QTM 2101 v3

8/8/2019 QTM 2101 v3

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Quantitative Techniques for Management

Aditya K Biswas

8/8/2019 QTM 2101 v3


Time Value of Money

Cash Flow over Time

Future Value is not same as the current Value

Future Value of a single amount (present)

Present Value of a single amount (future)

Future Value of an Annuity

Present Value of an Annuity

Intra Year Compounding and discounting

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Simple Interest

1) A bank employee has taken a loan of Rs 5 lac

for housing. The interest rate is 10 % simple .

± How much he will have to pay after 5 Years ?

± If he pays Rs 1 lac every year when will his loan be

paid in full ?

± What will be the approx EMI if the loan period is 15

years?

FV = PV ( 1 + r n)

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Compound Interest

1a) For the previous example what will be the amount

due after 5 years if the interest is @ 10 % compound

1b) If the employee pays Rs 1 lac per year what will be

his payment schedule? FVn = PV (1 + r ) n

ear O/ al Interest C/ al O/ al Interest C/ al

1 1,000 100 1,100 1,000 100 1,100

5 1,400 100 1,500 1,464 146 1,610

10 1,900 100 2,000 2,358 236 2,594

20 2,900 100 3,000 6,116 612 6,728

50 5,900 100 6,000 106,718 10,672 117,390

100 10,900 100 11,000 12,527,829 1,252,783 13,780,612

Simple Interest Compound Interest

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Growth Rate

Doubling period for deposits is given by

Log ( )= n Log(1+r)

± Thumb rule is n = 0. 5 + ( r)

Growth rate (g=CAGR) can be calculated as

G = (1+g)n

Calculate the CAG

R when a company salesgrows 10 times in 10 years ( 0. % ?)

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Present Value ± Discounting future

We have seen FVn = PV (1 + r ) n

Therefore PV = FVn [1 (1 + r ) n ]

ear 1

arning -10000 -5000 0 5000 5000 5000 5000 5000

Discount NPV

10,000.00

1 8,750.99

7,598.23

6,533.69

5,550.10

4,640.87

3,799.96

3,021.92

2,301.76

1,634.91

1 1,017.23

11 444.92

1 -85.50

1 -577.20

1 -1,033.11

1 -1,455.89

-4,000.00

-2,000.00

0.00

2,000.00

4,000.00

6,000.00

8,000.00

10,000.00

12,000.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

NPV

NPV Linear (NPV)

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Internal Rate of Return

PV is zero when discounting rate = IRR

Marginal attractive rate of return while analyzing

portfolio of projects

IRR for initial screening of projects

Cash inflow and outflow both discounted at the

same rate !

Compare PV at a given rate against IRR for different projects to select

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Depreciation

Measure against wear and tear of assets

Fund created to replace after the useful life

Books of accounts as per rules

Benefits in TAX calculation

Straight line method

Written Down Value (WDV)

Full year or part year

Asset types ± building machinery computers

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Problems - Depreciation

) Calculate book value of a computer

purchased at Rs 100 lac after years. The WDV

method was followed @ 0 % annual

± What are the tax benefits during year 1 & 5 if the company Tax is 0 %

) Expenditure of Rs 100 lac in the current year

will be amortized in next 5 years. What is the

loss in Tax benefits if the Tax bracket is 0 % allthe years. Depreciation is not applicable here.

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Annuity

Due in the beginning or end period (deferred)

FVAn = A(1+r) n-1 + A(1+r) n- « + A

FVAn

= A * FVIFAr n

Where FVIFAr n

= [(1+r)n ± 1] r

PVAn = A(1+r) -1 + «. A(1+r) -n

PVAn = A[{1-(1 (1+r) n} r]

PVAn = A * PVIFAr n (= A r in case of perpetuity)

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Annuity

) Deposited Rs 0 000 every year to PPF for 0

yrs interest being @11%. The Future value will

be 0 000 [(1+0.11) 0 ± 1] 0.11 = Rs 5 0 00

5) You deposit Rs 1000 every year for 5 yearsand get Rs 15 after 10 years. What is the

effective interest rate? (Is it 10 % ?)

