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8/8/2019 QTM 2101 v3
http://slidepdf.com/reader/full/qtm-2101-v3 1/31
Quantitative Techniques for Management
Aditya K Biswas
8/8/2019 QTM 2101 v3
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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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