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The Time Value of Money

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LOS a
Interest rate can be interpreted as - Required rate of return, Discount rate
or Opportunity cost

LOS b

International Fischer Relationship (approx.)

1
100

@ 10% p.a.

110

Consumption cost 107

True saving 3

Inflation

Real rate of
return

2

e

Nominal risk-free rate = Real risk-free rate + Expected inflation

Treasury bonds = Real RFR + Expected inflation

re

Corporate bonds = Real RFR + Expected inflation + Risk premium
Return on Non-investment grade bond > Return on Investment grade bond, because
risk of Non-investment grade bond > risk of Investment grade bond
Types of risks

nT

3

Liquidity
risk

Maturity
risk

Risk that borrower
will not make
promised payments
in a timely manner

Risk of receiving

less than FV for an
investment if it
must be sold for
cash quickly

Risk of volatility
of price of a bond
because of its
longer maturity

Default risk
premium

Liquidity risk
premium

Maturity risk
premium

Low default rate
=
Low DRP

Less liquidity
=
High LRP

Shorter maturity
=
Low MRP


Fi

Default
risk


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LOS c

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Calculation and interpretation of effective annual rate
The rate of interest that an investor actually earns as a result of
compounding is known as EAR
1 + (Int. rate/m)m - 1

Effective Annual Rate =

m = compounding frequencies per year
EAR on TI BA II Plus Professional - 2nd 2

LOS d

TVM with different compounding frequencies

Annual

Quarterly


Semiannual

N=1
I/Y = 13.25
PV = -100
FV = 113.25

N=1X2=2
I/Y = 13.25/2 = 6.625
PV = -100
FV = 113.68

LOS e

Monthly

N=1X4=4
I/Y = 13.25/4 = 3.3125
PV = -100
FV = 113.92

N = 1 X 12 = 12
I/Y = 13.25/12 = 1.104
PV = -100
FV = 114.08

Annuity

1


e

It is a stream of equal cash flows occurring at equal intervals.

Annuity due

re

Ordinary annuity

End mode
1

2

100

100

0

1

2

3

100


100

100

100

100

PV of perpetuity =

2

LOS f

3

nT

0

Beginning mode

1

Amortization schedule

Loan - 100,000

Fi


CF
Disc. rate

Int. rate - 10%

Tenure - 4 yrs.

Year

Opening
loan

Instalment

Interest
(Op. loan X
rate of int.)

Principal
repayment
(Inst - Int.)

Closing loan
(Op. loan Princ.
repayment)

1

100,000


31,547

10,000

21,547

78,452

2

78,452

31,547

7,845

23,701

54,751

3

54,751

31,547

5,475

26,071


28,679

4

28,679

31,547

2,868

28,679

-


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Using amort function in TI BA II plus professional
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ

Ÿ
Ÿ
Ÿ

2nd CLR TVM (FV)
PV = -100,000 N = 4 I/Y = 10 CPT PMT = 31,547
2nd AMORT (PV)
2nd CLR WORK (CE|C)
P1 = 1
P2 = 1
BAL = 78,452
PRN = 21,547
INT = 10,000
P1 = 2
P2 = 2

3
Eg.
CFs:

PV and FV of uneven CFs
Discount rate = 10%

N = 6 years

Year 1 = −1,000 Year 2 = −500 Year 3 = 0 Year 4 = 4,000 Year 5 = 3,500 Year 6 = 2,000

re

CF

2nd CLR WORK (CE|C)
CF0 = 0
CF1 = −1,000
CF2 = −500
CF3 = 0
CF4 = 4,000
CF5 = 3,500
CF6 = 2,000
I = 10 → Enter → ↓ (down key)
CPT NPV = 4711.91

Fi

nT

Ÿ
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ
Ÿ

e

Using CF function in TI BA II plus professional



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Discounted Cash Flow Applications

