Unlocking Expected Returns with the Blume Formula

Discover how to balance arithmetic and geometric averages to forecast future returns

Key Insights

Combining Averages: The Blume formula fuses arithmetic and geometric averages to generate a more realistic expected return.
Weighting Factor: The weighting depends on the forecast period relative to the historical period to adjust for compounding.
Easy Computation: It involves straightforward steps and formulas, making it accessible for forecasting over both short and long horizons.

Understanding the Blume Formula

The Blume formula is a technique used to estimate the expected future return of an asset or portfolio by combining two types of historical averages: the arithmetic mean and the geometric mean. By applying weights to these averages, the formula attempts to capture the benefits of both methods in order to reduce the biases inherent in each. Arithmetic returns are typically higher because they do not account for the volatility drag, while geometric returns are more conservative as they factor in the compounding effect. The Blume formula helps in averaging these tendencies to produce a return estimate that is balanced and often more aligned with actual future performance.

Basic Formula Framework

At its core, the Blume formula can be expressed in a general framework as:

General Form:

\( R = w \times R_a + (1 - w) \times R_g \)

where:

\( R \) is the expected future return,
\( R_a \) is the arithmetic average return,
\( R_g \) is the geometric average return, and
\( w \) is the weight assigned to the arithmetic average return. This weight is usually derived from the ratio of the forecast period to the historical period.

Step-by-Step Calculation

1. Gather Historical Data

Start by collecting annual return data for the asset or portfolio over a specific historical period. The more reliable and longer the historical data, the better the quality of the forecast.

For example, you might gather returns from the past 10, 20, or 40 years.

2. Calculate the Arithmetic Average Return

The arithmetic average is computed by summing the annual returns and dividing by the number of periods. Mathematically, it is expressed as:

Arithmetic Average Formula:

\( R_a = \frac{1}{N} \sum_{i=1}^{N} r_i \)

where:

\( N \) is the number of periods,
\( r_i \) represents each individual annual return.

For instance, if an asset returned 5%, 10%, and -3% over three years, the arithmetic average would be:

\( R_a = \frac{5 + 10 + (-3)}{3} = \frac{12}{3} = 4\% \)

3. Calculate the Geometric Average Return

The geometric average considers the effect of compounding returns. It is determined by taking the product of all (1 + return) values, extracting the nth root (where n is the number of periods), and then subtracting 1:

Geometric Average Formula:

\( R_g = \left( \prod_{i=1}^{N} (1 + r_i) \right)^{\frac{1}{N}} - 1 \)

Continuing with our previous example, the geometric average is:

\( R_g = \left((1+0.05)(1+0.10)(1-0.03)\right)^{\frac{1}{3}} - 1 \)

Calculating step-by-step:

Product of returns: \( (1.05) \times (1.10) \times (0.97) \approx 1.1234 \)
Taking the cube root: \( 1.1234^{\frac{1}{3}} \approx 1.039 \)
Subtracting 1: \( 1.039 - 1 = 0.039 \) or approximately 3.9%

4. Determine the Weighting Factor

The weight \( w \) typically depends on the ratio of the forecast period \( T \) and the historical period \( N \). A common method is to set:

Weighting Factor Formula:

\( w = \frac{T}{N} \)

By choosing this proportion, you are giving more weight to the arithmetic average if the forecast period is relatively long compared to the historical period, capturing more of the volatility that might respond to future shifts.

For example, if you have historical data covering 40 years and you want to forecast returns for the next 10 years:

\( w = \frac{10}{40} = 0.25 \)

5. Applying the Blume Formula

Once you have obtained the arithmetic average \( R_a \), the geometric average \( R_g \), and the weighting factor \( w \), substitute these values into the Blume formula:

Blume Formula Revisited:

\( R = w \times R_a + (1 - w) \times R_g \)

Following our example:

Arithmetic Return, \( R_a \): 4%
Geometric Return, \( R_g \): 3.9%
Weight \( w \): 0.25 (for a 10-year forecast based on 40 years of data)

Thus:

\( R = 0.25 \times 4\% + 0.75 \times 3.9\% \)

Calculating each component:

0.25 × 4% = 1%
0.75 × 3.9% ≈ 2.925%

Adding them yields:

\( R \approx 1\% + 2.925\% = 3.925\% \)

A Comprehensive Example

Scenario: Forecasting Future Annual Returns

Imagine you have the following historical annual returns for a portfolio observed over 10 years: 8%, 12%, 7%, -2%, 15%, 10%, 5%, 9%, 11%, and 6%. We want to forecast the expected return for a forecast period of 4 years using the Blume formula.

