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DEFINITIONS OF PROMISSORY NOTE, CHEQUE & BILL

Quick Revision of the chapter

PROMISSORY NOTE: (SECTION 4)

A Promissory note is an instrument in writing (not being a bank note or cy note) containing an unconditional undertaking, signed by the maker, certain sum of money only to, or to the order of, a certain person, or to of the instrument.

Parties:

Maker: The person who makes the promissory note and promises called the maker.

Payee: The person to whom the payment is to be made is called the Requisites of a Promissory Note:

The promissory note must be in writing.

It must contain an undertaking to pay. There must be an express promise

The instrument must contain a promise to pay money and money

currenpay a bearer pay is payee. promise to

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M AT H E M AT I C S O F F I N A N C E -

A N N U I T Y

C H A P T E R

Definition:

A sequence of payments, generally equal in size, made at equal intervals of times is called an annuity.

Monthly Rent; premiums of LIC; deposit into a recurring account in a bank; equal monthly payments got by a retired government servant as pension and loan instalments to houses or automobiles etc.

Some terms related with annuities

Periodic Payment: The size of each payment of an annuity is called the periodic payment of the annuity.

Annual Rent: The sum of all payments of an annuity made in one year is called its annual rent.

Payment Period/Interval: The duration between two successive payments of an annuity is called the payment period (or payment interval) of the annuity.

Term: The total duration from the beginning time of the first payment period to the end of the last payment period is called the term of the annuity.

Amount of an Annuity: The total Value of all the payments at the maturity time of an annuity is called the amount (or future value) of the annuity.

Present Value of an Annuity: Sum of the present values of all the payments of an annuity is called the present value or capital value of the annuity.

TYPES OF ANNUITIES

Ordinary Annuity: If the payments of an annuity are made at the end of payment interval is called An Ordinary annuity or Regular annuity.

9.2 MATHEMATICS OF FINANCE - ANNUITY

Annuity Due: If the payments of an annuity are made at the beginning of payment interval is called An Annuity Due or Annuity Immediate.

Perpetuity: A perpetuity is an annuity whose payments continue forever.

Note: In what is to follow, it is understood that the payment interval coincides with the interest period unless statement to the contrary is made.

ORDINARY ANNUITY OR ANNUITY REGULAR

Definition: Payments of an annuity are made at the end of payment interval.

Type-I

(To Find Amount)

WhereS = Amount of an Annuity

A= Value of each instalment

r = rate of interest

m = No. of conversion periods in a year

n = m.t = No. of instalments made in t years. i = m r 100 = Rate of interest of one conversion Period

Calculator Trick:

Step - I Find (1 + i)n by calculator i.e. Type r  100 m + 1 Then push  button then push = button (n - 1) times.

Step - II Then - 1

Step - III  r  100m

Step - IV Then  A push = button (We get the required value of Amount)

Example 1. Find the future value of an annuity of `500 is made annually for 7 years at interest rate of 14% compounded annually. [Given that (1.14) 7 = 2.5023] ( a ) `5365.25( b ) `5265.25( c ) `5465.25( d )None

Solution: Option ( a) is correct

Calculator Trick: (1)1 1005365.25 n i SA m r 

Step-I Find 7 14 1

As Type 14  100 + 1  Push = button 6 times.

Step-II Type - 1  14 then  100 (Because it is annually)

Step-III Then  500 = (we get the result)

Example 2: `200 is invested at the end of each month in an account paying interest 6% per year compounded monthly. What is the future value of this annuity after 10th payment? Given that (1.005) 10 =1.0511

( a ) `2544( b ) `2144( c ) `2544( d )None

Solution: (a) is correct.

Here A = 200 ; r = 6% compounded monthly n = 10 = No. of payments.

Calculator Trick:

Step-I Type 6  1200 + 1 Then push  button then push = button 9 times.

Step-II Type - 1 Then  6  1200

Step-III Then Type  200 = buttons we get the required amount.

Note: If (1 + i)n value is given in the question then use given value in the question otherwise answer may vary.

Type-II

To find the Value of Each Instalment

Example: If a bank pays 6% interest compounded quarterly what equal deposit have to be made at the end of the each quarter for 3 years if you want to have `1500 at the end of 3 years?

