Question

Your company has launched a new insurance product and is projecting the premium cashflows, for budgeting purposes. You expect to steadily acquire new clients over the first year at a constant rate, and are projecting having 1,000 clients by the end of the year. Each client pays an annual premium of $10 in the first year.

The company expects slower growth in clients in Years 2 and 3, with 500 new clients in Year 2, and 200 new clients in Year 3. They also expect some lapses, whereby 10% of those who paid the premium in their first year do not make the second year of payments. Premiums will remain fixed at $10 per year for the three year period.

Define ?(?) as the total premium paid between time 0 and time ?.

1. Derive the expression(s) for ?(?) for the period ? = 0 through to ? = 3.
2. If there is a constant force of interest of 5% over the three years, write down the expression for the present value of this stream of premium payments.

M(t) is the total premium paid between time 0 and time t

Answer 1

Year	1	2	3
Clients	1000	1000 (= to total clients at year 1)	1400(= to total clients at year 2)
New Clients added	0	500	200
Clients left	0	100 (=10% of 1000 ie clients in year 1)	0
Total clients	1000	1400	1600
Premium per client	10	10	10
Total premium received	10000 (=1000*10)	14000 (=1400*10)	16000 (=1600*10)

M(1): Total premium received till end of year 1 = 10000

M(2): Total premium received till end of year 2 = 10000 + 14000 = 24000

M(3): Total premium received till end of year 3 = 10000 + 14000 + 16000 = 40000

discount factor = exp^(-r*t); where

exp: natural exponential

r: constant force of interest = 5%

t: time

Present value of M(1) = 10000*exp(-5%*0.5) = 9753.10

**time taken as 0.5 year because we assume the premiums on an average are collected by mid of 1st year)

Present value of M(2) = 10000*exp(-5%*0.5)+14000*exp(-5%*1.5) = 22741.51

**time taken as 1.5 year in second term because we assume the premiums in 2nd year are on an average collected by mid of 2nd year)

Present value of M(3) = 10000*exp(-5%*0.5)+14000*exp(-5%*1.5)+16000*exp(-5%*2.5) = 36861.46

**time taken as 2.5 year in third term because we assume the premiums in 3rd year are on an average collected by mid of 3rd year)