Mathematical andStatistical Techniques

(As per the Revised Syllabus of F.Y. B.Com., 2012-13, Semester IUniversity of Mumbai)

Dr. (Mrs.) Abhilasha S. MagarM.Sc. (MATHS), B.Ed., Dip. C.P., Ph.D.

Department of Mathematics,A. Vartak College,

Vasai Road (W), Palghar – 401202.

Manohar B. BhagirathM.Sc. (MATHS), M.Phil., SET, PGDIT

Department of Mathematics,A. Vartak College,

Vasai Road (W), Palghar – 401202.

MUMBAI NEW DELHI NAGPUR BENGALURU HYDERABAD CHENNAI PUNE LUCKNOW AHMEDABAD ERNAKULAM BHUBANESWAR INDORE KOLKATA GUWAHATI

© AuthorsNo part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by anymeans, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of thepublisher.

First Edition : 2015

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PREFACE

We are very happy to present “Mathematical and Statistical Techniques” to the studentsof F.Y. B.Com. (Semester I). The book is based on the new revised syllabus. The text closelyfollows the chapters, topic, patterns prescribed in the syllabus by University of Mumbai.

We have tried to cover the topics in sufficient depth and in simple manner with easy tounderstand examples. We have also given tutorials in the end of the unit which is convenient forteachers as well as students.

We would like to thank our principal Dr. Keshav N. Ghorude, Dr. Rajendra P. Deore (Professor,Department of Mathematics, University of Mumbai) and Dr. Vinod M. Gaikwad for providing anenvironment which stimulates new thinking and innovation.

We would like to thank our seniormost colleague and Head, Dr. Santosh A. Shende andalso my loving friend Mrs. Arundhati Bane for the encouragement and continued support forcompleting this book.

Special thanks to our family for support this work.

Nothing is perfect in this world and there are always chances to improve. I shall alwaysremain obliged for any improvement suggestions.

A big thanks to Mr. Srivastav and all the staff of Himalaya Publishing House Pvt. Ltd. forbringing out this book in time.

Dr. (Mrs.) Abhilasha S. Magar(magar_a47@yahoo.com)

Prof. Manohar B. Bhagirath(manoharbhagirath@gmail.com)

SYLLABUSWorkload: Mathematics : 2 lectures per week

Statistics : 3 lectures per week

Tutorial : 1 per week (per batch)

Tutorial batch size : 25 students

SEMESTER ICourse: UBCOMFSI.6

Mathematical and Statistical Techniques-I

[A] Mathematics: (30 marks)UNIT I: SHARES AND MUTUAL FUNDS

(a) Shares: Concept of share, face value, market value, dividend, equity shares, preferential shares, bonusshares. Simple examples.

(b) Mutual Funds: Simple problems on calculation of net income after considering entry load, dividend, change inNet Asset Value (NAV) and exit load. Averaging of price, under the Systematic Investment Plan (SIP)

UNIT II: PERMUTATION, COMBINATION AND LINEAR PROGRAMMING PROBLEMS(a) Permutation and Combination: Factorial Notation, Fundamental principle of counting, Permutation as

arrangement, Simple examples, combination as selection, Simple examples, Relation between nCr and nPr,Examples on commercial application of permutation and combination.

(b) Linear Programming Problem: Sketching of graphs of (i) linear equation Ax + By + C = 0 and (ii) linearinequalities. Mathematical Formulation of Linear Programming Problems upto 3 variables. Solution of LinearProgramming Problems using graphical method upto two variables.

[B] Statistics: (45 marks)UNIT III: SUMMARIZATION MEASURES

(a) Measures of Central Tendencies: Definition of Average, Types of Averages: Arithmetic Mean, Median andMode for grouped as well as ungrouped data. Quartiles, Deciles and Percentiles. Using Ogive, locate medianand Quartiles. Using Histogram, locate mode. Combined and Weighted mean.

(b) Measures of Dispersions: Concept and idea of dispersion. Various measures Range, Quartile Deviation, MeanDeviation, Standard Deviation, Variance, Combined Variance.

