Interest & Annuity

In this part, we discus about Interest & Annuity and related topics

Simple Interest
Compound Interest
Present Value of money
Future Value of money
Annuity
Sinking Fund

Interest & Annuity – Meaning of Terms – Video in English

Simple Interest

Many of us need to borrow money to meet expenses, or to run business. On the other hand, some people save money for future contingencies. People needing the money would like to get the surplus money saved by other people. So, he would borrow the money (borrower) from the other people (lender) and repay the borrowed money to the lender after some time, along with some extra money called Interest.

So, for borrower, Interest is the extra amount paid (expense) for use of money. For lender, it is extra money earned for letting the borrower use his money.

So, Interest may be defined as price paid by the borrower for use of lender’s money, due to following major factors

Time Value of Money: Value of money today is more than the value of money in future. So, lender would charge Interest for the money paid today, for receiving back in future.
Opportunity Cost : When the lender invests, he loses the opportunity to gain from the alternative investment. So, the lender would charge interest for the opportunity foregone.
Inflation : Due to inflation, value of money falls over time. So, Interest is charged to compensate for the falling value of money due to inflation.
Risk : There is inherent risk of borrower defaulting to pay back. So, the Interest rate would be higher for riskier investment.

Simple interest is the interest computed on the principal for the entire period of borrowing. It is calculated on the outstanding principal balance and not on interest previously earned. No interest is paid on interest earned during the term of loan.

So, the amount of Simple Interest directly varies with Principal Amount, Time (period of loan) and Interest Rate.

Formula of Simple interest I=P x R x T A=P+I = P+(P x R x T)=P x [1+(R x T)] where A = Accumulated amount (final value of an investment including Interest ) P = Principal (initial value of an investment), R = Rate of Simple Interest for the unit period T= Period Units I = Amount of Simple Interest

Ex 1. Calculate Simple Interest earned on Rs. 5000 at 6% for 3 years Here P= 5000, R= $\frac{6}{{100}}$ (6%), T=3
I = 5000 x $\frac{6}{{100}}$ x 3 = 900 A= Amount at the end of the period = Principal (P) + Interest (I) = 5000+900 = 5900. So, the amount of Rs 5000 will become Rs 5900 after 3 years at Simple Interest Rate of 6% per annum

Ex 2. Tapan deposited Rs. 4,00,000 in his bank for 4 years at simple interest rate of 5%. How much interest would he earn? How much would be the final value of deposit? Here P= 4,00,000, R= $\frac{5}{{100}}$ (5%), T=4
I = 4,00,000 x $\frac{5}{{100}}$ x 4 = 80,000 A= Amount at the end of the period = Principal (P) + Interest (I) = 4,00,000+80,000 = 4,80,000. So, the amount of Rs 4,00,000 will become Rs 4,80,000 after 4 years at Simple Interest Rate of 5% per annum

Ex. 3 : What sum of money will produce Rs.28600 interest in 3 years and 3 months at 2.5% p.a. simple interest? Let P be the amount of Principal Step 1 : Here I=28,600, T=3.25, R=2.5. Step 2 : I= Px $\left( {\frac{{2.5}}{{100}}} \right)$ x (3.25), or 28600= P x $\left( {\frac{{8.125}}{{100}}} \right)$ = P x (.08125)
Step 3 : So, P= $\frac{{28600}}{{.08125}}$ = 3,52,000 Hence Rs.352000 will produce Rs.28600 interest in 3 years and 3 months at 2.5% p.a simple interest

Ex. 4 : An individual borrowed Rs. 2000 at the rate of 10% per annum simple interest. What amount will he pay to clear the debt after 4 years ?Step 1 : Here P= 2,000, R= $\frac{{10}}{{100}}$ (10%), T=4
Step 2 : I = [2000x $\frac{{10}}{{100}}$ x 4] = 800 Step 3 : A= P + I = 2,000 + 800 = 2800 So, he will have to Pay Rs 2800 to clear the debt after 4 years Ex.5 :The interest on a certain deposit at 4.5 p.a. is Rs. 202.50 in one year. How much will the additional interest in one year be on the same deposit at 6 % p.a. ? Let P be the Principal Amount.

