Interest on Principal
Money makes money, that's what lending does. There are typically two kinds of interests - simple interest and compound interest.
Scroll down to Continuous Compound Interest to skip the brief introduction
Simple Interest
A fixed amount (principal) is loaned and a fixed payment is made for keeping that amount of money (simple interest). The accumulation of simple interest is a linear function of the time taken to return the principal.
I = Interest
p = Principal
r = rate of interest (per annum)
n = number of years
I = p * r * n
Here, the principal and rate of interest are constant and interest is a linear function of time.
Compound Interest
The simple interest does not vary with time. It does not take into account the effects of inflation. In time, continuing to pay the interest will be much cheaper than repaying the principal. Taking a 5000 dollars loan and paying an interest of 1% (50 dollars) every year for 50 years since 2000 might be something in the year 2000 and hardly anything in 2050.
Compound interest introduced the concept of charging interest on unpaid interest in addition to the principal. This calculation creates a sense of urgency because more time means more interest.
A = Payable amount
p = Principal
r = rate of interest (per annum)
n = number of years
A = p * ( (1+r)^n)
Here, it is assumed that interest is paid in the end (only when the whole loan has been repaid). In other cases, the the paid amount is to be reduced from the current principal.
Continuous Compound Interest
In the current compounding scheme, the addition of unpaid interest to the principal amount happens at intervals that is agreed upon in the beginning of the transaction. It could be annually, quarterly, monthly, fortnightly, weekly, etc. In case of a monthly term, one can benefit or lose significant money by paying one day early or late. This leads to pointless delay (if the term is 30 days, then a delay of 29 days) and unjustified gain/loss to either parties. Hence, the continuous compound interest. Here is the theory behind it.
The interest is calculated every instant and it is added to the principal simultaneously. In this way, being late matters even if it is weeks or hours. This way, the importance of time is non-negotiable.
Calculation
A = Payable amount
p = Principal
r = rate of interest (per annum)
n = number of years
A = p * (e^(r*t))
e =
x! | ||
Inv | sin | ln |
π | cos | log |
e | tan | √ |
Ans | EXP | xy |
( | ) | % | AC |
7 | 8 | 9 | ÷ |
4 | 5 | 6 | × |
1 | 2 | 3 | − |
0 | . | = | + |
Euler's constant
Derivation
A = Payable amount
p = Principal
r = rate of interest (per annum)
n = number of years
di = instantaneous interest
dt = infinitesimal time period
di = p*r*dt
(Interest for time dt)
pn = p + di
(pn is the new principal after time dt)
Hence, the differential equation is:
dp/dt = p*r
dp/p = r*dt
∫ dp/p = ∫ r*dt
(Integrating from p from p to A and t from 0 to t)
A = p * (e^(r*t))
Graphs and examples
The data given in the example graphs below can be easily analysed to get a fair picture.
However, the same rates of interest cannot be used as the principal will increase quickly in the continuous form because the interest is continuously being added to the principal. Hence, some moderation is required in order to not affect the system. In the example, the graph of a normal compound interest at 7% p.a. aligned very well with continuous compound interest at 6.75% p.a.
Remember, this method is used not to make money faster but to maintain a degree of uniformity.
You could also trying tweaking the values to see how it affects the data easily using the digital graph using this link: https://www.desmos.com/calculator/qxn988aq9f
JoeTech
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