Louis’ Law: “One and one does not necessarily make 11”.

The Technical Bits

This section is for those who wish to delve deeper into the maths and to understand the formulae connected with most financial questions.

Firstly, it will be helpful to list some of the key financial formulae that can be used in computer programming or when using spreadsheets, many of which can also be worked on a normal hand calculator.

The principle questions about a loan or investment can be answered using a maximum of just five key mathematical variables: -

Loan repayments
When the payment made to a lender includes both capital and interest, it is referred to as a re-payment. To calculate the repayment figure (pmt) per period, given interest rate, initial & final capital and the total number of periods: -

pmt = pv x  i / (1 - (1 + i) ^-n ) - fv x i/((1 + i) ⁿ  -1) ---------------- (1)

where i is the fractional interest rate per period, n is the total number of periods and fv is any future value:  these variables are the same ones listed in the previous paragraph.  x means "multiply by".  If the loan amortises to zero, which is normal for a repayment mortgage, fv is zero so the second term can be ignored and then:

pmt = pv x  i / (1 - (1 + i) ^-n ) --------------------------------------- (2)

Note that (1 + i) ^–n is the same as 1 / (1 + i) ⁿ   

Instead of using the ÷ sign for division (which might be confused with a – or a minus sign) it is clearer to use  / as a division sign.

So a loan of £10,000 over five years @ 9 % per annum nominal interest compounding monthly (i.e. 0.75 % per month true rate over 60 months) requires a monthly payment of ; -

10,000 x 0.0075 / (1 - (1 + 0.0075) ^-60) = £207.58 per month. 

The payment calculated with this formula is always the payment made per period (in arrears) but which may not necessarily be the same as the desired payment frequency.

So, to calculate a loan where repayments are made monthly, but interest is compounding annually (i.e a typical Building Society mortgage), formula (2) above will calculate the annual payment.  This is because these formulae all require the payment frequency to be the same as the compounding frequency.  Having calculated the annual payment, you simply divide it by 12 to get the monthly payment. 

Suppose a Building Society lends £50,000 over 25 years and charges a borrower 6% pa compounding annually.  What is the monthly repayment? 

The annual payment is: -

50,000 x (0.06 / (1 - (1 + 0.06) ^-25) = £3,911.34 per annum.

Dividing by 12 gives us £325.94 per month. 

These answers can be checked out using the “Loan Comparator” spreadsheet supplied. 

Another general-purpose monthly repayment formula that takes a nominal rate and produces a monthly repayment is: -

pmt.pm = pv x  Nom / 12 / (1 - (1 + Nom/p) ^{– (y x p)} ) ------------ (3)

where p is the number of periods or rests per annum, Nom is the nominal rate as a fraction (ie the true rate per period x p),  y is the total term in years so
y x p is the total number of periods.  Whilst formula (2) is more fundamental, formula (3) is actually more useful for mortgages as it automatically deals with different compounding frequencies and always produces a monthly answer.

Calculating true rate per annum from the nominal rate
In general, the kernel within most formulae for compound growth calculations is (1 + i) ⁿ where i is the fractional interest rate per period and n is the total number of time periods and repayments. 

£1 will grow in 12 months @ 1% per month by the factor (1 + 1/100) ¹² which works out to 1.12682.  This is because we first add on 1%, i.e. 1.01 times the initial capital.  Next month we add on another 1% of the new, higher amount now making it 1.01 x 1.01 of £1.  In twelve months that will be 1.01 x 1.01 x 1.01 and so on 12 times.  This is the same as saying 1.01 to the power of 12, written as 1.01 ¹² .  If we were dealing with simple interest the growth factor would be 1.01 times 12 instead of 1.01 to the power of 12.

To convert a monthly true rate of interest to an annual true rate, use the formula below where m is the monthly rate expressed as a fraction: -

True Rate pa = (1 + m) ¹² - 1

The answer is expressed as a fraction so multiply by 100 to get the percentage.

Using this formula, 1% per month is equivalent to a true rate of 12.6825 % per annum.  If you do not have a calculator that performs powers, you can use an ordinary calculator and repeat the multiplication 12 times.

The more general formula used to calculate the true rate from any nominal rate is

True Rate = (1 + Nom / p) ^p – 1  -------------------------------- (4)

where Nom is the nominal rate per annum (expressed as a fraction) and p is the number of periods per annum.  This is the same formula used earlier but with Nominal rate per annum being used instead of the true rate per period.

Suppose a lender charged 8% per annum compounding quarterly so p = 4.  The true rate is (1 + 0.08 / 4) ⁴ – 1 = .082432 = 8.2432 % pa true.

This formula applies to any loan or investment where any payments, if made, are at the same frequency as the interest compounding frequency.  So interest-only loans, like endowment mortgages, qualify.

But the true rate calculation is different if repayments are made say monthly but interest is compounded annually.  It is then necessary to look at the actual cash flows and use a trial and error method, an iteration process or a special spreadsheet function, as used in the example in Figure 8.

Natural growth
Note that in the example shown in Figure 4, the true rate for 12% pa nominal is tending to maximise at 12.749685.  The higher the frequency you compound at, the nearer true rate will approach a maximum, but never quite reach it.  There is an analogy here to natural growth.  Living cells might grow continuously rather than in monthly spurts, and compounding continuously is perhaps the most natural way for all compound interest calculations.   But it is not so easy to write it all down so as to follow the calculations period by period, as required in a simple monthly statement.

For pure maths students it is interesting to note that when i is 100% and the period approaches infinity, the formula (1 + 1/p) ^p is approaching 2.718281.  This is one of the few magic, irrational numbers in mathematics like π (pie), the circumference of a circle divided by the radius.  This one is called e (After Euler- a great classical mathematician) and is essential in many areas of pure mathematics and is the base for natural logarithms. 

The growth analogy is relevant for cells which double continuously (ie a nominal growth rate of 100%).  e is like π and is an indeterminate number where the decimal figures go on for ever to whatever accuracy you desire. 

Interestingly e is linked to π by another strange number called i, which is the square root of –1 and is an imaginary number.  (This i should not be confused with the i for interest used in financial calculations).

The remarkable formula that connects all three with the two other special numbers, 0 and 1 is

e ^{π i} + 1 = 0.  But that is another, fascinating story.