###### Considerations and Biases in Sampling

Financial assets are primarily defined based on their return-risk characteristics. This helps when building a portfolio from all the assets available. Regarding returns, there are different ways of measuring returns.

Financial market assets generate two different streams of return: income through cash dividends or interest payments and capital gain or loss through financial asset price increases or decreases.

Some financial assets give only one stream of return. For instance, headline stock market indices typically report on price appreciation only. They do not include the dividend income unless the index specifies it is a “total return” series.

A holding period return is earned from holding an asset for a single specified period. The time period can be any specified period, such as a day, month, or ten years.

The general formula of the holding period return is given by:

$$R=\frac{\left(P_1-P_0\right)+I_1}{P_0}$$

\(P_0\) = Price of an asset at the beginning of the period (*t*=0).

\(P_1\) = Price of an asset at the end of the period (*t*=1).

\(I_1\) = Income received at the end of the period (*t*=1).

**Example: Calculating Holding Period Return**

An investor purchased 100 shares of a stock at $50 per share and held the investment for one year. During that period, the stock paid dividends of $2 per share. At the end of the year, the investor sold all the shares for $60 per share.

The holding period return is *closest to*:

**Solution**

In this case, we have:

$$\begin{align}P_0&=100 \text{ shares} \times \$50\ \text{per share} =\$5,000\\ I_1&=100\ \text{shares} \times\$2\ \text{per share} =\$200\\ P_1&=100 \text{ shares} \times \$60\ \text{per share} =\$6,000\end{align}$$

Therefore,

$$R=\frac{\left(P_1-P_0\right)+I_1}{P_0}=\frac{6,000-5,000+200}{5,000}=24\%$$

Holding period returns can also be calculated for periods longer than a year. For instance, if we need to calculate the holding period return for a five-year period, we should compound the five annual returns as follows:

$$R=\left[\left(1+R_1\right)\times\left(1\times R_2\right)\times\ldots\times(1\times R_5)\right]-1$$

When we have assets for multiple holding periods, it is necessary to aggregate the returns into one overall return.

Denoted by \({\bar{R}}_i\) arithmetic mean for an asset \(i\) is a simple process of finding the average holding period returns. It is given by:

$${\bar{R}}_i=\frac{R_{i,1}+R_{i,1}+\ldots+R_{i,T-1}+R_{iT}}{T}=\frac{1}{T}\sum_{t=1}^{T}R_{it}$$

Where:

\(R_{it}\) = Return of asset \(i\) in period \(t\).

\(T\) = Total number of periods.

For example, if a share has returned 15%, 10%, 12%, and 3% over the last four years, then the arithmetic mean is computed as follows:

$${\bar{R}}_i=\frac{1}{T}\sum_{t=1}^{T}R_{it}=\frac{1}{4}\left(15\%+10\%+12\%+3\%\right)=10\%$$

Computing a geometric mean follows a principle similar to the one used to compute compound interest. It involves compounding returns from the previous year to the initial investment’s value at the start of the new period, allowing you to earn returns on your returns.

A geometric return provides a more accurate representation of the portfolio value growth than an arithmetic return.

Denoted by \({\bar{R}}_{Gi}\) the geometric return for asset \(i\) is given by:

$$\begin{align}{\bar{R}}_{Gi}&=\sqrt[T]{\left(1+R_{i,1}\right)\times\left(1+R_{i2}\right)\times\ldots\times\left(1+R_{i,T-1}\right)\times\left(1+R_{iT}\right)}-1\\ &=\sqrt[T]{\prod_{t=1}^{T}{(1+R_t)}}-1\end{align}$$

Using the same annual returns of 15%, 10%, 12%, and 3% as shown above, we compute the geometric mean as follows:

$$\begin{align}\text{Geometric mean} &=\ \left[\left(1+15\%\right)\times\ \left(1+10\%\right)\times\ \left(1+12\%\right)\times\ \left(1+3\%\right)\right]^\frac{1}{4}\ -1\\&=\ 9.9\%\end{align}$$

Note that the geometric return is slightly less than the arithmetic return. Arithmetic returns tend to be biased upwards unless the holding period returns are all equal.

The harmonic mean is a measure of central tendency. It’s especially useful for rates or ratios such as P/E ratios. Its formula is derived from the harmonic series, which is a specific mathematical sequence.

$${\bar{X}}_H=\frac{n}{\sum_{i=1}^{n}\frac{1}{X_i}},\ X_i>0 \text{ for all } i=1,2,\ldots,n$$

The above formula is interpreted as the “harmonic mean of observations \(X_1,\ X_2,\ldots,\ X_n\).”

The harmonic mean is handy for averaging ratios when those ratios are consistently applied to a fixed quantity, resulting in varying unit numbers. For instance, it’s applied in cost-averaging strategies where you invest a fixed amount of money at regular intervals.

An investor is practicing cost averaging by investing in a particular stock over a period of three months. The investor decides to allocate different amounts of money each month. In the first month, the investor invests $2,000; in the second month, $3,000; and in the third month, $4,000. The share prices of the stock for these three months are $10, $12, and $15, respectively.

Calculate the average price paid per share for the three-month period.

**Solution**

Using the harmonic mean formula,

$${\bar{X}}_H=\frac{n}{\sum_{i=1}^{n}\frac{1}{X_i}}=\frac{3}{\frac{1}{10}+\frac{1}{12}+\frac{1}{15}}=12$$

Trimmed and Winsorized means seek to lower the effect of outliers in a data set.

The trimmed mean is a measure of central tendency in which we calculate the mean after excluding a small percentage of the lowest and highest values from the dataset.

For example, a data set consists of 10 observations: 12, 15, 18, 20, 22, 25, 27, 30, 35, and 40. We can calculate the trimmed mean after removing the highest and lowest values.

After removing these values, the remaining data set is: 15, 18, 20, 22, 25, 27, 30, and 35.

Now, let’s calculate the trimmed mean by taking the average of these remaining values:

$$\frac{15+18+20+22+25+27+30+35}{8}\ =\frac{192}{8}\ \ =\ 24$$

Therefore, the trimmed mean of the given data set is 24.

The Winsorized mean is a central tendency measure. It works by replacing extreme values at both ends of the data with the values of their closest observations. This process is similar to the trimmed mean. Essentially, it helps eliminate outliers in a dataset.

For example, consider a dataset of 12 observations: 8, 12, 15, 18, 20, 22, 25, 27, 30, 35, 40, and 50. We can calculate the Winsorized mean by replacing the lowest and highest values with those closest to the 10th and 90th percentiles, respectively. As such, the new values are **10**, 12, 15, 18, 20, 22, 25, 27, 30, 35, **37.5**, and 40, and the winsorized mean is:

$$\frac{10 +12+15+ 18 + 20 +22 + 25 +27 +30+35+37.5+40}{12} \approx 24.46$$

Question 1What are the arithmetic mean and geometric mean, respectively, of an investment that returns 8%, -2%, and 6% each year for three years?

A. Arithmetic mean = 5.3%; Geometric mean = 5.2%.

B. Arithmetic mean = 4.0%; Geometric mean = 3.6%.

C. Arithmentic mean = 4.0%; Geometric mean = 3.9%.

SolutionThe correct answer is

C.$$ \text{Arithmetic mean} = \frac {8\% + (-2\%) + 6\%} {3} = 4\% $$

$$ \text{Geometric mean} = [(1+8\%) × (1+(-2\%)) × (1+6\%)]^{1/3} – 1 = 3.9\% $$