Ex.5 :The interest on a certain deposit at 4.5 p.a. is Rs. 202.50 in one year. How much will the additional interest in one year be on the same deposit at 6 % p.a. ?
Let P be the Principal Amount.
Step 1 : R= $\frac{{4.5}}{{100}}$ (4.5%), T=1, I=202.50,
Step 2 : I = Px R x T, or 202.50 = Px $\frac{{4.5}}{{100}}$ x 1, or P= 202.50 x $\left( {\frac{{100}}{{4.5}}} \right)$ = $\frac{{20250}}{{4.5}}$ = 4500
So, the Principal Amount is Rs 4500
Step 3 : Interest @ 6% on 4500 for one year
I = [4500 x $\left( {\frac{6}{{100}}} \right)$ x 1] = 270 Step 4 : Additional Interest when Rate is increase from 4.5 % to 6% =Revised Interest (@6%) – Previous Interest (@4.5%)= 270 -202.50 = Rs.67.50

Simple Interest – Hindi Video

Simple Interest – Video in English

Simple Interest Hindi Video – Formula, Problems and Solutions

Compound Interest

Compound Interest is interest added to the principal so that the added interest also earns interest from then on (normally denoted by CI.)

Compound Interest is the Interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. Compound interest can be thought of as Simple Interest and additional “interest on interest”

Compound Interest (CI) = PxRⁿ, where P is the Principal, R is the Rate per period, n = Period

A= P + CI = P + (PxRⁿ) = P(1+Rⁿ)

Compounding Period : Compound Period is the unit period at the end of which the Interest is added to Principal to compute the Interest for the next period.

In compound interest, the amount yielded (Principal + Interest) become the principal for the next compounding period. So, the computation of Compound Interest depends on the period interval. So the Interest computed at the same rate, for the same principal, for the same period, would yield more if the compounding period is smaller (compounded more frequently)., so the interest is added more frequently.

So, the Interest on Rs 5000 at 12% per annum compounded Half Yearly or Quarterly would yield more than compounded annually

Ex.1 : Compute the compound interest on Rs. 5,000 for 3 years, at 10%, compounded annually.

Step 1 : A= P[1+( $\frac{R}{{100}}$ )]ⁿ

Here, P = Rs. 5,000. n = 3 years, R = $\frac{{10}}{{100}}$ (10%), A= Principal + Interest

Step 2 : A=5000 [ 1+ ( $\frac{{10}}{{100}}$ )]³

= 5000 x ( $\frac{{110}}{{100}}$ ) x ( $\frac{{110}}{{100}}$ )x ( $\frac{{110}}{{100}}$ )

= 5000 x $\left( {\frac{{11}}{{10}}} \right)$ x $\left( {\frac{{11}}{{10}}} \right)$ x $\left( {\frac{{11}}{{10}}} \right)$

= 5x 11x11x11 = 6655

Step 3 : So, Interest = 6655-5000 = 1655

Ex.2 : In what time Rs. 12,000 will amount to Rs. 13,230 at 5% compound interest, compounded annually.

Step 1 : A= P[1+( $\frac{R}{{100}}$ )]ⁿ

Here, P = Rs. 12,000 A = Rs. 13230 R = 5%= $\frac{5}{{100}}$

Step 2 : So, 13,230 = 12000 [ 1+ $\frac{5}{{100}}$ ]ⁿ

Or [ 1+ $\frac{5}{{100}}$ ]ⁿ= $\frac{{13230}}{{12000}}$

Or ( $\frac{{105}}{{100}}$ )ⁿ= $\frac{{441}}{{400}}$

Or ( $\frac{{21}}{{20}}$ )ⁿ= $\frac{{441}}{{400}}$ = [(21)²/ (20)²] = [latex]\frac{{21}}{{20}}[/latex]²

So, n=2. So, in 2 years, Rs. 12,000 at 5% p.a compound interest will amount to Rs. 13,230.

Computation of Interest Compounded Half Yearly

Ex. 3: Find the compound amount and compound interest on Rs.5000 at 6% compounded Half Yearly, for 6 years, and compounded Quarterly.

