Be it credit, loan or mortgage, before taking on any form of debt, it is very important to:
– ask yourself whether you absolutely need to do so and whether you can definitely afford to make the repayments. Always plan how you will pay money back before spending.
– work out exactly how much it will cost you in fees and interest so that you are not hit with any nasty surprises that will leave you financially weakened later on.
How lenders calculate:
When buying a house, you may borrow the purchase price from a bank (mortgage), and agree to pay the money back in regular equal instalments over a certain time. If the amount of the loan is A, the interest rate is r and the number of repayments is n, what is the amount of the monthly repayment R?
Let us look at the problem from the point of view of the bank (lender): Their loan A must be balanced by the present value (PV) of the repayments.
The PV of the first repayment R is R(1+r)-1
The PV of the second repayment R is R(1+r)-2
and, in general
The PV of the nth repayment R is R(1+r)-n
The total PV of all the repayments is therefore
R(1+r)-1 + R(1+r)-2 + ….. + R(1+r)-n
which is a geometric series of n terms with first term R(1+r)-1 and common ratio (1+r)-1. The sum of the series is:
{R(1+r)-1 [1 – (1+r)-n]}/1 – (1+r)-1
= R[(1+r)n-1]/r(1+r)-n
Equating this to A, we obtain
R = Ar(1+r)n/(1+r)n-1
If you, for example, take a loan of € 100,000 which is to be repaid over 25 years in equal monthly instalments at an annual interest rate of 7%.
The monthly repayment will be calculated as follows:
We have A = 100,000; r = 0.07/12 (since 7% is the annual interest rate); and n = 300 (since the repayments are monthly and 25 years has 300 months). So,
R = {100,000 x 0.07/12 x [1 + (0.07/12)]300}/[1 + (0.07/12)]300-1
= 706.78
The repayments are made up of two components: interest on the outstanding balance, and an amount of capital repayment. The table below shows the breakdown of the first three repayments.
Interest Capital Repayment Outstanding Capital
1 583.33 123.45 99876.55
2 582.61 124.17 99752.38
3 581.89 124.89 99627.49
Initially, the monthly repayment is largely interest, with only 17.46% of capital repayment. The interest component falls and the capital repayment rises as the payments are made.
After 20 years, what proportion of the capital has been repaid? The answer could be found by continuing the above table for 240 lines, an easy exercise on a spreadsheet. A more elegant solution is to realise that after 240 payments, the outstanding capital is precisely the present value of the remaining 60 payments:
R(1+r)-1 + R(1+r)-2 + ….. + R(1+r)-60
Summing the geometric series, we get: R/r[1-(1+r)-60]
= 706.78/(0.07/12){1-[1+(0.07/12)]-60}
= 35693.80
So more than a third of the capital is still outstanding, even after 80% of the repayments have been made.
Suppose that, for purely numerical neatness, it is decided to round the monthly repayment up from €706.78 to €710. How does this affect the number of repayments?
From the formula relating A, r, n and R
A = {R[(1+r)n-1]}/r(1+r)n
we obtain
n = [ln R – ln (R – Ar)]/ln(1+r)
and hence, putting R = 710, A = 100,000 and r = 0.07/12, we obtain
n= 296.35
So a substantial saving of more than 3 repayments is made with a very small “rounding up”. Instead of 300 payments of €706.78, totalling €212034, 296.35 payments of €710 are made, totalling €210408.50. The difference is €1625.50!
Rounding the repayments up to €720 per month saves a further 10 repayments at the end.