LOS a
NPV

PV of inflows − PV of outflows

IRR

Rate at which PV of inflows = PV of outflows
At IRR, NPV = 0

LOS b

Decision rule
IRR

NPV
+ve = Accept

If IRR > WACC = Accept

−ve = Reject


If IRR < WACC = Reject

Mutually exclusive projects Accept project with highest NPV

For a single project NPV and IRR rules lead to same accept/reject decision

e

If IRR > WACC, NPV =+ve
If IRR < WACC, NPV =−ve

LOS c

re

Holding period return (HPR)
Ending value − beginning value
Beginning value

Or

Ending value − 1
Beginning value

Total return

Ending value + CF received − 1
Beginning value

Time-weighted rate of return

(TWRR)

IRR

Geometric mean of HPR

Appropriate if manager has complete
control over inflows and outflows

!

Or

Money-weighted rate of return
(MWRR)

Fi

LOS d

nT

Ending value - beginning value + CF received
Beginning value

Provides better measure of manager’s
ability to select investments

TWRR is not affected by timing of the cash flows, therefore it is more preferred method of
performance measurement

! If funds are contributed to a portfolio just prior to a period of relatively poor performance,
MWRR < TWRR
! If funds are contributed to a portfolio just prior to a period of relatively high returns,


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LOS e & f
Effective earning yield
(1+3.09%)365/90- 1
= 13.13%

j
k
l

Or

3 mistakes analogy to
remember the formulas
(indicated in red)

Compounding
365 days
Investment
value as base

90


3.09%

365

12.53%

3.09
1 + 3.09%

1000
90 days

30
= 3.09%
970

Bank discount yield

12%

j
k
l

Money market yield

No Compounding
360 days
Face value as

base

3
1 − 3%

90

3.09%

360

12.36%

Fi

nT

re

e

360

No Compounding
365 days
Investment
value as base

T - bill
970


30/1000 =3%

j
k
l

Holding period yield

N = 90/365
PV = -970
FV = 1000
I/Y = 13.14 %

90

Bond equivalent yield

j
k
l

No Compounding
360 days
Investment value
as base


Statistical Concepts and Market Returns


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LOS a
1

Descriptive statistics

Inferential statistics

Used to summarize important
characteristics of large data

Used to make forecasts of
large data

Eg. Average of weekly tests

Eg. Forecast on pass or not

2

Sample

Set of all possible members of a stated group

Subset of population

CFA level 1 candidates globally


CFA level 1 candidates in class

Nominal

Ordinal

Interval

Higher level of
measurement than
nominal scales

Provides relative
ranking and
assurance that
differences
between scale
values are equal

nT

Contains least
information

Types of measurement scales

re

3


e

Population

Classification
has no
particular order

Observation is
assigned to a
category

Fi

Eg. MF’s star
rating

LOS b

Parameter

Measure used to
describe a
characteristic of a
population

Sample statistic

Weakness - Zero

doesn’t mean total
absence
Eg. Temperature
measurement

Ratio

Most refined level
of measurement
Provides ranking
and equal
differences
between scale
values
Has a true zero
point as origin

Frequency distribution Tabular presentation of statistical data

It is used to measure
a characteristic of a
sample

Data employed with a frequency
distribution may be measured using any
type of measurement scale


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LOS c

© 2017 FinTree Education Pvt. Ltd.

Relative frequency and cumulative relative frequency
Interval /
class

Frequency

Cumulative
frequency

Relative frequency

Cumulative
relative frequency

10 - 15

7

7

(7/50) 14%

14%

15 - 20


12

19

(12/50) 24%

38%

20 - 25

21

40

(21/50) 42%

80%

25 - 30

10

50

(10/50) 20%

100%

Total


50

100%

Histogram and Frequency polygon

LOS d

Frequency

Interval

Interval
midpoints

Frequency polygon

re

Histogram

e

Frequency

LOS e

Measures of central tendency
1


Weighted mean

AM = 10 + 14 + 4 + 8
4

WM = 10(20%) +
14(20%) +
4(35%) +
8(25%)

