**CFA® Quantitative Methods**

**Tools for valuation**

and

the basics of statistics

and

the basics of statistics

CFA level 1 Quantitative Methods can be divided into two parts:

**Time Value of Money**

**Probability and Statistics**

Generally, after covering readings concerned with the concepts of time value of money,

CFA level 1 candidates should understand and master problems of calculating present and future values for particular cases:

TVM principles by Khan Academy:

Central Limit Theorem by Khan Academy

Time value of money

Probability and Statistics

TVM techniques are tools that help financial specialists:

financial instruments and investments

1

3

value

make financial decisions

2

value

a company or an equity

about whether to invest in a project or not

TVM is

for a financial analyst

like

addition and subtraction

for a primary school pupil.

It is essential to master TVM before moving to more complex issues!

To solve TVM questions you can use:

direct formulas,

TVM functions on your calculator,

cash flow, NPVand IRR sheets on your calculator.

Useful Formulas

Normal Distribution

Chebyshev's Inequality

or

or

Draw a

time line

Example:

John is planning to set aside $100.000 today. He is also planning to take $7.000 from this account at the beginning of every year for the nearest 10 years. Additionally, as a happy father of a newborn daughter, he has decided to give her on her 17 birthday a sum of money. How much will he be able to give

to his daughter if the account pays an interest rate of

8 percent a year for all this time.

Future Value

Present Value

Uneven Cash Flows

use time line

Ordinary Annuity

Annuity Due

Future Value of a single sum of money

Although most prep providers show mainly how to make calculations using TVM sheet on your calculator, we strongly encourage you to become familiar with other methods mentioned above, as well.

Future Value of

and

Present Value of

a single sum of money

Present Value of a perpetuity

FV

PV

A

PV

A

A

A

...

FV

PV

Ordinary Annuity

Annuity Due

Present Value of

ordinary annuity

and

annuity due

ordinary annuity

annuity

due

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

For more complicated questions, you sometimes have to calculate the present value or future value

of uneven cash flows.

To manage such questions...

-100

+7

+7

+7

+7

+7

+7

+7

+7

+7

+7

+X

Solution

(using cash flow sheet and NPV sheet

on TiBA II Plus Professional)

=>CF

[2nd][CE/C]

clear cash flow sheet

CF0= -100+7=-93

[Enter]

->

CF1= +7

[Enter]

->

F01= 9 (because first 7 is added at the beginning)

[Enter]

=>NPV

I=8

[Enter]

->

NPV

[CPT]

NPV=-49.27 (this is the present value of the last payment=X)

X=49.27*1.08^17

X=182.31

Contest

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)

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for Contributors

how to prove, using cash flow sheet and NPV sheet on TiBA II Plus (Professional), that X=182.31 is the correct answer for this question

for CFA level 1 Candidates

This part of the presentation covers 3 topics:

1. measures of central tendency

2. measures of dispersion

3. skewness

4. kurtosis

Properties of Return Distributions

TVM on the exam

**Why does QM matter?**

1.

Properties of Return Distributions

2.

3.

Chebyshev's Inequality

Normal Distribution and

Central Limit Theorem

Positively skewed

mode < median < mean

Negatively skewed

mean < median < mode

For all k>1, the proportion of the observations within k standard deviations of the arithmetic mean is at least:

Chebyshev’s inequality is valid for any distribution with finite variance!

It states that:

Key properties

is described by two parameters: mean and standard deviation,

a linear combination of normal random variables is normally distributed,

is symmetric (skewness=0),

its kurtosis is equal to 3.

Normal distribution

1

2

both are used as tools that help financial specialists

describe and estimate risk.

For a set of numbers, three basic

measures of central tendency

are distinguished:

the mean

the median

the mode

There are different types of means and they are used in different situations.

Remember that:

the harmonic mean (

h

) is always less or equal to the geometric mean(

g

)

the geometric mean(

g

) is always less or equal to the arithmetic mean(

a

)

h

<

g

<

a

For an ordered set of elements, the value for which the number of elements higher than the value is equal to the number of elements lower than the value is called the median.

Depending on the number of elements in a given set, the median is calculated differently:

The mode is the element of the set repeated most frequently.

The median

If the set contains an odd number of elements, n, then the median is equal to the value of the element occupying the position:

If the set contains an even number of elements, n, then the median is equal to the value calculated as the mean of the elements occupying positions:

{inverse alphabetical order}

The mean

The mode

Remember

If a unimodal distribution is symmetric,

the mean

=

the median

=

the mode!

Kurtosis

is the measure of the peakedness of the return distribution.

Kurtosis

of a normal distribution equals 3.

Excess kurtosis

(by definition) of a normal distribution equals 0.

and

=

=

D

ispersion

is the variability around the central tendency.

The common measures of dispersion are:

the range

the Mean Absolute Deviation (MAD)

the variance

the standard deviation

range

= maximum value – minimum value

Mean Absolute Variation

(MAD):

Population variance:

Sample variance:

Standard deviation is the positive square root of variance.

Time value of money (TVM) is extremely important

because:

are important because:

and use it to mark all significant data used in the question.

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