Kimberly is told that she can receive a 250,000 death benefit from her husband's life insurance in annual installments of 25,000 at the beginning of each year for 11 years and a final payment of 16,265 at the beginning of the 12th year.
Alternatively, Kimberly may receive annual installments of 13,000 at the beginning of each year for life, with a certain period of 10 years.
Calculate the present value of a 10-year deferred life annuity-due of one dollar per annum at Kimberly's issue age.
Okay so ultimately, I figured the present value of a 10-year deferred life annuity-due would be the discounted rate of 10 years times the present value of a perpetuity due….
To get the discount rate I used the first payment option of an 11-year annuity due with annual payment of 25,000 plus a final payment of 16,265 at time t=11 since the annuity-due begins payment at time t=0, this means that at time t=11, the final payment will be made thus,
$$ 250000=25000\require{enclose} \ddot a_{\enclose{actuarial}{11}{i}} + 16265v^{11} $$
this formula generates interest rate i=3% thus, the discount rate d = .03/1.03 = .029 or 2.9%
Now, for perpetuity-due it is simply payment over discount which in this case = 13,000 / .029 = $448,275.86.
However, because it is deferred for 10 years and we only want it in terms of 1 dollar per annum, this becomes,
$$v^{10} * \frac {1}{.029}\\ \\ = (1.03)^{-10}*34.483 \\ \\= 25.6584 $$
However, my answer sheet gives a different value without explaining the solution so I have no clue how to interpret this question.. can somebody please explain this for me?
Best Answer
There's no perpetuity here. Kimberley is offered a life annuity.
From the first option, we can compute the interest rate. In the second option, Kimberley is offered a life annuity with period certain, also worth $250,000$. A life annuity with a $10$-year period certain is the sum of a a ten-year annuity certain, and a 10-year deferred life annuity. Since ww know the interest rate, we can compute the value of the annuity certain, and then subtraction gives the value of the life annuity. We just have to divide by $13,000$ to get the answer to the question posed.