) Borrowed Rs 1 0 0 000 for flat. Paying Rs1 0 000 per year. What will be the maturity

period if interest rate is 1 .5 %? (1.1 5n)= ;

n=11. years)

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Intra year compounding

FVn = PV (1 + r m ) mn

PV = FVn [1 (1 + r m ) ] mn

Effective interest rate for 1 % annual when

compounded half yearly is 1 . % and when

compounded quarterly it becomes 1 .55 %

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AP & GP

Series ± atural numbers odd numbers even

numbers squares of Fibonacci Prime etc.

th term by Induction : Tn = f(n)

Series with common difference or common ratio

Arithmatico- Geometric series ±

a (a+d)r (a+ d)r «. (a+(n-1)d)r n-1

S=a+(a+d)+(a+ d) « n terms .. AP series

S= a+ar+ar + « n terms « GP series

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Series SUM

Sn = n(a+l) = an + n(n-1)d for AP

Sn = a(r n ± 1) (r-1) for GP

Sn = 1 + + + + «« + n

= n(n+1)( n+1) [Hint consider n ± (n-1) ]

Similar for Sn = 1 + + « + n = [n(n+1) ]

S= +5x+ x^ +11x^ + «. |x| <1 = ( +x) (1-x)

S= *5+5* + *11 + « + ( n-1)( n+ ) .. n terms

= n (n+1)(n+ )

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Mean

AM = (X1+X + .. Xn) n ..

± Consider Bank interest ±and also the concept of

weighted averages

GM = (X1*X ..*Xn)1 n ..

± Interest rate over periods - multi period and intra year

± CAGR estimation

HM = n ( 1 X1 +1 X .. + 1 Xn)

± Consider 5 and see that AM >=GM >= HM

± Average speed when D distance travel with v1 and

then again D with v . Average P E ratios in equity.

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Permutation

n! = 1* * «*n .. This is factorial of n

Multiplication principle in permutation

± ne event has m possible results another event has

n possibility. Then together there are mn possibleoutcomes of the two events.

Arrange n distinct objects .. n !

Arrange r distinct objects out of n .. n! (n-r)!

Arrange n objects n1 alike n alike ..n! (n1! n !)

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Combinations

Choose r distinct objects from n .. o repetition

± n! [(n-r)!r!]

Choose r distinct objects from n .. Repetition ok

± n^r permutations (n+r-1)! [(n-1)!r!] combinations

Binomial theorem (x+y)^n = n (nCk)x^k*y^n-k

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Problems

) books on Maths Chemistry History and 1English. How many ways these books can be arranged

keeping same subjects together ( 1 )

) Tournament has Russians from US A fromGBand 1 from Brazil. Result will be in the order of

placement showing country. How many possibilities ?

(1 00)

) men and 5 women to from a committee of women

and men ( 50). What if men refuse to work together

( 00)

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Concept of Probability

A full deck of cards contain suits X 1 each.

Find the number of ways

± a) 5 cards can be selected from 5 cards

± b) cards will be Kings and Queens

P {X=n} = n -> infinity (n success in trials)

http: www.scribd.com doc 5 11 1 00-

Permutation-Combination-Prob-Lesson

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SET theory

SET objects membership (Fuzzy) Finite

Countable Uncountable Real numbers

Singleton Empty Subset Superset

rdered pairs Triples Quadruples

Complementary set Disjoint set

Venn diagram

Union Intersection and Difference of sets

Cardinality of Set Union of two sets

De Morgan¶s Law

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Problems

10) In a town of 500 people 5 read Hindu

1 read Indian Express & 1 read ToI. f

these 0 read Hindu and ToI read Hindu and

IE 5 read ToI & IE. 50 do not read newspaper.How many read only one paper? [ 1]

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More

Relations: R is binary relation if all elements are

ordered pairs. z R where z =(x y) xRy

± Domain R ={x| there exists y such that xRy holds}

± Range R={y| there exists x for each y that xRy holds}

Functions: Domain F=A Range F in B

± F:A->B <F(a)| a A>

± Injection Bijection Surjection examples

X = {1 } Y= {D B C A}

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Problems

11. How much you need to deposit (one time) in bank to

get Rs 100 000 after 5 years when the interest is 1 %

compounded monthly?

