CAT Quant: Simple and Compound Interest – Important Formulas and Concepts

Interest

Interest is money paid to the lender by the borrower for using lenders money for a specifies period or time. It is the cost of using a sum of money over a period of time. Various terms used and their representation are as follows :-

  • Interest – The money paid by the borrower for using lender’s money. Often denoted by “I”.
  • Principal – The original sum that was initially borrowed. Often denoted by “P”.
  • Time – The time/period for which the money is borrowed. Often denoted by “T/n”
  • Rate of Interest – The rate at which interest is calculated on the principal amount. It is expressed as a percentage or decimal fraction. Often denoted by “R/r”
  • Amount – The sum of Principle and Interest.. Often denoted by “A”

Simple Interest

  • When Interest is calculated every year (time period) on the original principal (i.e sum at the beginning of 1st year), such interest is called Simple Interest.
  • In case of Simple Interest, the principal remains the unchanged every year. Basically the interest for any year is the same as that for any other year.

$$SI = \frac{PRT}{100}$$

$$A = \frac{P+PRT}{100} = P{\displaystyle\LARGE[}1+\frac{RT}{100}{\displaystyle\LARGE]}$$

  • When the Amount under Simple Interest is “n times” that of Principle, The formula is derived as –

$$A = P + SI$$

$$nP = P + \frac{PRT}{100}$$

$$100(n-1) = RT$$

  • Overall Rate if Interest under SI – When the rate of interest for each year is different each year, the overall rate of interest is calculated as follows –

$$R = \frac{P_1R_1+P_2R_2+P_3R_3+….}{P_1+P_2+P_3+….}$$

Compound Interest

  • Under Compound Interest, the interest is added to the principal at the end of each period to arrive the new principle for the next period. Basically, the Amount at the end of 1st year becomes the Principle for the 2nd year and so on for the consecutive years.

$$A = P{\displaystyle\LARGE[}1+\frac{R}{100}{\displaystyle\LARGE]}^n = PQ^n$$

= Amount at the end of Period n

= Principle (P) at the beginning of Period (n+1)

  • The compound Interest for the given period can be calculated as follows –

$$CI = A – P = P[Q^n – 1]$$

Compounding at Different Rates

  • When the rate of Interest for each year is different each year we calculate the Amount for the defined period as follows –

$$A = P{\displaystyle\LARGE[}1+\frac{r_1}{100}{\displaystyle\LARGE]}^{n_1}{\displaystyle\LARGE[}1+\frac{r_2}{100}{\displaystyle\LARGE]}^{n_2}{\displaystyle\LARGE[}1+\frac{r_3}{100}{\displaystyle\LARGE]}^{n_3}$$

where, r1% for n1 years, r2% for n2 years and so on.

Compounding more then once a year

  • Compounding can also be done more frequently than once a year. For Example, If interest is added to the principle every 6 months, we say that compounding is done every “twice a year”, If interest is added to the principle every 4 months, we say that compounding is done “thrice a year” and so on.

$$A = P{\displaystyle\LARGE[}1+\frac{r/x}{100}{\displaystyle\LARGE]}^{nx} $$

where, x = times compounding is done every year.

  • when compounded, half yearly (x = 2), quarterly (x = 3), every 3 months (x = 4), monthly (x = 12)
  • When the number of times compounding done per year is increased to infinity, we say compounding is done “Every Moment”. The formula for the same is –

$$ A = Pe^{\frac{nr}{100}}$$

Difference between SI and CI

  • SI is same every year, CI increases every year by the same factor.
  • For the 1st year – CI = SI, For any other year – CI > SI
  • Difference between SI and CI for 2 years – The difference between the SI and CI on a certain sum for 2 years is equal to interest calculated for 1year on 1st year’s SI. Basically, 2nd year CI = SI + (Interest on SI)
  • Formula for calculating difference between SI and CI of 2nd year/ 2 years (the formula is same in both cases) –

$$CI_2 – SI_2 = P{\displaystyle\LARGE[}\frac{r}{100}{\displaystyle\LARGE]}^2$$

  • Formula for calculating difference between SI and CI of 3rd year/ 3 years (the formula is same in both cases) –

$$CI_3 – SI_3 = P{\displaystyle\LARGE[}\frac{R^2(300+R)}{100^3}{\displaystyle\LARGE]}$$

Important Points related to CI

  • [CI for “k” years] – [CI for “k+1” years] = Interest on Amount(A) for “kth” year (amount at the end of kth year)
  • [A for “k” years] – [A for “k+1” years] = Interest on Amount(A) for “kth” year (amount at the end of kth year)
  • [CI of “kth” year] – [CI of “(k+1)th” year] = Interest on CI of the “kth” year

Present Value

  • Present value is the current value of a future sum of money or stream of cash flows given a specified rate of return. Present Value can be looked at both under SI and CI. If the principal P is amounting to X in n periods, then
  • Present Value under Simple Interest (SI) – The present value P of an amount X coming (or due) after n periods under SI is given by –

$$P = \frac{X}{[1 + \frac{nr}{100}]}$$

  • Present Value under Compound Interest (CI) – The present value P of an amount X coming (or due) after n periods under CI is given by –

$$P = \frac{X}{[1 + \frac{r}{100}]^n}$$

Repayment in Equal Installment (CI)

  • If a sum P is borrowed and is repaid in n equal installments, compound interest (CI) being calculated at r% per period pf instalment, we can find out the Value of Each Installment using the given formula –

$$Each \ Installment = \frac{Pr}{100{\displaystyle\Large[}1-{\displaystyle\Large(}\frac{100}{100+r}{\displaystyle\Large)}^n{\displaystyle\Large]}}$$

Depreciation of Value

  • The value of an machine/asset is subject to wear and tear, decrease with time. This reduction is value of an asset is known as depreciation and is calculated as –

$$V_1 = V_0{\displaystyle\LARGE[}1-\frac{r}{100}{\displaystyle\LARGE]}^n$$

  • here, [V1] is value of asset after n years, [V0] is value of asset at the beginning of the period.

Read concepts and formulas for: Simple & Quadratic Equations

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