1.3 What does customer value mean?
In customer the field of portfolio management, one client’s value is represented by the customer lifetime value (CLV). CLV is the present value of all future purchases made by a customer over their lifetime in the firm’s portfolio, taking into account the attrition risk. CLV depends both on the purchase recency as well as on the purchasing rate and aims at identifying the most valuable customer groups. Formally, Gupta and Lehmann (2003) define CLV for customer \(i\) as follows:
\[\begin{equation} \text{CLV}_i = \sum_{t=0}^{T} \frac{(p_{i,t} - c_t)r_{i,t}}{(1+a)^t} - \text{AC}_i \tag{1.1} \end{equation}\]
with,
- \(p_{i,t}\) the price paid by customer \(i\) at time \(t\)
- \(c_t\) the marginal cost at time \(t\)
- \(r_{i,t}\) the probability that customer \(i\) be active at time \(t\)
- \(a\) the discount rate
- \(\text{AC}_i\) the acquisition cost of customer \(i\)
- \(T\) the duration of observation
An estimation of the portfolio’s overall value can be calculated through customer equity (CE) which amounts to the sum of all the CLVs. Since CE appears to be a good proxy of the firm’s value, the firm’s profit-maximization program can be written as:
\[\begin{equation} \begin{aligned} \max_{\mathrm{p}} \quad & \textrm{CE} = \sum_{i=1}^{N} \sum_{t=0}^{T} \frac{(p_{i,t} - c_t)r_{i,t}}{(1+a)^t} - AC_i\\ \textrm{s.t.} \quad & r_{i,t} \in [0, 1]\\ &p_t > c_t \\ \end{aligned} \tag{1.2} \end{equation}\]
where \(\mathrm{p}\) is the vector of prices over all periods that the firm needs to optimize.