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Applied Mathematics · Class 12 Commerce

Ch 9Financial Mathematics — Class 12 Applied Mathematics, concept-first.

This concept map shows how the chapter fits together. Financial mathematics here covers perpetuities and sinking funds, the valuation of bonds (by the present-value and relative-price approaches), EMI / amortization of loans (flat-rate and reducing-balance methods), the compound annual growth rate (CAGR), rates of retu…

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Key concepts

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Present Value Perpetuity

Imagine you own a magic tree that gives you exactly ₹100 every year, forever. No matter what happens, that ₹100 arrives each year, year after year, for eternity.

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In previous exams

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CA Foundation

3 concepts · 48 appearances

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Concept Map

This concept map shows how the chapter fits together. Financial mathematics here covers perpetuities and sinking funds, the valuation of bonds (by the present-value and relative-price approaches), EMI…

Introduction

Financial mathematics touches almost every transaction you make, from a bank fixed deposit to buying a home on loan.

7.1

Perpetuity and Sinking Fund

A perpetuity is a stream of equal payments that continues forever — think of an endowment fund that pays a fixed scholarship each year without ever exhausting its principal.

+Exercise 7.1i9 questions
  1. Q1Find the present value of a sequence of payments of ₹80 made at the end of each 6 months and continuing forever, if money is worth 4% compou…Free
  2. Q2Find the present value of an annuity of ₹1800 made at the end of each quarter and continuing forever, if money is worth 5% compounded quarte…Free
  3. Q3If the cash equivalent of a perpetuity of ₹300 payable at the end of each quarter is ₹24,000. Find the rate of interest compounded quarterly…Free
  4. Q4Find the present value of a perpetuity of ₹780 payable at the beginning of each year, if money is worth 6% effective.Preview
  5. Q5The present value of a perpetual income of ₹x at the end of each 6 months is ₹36000. Find the value of x if money is worth 6% compounded sem…Preview
  6. Q6If you need ₹20,000 for your daughter's education, how much must you set aside each quarter for 10 years to accumulate this amount at the ra…Preview
  7. Q7To save for child's education, a sinking fund is created to have ₹1,00,000 at the end of 25 years. How much money should be retained out of…Preview
  8. Q8A machine costs ₹1,00,000 and its effective life is estimated to be 12 years. A sinking fund is created for replacing the machine by a new m…Preview
  9. Q9Suppose a machine costing ₹50,000 is to be replaced at the end of 10 years, at that time it will have a salvage value of ₹5,000. In order to…Preview
7.1.1

Perpetuity

A perpetuity is a stream of equal cash flows that continues forever — there is no end date. Unlike an annuity, which stops after a fixed number of payments, a perpetuity keeps paying the same amount e…

+Worked Examplesi5 questions
  1. Example 1Find the present value of a sequence of payments of ₹60 made at the end of each 6 months and continuing forever, if money is worth 4% compou…Free
  2. Example 2At 6% converted quarterly, find the present value of a perpetuity of ₹600 payable at the end of each quarter.Free
  3. Example 3At what rate converted semi-annually will the present value of a perpetuity of ₹450 payable at the end of each 6 months be ₹20,000?Preview
  4. Example 4How much money is needed to endure a series of lectures costing ₹2500 at the beginning of each year indefinitely, if money is worth 3% compo…Preview
  5. Example 5The present value of a perpetual income of ₹x at the end of each six months is ₹40000. Find the value of x if money is worth 6% compounded s…Preview
7.1.2

Sinking Fund

A sinking fund is a disciplined way to set aside a fixed amount of money at regular intervals so that a large future debt or liability can be paid off in one go.

+Worked Examplesi3 questions
  1. Example 6A company establishes sinking fund to provide for the payment of ₹1,00,000 debt maturing in 4 years. Contributions to the fund are to be mad…Free
  2. Example 7In 10 years, a machine costing ₹40,000 will have a salvage value of ₹4,000. A New Machine at that time is expected to sell for ₹52,000. In o…Preview
  3. Example 8Mr X plans to save amount for higher studies of his son, required after 10 years. He expects the cost of these studies to be ₹1,00,000. How…Preview
7.2

Valuation of Bonds

9 Q

A bond is a written contract between a borrower (the issuer) and a lender (the bondholder). Through it the issuer promises to repay a fixed sum on a stated future date and, until that date, to pay int…

