TikTok's Math

Assume TikTok acquires a cohort of $N$ users, with an average customer acquisition cost of $CAC$ per user.

Then the total contribution profit generated by this cohort can be written as:

$$Profit = N \times (LTV - CAC)$$

Here, $LTV$ means the lifetime gross profit per user, excluding acquisition cost. It can be decomposed as:

$$LTV = \sum_{t=1}^{T} R_t \times (ARPDAU_t - CTS_t)$$

where $R_t$ is the retention rate of this cohort on day $t$, $ARPDAU_t$ is the average revenue per daily active user on day $t$, and $CTS_t$ is the cost to serve one active user on day $t$.

For an advertising product like TikTok, $ARPDAU_t$ can be further decomposed as:

$$ARPDAU_t = MPU_t \times AdLoad_t \times \frac{CPM_t}{1000}$$

where: $MPU_t$ is minutes per active user on day $t$, $AdLoad_t$ is the number of ad impressions per minute, and $CPM_t$ is the revenue per thousand ad impressions.

Therefore:

$$LTV = \sum_{t=1}^{T} R_t \times \left( MPU_t \times AdLoad_t \times \frac{CPM_t}{1000} - CTS_t \right)$$

And the cohort-level profit becomes:

$$Profit = N \times \left[ \sum_{t=1}^{T} R_t \times \left( MPU_t \times AdLoad_t \times \frac{CPM_t}{1000} - CTS_t \right) - CAC \right]$$

Since the number of active users from this cohort on day $t$ is:

$$DAU_t = N \times R_t$$

we can rewrite the same formula as:

$$Profit = \sum_{t=1}^{T} DAU_t \times (ARPDAU_t - CTS_t) - N \times CAC$$

Or fully expanded:

$$Profit = \sum_{t=1}^{T} DAU_t \times \left( MPU_t \times AdLoad_t \times \frac{CPM_t}{1000} - CTS_t \right) - N \times CAC$$