WM = 8.2

Geometric mean

Harmonic mean

GM =

HM =

√1.1 X 1.14 X 1.04 X 1.08 − 1

4
1/10 + 1/14 + 1/4 +1/8

GM = 8.94

HM = 7.32

4


Fi

AM = 9

nT

Arithmetic mean

Mean

ª Sum of deviations from arithmetic mean is always zero
ª To calculate portfolio return, weighted mean is used

ª

Geometric mean is used for calculating investment returns over multiple periods
ª Harmonic mean is used to calculate average of ratios
ª Arithmetic mean > Geometric mean > Harmonic mean


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Median

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It is the midpoint of a data set

Median = [(n+1) X 50%]th observation
Data needs to be arranged in ascending order
to calculate median using above formula

1

3

2

3

Mode

4

5

6

7

8

9

Median = [(9+1) X 50%] = 5th observation

Value that occurs most frequently in a data set
A data set can have more than one mode or

even no mode
If a data set has one/two/three modes it is
said to be unimodal/bimodal/trimodal

LOS f

Quartiles, quintiles, deciles, percentiles
Quartiles

e
Distribution is
divided into
tenths

Distribution is
divided into
fifths

[(n + 1) × 25%]th

[(n + 1) × 20%]th

re

Distribution is
divided into
quarters

LOS g


Percentiles

Deciles

Quintiles

[(n + 1) × 10%]th

Distribution is
divided into
hundreds

[(n + 1) × 1%]th

nT

Measures of dispersion

Mean absolute
deviation
(MAD)

Variance

Maximum value
− minimum
value

∑|(x − x)|
n


Population variance -

Fi

Range

∑ (x − μ)2
n

Standard deviation

Population SD -

Sample SD -

Sample variance ∑ (x − x)2
n−1
Variance = σ2

SD can be calculated directly on TI BA II plus professional.
Ÿ Use DATA (2nd 7) to enter data then,
Ÿ Use STAT (2nd 8) to see SD


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LOS h


Chebyshev’s inequality
Applies to sample or population data, normal or skewed distribution
Calculated as,
1 − 1/k2
60

Eg.

70

SD = 5

where k > 1
80

Chebyshev’s inequality = 1 − 1/22
= 1 − 1/4

K = 10/SD
K=2

= 75%

Interpretation: 75% observations lie within ±2 SD of mean

LOS i

Sharpe ratio


Coefficient of variation (CV)

It is used to measure excess
return per unit of risk

It is used to measure
the risk per unit of
expected return

aka reward-to-variability
ratio

e

CV = SDx
X

SR = Portfolio return − RFR
SD of portfolio

Lower the better

re

Higher the better

Sharpe ratio

Fi


RFR

10 km

Which is more economical ?

Wrong interpretation -

25/3 = 8.33

Correct interpretation -

15/3 = 5

Sharpe ratio =

km

km

15 km

Motorcycle takes 2.2 ltrs of petrol
to cover the entire distance

10

10
RFR


nT

Motorcycle takes 3 ltrs of petrol
to cover the entire distance

✘

20/2.2 = 9.09

✓

25 − 10
3

✓

10/2.2 = 4.54 ✘
Rp − RFR
SDp

20 − 10
2.2


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LOS j


Skewness

Negatively skewed/
left skew

Positively skewed/
right skew

No skew

Normal distribution

Skewed distribution

Symmetrical
distribution

Asymmetrical
distribution
Skewness: Extent to which data is not symmetrical

Negative skew in returns distributions indicates increased risk

LOS k

re

e

Locations of mean, median and mode


Mean,
median,
mode

LOS l

Median

Median

Mean > Median > Mode

Fi

1

Mean Mode

nT

Mean = Median = Mode

Mode Mean

Mean < Median < Mode

Kurtosis

Mesokurtic

distribution

Leptokurtic
distribution

Platykurtic
distribution

Kurtosis = 3

Kurtosis > 3

Kurtosis < 3

Excess kurtosis = 0

Excess kurtosis = +ve

Excess kurtosis = -ve

Kurtosis: Measures the peakedness of a distribution
Positive kurtosis in returns distributions indicates increased risk


