## Calculation of Interest and Annuities

JAIIB - Accounting & Finance
Module A : Business Mathematics and Finance

Calculation of Interest and Annuities

1.  Calculation of Simple Interest (SI) and Compound Interest (CI)

a.  Simple Interest (SI) / Flat rate interest is calculated only on the principal amount (initial amount).
Formula for calculating SI is :
SI = Principal (P) x Rate (R) x Time (T)  [N.B - here Rate is expressed in fraction, e.g., 0.05, instead of 5 %]

E.g., Principal amount of Rs. 5,000 with 10 % annual simple interest rate, will earn total interest in 3 years = 5,000 x 0.1 x 3 = Rs. 1,500;
whereas, it will earn total interest in 3 months = 5,000 x 0.1 x 3/12 = Rs. 125

b.  Compound Interest (CI) / Cumulative Interest is calculated both on the principal amount and prior interest earned.
Formula for calculating CI is:
CI = Amount (A) - Principal (P),
where, A = P(1+R/n)^nT  [N.B - here Rate is expressed in fraction, e.g., 0.05, instead of 5 %]

Note - n is the number of compounding per year, e.g., for yearly compounding n=12/12=1, for half-yearly compounding n=12/6=2, for quarterly compounding n=12/3=4, for monthly compounding n=12/1=12, etc.

E.g., Principal amount of Rs. 5,000 with 10 % annual compound interest rate, will earn total interest in 3 years = [ 5,000 x (1 + 0.1) ^ 3 ] - 5,000 = Rs. 1,655

But, if it is, say, quarterly compounding, then CI for 3 years is = [ 5,000 x (1 + 0.1/4) ^ (4 x 3) ] - 5,000 = Rs. Rs. 6724.44 - 5000 = Rs. 1724.44

2.  Calculation of Equated Monthly Installments (EMI)

Equated Monthly Installments (EMI) is the amount a borrower pays every month for his loan. It is a combination of interest and principal payment, where the total monthly amount is such calculated, that it remains constant (i.e., equated) throughout the repayment period.

Formula : EMI = [PR(1+R)^n] / [(1+R)^n - 1], (to remember EMI=PRX/(X-1), where X=(1+R)^n)
Here, P=Principal amount,
R=interest (compound) rate per month (NOTE: if annual intt rate is 10 %, then R=10/(12x100), i.e., conversion to monthly is necessary)
N=no. of monthly installments
X=compounding factor=(1+R)^n

For example, suppose a loan of Rs. 5 lakh is to be repaid over 30 years in EMI, with 10 % annual interest. Here, the monthly EMI for this loan will be calculated as follows -
i.  First calculate R (convert to monthly intt. rate) = 10/(12x100) % = 0.00833 (approx)
ii.  Then calculate no. of monthly installments, n = 30 years x 12 = 360
iii.  Calculate compounding factor X = (1+R)^n = (1+0.00833)^360 = 19.8138 (approx)
iv.  EMI = PRX / (X-1) = (500000 x 0.00833 x 19.8138) / 18.8138 = 4386.38 = 4387 (approx)

EMI for this loan will be approx Rs. 4,387.

3.  Fixed and Floating Interest Rates

a.  Fixed Interest rate on a loan, means that the interest rate will be fixed / constant, either for the entire term of the loan, or for part of the total term of the loan.

b.  Floating / Variable Interest rate on a loan, means that the interest rate can be changed (increase or decrease) depending on the market conditions.

Comparison -
• If market condition is favorable, then the borrower enjoys reduced rate of interest for Floating interest loans, whereas he cannot enjoy the benefit if he avails Fixed interest loans.
Similarly, if market condition deteriorates, he is not affected, if he avails Fixed interest loans, whereas he has to pay more for floating interest loans.
• Fixed interest loans bring a sense of certainty and security. It is good for those borrowers, who are good at budgeting.
• Often fixed interest rates are higher than the floating interest loans.

4.  Calculation of Annuities

Before starting with Annuities, first understand the time value of money.
Present Value (PV) of money is the value at the current point of time. Whereas, a given sum of money "worth" more at a specified time in the future, assuming a certain interest rate. This is known as Future Value (FV) of money.