( a ) `117.86( b ) `115.01( c ) `150.50( d )None of these

9.4 MATHEMATICS OF FINANCE - ANNUITY

Solution: (b) is correct

A = ` 150.01

Calculator Trick:

Step-I Type 6  400 + 1 Then push  button then push = buttons 11 times

Step-II Then push -1  6  400 buttons

Step-III Then push M+ button to save the typed value.

Step-IV Then type 1500 then  button then push “MRC” button 2 times then push = button.

[we get the required result]

Type-III

(To find Present Value for Ordinary Annuity) PV = Present value =

) (1 1

Calculator Trick:

Step-I Type (1 + i) value then push ÷ button

Step-II Then push = buttons “n” times

Step-III Push GT button

Step-IV Then type  A (value) then push = button

We get the required result.

Example: Find the present value of an annuity which pays 200 at the end of each 3 months for 10 years assuming money to be worth 5% converted quarterly?

( a ) `3473.86( b ) `3108.60( c ) `6265.38( d )None of these

Solution: Option (c) is correct

Here A = 200; m = 4; r = 5% 1/4 yearly

t = 10 years  n = mt = 4 × 10 = 40 year PV = ?

Calculator Trick:

Step-I Type 5  400 + 1 then push  button

Step-II Then push = buttons 40 times

Step-III Then Push GT button

Step-IV Then type  200 = buttons [We get the resulting value]

Type-IV

(To find instalment value if PV is given).

Example: Mr. A borrows 5,00,000 to buy a house.

If he pays equal instalments for 20 years and 10% interest on outstanding balance what will be the equal annual instalment?

( a ) `58239.84( b ) `58729.84( c ) `68729.84( d )None of these

Solution: (b) is correct

Here PV = `5,00,000; r = 10% yearly.

t = 20 years

 n = 20; A = ?

Calculator Trick:

Step-I Type 10  100 + 1 then push ÷ button

Step-II Push = buttons 20 times

Step-III Then Push GT button

Step-IV Then M+ buttons to save the result.

Step-V Type 5,00,000 then push  button then- MRC button 2 time and then = button.

(We get the required result)

Annuity Immediate/Due

Definition: An annuity due is an annuity the first payment of which is made at the beginning of the first payment interval.

9.6 MATHEMATICS OF FINANCE - ANNUITY

Type-V

(To find Amount)

FV = Amount

Calculator Trick (work as ordinary annuity)

Step-I Type r  100 m + 1 then push  button

Step-II Push = buttons n + 1 - 1 = n times then push - 1 button then push  button then push r value then push  100m value buttons.

Step-III Push - 1 button then  button and then type A value & then push = button (we get the required result)

PAST EXAM QUESTIONS WITH SOLUTIONS (MEMORY BASED)

Q.1. Suppose your mom decides to gift you ` 10,000 every year starting from today for the next sixteen years. You deposit this amount in a bank as and when you receive and get 8.5% per annum interest rate compounded annually. What is the present value of this money: [Given that P (15, 0.085) = 8.304236]

( a )83,042( b )90,100 ( c )93,042( d )10,100 [Dec. 2015]

Solution: (c) is correct PV = 10,000

Q.2. The future value of an annuity of ` 1500 made annually for 5 years at an interest rate of 10% compounded annually is _______ [Given that (1.1)5 = 1.61051]

( a )9517.56( b )9157.65

( c )9715.56( d )9175.65 [June 2017]

Solution:

Use Calculator tricks = 9157.65 ` option (b) is correct.

Q.3. What sum should be invested at the end of every year so as to accumulate an amount of ` 796870 at the end of 10 years at the rate of interest

10% compounded annually, [given that A(10 ; 0.1) = 15.9374]

( a )40,000( b )4,50,000

( c )4,80,000( d )50,000

[June 2017]

Solution:

Calculator Tricks:

= ` 50,000

option (d) is correct.

Q.4. A person invests ` 2,000 at the end of each month @ of interest 6% compounding monthly, find the amount of annuity after the 10th payment is:

( a ) `20,456( b ) `20,156

( c ) `20,256( d ) `20,356

[June 2018]

Solution : (a) is correct

= ` 20,456

Type 6  1200 + 1 then press  button then = button 9 times - 1  6  1200  2000 = button; we will get the required result.