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3


Sample skewness - Sk = ∑(X-X)
x 1
3
SD
n

Sk > 0.5 indicates significant level of skewness

4
x 1
Sk = ∑(X-X)
4
SD
n

Excess kurtosis > 1 is considered a large value

Sample kurtosis -

LOS m

Use of arithmetic mean and geometric
mean when analyzing investment returns

Arithmetic mean return is appropriate for forecasting single period returns in future periods

Fi

nT


re

e

Geometric mean return is appropriate for forecasting future compound returns over multiple periods


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Probability Concepts

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LOS a
Random variable

Outcome

Event

Mutually
exclusive
events

Uncertain quantity/
number

Observed value
of a random

variable

An outcome or
a set of
outcomes

Events that can
not happen
together

1

LOS b

All possible events

Two defining properties of probability

è

Probability is always between 0 & 1

è

If we have mutually exclusive and exhaustive events then sum of
probabilities of those events will always be 1

2

e


Probabilities

Objective

re

Subjective

Priori

Empirical

Eg. Historical pass
rates

Least formal
method of
developing
probabilities

Determined using
formal reasoning

nT

Established by
analyzing historical
data


Involves personal
judgement

Eg. Throwing a die
= 1/6

Probability of an event in terms of odds

Fi

LOS c

Exhaustive
events

If probability of an
event is 20%

Then for 10 experiments,
success = 2 failure = 8

2/10

Odds for

Odds against

2/8

8/2


Two-to-eight

Eight-to-two


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LOS d

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Unconditional and conditional probabilities
Unconditional

Conditional

µ

Refers to probability of an event
regardless of occurrence of other
events

µ

Occurrence of one event affects the
probability of occurrence of other
event

µ


Also known as marginal probability

µ

A conditional probability of an
occurrence is also called its likelihood

Eg.
µ
µ

LOS e

Eg.

P(heads) = 50%
P(head/rains) =50%

µ
µ

P(pass/study) = 80%
P(pass/studyc) = 50%

Multiplication, addition and total probability rules
Multiplication rule

e


Addition rule

Used to determine the probability
that at least one of two events
will occur

re

Used to determine the joint
probability of two events

Apply this rule when a
question says ‘and’

Apply this rule when a
question says ‘or’

P(AB) = P(A|B) × P(B)

P(A or B) = P(A) + P(B) − P(AB)

nT

P(A|B) = P(AB)
P(B)

Total probability rule -

Used to determine unconditional probability of an event,
given conditional probabilities


P(A) = P(A|B1) × P(B1) + P (A|B2) x P(B2) +....... P(A|Bn) × P(Bn)

Joint probability - Probability that all the events will occur at the same time

Fi

LOS f

For mutually exclusive events the joint probability is zero

For events that are not mutually exclusive, joint probability must be subtracted
from the total of unconditional probabilities to avoid double counting
Eg.

P(A) = 60%

P(B) = 30%

P(Both) = 60% × 30% = 18%
P(A)

P(B)

P(At least one) = 60% + 30% − 18% = 72%
P(None) = 1 − 72% = 28% Or

P(AB)

(1 − 60%) × (1 − 30%) = 28%



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LOS g

Dependent and independent events
Independent events

Dependent events

Ÿ

Occurrence of one event has no
influence on occurrence of other events

Ÿ

P (A|B) = P(A)

Ÿ

Getting 5 on a second roll of die is
independent of getting 5 on the first roll
of die

LOS h, i, j


1

Ÿ

Occurrence of one event is dependent
on occurrence of other events

Unconditional probability using total probability rule
Prob of good
economy and
rate increase