UNIT IV: ELEMENTARY PROBABILITY THEORY(a) Probability Theory: Concept of random experiment/trial and possible outcomes; Sample Space and Discrete

Sample Space; Events their types, Algebra of Events, Mutually Exclusive and Exhaustive Events,Complimentary events.Classical definition of Probability, Addition theorem (without proof), conditional probability.

(b) Random Variable: Probability distribution of a discrete random variable; Expectation and Variance of randomvariable, simple examples on probability distributions.

UNIT V: DECISION THEORYDecision making situation, Decision maker, Courses of Action, States of Nature, Pay-off and Pay-off matrix;

Decision making under uncertainty, Maximin, Maximax, Minimax regret and Laplace criteria; simple examples to findoptimum decision. Formulation of Payoff Matrix. Decision making under Risk, Expected Monetary Value (EMV); Decision Tree;Simple Examples based on EMV. Expected Opportunity Loss (EOL), simple examples based on EOL.

PAPER PATTERNSemester I End Exam (75 marks)

Instructions (i) All questions are Compulsory.

(ii) Attempt any THREE sub-questions from each question.

(iii) Each question is of 15 marks.

(iv) Simple non-programmable calculator is allowed.

Q. 1. (a) 5 Marks(b) 5 Marks

(c) 5 Marks

(d) 5 Marks

Q. 2. (a) 5 Marks

(b) 5 Marks

(c) 5 Marks(d) 5 Marks

Q. 3. (a) 5 Marks

(b) 5 Marks

(c) 5 Marks

(d) 5 Marks

Q. 4. (a) 5 Marks(b) 5 Marks(c) 5 Marks

(d) 5 Marks

Q. 5. (a) 5 Marks

(b) 5 Marks

(c) 5 Marks(d) 5 Marks

CONTENTS

ChapterNo.

Title Page No.

1 Shares and Mutual Funds 1 – 152 Permutations and Combinations 16 – 283 Linear Programming Problem (LPP) – I

[Formulation and Graphical Method]29 – 48

4 Measures of Central Tendency 49 – 1175 Measures of Dispersion 118 – 1536 Probability 154 – 1777 Random Variable 178 – 1848 Decision Theory 185 – 229

Tutorial 230 – 238

Introduction

A unit of ownership, that represents an equal proportion of a company’s capital is called asshares.

According to financial term, the share is regarded as unit of account that can representseveral monetary instrument such as stocks, mutual funds, REIT’s etc. A stock market or sharemarket is the aggregation of buyers and sellers of stock or shares. A stock exchange is a place totrade stocks. In India (i) Bombay stock Exchange (BSE) and (ii) National stock Exchange (NSE)these are the two stock Exchange.

Shareholder: Any Person who owns at least one share of a company’s stock is calledshare-holder or stock holder.

Types of sharesThere are two major types of shares are

(i) Ordinary shares: Ordinary shares are also called as equity shares or common stock,which entitle the shareholder to share in the earnings of the company as and when they occureand to cast a vote at the company’s annual general metting’s and other official mettings.

Equity shares are those shares which are ordinary in the course of company’s business.Equity share capital is owned capital and every share has fixed value. Equity share are easilytransferable and always irredeemable.

(ii) Preference shares: Preference shares are also called as preferred stock, which entitlethe shareholder to a fixed periodic income or interest but generally do not give him or her votingrights.

Preference shares are cummulative, non-cummulative, redeemable, convertible etc. various typesof preference shares are seen in the market based on their different structure, maturity term,dividend rate etc.

CHAPTER

1 SHARES AND MUTUALS FUNDS

UNIT I

Face value: Also called nominal value or dollar value. It is the origanal cost of the stockshown on certificate. The face value of equity share is normally ` 10 and for Preference share it is` 100.

