# Measuring Central Tendency in Multiples

## Averaging Multiples/The Harmonic Mean

The harmonic mean and the weighted harmonic mean are applied to average a group of price multiples.

### Example: Measuring Central Tendency in Multiples

Consider a portfolio that contains two stocks. Assuming the portfolio owns 100 percent of the shares of each stock.

$$\small{\begin{array}{l|c|c|c}& \textbf{Market capitalization} & \textbf{Earnings} & \textbf{P/E} \\ \hline\text{Stock A} & 515 & 41.5 & 12.41 \\ \hline\text{Stock B} & 485 & 23.5 & 20.63\\ \end{array}}$$

The P/E for the portfolio is computed directly by aggregating the companies’ market capitalizations and earnings.

$$\frac{515+485}{41.5+23.5}=\frac{1000}{65}=15.38$$

If the ratio of an individual holding is represented by $$X_i$$, the expression for the simple harmonic mean of the ratio is:

$$\text{X}_{\text{H}}=\frac{\text{n}}{\sum_{i=1}^{\text{n}}(1/\text{X}_{\text{i}})}$$

The expression for the weighted harmonic mean is:

$$\text{X}_{\text{WH}}=\frac{\text{1}}{\sum_{i=1}^{\text{n}}(ω_{\text{i}/\text{X}_{\text{i}}})}$$

Where $$(ω_{\text{i}}$$ are portfolio value weights summing to 1.

\begin{align*}\text{Arithmetic mean P/E} &= \frac{12.41+20.63}{2}\\&=16.52\\ \text{Weighted mean P/E}& = \bigg(\frac{515}{1000}×12.41\bigg)+\bigg(\frac{485}{1000}×20.63\bigg)\\&=16.39\\ \text{Harmonic mean P/E}&=\frac{2}{\bigg(\frac{1}{12.41}+\frac{1}{20.63}\bigg)}\\&=15.44\\ \text{Weighted harmonic mean P/E}&= \frac{1}{\bigg(\frac{515}{1000}×\frac{1}{12.41}\bigg)+\bigg(\frac{485}{1000}×\frac{1}{20.63}\bigg)}\\&=15.50\end{align*}

The following observations can be made:

• The simple harmonic mean (15.44) is lower than the arithmetic mean (16.52). The simple harmonic mean gives a lower weight to higher P/Es and a higher weight to lower P/Es.
• Using the median, rather than the mean, reduces the effect of outliers.
• The harmonic mean may also be used to reduce the impact of outliers. However, while the harmonic mean reduces the impact of significant outliers, it may worsen the impact of small outliers.
• For an equal-weighted index, the weighted harmonic mean and the simple harmonic mean are equal.

## Using Multiple Valuation Indicators

Analysts use valuation indicators to define their world of investments through a process called screening. Stock screens define criteria for including stocks in an investment portfolio and are a useful way of narrowing down a search for investments. Using more than one valuation indicator in stock valuation and selection is often used since different indicators provide different pieces of information.

An issue regarding the use of ratios in an investing strategy is look-ahead bias. Look-ahead bias is introduced in studies or simulations by using information or data that were unknown at the time the study or simulation was conducted. Investment analysts often use historical data to backtest an investment strategy that involves stock selection based on price multiples or other factors.

## Question

Consider a portfolio with 100 shares each of Stock A and Stock B:

1. Stock A is trading at $32 with an EPS of$4, resulting in a P/E of 8.
2. Stock B is trading at $27 with an EPS of$3, resulting in a P/E of 9.

The simple harmonic mean is closest to:

1. 8.47.
2. 8.50.
3. 8.55.

### Solution

\begin{align*}\text{X}_{\text{H}}&=\frac{\text{n}}{\sum_{i=1}^{\text{n}}(1/\text{X}_{\text{i}})}\\&=\frac{2}{\frac{1}{8}+\frac{1}{9}}\\&=8.47\end{align*}

Reading 25: Market-Based Valuation: Price and Enterprise Value Multiples

LOS 25 (q) Explain the use of the arithmetic mean, the harmonic mean, the weighted harmonic mean, and the median to describe the central tendency of a group of multiples.

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