+Worked Examplesi3 questions
  1. Example 9Find the purchase price of a ₹600, 8% bond, dividends payable semi-annually redeemable at par in 5 years, if the yield rate is to be 8% comp…Free
  2. Example 10A ₹2,000, 8% bond is redeemable at the end of 10 years at ₹105. Find the purchase price to yield 10% effective rate.Preview
  3. Example 11Consider a bond with a coupon rate of 10% charged annually. The par value is ₹2,000 and the bond has 5 years to maturity. The yield to matur…Preview
+Exercise 7.2i6 questions
  1. Q1What should be the price of the bond to yield an effective interest rate of 8% if it has a face value of ₹1,000 and maturity period of 15 ye…Free
  2. Q2Suppose a bond has a face value of ₹1,000, redeemable at the end of 12 years at 15% premium and paying annual interest at 8%. If the yield r…Free
  3. Q3An investor is considering purchasing a 5 year bond of ₹1,00,000 at par value and an annual fixed coupon rate of 12% while coupon payments a…Preview
  4. Q4Suppose that a bond has a face value of ₹1,000 and will mature in 10 years. The annual coupon rate is 5%, the bond makes semi-annual coupon…Preview
  5. Q5A bond with a face value of ₹1,000 matures in 10 years. The nominal rate of interest on bond is 11% p.a. paid annually. What should be the p…Preview
  6. Q6What is the value of the bond, considering a bond has a coupon rate of 10% charged annually, par value being ₹1,000 and the bond has 5 years…Preview
7.3

Calculation of EMI or Amortization of Loans

People meet many large expenses — buying a car or house, starting a business, funding a trip — and often cannot pay for them outright, so they borrow money (take a loan) and repay it to the lender ove…

+Exercise 7.3i5 questions
  1. Q1Mohan takes a loan of ₹5,00,000 with 8% annual interest rate for 6 years. Calculate EMI under Flat-Rate system.Free
  2. Q2XYZ company borrows ₹3,00,000 with 7% annual interest rate for 4 years. Calculate EMI under Reducing Balance method.Free
  3. Q3Rajesh borrows ₹6,00,000 with 9% annual interest rate for 5 years. Calculate EMI under Reducing Balance method.Preview
  4. Q4A person amortizes a loan of ₹1,50,000 for a new home by obtaining a 10 year mortgage at the rate of 12% compounded monthly. Find (i) The mo…Preview
  5. Q5A couple wishes to purchase a house for ₹12,00,000 with a down payment of ₹2,50,000. If they can amortize the balance at 9% per annum compou…Preview
7.3.1

Methods of Calculation of EMI or Instalment

When you take a loan, the bank doesn't just expect you to return the principal — it wants the interest too, spread evenly across your repayment period.

+Worked Examplesi3 questions
  1. Example 12Mr. X takes a loan of ₹2,00,000 with 10% annual interest rate for 5 years. Calculate EMI under Flat Rate system.Free
  2. Example 13A couple wishes to purchase a house for ₹10,00,000 with a down payment of ₹2,00,000. If they can amortize the balance at 9% per annum compou…Preview
  3. Example 14Mr. M borrowed ₹10,00,000 from a bank to purchase a house and decided to repay by monthly equal instalments in 10 years. The bank charges in…Preview
7.4

Nominal and Effective Rate of Interest

9 Q

When a bank or financial institution quotes an interest rate, it is almost always the nominal rate — the stated annual percentage.

+Worked Examplesi4 questions
  1. Example 15Mr X took a loan of ₹2,000 for 6 months. Lender deducts ₹200 as interest while lending. Find the effective rate of interest charged by lende…Free
  2. Example 16What effective rate is equivalent to a nominal rate of 8% converted quarterly?Free
  3. Example 17Mr. Y has two investment options - either at 10% per annum compounded semi-annually or 9.5% per annum compounded continuously. Which option…Preview
  4. Example 18Find the effective rate of interest equivalent to a nominal rate of 6% compounded (i) Semi-annually (ii) Quarterly (iii) Continuously.Preview
+Exercise 7.4i5 questions
  1. Q1What is the effective annual rate of interest compounding equivalent to a nominal rate of interest 5% per annum compounded quarterly?Free
  2. Q2Which is the better investment, 3% per year compounded monthly or 3.1% per year compounded quarterly?Free
  3. Q3What effective rate of interest is equivalent to a nominal rate of 8% converted quarterly?Preview
  4. Q4To what amount will ₹12000 accumulate in 12 years if invested at an effective rate of 5%?Preview
  5. Q5Which yields more interest: 8% effective or 7.8% compounded semi-annually?Preview
7.5

Compound Annual Growth Rate

8 Q

The Compound Annual Growth Rate (CAGR) is the single rate at which an investment would have grown if it had compounded smoothly each year, smoothing out the ups and downs of actual returns.