Dividend: A share is a Part of the company. When the company makes profit, you oftenreceive a part of it. This is the primary source of income. Dividend depends on the profit made bya company. Dividend is declared by the company on the face value of a share. Dividends aredeclared quarterly, half-yearly or annually depends company’s contribution.

Bonus shares: Bonus shares are additional shares given to the shareholder without anyadditional cost; based on the number of shares that a shareholder own. Normally, bonus sharesare declared in the ratio.

Brokerage: Brokerage is the commission charged by broker for buying and selling of sharesfrom buyer or seller. If you are buying shares, brokerage is added to the market price and if youselling shares, brokerage is subtracted from the market value.

Some useful formulae

1. Number of shares purchased =shareaofpriceMarket

investedamountTotal

2. Total dividend =100

shares of No. value Face dividend of Rate

3. Percentage gain = 100InvestedAmountgainNet

4. Net gain = Total sale + Total dividend – Total investmentOR

Net gain = Total sale of – Total Purchase.

5. Rate of Return = 100shareoneinInvestment

shareperreceivedDividend

6. Number of shares sold =shareaofpriceSelling

investedamount Total

Examples on shares

Example 1: Find how many shares of market value of ` 50 with face value of ` 10 can bepurchased for ` 10,000

Solution: Given: market value of 1 share = ` 50Face value = ` 10. Total invested amount = ` 10,000

No. of shares Purchased =share1ofvalueMarket

Investment Total

=50

10,000 = 200 shares

Example 2: How many shares of market price of ` 130 each, can be purchased for ` 50150with brokerage being 0.5%?

Solution: Given; market price of 1 share = ` 130Total Investment amount = ` 50,150

Brokerage = 0.5

Brokerage per share = 0.5% of market price

= 65.01301000.5

Purchased price of 1 share = M.V. + Brokerage

= 130 + 0.65 = 130.65.

No. of shares purchased =share1ofpricePurchase

InvestmentTotal

=65.130

50150 = 383.84 shares

Example 3: Mr. Sham owns 500 shares of ABC company. The company declares a dividend of10% the face value of each share is ` 10. Find total dividend which Mr. Sham will get.

Solution: Given: No. of shares = 500Rate of dividend = 10%F.V. = ` 10

Total dividend = Rate of dividend × F.V. × No. of shares

= 5001010010

= ` 500

Example 4: Find Face value of a 15% share, if ` 12,500 were invested to purchase shares ata market value of ` 150 and total dividend of ` 125 was received.

Solution: Given: Investment (Total) = ` 12,500Rate of divident = 15%Market value of each share = ` 150

F.V. = ?

No. of shares =share1ofM.V.

InvestmentTotal

=150

12500 = 83.33 shares

Total dividend = Rate of dividend × F.V. × No. of shares.

125 =10015 × F. V. × 83.33

33.8315100125

= F. V.

FV = ` 10

Example 5: From the following which investment is better:7% of ` 100 shares at ` 120 or

8% of ` 10 shares at ` 13.50?

Solution: Let the investment amount be ` XCase I Given: Rate of dividend = 7%F.V. = ` 100, Market value = ` 120.

No. Of shares =120

xshare1of.V.MamountInvestment

Total dividend = Rate of dividend × F.V. × M. V. of 1 share

= X05833.0120

x100100

7

Case II Given: Rate of dividend = 8%F.V. = ` 10, Market value = ` 13.50

No. of Shares =50.13

xshare1of.V.MamountInvestment

Total dividend = Rate × F.V. × M.V.

=50.13

X10100

8 = 0.05925 X

As dividend for Case II is more than case I.

Case II is more profitable to invest amount.

Example 6: Mr. Saptak invested to ` 80,000 to purchase equity shares of TATA company atmarket value of ` 210 each through a broker, charging 1% brokerase. The face value of ashare is ` 10. How many shares did saptak purchase?

Solution: Given: Total Investment = ` 80,000M.V. ` 210, F.V. = ` 10, Brokerage = 1%

Now, Brokerage per share = 1% of M.V.

