Apr 2004 #8 - single prem policy

[MindFull]

"A company issues a block of 5-year single premium investment policies to lives each aged 60 exact at commencement of a policy. It guarantees simple annual reversionary bonuses of 8% per annum of the single premium, with the possibility of a terminal bonus at maturity. The death benefit is 5 times the single premium. All premiums received under this policy are invested in an asset class where 5-year returns have a normal distribution with a mean of 50% and standard deviation of 25%. The company intends to declare terminal bonuses on maturity such that the proceeds of the policy are the greater of the guaranteed amount and 90% of the underlying asset value."

The ans. for this says that a loss will be made if the assets are worth less than 1.4 assuming that the single prem. is 1. If the total of the bonuses is 0.4, why is it that a loss would be made for assets less than 1.4 and not 0.4?

Thanks.

 

[Hamilton]

hope this helps

If we multiply 140% by 10 / 9 we get the figure of 155.56

So if the assets exceed the 155.56% level at the end of the 5 year term , our terminal bonus kicks in to guarantee our costumers at least 90% of this.

 

[MindFull]

But why would it be 140% and not 40%? Since it's a single prem. policy, the ins. co. receives the 1 in prems. and then a bonus of 8 cents would be given out every yr for 5 yrs which gives 40 cents. So why is it that that a loss would come if the asset val. is > than 1.4?

 

[Hamilton]

hum

not sure what you mean ? You mean part (b) right?
I do have to admit the question is basically impossible .No way in a million years would I have got this out ,but I think I understand the answer ,especially to part (b) part (a) is less clear to me .

important to realise this is mearly 2 separate probability questions

part (a) asks about the probability of making a loss and its the sum of two mutually exclusive cases ,of death (death benefit is 5.Prem so definite loss)
or
survival and poor investment performance.

Now part (b)
asks probability of the company paying a terminal bonus

they have promised 8% simple interest return for 5 years ie they guarantee 140% of your money back if you live
So when will you get a terminal bonus ?
when 90% of the assets return is greater than the guaranteed 140% . times the probability your alive.

the company certainly is'nt making a loss in this case quite the opposite they are only promising 90% of the assets to the customer lovely 10% left for themselves at least 15.56 . ( plus 1/10 of any higher return)
if your draw a graph of return on assets vs company profit
its a slope 1 (45 degree) line with a kink in it at asset return of 155.56% where the slope changes to reflect the terminal bonus ie the company starts giving away 90% of every extra pound it earns past the 155.56% asset return point.

so the answer multiplies your survival probability and the chance a normal (50 , 25) will exceed 55.56% return . this is when a terminal bonus is paid.

Ok now im starting to think you actually mean part (a) and well the same reasoning applies i.e.
"they have promised 8% simple interest return for 5 years ie they guarantee 140% of your money back if you live "
so the company will lose out if you either die or if you live and the assets do bad i.e.
the chance a normal (50 , 25) is less than 40
the question also states its for a particular policy not for the portfolio!
which would be different , pretty sure there is enough data to figure that out but i dont want to think about that.

 

[MindFull]

wowser

Well ok, Im uhh still a little confused and I actually am referring to part (a) and so i still don't get why you are promised 140% of your money back, instead of 40% or is it that the policy holder gets back the $1 at the end of the 5 yrs. along with the bonus payments...

Thanx

 

[didster]

Just to add to Hamilton's response, in case your problem is not understanding the description of the policy itself.

It's an investment policy. So the policyholder is going to want his original premium back with some interest at the very least.
Bonus guaranteed at 8%pa simple, this is your interest.
Policy also has a death benefit. This is more of an add on, since the policy is primarily an investment one.

Question looks daunting, but it's not as bad as it seems at a first glance. Eg you don't need to calculate the expected profit. You just need to look at which cases give you a loss or a terminal bonus and relate probabilities to these.

 

[MindFull]

THANK YOU... You're right. I didn't understand the policy type, but it makes sense now that you say it. Thanks again.