Amberle Dec 1918

The equal annual repayment has to cover both the principal and the interest.

If money is borrowed at 8% then the PV of the repayments and the interest will always be the amount borrowed. (This is true however the borrowing is repaid – for example they could repay 17,500 a year and then all of the interest at the end of 4 years, or they could just pay interest each year and then the principal at the end of 4 years, and so on. However they repay the PV of the payments will be equal to the PV.)

So if they are paying X per year (to cover interest and principal) for 4 years, the PV = X x the annuity factor for 4 years at % and must be equal to $70,000. Therefore the equal amount per year X = 70,000 / the 4 year annuity factor at 8%.

During the first year the amount owing is 70,000 and so the interest is 8% x 70,000 = 5,600.
At the end of the first year they will repay 21,135 (the equal annual repayment) and so the amount owing at the start of the second year is 70,000 + 5,600 – 21,135 = 54,465. So the interest in the second year is 8% x 54,465 = 4,357.

And so on each year.

(The (3) at the end of the 4th year is simply due to rounding and is irrelevant. Without rounding it would be 0 because 21,135 each year will repay the principal together with interest.)

Suppose you borrow $1,000 with interest at 10% for 3 years.

What you might do is repay nothing for 3 years and then pay $1000 with 3 years interest, which is $1,331 in 3 years time.

Another way is that you might just pay interest of $100 at the end of each of the 3 years and repay the principal of $1,000 at the end of 3 years.

These are just two ways in which you could repay the money (and interest has obviously to be paid at some stage). If you discount the repayments (principal and interest payments) then the PV will always be the amount borrowed of $1,000. Check it yourself and see 🙂

Given that overall the repayments have to cover both the principal and the interest, if they pay in equal instalments over 3 years, then the equal instalments are part interest and part repayment of principal.

Using the same logic as I gave in my previous reply, equal amounts each year are by definition an annuity and therefore the equal amount repaid each year for 3 years is in this example $1,000 divided by the 3 year annuity factor at 10%. (If you have forgotten about discounting annuities then look back at the Paper MA lectures on discounting, because annuity factors are revision from Paper MA (and Paper FM).

As far as adding the interest in calculating the balance owing at the start of the following year, then this is nothing to do with discounting but is basic accounting. Always interest on any borrowing is calculated on the amount that was owing at the start of the year, and the amount owing at the end of the year is the amount that was owing at the start, plus interest for the year, less however much had been repaid at the end of the year.

(You wrote in your last post that you understood that the present value of the repayments of the principal will equal to amount borrowed. This cannot possibly be true! The whole point of discounting is to remove interest, and the PV of just the repayments of $1,000 in my short example here (or the repayments of the $70,000 in the original example) cannot possibly be equal to the amount borrowed. Check for yourself and see 🙂 )

If you borrow 1,000 for 3 years then the interest is automatically compound interest. So the amount owing at the end of 3 years is 1,000 x (1.10^3) = 1,331.

The PV of 1,331 in 3 years time is 1,331 x (1/1.10^3) = 1,000.

You must watch the Paper MA (was F2) lectures on interest and on investment appraisal, because this is revision from Paper MA.