Interest rates
increase
Prob = 75%
Good economy next year

Interest rates
decrease

Prob of good
economy and
rate decrease

Prob = 25%

0.4 x 0.25 = 10%

re


e

Prob = 40%

Unconditional
probability

Conditional
probability

od)

P(Go

EPS = 10

0.4 × 0.6 × 10 = 2.4

40%

EPS = 8

0.4 × 0.4 × 8 = 1.28

70%

EPS = 6

0.6 × 0.7 × 6 = 2.52


%

Go

Fi

60%

= 40

Economy

P(

Joint
probability

Expected value

nT

2

0.4 x 0.75 = 30%

od c
)=

60


%

30

%

EPS = 3

0.6 x 0.3 x 3 = 0.54

Expected value of EPS = 2.4 + 1.28 + 2.52 + 0.54
= 6.74


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LOS k

Covariance and correlation
Correlation

Covariance
µ

It is a measure of how two assets move
together

µ


Covariance of return with itself is its
variance

µ

Expressed in terms of square units

µ

Population Cov(x,y) = ∑(X − X) (Y − Y)
n

µ

Sample Cov(x,y) = ∑(X − X) (Y − Y)
n−1
Cov(x,y) = r × σx × σy

µ

Range = −∞ to +∞

µ

Only +ve and −ve sign matters for
determining relationship b/w the variables

LOS l


Standardized measure of covariance

µ

Measures strength of linear relationship
between two random variables

µ

Does not have a unit

µ

r = Cov(x,y)
σx × σy

µ

Does not exhibit causal relationship

µ

Range = −1 to +1

µ
µ
µ

r = 1 means perfectly +ve relation
r = 0 means no correlation

r = −1 means perfectly −ve relation

e

µ

µ

Expected value, variance and standard deviation of portfolio
Expected value = W1E(R1) + W2E(R2) +W3E(R3) +.......+WnE(Rn)

re

1

Variance = (W1σ1)2 + (W2σ2)2 + 2W1σ1W2σ2 × r
(W1σ1)2 + (W2σ2)2 + 2W1W2 × Cov(x,y)

Standard deviation = √Variance

LOS n

When r = −1,
Sdp = (W1σ1) − (W2σ2)
Sdp = Lowest

When r = 0,
Sdp= √(W1σ1)2 + (W2σ2)2

nT


2

When r = 1,
Sdp = (W1σ1) + (W2σ2)
Sdp = Highest

Bayes’ formula

It is used to calculate updates probability
P(Ac|Bc) = 40%

P(A|B) = 30%

Fi

Eg.

P(B)

P(A)

P(B|Ac) = ?

B A = 9%

%

= 30


P(A c
) =7

P(

B )c

%
=30

P(B) = 30%

=

70

%

P(A)
P(

A )c

=

0%

B Ac = 21%

=60%


Bc A = 42%

40

%

Bc Ac = 28%

B Ac =

21
21+28

= 42.86%


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Counting problems

LOS o
Labeling

Permutation

n!
n1! x n2! x .... x nk!


n

Combination

n

Pr

Cr = nCn-r

Eg. A person has 8 cars. He uses 3 cars
for work, 3 other for long distance trips
and 2 other for commute other than
work. Calculate the no. of different
ways to label them.

Eg. How many different
ways are there to select
3 players from 5, if the
order of selection is
important ?

Eg. How many different
ways are there to select
3 players from 5, if the
order of selection is not
important ?

8!

3! x 3! x 2!

5 → 2nd(−) nPr → 3

5 → 2nd(+) nCr → 3 or 2 (5−3)

= 60

= 10

= 560

Permutation is used when order of selection is important

Fi

nT

re

e

Combination is used when order of selection is not important


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Common Probability Distribution


LOS a
Probability distribution - Describes the probabilities of all possible outcomes
of a random variable. Probabilities of all outcomes
should equal to 1
Discrete random variable - There is a finite number of possible outcomes. Eg.
number of stocks in portfolio

No. of stocks
100

200

300

% return

0.001 %

Continuous random variable

re

Discrete random variable

LOS b

e

Continuous random variable - There is an infinite number of possible outcomes. Eg.

