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Net and gross - tax allowances
If interest is paid on a deposit then tax is usually payable on the income. Normally interest rates are quoted gross, i.e. without taking tax into account.
For example, if the deposit of $100 earns 10% effective per annum then $10 is earned in the first year, but T% of this has to be paid in tax, i.e. reducing it by T%.
Thus the interest actually received at the end of one year is:
tax paid interest=PV*I*(1-T)
You should be able to see that the effect of having to pay tax reduces the effective rate from I% to I*(1-T)%.
An interest rate that doesn’t take tax into account is called a gross rate and one that does is called a net rate.
To convert from gross to net use the formula:
net = gross*(1-T)
to convert from net to gross use:
For example, if a building society offers a gross rate of interest of 4% and tax is paid at 20%, then this is equivalent to receiving 0.04*(1-0.20) or 3.2% gross.
There is always the question of which rate should be quoted in connection with an interest bearing investment - gross nominal, net nominal, gross effective or net effective.
The bank or building society always prefers to quote the highest possible rate and this is, of course, the effective gross rate, also called the gross CAR (Compound Annual Rate). In all situations the net nominal rate should be converted to the net effective rate before being converted to the gross effective rate or gross CAR.
Of course what matters to the investor is the net effective rate because this reflects what is actually received on the investment. However, in comparing interest rates all that really matters is that the same type of rates are compared.
Calculating and compounding
There is one last complication to be considered.
When a balance on deposit can vary from day-to-day calculating the interest to be added for a longer compounding period presents something of a problem.
For example, suppose a bank current account offers 1% per month with interest added monthly - what balance should be used to calculate the monthly interest?
As the balance can change each day it seems more reasonable to calculate the interest daily using a rate of I%/365 but only add the interest once a month.
This is how nearly all bank accounts work, including overdrafts, with interest calculated on the balance daily but only being added monthly or at a lesser frequency. (Of course the interest rate for an overdrawn account will be different from the one applied to one in credit.)
This doesn’t alter any of the calculations described earlier and the effective rate is still given according to the compounding period and not according to period used to calculate the interest.
It is only when the interest is added to the account that matters. If the balance in the account is constant then the daily calculation of interest produces the same result as calculating and adding the interest at the end of each compounding period.
In the case of banks, the daily balance may not be what appears on the statement. The reason is that a cheque is credited to your account when it is paid in but only cleared for interest calculations three or so working days after that. Trying to keep an exact check on a daily bank balance can be very tricky as only the bank will actually know when a cheque has cleared for interest.
This use of daily interest calculation also raises the question of how many days there are between any two given dates. Dates and date arithmetic is the subject of a later chapter.