= 210100

1 = 2.10 per share

Purchase price of 1 share = M.V. + Brokerage= 210 + 2.1 = ` 212.10

No. of shares purchased =share1ofpricePurchase

InvestmentTotal

=10.212

000,80 = 377.18 shares.

Examples 7 : Farhan purchased shares of IDBI company at ` 100 and sold them on next day at` 122 due to some personal problem. The brokerage charged was 0.5% on sale. If net profitreceived by Farhan was ` 12,000 in both the transaction. Find number of shares purchased onthat day.

Solution: For salesGiven: M.V. of each share = ` 122

Brokerage = 0.5% Actual selling price of 1 share = MV – Brokerage

=

122100

5.0122 = 122 – 0.61

= 121.39

Profit per share = 121.39 – 100 = 21.39

Given: Net Profit = 12,000.

No. of shares purchased =shareperProfit

profitNet

= shares56139.21

000,12

Example 8: Asha invested ` 50,000 in equity shares of face value ` 10 and market price of `95 each. After receiving 10% dividend, she sold all the shares at market price of ` 120 each.She paid 2% brokerage on both the transaction. Find her percentage profit or loss on boththe transaction.

Solution: For purchaseGiven: M.V. = ` 95, F.V. = ` 10

Investment amount = 50,000, Brokerage = 2%

Purchase price of one share = M.V. + Brokerage

= 9.9695100

295

No. of shares purchased =share1ofpricePurchased

InvestmentTotal

=9.96

000,50 = 515.99 shares.

For salesM.V. = ` 120, Brokerage = 2%

F.V. = ` 10, Rate of divi. = 10%

Selling price of 1 share = M.V. – Brokerage.

=

120100

2–120 = 117.6

Selling price of all shares = 117.6 × 515.99= ` 60,680.42

Total dividend = Rate of dividend × F.V. × No. of shares

=10010 × 10 × 515.99 = 515.99

Total gain = sale + dividend – purchase price

= 60,680.42 + 515.96 – 50,000

= 11,196.384.

Percentage gain or profit = 100pricePurchase

gainTotal

= 100000,50

384.196,11 = 22.39%

Example 9 : Nilakshi purchased 150 shares of ICICI at ` 210 on 3rd Jan 2014. On 13th Feb.2014, the company decided to split all the shares of company so that the face value of theshare become ` 10 from ` 20 per share. The market price as on 30th July, 2014 is ` 300.Find number of shares held by Nilakshi as on 30th July 2014. Also Find gain as on 30th July2014.

Solution: Given: Total number of shares = 150M.V. = ` 210, F.V. = ` 20.

Total amount invested = No. of shares × M.V.

= 150 × 210

= 31,500

No. of shares held on 30th July 2014.

=10

20150 = 300 shares

Actual value of shares on 30th July 2014

= 300 × 300

= 90,000

Profit (Gain) = 90,000 – 31,500

= 58,500.

Example 10: Mr. Saket sold 200 shares of AIR INDIA corporate at ` 450 per share with facevalue of ` 100. Company paid 10% dividend per annum. After 10 days he invested theamount in purchasing the shares of CIPLA company at ` 500 per share with face valueof ` 100. Company paid a divident of 12% per annum. Find the no. of shares purchased andtotal dividend.

Solution: For saleNo. of share sold = 200, F.V. = ` 100

M.V. = ` 450, Dividend rate = 10%

Total Dividend received = Rate × F.V. × No. of shares

= 000,220010010010

Total S.P. = 450 × 200 = 90,000Total amount received = S.P. + Dividend= 90,000 + 2,000 = 92,000For PurchaseM.V. = ` 500, F.V. = ` 100, rate of dividend = 12%

No. of share purchased =share1ofM.V.

investedAmount

=500

000,92 = 184 share

Total dividend = rate × F.V. × No. of shares.

