Lifetime Value

Study guide: Exam 5

Lifetime Value (LTV) is the present value of the expected profit from a customer relationship over its whole life, rather than over a single policy period. It is the framework Werner uses for the retention and acquisition questions that sit on top of the actuarial indication.

$\text{LTV} = \sum_{t=1}^{T} \frac{\left(P_t - L_t - E_t\right) \times {}_{t}p}{(1 + r)^{t}}$

$\text{Retention Ratio} = \frac{\text{Policies renewed}}{\text{Policies eligible to renew}}$

${}_{t}p$ is the probability the customer is still on the books at time $t$ and $r$ the discount rate. Retention is what makes the sum finite: a book renewing at $85\%$ has an average customer life of about $1/(1-0.85) \approx 6.7$ years, at $90\%$ about ten years.
LTV justifies first-year pricing below full cost where acquisition costs are large and renewal margins are reliable — but only if the retention and loss-improvement assumptions hold. The "seasoning" effect (a policy's loss ratio improving with tenure) is real in personal lines and is one of the strongest arguments for the approach.
The corresponding risk is adverse selection on price: a below-cost new-business rate attracts the most price-sensitive shoppers, who are also the least likely to renew. If the customers acquired are not the ones the retention assumption describes, the LTV never materializes.
Price elasticity and retention modelling are what make LTV operational. Modelling renewal probability as a function of the rate change offered lets an insurer see the trade-off directly: a larger increase raises premium per policy and lowers the number of policies, and the optimum is rarely at either extreme.
The professional boundary matters: the indication is an actuarial estimate of expected cost; LTV is a business decision layer applied to it. Departing from the indication for retention reasons is legitimate and must be documented as a departure (Rate Change, Ratemaking Constraints) — not folded back into the cost estimate.

LTV Justifying a First-Year Loss Acquiring a homeowners customer costs $\$200$ . The first-year underwriting result is $-\$50$ ; expected renewal profit is $\$120$ a year for four further years, and the discount rate is $8\%$ . Assume renewal is certain for those four years.

Compute the lifetime value.
Answer
 $\begin{align*} \text{PV of renewals} &= \$120 \times \frac{1 - 1.08^{-4}}{0.08} \\ &= \$120 \times 3.3121 \\ &= \$397.45 \\[6pt] \text{LTV} &= -\$200 - \$50 + \$397.45 \\ &= +\$147.45 \end{align*}$ 
The relationship is worth $\$147$ despite losing money in year one, so the acquisition is justified — conditional on the customer renewing four times.

The Same Customer at a Realistic Retention Rate Repeat the calculation with an annual retention rate of $80\%$ , and then with the $65\%$ retention typical of price-shopped new business.
Answer With retention $p$ , the year- $t$ profit is received with probability $p^{t}$ :

$\text{PV} = \$120 \sum_{t=1}^{4} \frac{p^{t}}{1.08^{t}}$

At $p = 0.80$ the per-year discount factor is $0.80/1.08 = 0.7407$ :
 $\begin{align*} \text{PV} &= \$120 \times (0.7407 + 0.5487 + 0.4064 + 0.3011) \\ &= \$120 \times 1.9969 = \$239.63 \\[4pt] \text{LTV} &= -\$250 + \$239.63 = -\$10.37 \end{align*}$ 
At $p = 0.65$ , the factor is $0.6019$ :
 $\begin{align*} \text{PV} &= \$120 \times (0.6019 + 0.3622 + 0.2180 + 0.1312) \\ &= \$120 \times 1.3133 = \$157.60 \\[4pt] \text{LTV} &= -\$250 + \$157.60 = -\$92.40 \end{align*}$ 
The acquisition swings from $+\$147$ to $-\$92$ on the retention assumption alone.

This is the trap in LTV-justified pricing. The customers attracted by a below-cost first-year rate are precisely those shopping on price, and they are the least likely to renew at the full rate — so the low-retention column is the relevant one, not the certain-renewal column used to justify the discount. Testing the LTV at the retention rate the discount itself produces, rather than at the book's average, is what separates a sound acquisition strategy from a slow leak.