Let's tackle both parts step by step. **(Please double-check the numbers and methodology if you use this for an assignment or business plan!)** --- ## **Given Data** | | Silver | Gold | Platinum | |----------------|--------|------|----------| | Visits/Month | 2.6 | 3.9 | 7.3 | | Revenue/Visit | $3.25 | $3.76 | $4.29 | | Annual Retention Probability | 0.30 | 0.50 | 0.75 | - **Gross Margin:** 45% - **Acquisition Cost:** $10 (one-time) - **Discount Rate:** 10% - **No one defects in first year** - **No annual marketing expense** - **CLV time horizon:** 5 years --- ## **(a) 5-Year Individual Customer CLV for Each Class** ### **Step 1: Calculate Annual Gross Profit per Customer** \[ \text{Annual Gross Profit} = (\text{Visits/Month}) \times 12 \times (\text{Revenue/Visit}) \times (\text{Gross Margin}) \] #### **Silver** \[ GP_{Silver} = 2.6 \times 12 \times 3.25 \times 0.45 = 31.2 \times 3.25 \times 0.45 \] \[ = 101.4 \times 0.45 = \$45.63 \] #### **Gold** \[ GP_{Gold} = 3.9 \times 12 \times 3.76 \times 0.45 = 46.8 \times 3.76 \times 0.45 \] \[ = 175.968 \times 0.45 = \$79.19 \] #### **Platinum** \[ GP_{Plat} = 7.3 \times 12 \times 4.29 \times 0.45 = 87.6 \times 4.29 \times 0.45 \] \[ = 375.804 \times 0.45 = \$169.11 \] --- ### **Step 2: Calculate 5-Year CLV for Each Class** #### **Key Points** - Customers never defect in Year 1. - From Year 2 onward, annual retention probability applies. - Use discount rate \( r = 10\% \), so discount factor for year \( t \): \( (1+r)^t \) - Acquisition cost is subtracted at time 0. #### **General Formula** Let: - \( GP \) = annual gross profit - \( p \) = annual retention probability - \( r \) = discount rate \[ \text{CLV} = -\text{Acquisition Cost} + \sum_{t=1}^5 \frac{GP \times s_t}{(1+r)^t} \] Where: - \( s_1 = 1 \) (everyone stays for the first year) - \( s_2 = p \) - \( s_3 = p^2 \) - \( s_4 = p^3 \) - \( s_5 = p^4 \) --- #### **Silver (\( GP = \$45.63, \ p = 0.3 \))** \[ \begin{align*} \text{CLV}_{Silver} = & -10 \\ &+ \frac{45.63 \times 1}{1.1^1} \\ &+ \frac{45.63 \times 0.3}{1.1^2} \\ &+ \frac{45.63 \times 0.09}{1.1^3} \\ &+ \frac{45.63 \times 0.027}{1.1^4} \\ &+ \frac{45.63 \times 0.0081}{1.1^5} \end{align*} \] Calculate each term: - \( 1.1^1 = 1.10 \) - \( 1.1^2 = 1.21 \) - \( 1.1^3 = 1.331 \) - \( 1.1^4 = 1.4641 \) - \( 1.1^5 = 1.61051 \) - Year 1: \( 45.63 / 1.10 = 41.48 \) - Year 2: \( 45.63 \times 0.3 / 1.21 = 13.689 / 1.21 = 11.32 \) - Year 3: \( 45.63 \times 0.09 / 1.331 = 4.107 / 1.331 = 3.09 \) - Year 4: \( 45.63 \times 0.027 / 1.4641 = 1.231 / 1.4641 = 0.84 \) - Year 5: \( 45.63 \times 0.0081 / 1.61051 = 0.3696 / 1.61051 = 0.23 \) Add up: \[ \text{Sum} = 41.48 + 11.32 + 3.09 + 0.84 + 0.23 = 56.96 \] \[ \text{CLV}_{Silver} = 56.96 - 10 = \$46.96 \] --- #### **Gold (\( GP = \$79.19, \ p = 0.5 \))** - Year 1: \( 79.19 / 1.10 = 71.99 \) - Year 2: \( 79.19 \times 0.5 / 1.21 = 39.595 / 1.21 = 32.73 \) - Year 3: \( 79.19 \times 0.25 / 1.331 = 19.7975 / 1.331 = 14.88 \) - Year 4: \( 79.19 \times 0.125 / 1.4641 = 9.8988 / 1.4641 = 6.76 \) - Year 5: \( 79.19 \times 0.0625 / 1.61051 = 4.9494 / 1.61051 = 3.07 \) \[ \text{Sum} = 71.99 + 32.73 + 14.88 + 6.76 + 3.07 = 129.43 \] \[ \text{CLV}_{Gold} = 129.43 - 10 = \$119.43 \] --- #### **Platinum (\( GP = \$169.11, \ p = 0.75 \))** - Year 1: \( 169.11 / 1.10 = 153.74 \) - Year 2: \( 169.11 \times 0.75 / 1.21 = 126.8325 / 1.21 = 104.89 \) - Year 3: \( 169.11 \times 0.5625 / 1.331 = 95.202 / 1.331 = 71.56 \) - Year 4: \( 169.11 \times 0.422 / 1.4641 = 71.45 / 1.4641 = 48.81 \) - Year 5: \( 169.11 \times 0.316 / 1.61051 = 53.45 / 1.61051 = 33.22 \) Add up: \[ \text{Sum} = 153.74 + 104.89 + 71.56 + 48.81 + 33.22 = 412.22 \] \[ \text{CLV}_{Plat} = 412.22 - 10 = \$402.22 \] --- ### **Final Answers for (a):** **5-year CLV per customer:** - **Silver:** \$46.96 - **Gold:** \$119.43 - **Platinum:** \$402.22 --- ## **(b) How many customers per store must convert from Gold to Platinum for breakeven?** ### **Given:** - Investment: \$40,000,000 - Stores: 4,500 - All conversions last the 5-year period ### **Step 1: Incremental CLV per conversion** \[ \Delta CLV = CLV_{Platinum} - CLV_{Gold} = 402.22 - 119.43 = \$282.79 \] ### **Step 2: Total number of conversions needed** \[ \text{Number of conversions needed} = \frac{40,000,000}{282.79} = 141,528 \] ### **Step 3: Conversions per store** \[ \text{Per store} = \frac{141,528}{4,500} = 31.45 \] --- ## **Final Answers (Boxed):** --- ### **(a) 5-year individual customer CLV:** - **Silver:** \(\boxed{\$46.96}\) - **Gold:** \(\boxed{\$119.43}\) - **Platinum:** \(\boxed{\$402.22}\) --- ### **(b) Number of customers per store to convert from Gold to Platinum for breakeven after five years:** \[ \boxed{32} \] (rounding up to the next whole number) **per store** --- **Let me know if you need the full step-by-step calculations in table format!**