Here P-5000, n = 6 Year, i=6% per annum

1. When Compounded Half Yearly

So, for Half Year, the Rate of Interest I= $\frac{{\left[ {\left( {\frac{6}{2}} \right)} \right]}}{{100}}$ = .03. n = 6 Years = 12 Half Years,

Compound Amount A= Px ( 1 + i)ⁿ = Px(1+.03)¹²

= 5000 x 1.42576 = 7,128.80

H/Y Compound Interest = 7,128.80 – 5000.00 = 2,128.80

2. When Compounded Quarterly

So, for a Quarter, the Rate of Interest I= $\frac{{\left[ {\left( {\frac{6}{4}} \right)} \right]}}{{100}}$ = .015. n = 6 Year = 24 Quarters,

Compound Amount A= Px ( 1 + i)ⁿ = Px(1+.015)²⁴

= 5000 x 1.42950 = 7,147.51

Quarterly Compound Interest = 7,147.51 – 5000.00 = 2,147.51

So, we see Quarterly Compounding Interest is more than Half Yearly Compounding Interest

Compound Interest – Hindi Video

Compound Interest – Video in English

Effective Rate of Interest

We have seen that in case of compounding, the Interest depends on the period interval. So, if the Interest is compounded more frequently than a year (e.g Monthly, Quarterly, Half Yearly), then the Interest would be more if compounded annually, for the same Annual Interest Rate. So, the Effective Rate is more for smaller period than the Annual Rate.

The effective rate is derived as E=(1+i)ⁿ-1, where E= Effective Rate, i= interest Rate and n=number of compounding period .

Example : Rs 5000 invested at Interest rate 6% per annum, compounded quarterly. What would be the Annual Interest and effective rate of Interest

Here, I=P[(1-i)ⁿ -1]. Here P=5000, i=6% pa=(6/4)% per quarter = 1.5% per quarter = .015 / quarter. I=Compound Interest
So, I=5000[(1+.015)⁴ – 1)] = 5000 x .06136355 = 306.82
E= 306.82/5000= .0613 or 6.136%
You can also compute from the formula, E=(1+.015)⁴ – 1)] = .06136= 6.136%

Effective Rate of Interest – Video in English

Annuity

Annuity is a series of specified number of Fixed Amount regular payment at same interval of time. For example school fees paid every month, Monthly repayment of loan installment etc

Annuity Due (or Immediate) : Payment of instalment is made at the beginning of each annuity period (e.g 1st day of month, 1st day of year etc)
Annuity Regular : Payment of instalment is made at the end of each annuity period (e.g end of month, end of year etc)

Future Value

The value of a particular sum of money at some time in future is called as future value. It is denoted by ‘ FV ’.

FV = CF(1 + i)ⁿ, where, FV = future value, CF = Cash flow or principal, i = rate of compound interest, n = Number of conversion period
Future value of an Annuity Regular: $FVR~=~CF~[\frac{{\{1+i)n~-~1}}{i}\}~]$
Future value of an Annuity Immediate : $FVI~=~FR~\times ~\left( {1+i} \right)=\left[ {CF~} \right[\frac{{\left\{ {\left( {1+i} \right)n~-1} \right\}}}{i}~\left] {-1} \right]]$

Future Value is value of today’s money in future, compounded at a specified rate of Interest. For example Rs 1000 is invested @9% Interest. So, at the end of year, Rs 1000 invested now would become 1090 at the end of year. So, FV (Future Value) of Rs 1000 after 1 year is Rs 1090

The Future Value of Rs 1000 after 2 years would be 1000x(1+.09)²=1000x 1.188=1188

Present Value

Present value of money is the reciprocal of future value of money. It is actually the today’s value of future money discounted at the interest rate (denoted by PV)

PV = FV /(1 + i)ⁿ, where I = rate of compound interest, n= Number of conversion, period, FV = Future value
${{P}_{{VR}}}=\frac{{{{F}_{{VR}}}}}{{({{{\left( {1+i} \right)}}^{n}})}}=\frac{{CF\left[ {{{{\left( {1+i} \right)}}^{n}}-1} \right]}}{{i{{{\left( {1+i} \right)}}^{n}}}}=\frac{{CF[1-{{{\left( {1+i} \right)}}^{{-n}}}}}{i}$
where F_{V R} = Future Value of Annuity Regular, P_{V R}= Present Value of Annuity Regular. CF = Cashflow, I = rate of compound interest, n= Number of conversion,
Present value of an Annuity Immediate : P_VI = P_VR + CF, where CF = Cash flow/period.

