This appendix provides mathematical details for assessing the impact of retention rates on the long-run market share and profits of a firm.
Consider the retention-defection or switching matrix for a two-brand market (Table B.1). Here brand A has a 95% retention rate, and brand B has 80% retention rate. This is the same as scenario 3 discussed in Chapter 3. Assume that at time period 0, both brands start with an equal share of 50% (you can easily verify, and it will be clear at the end of this discussion, that results do not change if starting shares are different). If the market consists of 100 units, then both brands have 50 units at time 0.
In the next time period, period 1, brand A retains 95% of its 50 unit sales, or 95% x 50 = 47.5 units. In addition, 20% of brand B's customers defect to brand A. Therefore brand A gets 20% x 50 = 10 units from brand B's defection. Therefore, the total sales (or share, since the market consists of 100 units) of brand A is 47.5 + 10 = 57.5 units. Brand B's sales and share in this period is 100 – 57.5 = 42.5 units. These steps are repeated each period as shown in Table B.2. Equilibrium is attained when brand A reaches an 80% share. The reason for this is that with 95% retention, or 5% defection rate, and 80% share, brand A loses 5% x 80 = 4 units (or share points). At the same time, since brand B has 20% defection rate and 20% share, Brand A gains 20% x 20 = 4 units from brand B. Hence, Brand A gains exactly the same number of units from brand B that it loses to that brand.
Table B.2. Evolution of Market Shares
Sales (or Share) over Time | ||
|---|---|---|
A | B | |
Time 0 | 50 | 50 |
Time 1 | 50 x 95% + 50 x 20% = 57.5 | 42.5 |
Time 2 | 57.5 x 95% + 42.5 x 20% = 63.125 | 36.875 |
Time 3 | 63.125 x 95% + 36.875 x 20% = 67.344 | 32.656 |
… | … | … |
Time n | 80 x 95% + 20 x 20% = 80 | 20 |
In general, the long-run market share of a brand can be derived as follows.
where
mA = long-run market share of brand A (80% in our example)
rA = retention rate of brand A (95% in our example)
mB = long-run share of brand B (20% in our example)
dB = rate of brand B customers defecting to brand A (20% in our example)
Since mA + mB = 100% and rA = 100 – dA, we can simplify the above equation to obtain the long-run share of brand A as:
We can now use equation (B.2) to estimate the long run share of a brand. For example, in scenario 1 in Chapter 3, dA = dB = 50%. Substituting in equation (B.2), we get mA = 50%. In scenario 2, dA = 10% and dB = 20%. Therefore, mA = 20/(20 + 10), or 66.67%. In scenario 3, dA = 5% and dB = 20%. Hence, mA = 20/(20 + 5), or 80%.
Retention elasticity measures the impact of a 1% change in retention on the percentage change in customer lifetime value (CLV). From Chapter 2 we know that
where m is the annual margin, i is the discount rate, and r is the retention rate. Differentiating equation (B.3) with respect to r gives us change in CLV due to a change in r,
Therefore, retention elasticity ηr is
How does this compare to the margin elasticity ηm? Using equation (B.3), we can derive the margin elasticity as follows:
Since the margin multiple is always positive, retention elasticity is always greater than margin elasticity. For example, at 90% retention rate and 12% discount rate, the margin multiple is approximately 4. In this case, retention elasticity is 5, or five times the size of the margin elasticity.