Groovy XM$,$ a satellite radio and streaming service offers three subscription plans. They are the

Music for $$13$ per month, Entertainment for $$20$ per month, and Platinum for $$32$ per month.

They expect that the plans will sell in the mix ratio of $7$:$5$:$2$ respectively. The variable costs

associated with each plan are $$4,$$$6,$ and $$9,$ respectively. Can make an impact in this market

they estimate that their promotions will cost $$5$ million.

a$.$ How many plans of each type do they need to sell to break even? $[3]$

b$.$ Assume that they could get a total of $1,250,000$ customers with the marketing budget

mentioned above, and they can achieve retention rates in the three segments of $80\%,73\%,$

and $65\%,$ respectively. Using a monthly discount rate of $4\%,$ please calculate the CLV for

each segment. $($Hint: $M=$ total contribution for each segment from part a$.$ For simplicity,

assume that $AC=F\frac{C}{?}$ total number of customers$)($Note: Just use the same AC for each

segment$)[6]$

$CLV=\frac{Mr}{(1+d-r)}-AC$

c$.$ Suppose that next year they have $$3,000,000$ in their marketing budget that they can spend

on one of the three segments. Help them decide by calculating what the retention rate

target should be for each segment if the entire $$3,000,000$ is spent on that segment. $($Hint:

Rearrange the formula in terms of $r,$ and calculate the new $M.$ To find the size of each

segment, use the sales mix, e$.$g$.$ Music would be $7/14$ of the total number of customers.$)[3]$

New $r=\frac{(CLV+AC)(1+d)}{(\text{N}\text{e}\text{w}M+CLV+AC)}$

d$.$ If they instead spend that $$3,000,000$ on advertising to $40,000$ households in their highest

CLV segment, how many new customers would they need to acquire too breakeven? $($Hint:

Use the CLV from part b$).[2]$

Breakeven Acquisition Rate $=\frac{\text{A}\text{c}\text{q}\text{u}\text{i}\text{s}\text{i}\text{t}\text{i}\text{o}\text{n}\text{S}\text{p}\text{e}\text{n}\text{d}\text{i}\text{n}\text{g}}{CLV}$

e$.$ Upon further research, they expect that after the first year, they will be able to increase

margins from their original batch of customers $($part b$)$ year to year by $3\%,5\%,$ and $6\%,$

respectively. Please recalculate the CLV for each segment using the growth rates. $[6]$

$CLV=\frac{Mr}{(1+d)-r(1+g)}-AC$