Assuming that the interest rate and number of periods remain constant, PV and FV vary jointly. For example, if PV increases, FV will increase too, and vice versa. There is a relationship between these two (with the above assumption), PV = FV x DF

here DF is known as Discounting Factor. Note that to get the present value of a certain sum of money, we have to discount (e.g., filter out the interests gained) the future value of that sum of money. This process is known as Discounting.

Formula for DF is DF = 1 / (1+R)^n
where R=rate of interest, and n=no. of period
[ Note, that we have taken Compounding Factor X=(1+R)^n while calculating EMI. Discounting Factor (DF) is reciprocal of the compounding factor]

Therefore, now we have the relations,
PV = FV x DF FV = PV / DF, where DF = 1 / (1+R)^n
or, PV = FV / X; FV = PV x X, where X = (1+R)^n

(take any one of the above formula for remembering)

Now, that you have clear concept of Present Value (PV) and Future Value (FV) of money, we can start with Annuity.

An Annuity is a series of equal/fixed payments/receipts at a regular interval, over a fixed period. For example, recurring deposit of Rs. 1000 every month, or LIC payment of Rs. 1000 every year for 10 years, etc.

Note that, payments can be made weekly, monthly, quarterly, yearly, or at any other interval of time.

Types of Annuity - There are 2 types in terms of valuation of annuity:
(NOTE - Basic classification of Annuity is - Ordinary Annuity & Annuity Due)

a.  Annuity Certain / Guaranteed Annuity - If the number of payment is known in advance, then it is known as Annuity Certain.

Annuity Certain is of 3 types -

i.  Ordinary Annuity / Annuity Immediate - If the payments are made at the end of the time period. In this case, before the payment is made, interest is accumulated or earned.

Periods:    0___1___2___ ... ___n  (Payments in bold, i.e., 1, 2, ... n)

Formula for Ordinary Annuity,
FV of Ordinary Annuity = C(X-1)/R, where X=(1+R)^n
Here C=cash flow per period, R=interest rate and n=number of payments

To get PV of Ordinary Annuity, use the formula already discussed, PV=FV/X (i.e., divide FV by compounding factor), or PV=FVxDF (i.e., multiply FV by discounting factor)

ii.  Annuity Due - If the payments are made at the beginning of each period. In this case, each annuity payment is allowed to compound for one extra period (note - paying before time!)

Periods:    0___1___2___ ... ___n  (Payments in bold, i.e., 0, 1, 2, ... n)

Note that, since each payment in this series is made 1 period earlier, we have to discount the formula for 1 period. We get the following formula for Annuity Due,

FV of Annuity Due = FV of Ordinary Annuity x (1+R) = [C(X-1)/R] x (1+R)
Similarly, PV of Annuity Due = FV of Annuity Due / X = FV of Annuity Due x DF

Note -

• FV will be greater than PV (assuming interest is earned). Therefore, to calculate PV from FV, we have to divide FV by compounding factor, (i.e., multiply FV by discounting factor). It will make PV lesser than the FV. And reverse while calculating FV from PV.
• Try remembering the formula, FV of Ordinary Annuity = C(X-1)/R, and then use the following formula to get the others -
• PV of Ordinary Annuity = FV of Ordinary Annuity / X
• FV of Annuity Due = FV of Ordinary Annuity x (1+R) (multiply by 1+R, since payment made 1 period earlier)
• PV of Annuity Due = FV of Annuity Due / X

iii.  Perpetuity - If the payments continue forever

b.  Life Annuities - If the payment is paid till the annuitant is alive. For example, life insurance, etc.

5.  Amortization of a Debt
Amortization is the process of decreasing an amount over a period. Amortization of a debt/loan is the process by which loan principal decreases over the life of a loan, typically an amortizing loan.

With each payment is made, a portion of the payment is applied towards reducing the principal, and another portion towards paying the interest on the loan.

6.  Sinking Funds
Sinking Fund is a fund into which a company sets aside money over a period of time, in order to fund a future capital expense, or repayment of a long-term debt.

Suppose a company wants a fixed amount of Rs. 50 crore in 5 years. It can deposit an amount (say, C) every year (i.e., annuity) with a bank, which gives interest rate of 8 % annually. This becomes Rs. 50 crore after 5 years, and can be used for repaying a debt, or any other purpose. This type of fund is known as Sinking Fund.

Note we can calculate C, as interest rate (R), FV(=Rs. 5 crore), and number of period are known.