= 18410010012

= 2,208

Exercise1. Dipali invested in shares of BIRLA Industries of face value ` 100. She received ` 1,425 as total dividend at

9.5%. Find the number of shares she bought.Ans: 150 shares

2. Find the total dividend of ` 10,000 at 10% rate of market value ` 130 and face value of ` 10Ans: ` 76.92

3. For the following data which company is betterCompany A: F.V. ` 10, market value = ` 250Rate of dividend = 25%Company B: F.V. ` 10, market value = ` 250Rate of dividend = 18%

Ans: Company B is better4. Find the percentage gain or loss if 200 shares of F.V. ` 10 were purchased at ` 350 each sold

later at ` 352/- the brokerage being 0.5% on both the transaction.Ans: Loss of 43%.

5. Mr. Abhishek purchased 30 shares of F.V. ` 10 on 18th Dec. 2014, at ` 36 Per share. On 13th Feb. 2015the company decided to split all the shares from F.V. ` 10 to ` 2 per share. On 2nd March 2015 themarket price of 1 share was ` 8 per share. Find gain or loss if he sells all the shares on 2nd March 2015

Ans: Gain = ` 120.6. Sneha invested ` 75,000 in equity shares of ` 10 F.V. and market value of ` 85 each After receiving

15% dividend she sold all the shares at ` 70 each. She paid brokerage of 1.5% on both thetransaction. Find her profit or loss.

Ans: Loss = ` 13,756.917. Rashmi purchased 150 shares of IDBI at ` 190 on 13th June 2013. On 25th Nov. 2014 she sold all

the shares at ` 205 each she paid brokerage of 1% on purchase and 2% on sale. Find her netprofit or loss.

Ans: Profit = ` 1,4858. Pankaj owns 500 shares of ` 10 each. Find his income, if 14% dividend is declared on these

shares in a certain year.Ans: ` 700.

9. What is the income on the investment of ` 32,300 in 8% stock at ` 95?Ans: ` 2,720

10. ABC company issued 2500 shares of F.V. ` 10 each. A dividend of ` 1,00,000 was declared by company.Find the rate of dividend for the company.

Ans: 4%

Mutual Funds

Introduction:An investment programme funded by shareholder’s, that trades in diversified holding and

professionally managed is called as mutual funds.Mutual fund can be considered basket of investments. Each basket hold dozens or hundreds

of stocks or bonds. Mutual fund can be summarized in simplicity, diversity and accessibility. Amutual fund is basically at less risk as compared to shares. All the profits or losses of the fund areshared by all the investors in the same propertion as the amount of contribution made by them.Mutual fund investment are subject to market risk.

There are thousands of Mutual funds, but they can be divided into stock funds and bondfunds. All mutual funds fall into two broad categories : open ended funds and closed ended funds.

Open-ended Fund: In open ended fund, there is no limit to the number of units purchasedand sold. It is collective investment scheme which can be issue and redeem share at any time.This scheme do not have a fixed maturity period.

In order to determine the value of a share in open-ended fund at any time, a number calledNet Asset Value (NAV) is used. Investor can buy and sells his units at Net Asset Value (NAV)which are declared on a daily basis. The majority of mutual funds are open-ended.

Closed-ended Fund: In closed-ended fund will issue a fixed number of shares are issued tothe public. The price of share in a closed-ended fund is determined by market-demand. Closed-ended funds are not redeemable and usually listed on a recognized stock exchange. This fundalso known as “Closed-end investment” or “closed-end mutual fund.”

These mutual funds scheme disclose N.A.V. generally or weekly basis.

Net Asset Value (N.A.V.)A mutual funds price per share or unit is called Net Asset Value (N.A..V.) All mutual funds

buy and sells, orders are processed at the N.A.V of the trade date. N.A.V. may represent thevalue of the total equity.

Entry Load and Exit loadIn a Mutual Fund, some Asset Management companies have sale charges or loads on their

funds, i.e., entry load and Exit load to compensate for distribution cost.