Q.5. Determine the present value of perpetuity of ` 50,000 per month @ Rate of interest 12% p.a. is _______

( a ) ` 45,00,000

( b ) ` 50,00,000

( c ) ` 55,00,000

( d ) ` 60,00,000 [June 2019]

Solution: (b) is correct

= ` 50,00,000

(b) is correct.

Q.6. A person wants to lease out a machine costing ` 5,00,000 for a 10 year period. It has fixed a rental of ` 51,272 per annum payable annually starting from the end of first year. Suppose rate of interest is 10% per annum, compounded annually on which money can be invested. To whom this agreement is favourable?

( a )Favour for lessee

( b )Favour for lessor

( c )Not for both

( d )Can’t be determined [June 2019]

Solution: (a) is correct

Cost = ` 5,00,000.

9.8 MATHEMATICS OF FINANCE - ANNUITY

So; GST = PV of Instalments made = PV = 51,272

Calculator Tricks:

Type button 1 100 10     10 times then press GT button then 51,272  = button = ` 3,15,044.25.

Which is less than ` 5,00,000.

So, Leasing is preferable.

(a) is correct.

Q.7. Let a person invest a fixed sum at the end of each month in an account paying interest 12% per year compounded monthly. It the future value of this annuity after the 12th payment is ` 55,000 then the amount invested every month is?

( a ) ` 4,837( b ) ` 4,637 ( c ) ` 4,337( d ) ` 3337 [June 2019]

Solution: (c) is correct

Calculator Tricks:

Value of each instalments

*Type times 11 1 1200 12     button. 1200 12 1  

Then press (m+) button.

*Type 55000  button then press MRC button then = button. We get ` 4337.

Q.8. Find the future value of annuity of ` 500 is made annually for 7 years interest rate of 14% compound at annually. Given that (1.14) 7 = 2.5023

( a ) ` 15635.35( b ) ` 10,730.71 ( c ) ` 16535.35( d ) ` 16355.35 [Dec. 2019]

Solution: (b) is correct 7 14

= ` 10,730.71

Q.9. Determine the present value of perpetuity 10 per month for infinite period at an effective rate of interest of 14% p.a.?

( a ) 657( b ) 757

( c ) 857( d ) 957 [Dec. 2020]

Solution: i =

Calculator Trick:

= 857.14 = 857.

Q.10. Which of the following statement is true?

( a )F.V of ordinary annuity < F.V of annuity due

( b )F.V of ordinary annuity > F.V of annuity due

( c )P.V of ordinary annuity > P.V of annuity due

( d )None of these [Dec. 2020]

Solution: (a) is correct.

Q.11. Suppose you deposit 900 per month into an account that pays 14.8% interest compounded monthly. How much money will you get after 9 months?

( a ) 8,511( b ) 9,000

( c ) 9,200( d ) 1,000 [Dec. 2020]

Solution: (a) is correct

FV = R

14.8 ÷ 1200 + 1 × = button 8 times -1

÷ 14.8 × 1200 × 900 = button. We get FV 8511.

(a) is correct.

Q.12. 2,500 is paid every year for 10 years to pay off a loan. What is the loan amount if interest rate be 14% per annum compounded annually?

( a )13,040.27( b )15,847.90

( c )14,674.21( d )16,345.11 [Dec. 2020]

Solution: Calculator Tricks: Loan amount = PV = R 1(1)

= 2500 10 4 11 100

Calculator Tricks:

Type 14 ÷ 100 + 1 ÷ = button 10 times (Press)

= 8511.31 = 8511

Then press GT button then × button. Type 2500 then = button. (Press)

We get PV = 13,040.28

(a) is correct.

Q.13. Assuming that the discount rate is 7% p.a. how much would pay to receive 200 growing at 5% annually for ever?

( a ) 2,500( b ) 5,000 ( c ) 7,500( d ) 10,000 [Jan. 2021]

9.10 MATHEMATICS OF

Solution: (d) is correct

Discount rate =i = 7% = = 0.07

Growing rate = g = 5% = 0.05

R = Value of each payment received = 200

 PVA =  = 10,000

Q.14. 800 is invested at the end of each month in an account paying interest 6% per year compounded monthly. What is the future value of this annually after 10th payment?

( a ) 4,444( b ) 8,756

( c ) 3,491( d ) 8,182 [Jan. 2021]

Solution: (d) is correct.

Monthly Instalment = A = 800 rate of interest = r = 6% p.a. compounded monthly

n = No. of Payments = 10