# How to calculate actuarial present value

1. Oct 16, 2011

### kingtaf

I am not sure if this was the right forum. if not let me know.

An auto repair shop plans to sell a new brand of car battery with a 4-year warranty. Given:

(i) If the battery fails within 4 years, the shop will refund a pro-rata share of the purchase price at the moment of failure.
(ii) The purchase price of the battery is equal to the sum of the manufacturing cost, loading for profit, and the actuarial present value of the warranty.
(iii) The manufacturing cost of the battery is 50, and profit is equal to 10% of the manufacturing cost.
(iv) The force of failure of the battery is mu(t) = 1/(10-t), 0 <= t < 10.
(v) delta (d) = 0.05

Calculate the purchase price of the battery.

I cant figure out how to calculate the actuarial present value in this case. Help please

2. Oct 16, 2011

### Locrian

You need a few things for this:

1) The loss to the seller of a failure at some time t
2) A function that gives the present value of that loss (what does delta tell you about whether it is continuous or discrete?)
3) The probability of failure at any time t

Once you have all three, you'll need to integrate. What are your limits?

Which, if any, of those have you figured out?

Out of curiosity, where is this problem coming from? Sounds a bit like FM/2.