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# System prepended metadata

title: CBCV short notes with maths

---

# CBCV short notes with maths

https://docs.google.com/document/d/1TgWXffAKWjDgfnVmd7TwYwCiLiV2mtF_wVYd_qs2ETw/edit#

Symbols so we agree

* $\rho$ annual churn rate in % (e.g. 50%)
* $T$ average customer lifetime in years with $T=\rho^{-1}$
* $\gamma$ customer growth rate in %
* $N$ number of customers, $N_t$ at time $t$ with $N=N_0$ current.

Other

* ARPU = average revenue per user in $
* RRR = revenue retention rate in %
* CAC = customer acquisition cost
* WACC = weighted average cost of capital in % (_rate a company is expected to pay to its security holders to finance assets_)

## Derived values

### General notes about assumptions we do

We do things a bit differently than classical valuation

* Redpoint capital, VC fund in California, also writes on stuff; Tomasz
* True enough that cost of maintenance per customer and taxes are negligible (for software stuff)
* CAPEX
    * no need for "new machines"
    * with software, more like royalty from a book
    * anything that would have been CAPEX ~ revenue to shareholder
* we ignore non-operating assets, and only take debt into account through the WACC (for now) (we could add that stuff later)
* we are trying to arrive at the market price of the company, not its supposed intrinsic value, so we want our model to produce results that are close to its market valuation (with a 20% difference seemingly acceptable)


### Future Revenue

$$
FR_{\mathrm{customer}} = ARPU_0 \times \sum_{t=0}^\infty {RRR^t \times (1-\rho)^t \over (1+WACC)^t}
$$

Average future revenue per individual current customer. 

There is a $\tau$ such that we can write that sum as

$$
FR_{\mathrm{customer}} = ARPU_0 \times \sum_{t=0}^\tau {RRR^t \times (1-\rho)^t \over (1+WACC)^t} + \epsilon
$$

with $\epsilon \approx 0$. 

Let's say $\tau \in \mathbf N$ then

$$
FR_{\mathrm{customer}} = ARPU_0 \times \left( {1 - \zeta^\tau\over 1-\zeta} \right)
$$

$$
\zeta = {RRR\times (1-\rho) \over (1+WACC)}
$$

Ok let's plug in numbers

* 30% churn
* RRR = 107
* ARPU = 2500
* WACC = 10%

$$
\zeta = 1.07 \times 0.7 / 1.1 = 0.681
$$

$$
1/(1-\zeta) = 3.1
$$

$$
\tau = 1/0.3 = 3.33
$$

$$
\zeta' =  1.07 / 1.1 = 0.97
$$

$$
\left( {1 - \zeta'^\tau\over 1-\zeta'} \right) = 3.2
$$


Other example

* 30% churn
* RRR = 5000
* ARPU = 2500
* WACC = 10%

$$
\zeta' =  50 / 1.1 = 45.5
$$

$$
\left( {1 - \zeta'^\tau\over 1-\zeta'} \right) = 7444
$$

$$
\zeta = 31.8
$$

$$
1/(1-\zeta) = -0.03
$$

---

Infinite sum

$$
FR_{\mathrm{customer}} = ARPU_0 \times \sum_{t=0}^\infty {RRR^t \times (1-\rho)^t \over (1+WACC)^t} = ARPU_0 \times {1 \over (1-\zeta)}
$$

---


$$
FR = ARPU \times {\left[ { (1 - RRR^T) \over (1 - RRR) } \right] \over \left[ {(1 - WACC^T) \over (1-WACC)}\right]}
$$

* $T$ potentially not integer so sum of geometric sequence is?
* should be gradual discounting, not end-of period but close enough?
* could have more sophisticated computation of discounted cashflows


### CLTV

Customer Lifetime Value

$$
CLTV = FR - (CS + CAC)
$$

* $CS$ cost of service (undefined for now)


### Future Customers

$$
FC = N \times (1 + \gamma)^{T-1}
$$

### Customer Growth Costs

$$
    CGC = (FC - N) \times CAC
$$

### Valuation

$$
    V = FR \times FC - CGC
$$

### ARR

$$
    ARR = N \times ARPU
$$

### ARR Multiple

$$
    ARRM = {V \over ARR} = {V \over N \times ARPU}
$$


### Derivation of T

$$
    T = 1 + (1 - \rho) + (1 - \rho)^2 + ...
$$

$$
    T  = {1\over 1 - (1 - \rho)} = {1 \over \rho}
$$