Entry load is charged at the time, an investor purchase the units. Whereas, exit load ischarged at the time of sale of all units.

e.g. If the N.A.V. is ` 100 and entry load is 1% then the purchased price of 1 unit is ` 101.Similarly, if the N.A.V. is ` 100 and exit load is 1% then selling price of 1 unit is ` 99.

i.e. If units are purchase the entry load are added and for selling units exit load aresubtracted.

Examples on Mutual Funds

Example 1: Mr. Raj invested ` 90,000 in TATA M.F. When N.A.V. of ` 150. If there is noentry load, Find how many units, has he bought?

Solution: Given: Amount Invested = ` 90,000N.A.V. = ` 150 each

Note that here there is no entry load

Number of unit purchased =unit1ofN.A.V.

InvestedAmount

=150

000,92 = 600 units.

Example 2: Ms. Nilu invests ` 65,000 in XYZ M.F. where N.A.V. of ` 98. How many unitsshe bought, if entry load is 2%.

Solution: Given: Amount Invested = ` 65,000N.A.V. = ` 98, Entry Load = 2%

Actual Purchase amount of 1 unit = NAV + Entry load

=

98100

298

= 99.96

No. of Unit purchased =unit1ofN.A.V.

InvestedAmount

=96.99

000,65 = 650.260

Example 3: If a HDFC funds N.A.V was ` 58 at the beginning of the year and ` 70 at the endof the year. Find (i) Absolute change in N.A.V. (ii) Percentage change in N.A.V.

Solution: We know that, Absolute change in N.A.V. = N.A.V. at the end of year – N.A.V atbeginning of year.

= 70 – 58 = ` 12.

Percentage change in N.A.V. = 100yearofbeginningatNAV

NAVinchangeAbsolute

= 20.689%

Example 4: If a RELIANCE M.F. had a N.A.V. of ` 60 at the beginning of the year and ifpercentage increase in NAV during the year was 15% Find absolute change in N.A.V.

Solution: Given:NAV at the beginning of the year = ` 60

% change in NAV = 15%

Absolute change in NAV = (% change in NAV) × (NAV at beginning of year)

= 96010015

NAV at the end of the year = (Absolute change in NAV) + (NAV at beginning of the year)

= 9 + 60 = 69

Example 5 : Narendra Purchased 200 units of SBI M.F. on 2nd May 2012, when N.A.V. of `45. His NAV on 30th Dec. 2012 was ` 50 and he sold all the units. The fund had 2% entryload and 2.5% exit load. Find amount invested and his net profit,

Solution: For purchase: No. of unit purchase = 200N.A.V. = ` 45, Entry load = 2%

Purchase price of 1 unit = NAV + Entry load

= 45 +

45100

2

= ` 45.90

Purchase Price of 200 units = 200 × 45.90= ` 9,180. Total amount investment = ` 9,180

For saleN.A.V. = ` 50, Exit load = 2.5%

Selling price of 1 unit = NAV – Exit load

= 50 –

50100

5.2

= ` 48.75

Selling price of all 20 units = 200 × 48.75 = ` 9,750

Net profit = Selling price – Purchase Price

= 9,750 – 9,180 = 570

Example 6: Kishor invested ` 25,000 in KBX M.F. with entry load 1%. Find the N.A.V. if thenumber of units purchased was 120.

Solution: Let N.A.V. of 1 unit = ` XPurchase price of 1 unit = NAV + Entry load

=

X100

1X

= X + 0.01 X

= 1.01 X -----------------------------------------------------------------(1)

GivenAmount investment = ` 25,000

No. of unit purchase = 120.

Purchase price of 1 unit =purchaseunitofNo.

InvestedAmount

=120

000,25 = 208.33 -----------------------------------------------------------------(2)

From (1) and (2)1.01 X = 208.33

01.133.208X = 206.27

I.e. N.A.V. (X) = ` 206.27

Example 7: Mr. Amin sold all units of BAJAJ M.F., when NAV of ` 110 with exit load of 1%. Hereceived ` 48,500; find the number of unit sold.