Step 1: Calculate the Arithmetic Average

Sum the returns: \(8 + 12 + 7 + (-2) + 15 + 10 + 5 + 9 + 11 + 6 = 81\%\). Since there are 10 observations:

\( R_a = \frac{81\%}{10} = 8.1\% \)

Step 2: Calculate the Geometric Average

Use the formula:

\( R_g = \left( \prod (1 + \text{return}) \right)^{\frac{1}{10}} - 1 \)

First, convert percentages to decimals and multiply:

Year	Return (%)	(1 + r)
1	8	1.08
2	12	1.12
3	7	1.07
4	-2	0.98
5	15	1.15
6	10	1.10
7	5	1.05
8	9	1.09
9	11	1.11
10	6	1.06

Multiply all the (1 + r) factors:

\( \text{Product} = 1.08 \times 1.12 \times 1.07 \times 0.98 \times 1.15 \times 1.10 \times 1.05 \times 1.09 \times 1.11 \times 1.06 \)

Assume the product comes to approximately 1.90 after rounding. Then,

\( R_g = (1.90)^{\frac{1}{10}} - 1 \)

Evaluating the 10th root:

\( (1.90)^{\frac1{10}} \approx 1.067 \) (i.e., a 6.7% return), so the geometric average is approximately:

\( R_g \approx 6.7\% \)

Step 3: Determine the Weighting Factor

The weighting factor \( w \) is computed based on the ratio of the forecast period \( T \) to the historical period \( N \). Here, \( T = 4 \) years and \( N = 10 \) years:

\( w = \frac{4}{10} = 0.4 \)

Step 4: Apply the Blume Formula

Substitute the computed values into the formula:

\( R = w \times R_a + (1 - w) \times R_g \)

Plugging in our values:

\( R = 0.4 \times 8.1\% + 0.6 \times 6.7\% \)

Calculating each component:

Arithmetic part: \( 0.4 \times 8.1\% = 3.24\% \)
Geometric part: \( 0.6 \times 6.7\% = 4.02\% \)

Adding them gives:

\( R \approx 3.24\% + 4.02\% = 7.26\% \)

Therefore, based on Blume’s formula, the expected annual return for the upcoming 4-year period is approximately 7.26%.

Additional Considerations

Practical Aspects of Using the Blume Formula

While the Blume formula is a useful tool in expected returns estimation, it is important to remember that it relies heavily on historical data. Historical averages may not always be perfect predictors of future behavior, particularly in volatile markets or when structural changes occur in the market dynamics.

Additionally, the process of determining the appropriate historical period and forecast period should be done carefully, as different lengths can lead to varying weights and potentially different return forecasts.

Despite these limitations, its strength lies in its simplicity and its ability to temper the often optimistic arithmetic mean with the more conservative geometric mean.

Comparison with Other Models

CAPM-Based Approaches

In some contexts, models such as the Capital Asset Pricing Model (CAPM) are used to estimate expected returns where the beta of the asset plays a key role (as seen with the formula \( E(R) = R_f + \beta(E(R_m) - R_f) \)). However, the Blume formula is specifically geared towards reconciling the differences between arithmetic and geometric measures, focusing more on return estimates derived from historical averages. It should therefore be seen as complementary rather than a replacement for risk-based models.

Summary through a Comprehensive Table

Component	Description	Formula/Example
Arithmetic Mean \( R_a \)	Average of annual returns	\( R_a = \frac{1}{N} \sum_{i=1}^{N} r_i \) (e.g., 8.1%)
Geometric Mean \( R_g \)	Compounded average of returns	\( R_g = \left(\prod_{i=1}^{N} (1 + r_i)\right)^{\frac{1}{N}} - 1 \) (e.g., 6.7%)
Weight \( w \)	Ratio of forecast period to historical period	\( w = \frac{T}{N} \) (e.g., \( \frac{4}{10} = 0.4 \))
Expected Return \( R \)	Combination of arithmetic and geometric means	\( R = w \times R_a + (1 - w) \times R_g \) (e.g., 7.26%)