1 . what will be equivalent monthly payment if you donot want one time deposit.

1 . A man has total Rs 5 00 000 which he will invest for

his son and daughter who will get the amount with

interest when they attain age 1 in the ratio :1. Son is

now aged 10 and the daughter is . The interest is 10%simple. How much to invest for son?

1 . Sum squares of n natural numbers.

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Problems

15. If (ab+bc) is the geometric mean of (a^ +b^ ) and

(b^ +c^ ) then show that b is the GM of a and c.

1 . numbers are in AP and the total is . The product

of first and last number is - 5. Find the numbers. 1 . Examination question paper contains 1 questions.

Parts I consisting of 5 and part II consisting of

questions Student has to attempt questions choosing

at least from each part. How many ways the selection

can be made.

1 . In hostel there are 100 students. 0 drink tea 50

drink milk and 0 drink both. How many do not take any?

Use Venn diagram.

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Problems

A = { 10} B={1 5 10}

1 . Explain Union and Intersection of A and B

0. What will be set difference of A related to B

1. Explain the symmetric difference

A = {1 } B= { } U={1 5 }

. State and explain De Morgan¶s law involving A and B

which are subsets of the universe U.

. Also explain the cardinality of Sets after Union.

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Problem

. If you invest Rs 500 000 in immediate Annuity (to

begin after 1 year) for 10 years. What amount will you

receive annually if the rate of return is % annual.

5. What will be the Annuity for 10 years if it is deferredby 10 years (annuity investment after 10 years) and the

interim rate of return is 10 %

. Book value of a computer system after years

following WDV for depreciation ( 0 %) is 0 % of book

value of another asset with same procurement cost andtime but following straight line depreciation. What is the

estimated life of this Asset?

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Problems

. Consider Patients Beds and Doctors in a Hospital.

How relationship can be established through Set

concepts & Cardinality.

. Analyze given yearly cash flows for projects for different discounting rates and comment.

± Project A: -1 000 +5000 +5000 +5000 +10000

± Project B: - 0000 +5000 +10000 +10000 +0

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Mathematics for Economics

Linear function: y=f(x)= a+bx where a & b known

Graph of a linear function

Equation of a straight line: y=mx+c;

y=c when x=0; slope of the line is m= dy dx

Line passing through a point (x1 y1) given m

y1=mx1+c will be (y-y1)=m(x-x1)

Intersection of lines:a1x+b1y=c1 & a x+b y=cwill be the solution of these equations. Solve by

Gauss Jordan method of elimination etc..

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Mathematics for Economics

Parallel shifts: ax+by=c1 c c «

Lines through same point on Y axis but different

intercepts on X axis: y=mx+c; m=m1 m «

Convex and Concave functions: Consider a pointin between f(x1) and f(x ) [uni modal?]

f[*X1 + (1- )x2] > f(X1 ) + (1- ) f(x2) : concave

f[*X1 + (1- )x ] < f(X1 ) + (1- ) f(x ) : convex

For all 1> > 0

First order derivative dY dX when Y=a+bX+cX is

b+ cX : rate of change tangent at x1=b+ cX1

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Linear and non linear

-10

-5

0

5

10

15

20

25

0 1 2 3 4 5

Y1 Y2 Y3 Y4 Y5

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Problems

. Profit from selling product A is Rs 5 and from Product

B is Rs 10. Show that total profit is represented by iso

profit parallel lines depending on number of A and B

products sold as X1 and X

0. When Y= +5X-X

± What is the rate of change of Y at X=

± where is an optimum ?

1. Find intersection of x+ y-1 =0 and x+ y-1 =0 . Y= +5X-X is a convex function or concave?

Consider between X=0 and X=

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QTM 2101 v3