SIMPLE INTEREST CALCULATIONS

In financial theory one often encounters one of two concepts in the calculation of interest payments. These are the concepts of 'Simple Interest' and ' Compound Interest'. They use different approaches to calculating the amount of interest that is due on a given loan. In order to help describe these two methods, we must first define some commonly used terms.

Principal
The amount initially borrowed / lent between two counterparties is termed the principal amount and is often denoted by the letter 'P' in mathematical notation.

Present value
The amount of value that one attaches today to an amount due to be received at some point in the future is termed the 'present value' of the amount due. This amount is sometimes referred to by the letter 'P' in mathematical notation which can cause confusion with the letter 'P' which is also sometimes used for 'principal'. At other times the letters 'PV' are used to denote present value. As an example, a man who is due to receive £110 from a borrower in one year's time may give that amount a present value of £100. In other words he would be indifferent to having £110 in one year's time and £100 today. Of course this preference implies a rate of interest (in this case 10%) that is sufficient to recompense the man for parting with his holding of £100 cash for one year.

Future value
The amount of value that one attaches to a given sum available today at some future time is termed the future value of that given sum. In the previous example, the future value is £110.

Interest
The amount of money charged by a lender / paid by a borrower for the use of money over a given period of time is referred to as interest. Thus a man who borrows £100 for one year and repays the £100 plus an extra £10 to the lender, has paid £10 interest. Interest is often expressed as a percentage of the principal. Here the rate of interest would be 10% per year since £10/£100 = 10%.

Discount rate
If one was to issue a lender of money with a piece of paper promising to pay him £110 in one year's time, the lender may give one £100 today as a loan in trust that the promise will be honoured. The lender has loaned £100 at 10% interest but there is another way of looking at the loan. One might say that the lender had bought the piece of paper at a 'discount' to the face value of £110. The amount of discount is £10 and the rate of discount is thus :

£10 / £110 9.091%.

The lender pays approximately 9.091% less than face value for the paper, in other words the discount rate agreed by the lender is 9.091%. We can see that interest rates and discount rates are not the same even though the amount of interest paid on the loan is the same in each case. The concept and terminology used in discounting is widely used throughout financial markets and we shall return to it in due course.

A note on interest calculations
It is important to remember for the purpose of mathematical calculation that an interest rate of, for example, 10% is represented by a fraction of 0.1 For example, when calculating the total repayment amount on a loan of £100 at 10% over one year, the principal must be multiplied by 1 + the interest rate as a fraction in order to calculate the total repayment (i.e. total repayment = £100 * 1.1 = £110).

Simple interest
When only the principal earns interest over the life of a loan, then the transaction is said to be a simple interest transaction. Thus :
F = P (1 + rt)
where,
F = Future value
P = Principal
r = interest rate as a fraction
t = number of periods of loan transaction.

So in simple interest, a loan of £100 for two years at 10% per year has a repayment value (or a future value) of F being calculated as :
F = £100 (1+.1 ´ 2)
= £100 (1.2)
= £120

Since
F = P (1 + rt)
we can see that,
P = F / (1 + rt)

So,
100 = 120 / (1.2) in the above example.

This description involved an assumption that simple interest was charged on the loan.

Where discount rates are used under assumptions of simple interest, then an amount borrowed on a discount basis would have the following present value:
P = F (1 - dt)
where d is the discount rate on repayment (i.e. face) value.

So, a bill promising to repay £100 in 3 months' time with a discount rate of 10% would have a theoretical present value of :
P = £100 (1 - .1 3/12)
= £100 (.975)
= £97.50

The interest rate equating to this discount rate is :
2.5 / 97.5 or 0.025641 or 2.5641% (over three months). Expressed as an annual rate, this becomes 4 * 2.5641% or 10.25641% per year

Ways of calculating simple interest
There are nine theoretical ways of calculating an amount of interest, for example on a deposit or accrued interest on a loan. Of these nine, only five are commonly used in practice. At this stage we will merely examine their application in simple interest markets. The different methods arise because of different assumptions as to how many days there are in a month and how many days there are in a year. This may seem a surprising source of uncertainty, nevertheless it is argued that some conventions developed out of a desire for ease of calculation whilst others were maintained out of a desire for accuracy.

1) Assumed number of days in a year
a) Exact Simple Interest or '365' year basis calculates interest using a 365 day year. Thus if I borrow £100 at 10% per year for 50 days, the interest payable is ;£100 (0.1 ´ 50/365) = £1.37
b) Ordinary Simple Interest or '360' year basis calculates interest as if a year has only 360 days. Thus if I borrow £100 at 10% per year for 50 days under this convention, interest payable is ;£100 (0.1 ´ 50/360) = £1.389
c) Actual Simple Interest or 'ACTUAL' year basis calculates interest using the actual number of days in a loan period multiplied by the number of interest payments in the year to produce the assumed number of days in a year. It is commonly used in certain bond market calculations. For example, a semi-annual bond (one paying two coupons per year) can display a period between coupons of between 181 and 184. Hence in this case the assumed number of days in a year can vary between 362 and 368 days.

2) Number of days in the loan period
a) Exact Time or 'ACTUAL' day count, counts the exact number of days for which an amount is borrowed. Only one of the start and end dates of the loan is counted. Thus, 27th August to 5th September is 9 days.
b) Approximate Time or '30' day count, assumes that each month has only 30 days. Only one of the start and end dates of the loan is counted. Any dates of the 31st that appear within the loan period are disregarded with the one exception stated below. Thus 27th August to 5th September is 8 days under this convention. If a loan begins on a 31st, change this date to a 30th. If the loan ends on a 31st, do not change this date unless the loan started on a 30th or a 31st in which case change the end date to a 30th.
c) '30E' day count, works in the same manner as the '30' day count except for the treatment of start and end dates which are on the 30th or 31st. Here, everything is as with the '30' day count treatment except that if the last day of the loan period is a 31st change this date to a 30th.

