Weekly growth to monthly growth conversion formula

Week is an useful  unit of measure in Product development. Unlike a month – it has a fixed number of days always. And the behavioral patterns are more repetitive. All Sundays follow a common buying pattern, all tuesdays have another buying pattern etc.
But most importantly it is just the right amount of time for giving an experiment enough time to run, while not being too long that the idea enters development hell. It is an ideal time interval for measuring the success rates of various experiments. So while developing products, I typically prefer to measure week on week progress.
But while looking at the overall progress of the company and presenting it to stakeholders, a month on month measure is more appropriate.
So we need a formula to convert weekly growth to monthly growth.

So if we decide to target a fixed week on week growth in some KPI, say 10%, how much will this translate to as a monthly growth rate? Let’s derive a formula now.
Let the KPI be number of transactions per week $WT$
Let the value of the KPI at Week 0 be $WT_0$
Let the rate of weekly growth of the KPI be $r$.
Then, the value of the KPI at Week 1 is found as
$$WT_1 = WT_0 + WT_0 * r$$
$$\implies WT_1 = WT_0 * (1+r)$$

Similary number of Transactions for  Week 2 will be
$$WT_2 = WT_1 * (1+r)$$
$$\implies WT_2 = WT_0 * (1+r)^2$$

So, number of transactions for Week n would be,
$$WT_n = WT_0 * (1+r)^n$$

Now, the number of transactions for month 1, $MT_1$ would be the sum of the weekly transactions of weeks 1-4 (assuming a month corresponds to 4 weeks)
$$MT_1 = WT_1 + WT_2 + WT_3 + WT_4$$
$$\implies MT_1 = WT_0 * (1+r) + WT_0 * (1+r)^2 + WT_0 * (1+r)^3 + WT_0 * (1+r)^4$$
$$\implies MT_1 = WT_0 * (1+r) * (1 + (1+r)+  (1+r)^2 +  (1+r)^3 )$$

Similary $MT_2$ will be
$$MT_2 = WT_5 + WT_6 + WT_7 + WT_8$$
$$\implies MT_2 = WT_0 * (1+r)^5 + WT_0 * (1+r)^6 + WT_0 * (1+r)^7 + WT_0 * (1+r)^8$$
$$\implies MT_2 = WT_0 * (1+r)^5 * (1 + (1+r)+  (1+r)^2 +  (1+r)^3 )$$

Generalizing $MT_n$ will be,
$$MT_n = WT_0 * (1+r)^{4*(n-1)+1} * (1 + (1+r)+  (1+r)^2 +  (1+r)^3 )$$

$$\implies MT_n = WT_0 * (1+r)^{4*(n-1)+1} * (r^3+4r^2+6r+4 )$$

$$\implies MT_n = WT_0 * (1+r)^{4n-3} * (r^3+4r^2+6r+4 )$$

So given the number of transactions in Week 0 as $WT_0$, and a weekly growth rate of $r$ we can find the transactions for any nth month starting from the week after that, using the formula.

$$MT_n = WT_0 * (1+r)^{4n-3} * (r^3+4r^2+6r+4 )$$

Now to find the monthly growth rate $m$,
$$\implies m=\frac{(1+r)^{4n-3}-(1+r)^{4(n-1)-3}}{(1+r)^{4(n-1)-3}}$$
Simplifying we get the formula for monthly growth rate $m$, given a weekly growth rate $r$ as,
Inversely, if we were given the monthly growth rate m, we can find the required weekly growth rate as
$$r=(1+m)^{\frac{1}{4}} – 1$$
$$\implies r=(1+m)^{0.25} – 1$$

Applying some numbers to the formulae,
If we achieve a weekly growth rate of 10%, ie $r=0.1$, then we can expect a monthly growth rate of $1.1^4-1=0.4641 = 46.41\%$
Conversely if we wanted to have a monthly growth rate of 30%, ie $m=0.3$, then the weekly growth rate we have to attain would be , $$1.3**0.25-1 = 0.0678 = 6.78\% $$

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