+Worked Examplesi3 questions
  1. Example 19Assume an investment's starting value is ₹10,000 and it grows to ₹60,000 in 4 years. Calculate CAGR.Free
  2. Example 20Calculate CAGR of unit sales on the basis of given information: Year — 2012, 2013, 2014, 2015, 2016; Sales — 53,000, 60,786, 73,450, 86,000,…Preview
  3. Example 21Suppose a person invested ₹15,000 in a mutual fund and the value of investment at the time of redemption was ₹25000. If CAGR for this invest…Preview
+Exercise 7.5i5 questions
  1. Q1An investment has a starting value of ₹5000 and it grows to ₹25,000 in 4 years. What will be its CAGR?Free
  2. Q2An investment has a starting value of ₹2000 and it grows to ₹18,000 in 3 years. What will be its CAGR?Free
  3. Q3Calculate CAGR from the following data: Year — 2015, 2016, 2017, 2018; Revenue (₹) — 3,00,000, 3,50,000, 4,00,000, 4,50,000.Preview
  4. Q4Mr. Kumar has invested ₹20,000 in year 2014 for 5 years. If CAGR for that investment turned out to be 11.84%. What will be the end balance?Preview
  5. Q5Mr. Naresh has bought 200 shares of City Look Company at ₹100 each in 2015. After selling them he has received ₹30,000 which accounts for 22…Preview
7.6

Stock, Shares and Debentures

9 Q

Starting a large business needs more money than one or two people can raise, so people join together to form a company.

+Worked Examplesi4 questions
  1. Example 22Find the cost of (i) ₹7200, 8% stock at 90 (ii) ₹4500, 8.5% stock at 4 premium (iii) ₹6400, 10% stock at 15 discountFree
  2. Example 23Which is better investment 7.5% stock at 105 or 6.5% stock at 94Free
  3. Example 24Find the cost of 96 shares of ₹10 each at $\frac{3}{4}$ discount, brokerage being ¼ per share.Preview
  4. Example 25A man sells ₹5000, 12% stock at 156 and invests the proceeds partly in 8% stock at 90 and 9% stock at 108. He thereby increases his income b…Preview
+Exercise 7.6i5 questions
  1. Q1Find the cash required to purchase ₹3200, 7 ½ % stock at 107 (brokerage ½ %)Free
  2. Q2Find the cash realised by selling ₹2440, 9.5 % stock at 4 discount (brokerage ¼%)Free
  3. Q3Which is better investment 11% stock at 143 or 9 ¾% stock at 117Preview
  4. Q4Find the income derived from 88 shares of ₹25 each at 5 premium, brokerage being ¼ per share and the rate of dividend being 7 ½ % per annum.…Preview
  5. Q5A man buys ₹25 shares in a company which pays 9% dividend. The money invested is such that it gives 10% on investment. At what price did he…Preview
7.7

Depreciation

12 Q

Depreciation is the fall in the value of assets — buildings, machinery, equipment of every kind — as they are used over time. A few terms describe how this loss is measured.

+Worked Examplesi4 questions
  1. Example 26On 1st April, 2020, Ram purchased a machinery costing ₹40,000 and spent ₹5,000 on its erection. The estimated effective life of the machiner…Free
  2. Example 27A machine costing ₹30,000 is expected to have a useful life of 4 years and a final scrap value of ₹4000. Find the annual depreciation charge…Free
  3. Example 28An asset costing ₹10,000 is expected to have a useful life of 4 years and a scrap value of zero. Find the annual depreciation charge using t…Preview
  4. Example 29A machine costing ₹50,000 depreciates at a constant rate of 8%. What is the depreciation charge for the 8th year. If the estimated useful li…Preview
+Exercise 7.7i8 questions
  1. Q1A machine costing ₹30000 is expected to have a useful life of 13 years and a final scrap value of ₹4000. Find the annual depreciation charge…Free
  2. Q2An asset costing ₹15,000 is expected to have a useful life of 5 years and a scrap value of ₹3000. Find the annual depreciation charge using…Free
  3. Q3A piece of machinery costing ₹10000 is expected to have a useful life of 4 years and a scrap value of zero. Find the annual depreciation cha…Free
  4. Q4A machine, the life of which is estimated to be 15 years, costs ₹40,000. Calculate the scrap value at the end of its life if it is depreciat…Preview
  5. Q5A machine costing ₹5000 depreciates at a constant rate of 5%. What is the depreciation charge for the 5th year?Preview
  6. Q6A firm bought a machinery for ₹7,40,000 on 1st April, 2018 and ₹60,000, is spent on its installation. Its useful life is estimated to be of…Preview
  7. Q7Shiv & Co. purchased a mobile phone for ₹21,000 on 1st April, 2019. The estimated life of the mobile phone is 10 years, after which its resi…Preview
  8. Q8On 1st April, 2015, Dreams Ltd. purchased an AC for ₹3,00,000 and incurred ₹21,000 towards freight, ₹3,000 towards carriage and ₹6,000 towar…Preview