Return earned in portfolio

Probability Function - P(X) = P(X=x)
X =1,2,3,4 P(X) = x else
10

nT

Eg.

P(X) = 0

The above function satisfies both the conditions of probability which are ;
a) 0 ≤ P(x) ≤ 1

LOS c

b) ∑ P(x) = 1

Discrete uniform
variable

Fi

Discrete non - uniform
variable

1

0


110

Number of people present in class

1

2

3

4

5

6

Probability distribution of a roll of a die


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Continuous non - uniform
variable

P(X) = 0, P(x) is between x1 and x2


LOS c & d

Continuous uniform
variable

Eg. Puncture on tyre. There are infinite
number of points on a tyre where
puncture can happen. Each point has
equal probability of occurance

Cumulative distribution function (cdf)
Eg.

X =1,2,3,4 P(X) = x
10
Cumulative
distribution function

P(1) = 1/10 = 10%
P(2) = 2/10 = 20%
P(3) = 3/10 = 30%
P(4) = 4/10 = 40%

F(1) = 10%
F(2) = 30%
F(3) = 60%
F(4) = 100%

F(-1) = 0.1587


re

e

Probability
density function

0.1587

LOS e & f

0

+1

nT

−1

Binomial random
variable

Fi

Outcome can be either
‘success’ or ‘failure’

When number of trials is 1,
it is called Bernoulli

random variable

Px x (1−P)n-x x nCr
P = Probability of success
n = No. of trials
X = No. of successes

−1

0

Eg.
P(win) = 70%, 4 matches, exactly 2 wins
Px x (1-P)n-x x nCr
(70%)2 x (1−0.7)4-2 x 4C2
= 26.46%

Ÿ Mean of binomial distribution = np
Ÿ Variance of binomial distribution = npq
Ÿ q=1−P


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LOS g

Binomial tree
Stock price (S) = 850 Uptick (u) = 1.2 Downtick (d) = 1/U = 1/1.2


1020 x 1.2
= 1224

Suu

1020 x 1/1.2
= 850

Sud

708 x 1.2 =
850

Sdu

708 x 1/1.2
= 590

Sdd

Su
850 x 1.2
= 1020

S

850
850 x 1/1.2
= 708


Sd

LOS h

Tracking error = Return on portfolio − Return on benchmark

LOS i

Continuous uniform distribution
Properties of continuous uniform distribution

e

ª For all a < x1 < x2 < b
(i.e. for all x1 and x2 between the boundaries a and b)

re

ª P(X < a or X > b) = 0
(i.e. probability of X outside the boundaries is zero)
ª P(x1 < X < x2) = (x2 - x1)/b - a
(This defines the probability of outcomes between x1 and x2)

ª Continuous uniform distribution will always have lower and upper bound (a,b)
ª Probability of X taking any value below ‘a’ or above ‘b’ will be zero

nT

Eg. X is uniformly distributed between 2 & 20. Calculate the probability that X will be between 6 & 15.

6

15

2

20

P(4<8) = 15 − 6
20 − 2

Fi

P(x) = 0 Because it is a continuous distribution

LOS j

è
è
è
è
è
è
è

Normal distribution

It is a continuous non-uniform distribution
Mean and variance needs to be defined
Skewness = 0

Kurtosis = 3
Mean = median = mode
A linear combination of a normally distributed random variable is also normally distributed
The probabilities of outcome further above and below mean get smaller and smaller but do
not go to zero (i.e. the tails get very thin but extend infinitely


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LOS k

Univeriate
distribution

Multiveriate
distribution

Single variable

More than one variables

A multivariate distribution with 10 variables has - 10 means 10 Variances 45 Correlations

n x (n-1) = 10 x 9 = 45
2
2

LOS l


Confidence interval
Eg.

