IFT Notes for Level I CFA^{®} Program

The arithmetic mean is the sum of the observations divided by the number of observations. It is the most frequently used measure of the middle or center of data.

**The Population Mean**

The population mean is the arithmetic mean computed for a population. It is expressed as:

where:

N is the number of observations in the entire population and

X_{i} is the i^{th} observation

Consider the stock returns of a company over the last 10 years: 2%, 5%, 4%, 7%, 8%, 8%, 12%, 10%, 8%, and 5%.

For this data set, the population mean can be computed as:

**The Sample Mean**

The sample mean is calculated like the population mean, except that we use the sample values. It is expressed as:

where: n is the number of observations in the sample.

If the sample data is: 8, 12, 10, 8 and 5, the sample mean can be calculated as:

The median is the midpoint of a data set that has been sorted into ascending or descending order.

For odd number of observations: 2,5,7,11,14 → Median = 7

For even number of observations: 3, 9, 10, 20 → Median = (9 + 10)/2 = 9.5

As compared to a mean, a median is less affected by extreme values (outliers).

The mode is the most frequently occurring value in a distribution.

For the following data set: 2, 4, 5, 5, 7, 8, 8, 8, 10, 12 → Mode = 8

A distribution can have more than one mode, or even no mode. When a distribution has one mode it is said to be unimodal. If a distribution has two or three modes, it is called bimodal or trimodal respectively.

**The Weighted Mean**

Here different observations are given different proportional influence on the mean. The formula for the weighted mean is:

where: the sum of the weights equals 1; that is

**Example **

Consider an investor with a portfolio of three stocks. $40 is invested in A, $60 in B, and $100 in C. If returns were 5% on A, 7% on B, and 9% on C, compute the portfolio return using the weighted mean.

**Solution:**

An arithmetic mean is a special case of a weighted mean where all observations are equally weighted by the factor 1/n.

**The Geometric Mean**

The most common application of the geometric mean is to calculate the average return of an investment. The formula is:

**Example **

The return over the last four periods for a given stock is: 10%, 8%, -5% and 2%. Calculate the geometric mean.

**Solution:**

Given the returns shown above, $1 invested at the start of period 1 grew to: $1.00 x 1.10 x 1.08 x 0.95 x 1.02 =$1.151. If the investment had grown at 3.58% every period, $1.00 invested at the start of period 1 would have increased to: $1.00 x 1.0358 x 1.0358 x 1.0358 x 1.0358 =$1.151. As expected, both scenarios give the same answer. 3.58% is simply the average growth rate per period.

**Instructor’s Note**

In the reading on Discounted Cash Flow Applications, we used the geometric mean to calculate the time-weighted rate of return.

**The Harmonic Mean**

The harmonic mean is a special type of weighted mean in which an observation’s weight is inversely proportional to its magnitude. The formula for a harmonic mean is:

where: X_{i} > 0 for i = 1, 2 … n, and n is the number of observations

The harmonic mean is used to find average purchase price for equal periodic investments.

**Example **

An investor purchases $1,000 of a security each month for three months. The share prices are $10, $15 and $20 at the three purchase dates. Calculate the average purchase price per share for the security purchased.

**Solution:**

The average purchase price is simply the harmonic mean of $10, $15 and $20.

The harmonic mean is:

A more intuitive way of solving this is total money spent purchasing the shares divided by the total number of shares purchased.

Total money spent purchasing the shares = $1,000 x 3 = $3,000

Total shares purchased = sum of shares bought each month

=

=

Average purchase price per share =

**Comparison of AM, GM and HM**

- If the returns are constant over time: AM = GM = HM.
- If the returns are variable: AM > GM > HM.
- The greater the variability of returns over time, the more the arithmetic mean will exceed the geometric mean.

A quantile is a value at or below which a stated fraction of the data lies. Some examples of quantiles include:

**Quartiles:**The distribution is divided into quarters.**Quintiles:**The distribution is divided into fifths.**Deciles:**The distribution is divided into tenths.**Percentile:**The distribution is divided into hundredths.

The formula for the position of a percentile in a data set with n observations sorted in ascending order is:

where:

y is the percentage point at which we are dividing the distribution.

n is the number of observations.

L_{y} is the location (L) of the percentile (P_{y}) in an array sorted in ascending order.

Some important points to remember are:

- When the location, L
_{y}, is a whole number, the location corresponds to an actual observation. - When L
_{y}is not a whole number or integer, L_{y}lies between the two closest integer numbers (one above and one below) and we use linear interpolation between those two places to determine P_{y}.

**Example**

Consider the data set:

47 35 37 32 40 39 36 34 35 31 44

- Find the 75
^{th}percentile point - Find the 1
^{st}quartile point - Find the 5
^{th}decile point - Find the 6
^{th}decile point.

**Solution to 1**:

First arrange the data in ascending order:

31, 32, 34, 35, 35, 36, 37, 39, 40, 44, 47

Location of the 75^{th} percentile is the:

L_{75} = (11 + 1) (75/100) = 9^{th} value. i.e. P_{75 }= 40

With a small data set, such as this one, the location and the value is approximate. As the data set becomes larger, the location and percentile value estimates become more precise.

**Solution to 2:**

Location of the 1^{st} quartile is the:

L_{25} = (11 + 1) (25/100) = 3^{rd} value. i.e. P_{25 }= 34

**Solution to 3:**

Location of the 5th decile is the:

L_{50} = (11 + 1) (50/100) = 6^{th} value. i.e. P_{50 }= 36

**Solution to 4:**

L_{60} = (11 + 1) (60/100) = 7.2

Use linear interpolation, which estimates an unknown value on the basis of two known values that surround it.

In this case, the 7^{th} value is 37 and the 8^{th} value is 39. The 6^{th} decile is: P_{60 }= 37+ 0.4 (0.2 times the linear distance between 37 and 39). P_{60 }= 37.4