Plekhanov Russian University of Economics
Mathematics in Finance
Topic: Simple Interest and Discount
Author: Alexander V. Bezrukov
Statistics dept., Ph.D.
INTEREST. TIME VALUE OF MONEY
By lending an asset or capital for a certain period of time, the owner of this capital
(the lender) expects to receive a reward from the borrower for using the assets.
The lending capital can be a loan, a deposit, or any other kind of borrowed or
invested value of money or finance.
The sum of this reward (which is the assets owner’s profit) depends on three
factors: the amount of assets lended; the term (the time period) of lending, and the
lending rate or percent.
Hence the interest rate can be regarded as the reward for the use of an asset or
capital, paid by the borrower to the owner of the capital.
SIMPLE INTEREST ACCUMULATION
Let us assume the following notations:
P – the principal value of assets;
S – the future (accumulated) value of assets; the value at date of maturity;
i – simple interest (decimal fraction);
I – the accumulated interest value; the assets owner’s sum of profit;
n – due period.
In case of simple interest, it is logical that the accumulated interest value can be
found as
I = Pni
Hence, the future value is calculated as
S = P + I = P + Pni = P(1 + ni)
SIMPLE INTEREST METHODS
For the simple interest, there are used three different methods:
1) exact interest with exact maturity date (365/365); most commonly used by many
countries' central banks and large commercial bank enterprises; widely used, for
example, in the U.S. and Great Britain (ACT/ACT)
2) Ordinary interest with exact maturity date (ACT/360 or 365/360), the so-called
Banker's Rule, most commonly used in external loans; for internal loans widely
used in France, Belgium, Switzerland. This method gives a slightly increased result
over the exact rule.
3) Ordinary interest with approximate maturity date (360/360). Most commonly
applied when there is no apparent need for exact calculations, e.g. in preliminary
calculations. The commercial banks of, for example, Germany, Denmark and
Sweden use it in their practice.
DISCOUNT TRANSACTIONS
In the financial practice there is commonly encountered a task reversal to
interest rate.
There is given the value to be paid out S (the future value) after a period n,
and it's necessary to find the amount of loan (P).
This situation can arise, for example, in the process of contract terms
development.
It also becomes necessary in the situations when the accumulated interest is
paid ahead, i.e. at the moment of granting the loan.
In these cases it is said that the value S is discounted; and the deducted
value is called the discount.
MATHEMATICAL DISCOUNT
There are two discount rules, based on the rate used:
1) mathematical rule of discount;
2) banking (commercial) rule of discount.
Mathematical rule of discount is the solving of the task reversal to interest
rate accumulation. The task is formulated as follows: what present loan value
must be granted to receive at the date of maturity the value S, given the
interest rate i.
Therefore,
P = S / (1+ni)
BANK DISCOUNT
The essence of this operation is the following. The bank or any other financial
institution before the date of maturity for a note, or any other payment obligation,
buys it from the owner by the value which is less than the maturity value for the
note. Thus, the bank discounts the note, buying it at a discounted value. Upon the
date of maturity the bank receives the money for the note, containing the profit
which is the discount
The percent value for using the loan, which is the discount, is charged for the
maturity value S.
In this operation the discount rate d is used.
Obviously, the sum value of discount equals S*n*d; if d is the discount per annum
then n is measured in years.
Hence,
P = S – Snd = S ( 1 – nd)
Thank you for attention!
Questions