Ans: Given: NAV = ` 110 Exit load = 1%

Total amount received = ` 48,500

Now, selling price of 1 unit = NAV – Exit load

= 110 –

110100

1 = ` 108.90

No. of unit sold =unit1of.P.S

InvestedAmount

=90.108

500,48 = 445.36 units.

Example 8: Vijayanand purchased 500 units of max M.F. at ` 8,000 and sold all the unitsafter 2 months when NAV of ` 35. The short term gain tax (STGT) was 10% of the profit.Find his net profit or loss.

Solution: Given: No. of unit purchased = 500Purchase Amount = ` 8,000

Now selling Price of all the units

= 35 × 500 = 17,500

Profit = SP – Pur. Price = 17,500 – 8,000 = 9,500

Short term gain tax (STGT) = 10% of profit

= 500,910010

= ` 950

Net profit = 9,500 – 950 = ` 8,550

Example 9: Prajakta invested ` 50,000 in a M.F. on 30th Jan. 2014. She redeemed all theunits on 2nd Sep. 2014 at 5% rate of returns when N.A.V. was ` 75. Find NAV on 30th Jan.2009.

Solution: Let N.A.V. = ` X

Rate of return = 100NAVPurchase

NAVPurchase–NAVRedemption

5 = 100X

)X–75(

5x = 7,500 – 100 x

105 x = 7,500 X =105500,7 = 71.428

i.e. NAV on 30th Jan. 2014 = ` 71.428.

Example 10: Sachin invested ` 25,050 in ABC M.F. when NAV of ` 65. He received adividend of ` 8 per unit. After 1 year he sold all the units when NAV of ` 78. If there was 2%load on both the transaction find his gain.

Ans: Purchase price of 1 unit = NAV + Entry load

= 65 +

65100

2 = ` 66.30

No. of unit purchased =30.66

050,25 = 377.828

Given dividend of ` 8 per unit.Total dividend = 377.828 × 8 = ` 3,022.624

Now selling price of 1 units = NAV – Exit load

= 78 –

78100

2 = ` 76.44

Selling price of all units = 377.828 × 76.44 = 28,881.172

Gain = S.P. + dividend – Purchase price= 28,881.172 + 3,022.624 – 25,050 = ` 6,853.796

Exercise

1. Sai invested ` 50,000 in ONGC M.F. when N.A.V. of ` 220. Find number of unit purchase, if thereis no entry load.

Ans: ` 227.2722. A sum of ` 75,500 was invested in a M.F. When NAV of ` 82. Find the number of unit purchased if

the entry load is 1.5%.Ans: 918.156

3. If DHFL M.F. ‘S NAV was ` 28 at the beginning of the year and ` 41 at the end of the year find(i) Absolute change in NAV (ii) percentage change in NAV.

Ans: (i) ` 13 (ii) 46.428%4. If a ABC M.F. had N.A.V. of ` 38 at the beginning of the year and if percentage increase in N.A.V.

during the year was 18%. Find the absolute change in N.A.V.Ans: ` 44.84.

5. Willy invested ` 20,200 in a M.F. when N.A.V. of ` 68. After one year, he sold all the units whenNAV of ` 80. The entry load and exit load was 1% on both the transaction. Find his profit.

Ans: ` 3094.11766. Devendra invested ` 1,20,000 in VCPL M.F. On 3rd Jan 2015 when NAV of ` 540.

On 5th Feb. 2015, he sold all the units when NAV of ` 560. Due to short-term he paid 8% STGT ofthe profit. Find his net profit

Ans: ` 4088.88857. Chetana invested ` 60,250 in MAX GOLD M.F. with 2% Entry load. Find the N.A.V., if the number

of units purchased was 175.Ans: N.A.V. = ` 337.535.

8. Sakharam purchased 100 units of KBX M.F. on 21st oct. 2010, when NAV. of ` 125. His NAV on5th Mar 2011 was ` 138 and he sold all the units. The fund had 2% Entry load and 2.25% exit load.Find total amount invested and his net profit.

Ans: Total Amount Invested = 12,750Profit = 739.5.