X = 700 σ = 200
Calculate 90%, 95%, 99%
confidence interval

34% 34%

X ± (z-value)σ
13%

3%

90% - 700 ± (1.65)200
= 370-1030

3%

-2σ

-1σ

1σ

2σ

3σ


re

-3σ

e

13%

68%
95%
99%

95% - 700 ± (1.96)200
= 308-1092
99% - 700 ± (2.58)200
= 184-1216

LOS m

nT

Interpretation: We are 99% of the time
confident that the expected outcome will
lie between 184 and 1216

Standard normal distribution

Standard normal It is a normal distribution that is standardized so that its
distribution mean = 0 and standard deviation = 1


Fi

Z-value = Observation − Population mean
standard deviation

Z score =

Eg.

X = 400 σ = 200 μ = 700

400 − 700 = −1.5 At 1.5 Z-value, Probability = 93.32%
200
Therefore probability of value less
than 400 = 1 − 0.9332 = 6.68%


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LOS n

Shortfall risk and Safety first ratio
Shortfall risk

Safety first ratio

Probability that portfolio value
or return will fall below a

particular value or target over a
given period of time

Excess return per unit of risk
over minimum acceptable
return/threshold level.
SF ratio = Rp − Threshold return
Sdp
Higher the better

Lower the better

Eg.

Average return = 20%

SD = 3%

Threshold level = 15%

Z-value = 15 − 20 = −1.66
3

Shortfall
risk

Probability at −1.66 z-value = 95.15%
Therefore shortfall risk ;
1 − 0.9515 = 4.85%


LOS o

20%

= 1.66

e

15%

SF ratio = 20 - 15
3

nT

re

Normal and lognormal distribution

Normal distribution

Ÿ
Ÿ

No skew
Not bounded by zero

Lognormal distribution
Ÿ
Ÿ

Ÿ

Skewed to the right
Bounded by zero
Useful for modeling asset prices,
because they can not take
negative values

Fi

The logarithms of lognormally distributed random variables are normally distributred

LOS p

Discrete and continuous compounding

Discrete compounding - Annual, semi-annual, quarterly, monthly etc.
Continuous compounding - No. of compounding periods within a given time period


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Continuous compounding calculations

100

20%

0


117.35

100

0.8

0

100 x e0.2 x 0.8 = 117.35

LOS q

20%

ln 117.35
100

117.35

100

0.8

0

0.8 → 16%

20%


117.35
0.8

117.35 x e-0.2 x 0.8 = 100

1 → 20%

Monte Carlo simulation
Technique based on repeated generation of one or more risk factors that
affect security values, to generate a distribution
It is used to

Its limitations are

LOS r

It is complex
It is subject to model risk and input risk
Simulation is not an analytic method, but a
statistic one.
Ÿ Increased complexity does not necessarily
ensure accuracy
Ÿ
Ÿ
Ÿ

e

Value complex securities
Simulate profits/losses from a trading strategy

Calculate estimates of VaR to determine the
riskiness of a portfolio
Ÿ Simulate pension fund assets and liabilities to
examine the variability of the differences
between the two
Ÿ Value portfolios of assets that do not have
normal returns distribution
Ÿ
Ÿ
Ÿ

re

Historical simulation

It is based on actual change in value or actual change in risk factor for some prior period
Each iteration of simulation involves randomly selecting one of these past changes for
each risk factor and calculating the value of the asset or portfolio in question, based
on those changes in risk factor

nT

Its advantage is that it uses actual distribution of risk factors, which need not be estimated.

Fi

Its limitations are :
Past changes in risk factor may not be a good indication of future changes
It can not address the sort of ‘whatif’ questions that Monte Carlo simulation can



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LOS a

Simple random sampling and sampling distribution
1

2

Sampling and Estimation

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Simple random sampling

Systematic sampling

Method of selecting a sample in
such a way that each item in the
population has same likelihood
of being included in the sample.

