Interpreting ROI Curves

The ROI curves are generated for a wholesaler, brand, investment vehicle and time (weekly). These are saturating curves that have diminishing returns. They all have different saturation points for different wiibles. These graphs constitute of 2 axes, the x axis which is the spends, the y axis is the impact that can be generated from the given spend amount. From these curves, we see that the there is negligible incremental impact from investments after a certain point and hence we need to invest in more profitable region.

These ROI curves are further used for optimization to find the model optimal allocation. Note that these curves are convex in nature.

Modelling Granularity: Wholesaler-Brand-Week

At the defined granularity, for each unique key, the model learns coefficients for the hierarchical regression equation based on historically available data during the training process. (minimizing for error on historical data points)

The latest methodology also allows us to capture interaction effects by sharing coefficients across terms in the equation.

For example, let us consider a simple case wherein we consider only 2 signals:

- Temperature
- Investments

In this case, a simplified version of the hierarchical regression equation would be:

\[ Y = \text{Base} \cdot f1(\text{temp}) + f2(\text{investments}) \cdot f3(f1(\text{temp})) \]

where

\(f3\) captures interaction effects
\(Y\) refers to Net Revenue for a particular key at the specified granularity

In the above example, the temperature signal contributes to NR impacts through the base as well as through interaction with the investment vehicle. The interaction effect can be understood as follows in a business context: Changes in temperature (or any other uncontrollable factors) can have an effect on the efficiency or ROI of investments. Eg: In summers, we know the consumers tend to consume greater quantities of cold beer. The resulting higher temperatures would also make consumers more susceptible to sales & marketing efforts undertaken by ABI, thereby increasing the ROI of our S&M investments.

Plotting the ROI Curve

Having learned all the required coefficients through the training process, we arrive at a unique equation for each key at the specified granularity. Note that each equation will consist of numerous terms corresponding to each investment vehicle. In order to visualize the curve for a particular investment vehicle, we must consider only the corresponding term. In the above example, to visualize the ROI curve for the investment vehicle we would consider only term 2:

\[ Y_1 = f_2(\text{investments}) \cdot f_3(f_1(\text{temp})) \]

where

\(Y_1\) is NR impact from investments signal

Plotting Y1 for different values of investments will give us the required ROI curve for a particular key at the specified granularity.

Example: Week Considered: 11th - 17th June 2023

roi_1

Note: The above equation is also dependent on temperature. However, given that we hope to answer the question of when investments would saturate, we shall plot the curve treating only the investment vehicle as the X variable. The temperature signal will be a constant having a value corresponding to the particular state-brand-wholesaler-week we are considering.

The temperature term would change for each different key as a result of different temperature each week and in each state, thereby giving us a different curve for each key. (This would also lead to different curves for forecasted years depending upon the forecasts for different input signals)

Example:

Week ending 10th June 2023

roi_2

Week ending 17th June 2023

roi_3

In this case, we plot the curves for the same Brand x region x investment vehicle for 2 weeks taking weekly spends as the variable (on the x-axis) and estimating corresponding Sales-to-Retailer impact (Y-axis).

Note that for the 2 weeks, the curves differ in terms of their values. At an investment of 4 million, we see a maco impact of 1.409 million in the first case and 1.403 million in the second case (A difference of 300k USD). This is due to the values of other signals changing across the 2 weeks. (such as temperature)

Aggregating Curves to a higher granularity

Because each vehicle x wholesaler x brand has a different ROI curve, the optimal structure of the roi curve for total investments for a set of optional investments is not just the sum of the investments. There are two options:

- If we want to keep the % allocation constant, we can hold the % allocation constant (for instance, for different wholesalers in the nation for national media), and calculate the ROI curve via the impacts of various totals allocated in the same way
- If we want to allow the model to optimize an allocation of multiple curves, for instance between brands within a family and between investment vehicles, we could calculate the optimal allocation for each spend amount and merge the curves by calculating a new curve of all the optimal plans