9. Mr. Hiranand invested ` 20,400 in a M.F. at ` 10 per unit. After 8 months, he sold all the unitswhen NAV of ` 15. If entry load 2% and exit load 0.5%, find number of unit purchased and amountreceived after sale.

Ans: No. of units purchased = 200Amount received = 29,850

10. Krishana invested ` 15,000 in M.F. when N.A.V. of ` 20. The current value of investment is `1885.Find the entry load.

Ans: Entry load = 3.0927%

Systematic Investment Plan (SIP)

This is plan where investors make regular, equal payment into a mutual fund. In this plan,investor can benifit by buying more units when the price falls and less units when the price rises.This scheme helps reduce the average cost per unit of investment, through a method calledAverage Rupee cost or Rupee cost Averaging. Also, in this plan investor may choose to increaseor decrease the investment amount.

In india, a recurring payment can be set for SIP using Electronic Clearing Services (ECS).SIP is like a recurring deposit in a bank, where you put a fixed amount every month.

Examples on Systematic Investment Plan

Example 1 : Nana invested ` 2,000 every month for 4 month in SBI Gold systematicInvestment plan (SIP). The N.A.V.’s on these particular dates were ` 25, 28, 26.75 and ` 30respectively. After 1 year, he sold all the units when N.A.V. was ` 29. Find average (mean) priceof one unit and his net profit or loss.

Note: there was no entry and exit load.

Solution: Amount Investment = ` 2,000 per month.No Entry load-------Given

Month Amount invested N.A.V. No. of unit purchased1 2000 25 2000/25 = 802 2000 28 71.42853 2000 26.75 74.76634 2000 30 66.6666

Total Amount Invested = 8000 Total no. of units = 292.861

Average (mean) price of 1 unit

=purchasedunitofno.TotalInvestedAmountTotal

=861.292

000,8 = 27.3166

Given: selling price of 1 unit = ` 29

S.P. of all units = 29 × 292.861

= 8,492.9806Net profit = S.P. – purchase price

= 8,492.9806 – 8,000 = 492.9806

Example 2: Ms. Lalita invested ` 5,000 every month for 5 months in BIRLA systematicinvestment plan (SIP) with N.A.V. of ` 35, 41, 40, 45 and ` 50 respectively the entry load was2%. After 9 months she sold all the units when N.A.V. of ` 48.

Solution: Given; Amount invested every month = ` 5,000.Entry Load = 2%. Exit Load = Nil.Selling price of 1 unit = ` 48

month N.A.V. Entry load Purchasedprice of 1 unit

No. of unit purchased

1 35 0.70 35.7070.35

000,5 = 140.056

2 41 0.82 41.82 119.5603 40 0.80 40.80 122.5494 45 0.90 45.90 108.9325 50 1 51 98.039

Total 215.22 589.136

Average price of 1 unit =purchasedunitofno.TotalInvestedAmountTotal

=136.589

000,25 = 42.435

Average (mean) price of 1 unit =5amountTotal

=5

215.22

= 43.044Total S.P. = 48 × 589.136= 28,278.528Gain = S.P. – Purchase price

= 28,278.528 – 25,000 = 3,278.528

Exercise

1. Mr. P.K. Invested ` 4,000 every month for 6 months in S.I.P. The N.A.V.’s on these particulardates were ` 50, 53, 56, 55 and ` 60 respectively. After one year he sold all the units when NAVwas ` 58. Find his net profit or loss.

Ans: Total No. of unit purchased = 440.366Profit/Gain = 1,541.228

2. Priya invested ` 3,000 every month for 4 months in UBI systematic Investment plan (SIP) withN.A.V. of ` 42, 40, 43 and ` 45 respectively. The entry load was 1%. After 10 months she sold allthe units when NAV of ` 50 find average (mean) price of 1 unit. Also find her net profit or loss.

Ans: Average price of 1 unit = 42.847

Average (Mean) price of 1 unit = 42.925Profit = ` 2,003