select the underlying loan parameters, including expected prepayment speed:

A prepayment speed of 1% per annum means that the borrower is expected to repay an additional 1% of the scheduled payment each year. A negative prepayment speed means that the borrower is expected to pay less than the scheduled payment, for example if the borrower is expected to miss payments or pay interest only for a period of time.

The selected loan details imply:

Scheduled payment: ; Scheduled term: yrs.
Expected payment: ; Expected term: yrs.

The expected future balance of the loan:

Select the commission rate payable on the balance and the maximum term of the commission (they are often limited to the first few years after origination)::

The discount rate reflects an opportunity cost of the funds, it should be selected according to the purpose of the calculation and the benchmarks for the revenue stream. Some reasonable values are:

the risk free rate (eg Australian government bonds);
the cost of foregoing receiving the commission upfront (which might be the returns on reinvesting funds, eg capital or alternate funding cost);
the effective interest rate of the loan.

The selections above, including the expected prepayment rate, imply:

Nominal value of commissions (ie undiscounted): .
Present value of commissions (ie discounted): .