How do you calculate for interest under leases using sum of digit method?

In effect, a finance lease is a means of acquiring an asset by “borrowing” money from the lessor, and paying it back over a fixed term, with interest.

Here is an example I can use to illustrate the process.

On1 January2010, XYZ leases an asset valued at $250,000 at the start of the lease term. The lease term is5 years, and there will be5 payments of $60,000 each. XYZ’s year end is31 December. The lease payments will be made (a) annually in advance or (b) annually in arrears.

First determine the amount of interest in total. This is the total amount of the lease payments, minus the value of the asset.5 lease payments of $60,000 per year makes a total of $300,000. The asset is valued at $250,000 so the total interest is $50,000.

Next, determine how many finance periods there are – that is the number of payments to be made using “borrowed money”. This determines how you allocate the total interest of $50,000 over the years of the lease, and will be different depending on whether payments are to be made in advance or in arrears.

For (a), payments are to be made on1 January2010,2011,2012,2013 and2014. The lease comes to an end on31 December2014, but as the final payment is to be made on1 January2014, no money will be being borrowed during that accounting period, and no interest will be accruing in the final year. Therefore, money is only being borrowed for4 years – so there are4 “finance periods”.

Next calculate the “sum of the digits”. The formula is n(n+1)/2, where n is the number of finance periods. The calculation is therefore4(4+1)/2 =10. For a short lease, you could also calculate it as4+3+2+1 =10, but that would be a very laborious method with a longer lease, hence the formula would help.

Next calculate the interest to be allocated to each financial year. This is the digit for that year, divided by the total digits. The first year has the highest amount of interest, and the last year has the lowest amount. Thus, in2010 the calculation is $50,000 x4/10 = $20,000; in2011 it is $50,000 x3/10 = £15,000; in2012 it is $50,000 x2/10 = $10,000; and in2013 it is $50,000 x1/10 = $5,000. Adding those together you can check that they total $50,000. In the final year of the lease, there is no interest allocated.

Your question only asked about the calculation of interest, but you will also need to calculate the amount of capital of the loan that has been repaid during the period, and the outstanding balance at the end. The capital repayment is $60,000 minus the amount of interest you have allocated to that year – so in2010 capital repaid is $40,000 ($60,000-$20,000) and in2011 it is $45,000 ($60,000-$15,000) and so on.

For (b), where payments are made in arrears (i.e. at the end of each year), money will be being borrowed throughout the whole term, thus there will be5 lease periods. This will affect the calculation of the sum of digits, and therefore the allocation of interest. Using the formula given above, the sum of digits will now be5(5+1)/2 =15, and the proportions to be used in each of the5 periods will be5/15,4/15,3/15,2/15 and1/15 (of $50,000). So interest in2010 will be5/15 of $50,000 = $16,667, and capital repaid will be $43,333 ($60,000-$16,667).