Another way to form an
approximately random sample.

Eg. Drawing a sample of 5
apples from 50 to calculate
average weight.


Eg. Selecting every nth item
from the population

Sampling distribution - It is a probability distribution of all possible sample statistic
computed from samples drawn from the population

Sampling error = Sample statistic − Population parameter

re

LOS b

e

Sampling distribution does not have to be normal distribution

Mean, Variance,
Standard Deviation of
sample

Stratified random sampling - Uses a classification system to separate the
population into small groups, based on one
or more distinguishing characteristics. Each
subgroup is called as stratum.

nT

LOS c

Mean, Variance,

Standard Deviation of
population

Fi

Eg. Avg. calorie intake of a nation

Sample3

Sample4
Sample5

Results of these samples
are then pooled to form a
combined sample

N
W

C

E

Sample1

S

Sample2

It is often used in bond indexing because of the difficulty and cost of replicating entire

population of bonds.


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LOS d

Time-series and Cross-sectional data
Time-series data

Cross-sectional data

It consists of observations taken
over a period of time

It consists of observations taken
at a single point in time

Time-series and cross-sectional data can be pooled in the same data set.
Longitudinal data - Observations over time of multiple characteristics of the
same entity. Eg. Unemployment, GDP growth rates,
inflation of a country over 10 years.
Panel data - Observations over time of same characteristic of the
multiple entities. Eg. analysis of D/E ratio of 20
companies over 8 quarters.
Panel and longitudinal data are typically presented in table or spreadsheat form.

Central limit theorem


LOS e

ª

ª

Variance equals ‘σ2/n’ as sample size becomes large

If sample size n, is sufficiently large (n ≥ 30), the sampling distribution of the sample
means will be approximately normal

re

ª

e

ª Sample mean(x) approaches population mean(μ) as sample size becomes large

If central limit theorem works, population mean(μ) = mean of sampling distribution
Standard deviation of sampling distribution = σ/√n (standard error)

nT

ª

LOS f

Standard error of sample mean

Population variance unknown

σ
√n

s
√n

Fi

Population variance known

LOS g

Describe properties of an estimator

ª Unbiasedness - It is one for which the expected value of the estimator is equal to the
parameter you are trying to estimate

ª Efficiency - Unbiased estimator is also efficient if the variance of its sampling distribution
is smaller than other unbiased estimators of parameter you are trying to estimate
ª

Consistency - An estimator for which the accuracy of the parameter estimate increases as
the sample size increases


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LOS h

Point estimate and confidence interval estimate
Point estimate - It is a single sample value used to estimate population
parameter
Confidence interval It is a range of values in which population parameter is
estimate - expected to lie.

LOS i
Properties of t-distribution

Student’s t-distribution
It is a bell-shaped probability distribution
è
è

It is symmetrical about its mean

It is appropriate to use when n < 30,
population variance is unknown and
distribution is normal

LOS j

ª It is defined by degrees of freedom(DoF) (n − 1)
ª It has more probability in the tails (fat tails)
ª As DoF increase, t-distribution approaches
standard normal distribution (z-distribution)
ª t-distribution is flatter and has fatter tails than

normal distribution
ª As number of observations increase, distribution
becomes more peaked and tails become thin i.e.
it converges to z-distribution

Computation and interpretation of confidence interval

e

è

1

Significance level (α) = 1 − Confidence interval

re

90% confidence level = 10% significance level = 5% in each tail

Construction of confidence interval

2

nT

Point estimate ± (Reliability factor × Standard error)

3

Selection of test for

reliability factor

Fi

Population
variance is
known

Non -normal
distribution

Normal
distribution

Population
variance is
unknown

Non -normal
distribution

Normal
distribution

Z - distribution

t - distribution

n ≥ 30


n < 30

n ≥ 30

n < 30

Z - distribution

No

t/z distribution

No