Actuarial present value

[rad6210]

An appliance store sells microwave ovens with a 3-year warranty against failure. At the time of purchase, the consumer may buy a 2-year extended warranty that would pay half of the original purchase price at the end of the year of failure. The extended warranty period begins exactly 3 years after the time of purchase, but only if the oven has not failed by then. Any failure is considered permanent. Delta (d) = 4%. Failure of the ovens follows the mortality table below:

Age(x) qx

0 0.008
1 0.015
2 0.026
3 0.042
4 0.063
5 0.089

Calculate the actuarial present value of the extended warranty as a percent of the purchase price.

I know the answer is 0.0405, I just obviously need help figuring out how to get to it. Thanks for any help!

 

[tkhunny]

It's not real clear to me what to do with your "delta", since $\displaystyle \delta$ and 'd' don't mean the same thing. As for the rest.

1) Pr(Dies in first year) = 0.008 and pays 0% at the end of year 1
2) Pr(Dies in second year) = Pr(survives 1 year)*pr(dies in year 2) = (1-0.008)(0.015) = 0.01488 and pays 0% at the end of year 2
3) Pr(Dies in third year) = Pr(survives 2 years)*pr(dies in year 3) = (1-0.008)(1-0.015)(0.026) = 0.025405 and pays 0% at the end of year 3
4) Pr(Dies in fourth year) = Pr(survives 3 years)*pr(dies in year 4) = (1-0.008)(1-0.015)(1-0.026)(0.042) = 0.039972 and pays 50% at the end of year 4
5) Pr(Dies in fifth year) = Pr(survives 4 years)*pr(dies in year 5) = (1-0.008)(1-0.015)(1-0.026)(1-0.042)(0.063) = 0.05744 and pays 50% at the end of year 5

If you meant 'd = 0.04', this translates to 0.040392

If you meant '$\displaystyle \delta = 0.04$', this translates to 0.040545

If you REALLY meant i = 0.04, then this translates to 0.04069