3) Nine different bases for interest calculation
The three different methods of day count combined with the three different methods of year count, produce nine theoretical ways of calculating the interest amount due on a loan. The methods are sometimes summarised with the day count as a numerator and the year basis as a denominator. The asterisks show those methods used in practice;

ACTUAL/365*   30/365   30E/365
ACTUAL/360*   30/360 *    30E/360 *
ACTUAL/ACTUAL*   30/ACTUAL    30E/ACTUAL

Which of the above methods is used depends upon the instrument and currency concerned. In our earlier simple examples then, we should now appreciate that the calculation of interest payable will be slightly more complicated than previously described. For instance a £100 one year loan at 10% in a leap year under the Actual/360 convention will yield an interest amount of £100 ´ .1 366/360 = £10.166.

COMPOUND INTEREST CALCULATIONS

Under compound interest, at stated intervals the interest due on a loan is converted into principal and thereafter interest is charged on this extra amount of principal as well as the original principal amount. The process of converting interest into principal is known as 'compounding'. The period of time that elapses between each compounding is known as the 'compounding period'. Thus a 'semi-annual compounding period' describes a twice yearly conversion of interest into principal.

Nominal and effective rates of interest
The use of the compound interest concept gives rise to the phenomenon of nominal and effective rates of interest. The nominal rate of interest to be charged on a loan is the stated rate that applies, always expressed as an annual rate. The effective rate is the actual amount of interest paid, also expressed as an annual rate.

For example, a 10% nominal rate with a semi-annual compounding period, when applied to a principal amount of £100 produces the following interest amount due after one year (ignore day and year count bases):

i) Interest calculated after 6 months is 5% of £100 or £5
ii) Compound interest of £5 into principal. Principal outstanding after 6 months is £100 + £5 or £105
iii) Interest calculated after second 6 months on outstanding principal of £105 is 5% of £105 or £5.25. Hence, the total interest payable is £10.25. The effective rate of interest here is 10.25% and the nominal rate of interest is 10%.

A simpler way of calculating the effective rate from a nominal rate, given the compounding period, is as follows :
r_e = (1 + rⁿ / c)^c - 1
re = effective rate of interest
rn = nominal rate of interest
c = number of compounding periods in nominal rate rn

For the above example,
re = (1 + 0.1 / 2)² - 1 = (1 + 0.05)² - 1 = 0.1025 or 10.25%

Continually compounded rate of interest
The greater the number of compounding periods for any one nominal rate, the higher the effective rate of interest. The maximum effective rate obtainable for any one nominal rate is given by ;
r_e = er - 1

Here, r_e is the maximum effective rate obtainable from a given nominal rate r, sometimes referred to as the continually compounded rate of interest and 'e' is the constant approximately equal to 2.7183. This procedure assumes that the number of compounding periods is infinitesimally large. The continually compounded rate of interest for a nominal rate 10% is :e^0.1 - 1 = 2.7183^0.1 - 1 = 0.10517 or 10.517%

PRESENT VALUE, FUTURE VALUE AND DISCOUNTING

In financial mathematics exams, one is often asked questions of the type 'how much will I have in the bank in two year's time if I deposit £100 at 10% per year today?' Well, on an Actual/365 basis, assuming that a) the 10% is a guaranteed per year rate for this year and next year and b) that interest is paid once yearly and c) that neither year is a leap year, then on today's date two years hence I should have in my bank account:

£100 * 1.1 * 1.1 = £121

£121 is the future value of £100 principal in two years' time at 10% per annum compounded annually.

The rate '10% per annum compounded annually' is more usually referred to as 10% annual. The £100 principal is itself the present value of £121 in two years' time at 10% annual. The process of finding the future value of the principal is sometimes referred to as 'future valuing', whist the process of finding the present value of an amount to be received in the future is referred to as 'present valuing' or 'discounting'. These processes can be described by formulas in a similar manner to those outlined in the simple interest section.

Future value under compound interest
F = P(1 + r)^t

Where,
r = nominal rate of interest per compounding period
t = number of compounding periods between present and future value date
F and P are defined as before

Thus our above described loan payoff can be calculated as follows
F = £100(1.1)² = £100(1.21) = £121

The same loan but with a 10% semi-annual compounding period would give F as:

F = £100 (1.05)⁴ = £100 (1.2155) = £121.55

Notice the formula uses four periods of 5%.

Present value under compound interest

Given that F = P(1 + r)^t we can say that :

P = F/(1 + r)t
= F (1 + r)-t   since 1/a = a -1

So we can calculate the present value of £121 in two years' time at 10% annual as ;

P = 121 (1 + .1)^-2
= 121 (1.1)-2
= 121 (1 / 1.12 )
= 121 (0.82644)
= 100

The amount 0.82644 is a 'two period discount factor at 10%' since it is the amount by which we must multiply the future value in order to obtain the present value of £100. Each different period and interest rate combination has an associated discount factor. The discount factor is simply 1 minus the discount rate for the entire period. Here a period of two years at 10% interest gives £21 interest. The future value is £121 and thus the discount rate for two periods is 21/121 0.173553. The discount factor is then 1 - 0.173553 0.8264.