Annuity – Video in English

Annuity – Problems

Future Value

Ex 1: Sourav invest Rs.7000 in a three year investment that pays you 12% per annum.

Compute the future value of the investment.

FV= CF(1 + i)ⁿ,

Step 1 : Here CF = 7000, i= 12% = .12, n= 3, FV is the Future Value

Step 2: FV= 7000(1+0.12)³=7000 X 1.4049 = 9834.30

Ex 2 : An overdraft of Rs. 70,000 is to be paid back in equal annual instalments over a period of 20 years. Find the value of the instalment, If interest is compounded annually at 14% per annum

Let C be the required instalment

$F=~\frac{{C~\left[ {~\left( {\left( {1~+~i} \right)n~1} \right)} \right]}}{i}~$
$Here,~F=70,000,~i=\frac{{14}}{{100}}~=~0.14,~n=20$
$So,~70000=~\frac{{C~\left[ {\left( {1+.14} \right)20~-1} \right]}}{{.14}}~$
$Now~the~RHS~=~\frac{{C\left[ ~ \right[\left( {1+.14} \right)20~-1]}}{{.14}}~]$
$\left( {Note~:~x=1.1420.~log~x=~10~log~1.14~=~20~\times ~0.0569~=1.138.~x=antilog~1.138=13.74} \right)$
$RHS~=~C\left[ {\frac{{\left( {13.74~-1} \right)}}{{.14}}~} \right]~=~C\left( {\frac{{12.74}}{{.14}}~} \right)~$
$So,~C\left( {\frac{{12.74}}{{.14}}~} \right)~=~70,000,~or~C~=~70,000~\times ~\left( {\frac{{.14}}{{12.74}}~} \right)~=~769.23$
So, Value of each instalment is Rs 769.23

Present Value

Ex 2: Determine the present value of Rs. 8,000 due at end of 4 years if money is worth 7 % effective.

$PVR=\frac{{FVR}}{{\left( {\left( {1~+~i} \right)n} \right)}}~$
$Here~FVR~~=~8000,~i=~\frac{7}{{100}}~=~.07,~n=~4$
$PV~R~=~\frac{{\left( {8000} \right)}}{{\left( {1~+~0.07} \right)4}}~=~~8000~\times ~0.76289~=~6103.12~.~Present~value~is~Rs.~6103.12~\left( {appx} \right)$

Annuity Practical Problems – Hindi Video

Annuity – Problems & Solutions – Video in English

Sinking Fund

Sinking Fund is a fund created for a specific purpose by setting aside a fixed amount periodically allowing it to grow earning compound interest.

$A=\left[ {\frac{{P\left\{ {\left( {1~+~i} \right)n~~1} \right\}}}{i}~} \right],~or~P~=\frac{{\left( {A~x~i} \right)}}{{\{\left( {1~+~i} \right)n~~1}}~,~$
where A is the amount of Sinking Fund to be created, P = amount to be set aside periodically, i= Rate of Compound Interests, n= number of periods

Ex. :A firm required Rs.5,00,000 at the end of 10 years to replace one of its asset. Find the yearly investment yielding 10% interest, required to create a sinking fund
$P~=~\frac{{\left( {A\times ~i} \right)~}}{{\left\{ {\left( {1~+~i} \right)n~~1} \right\}}}.~Here~A=5,00,000,~i=~\frac{{10}}{{100}}~=~.10,~n=~10$
$P=~\left[ {\frac{{\left( {5,00,000~\times ~0.1} \right)}}{{\left\{ {\left( {~1+.1} \right)10~-~1} \right\}}}~} \right]~=~\frac{{50000}}{{2.59374-1}}~=~\frac{{50,000}}{{1.59374}}~=~31372.74$
So, Rs 31372.74 has to be set aside yearly