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My question is about: Consider a death benefit of a life insurance policy. It can be obtained in four ways that all have the same present value:

  • A present value of a perpetuity of $ 300 at the end of each quarter;

  • Annuity payments of $ 600 at the end of each quarter for n years; first payment; one quarter after the death;

  • A present value of a payment (lump sum) of $ 60,000 at the end of the n -years after the moment of death;

  • A present value of a payment (lump sum) of $ B at the moment of death.

Calculate B.

Lorenz Clark
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    What does this have to do with the title? – Robert Israel Sep 21 '18 at 18:37
  • I edited your title... at least now it somehow related to the question. You can edit it again if you think this needs another title. – Yanko Sep 21 '18 at 18:41
  • Why not do what the hint says? Isn't that the only possible approach? – MPW Sep 21 '18 at 18:53
  • It's the only approach given, however, I'm still confuse on how to execute that approach without a Interest rate. Thanks for changing the title. – Lorenz Clark Sep 21 '18 at 19:05
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    You should not have the phrase "present value" in your bullets. The bullets give four options of payout. The problem says all four have the same present value. You need to use the equality to assess the discount rate being applied, then discount the $$60,000$ by that factor for $n$ years. – Ross Millikan Sep 21 '18 at 19:31
  • I've tried to equilise them but I think I'm making a mistake, please advice how. – Lorenz Clark Sep 21 '18 at 20:30

1 Answers1

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If you don't know the interest rate, then you simply leave it as an unknown. Since the smallest increment of compounding is a calendar quarter, let $j$ be the effective quarterly ($3$-month) rate of interest.

For a perpetuity-immediate of $1$ paid quarterly--I have assumed that, as in the other cases, payment of the benefit begins one quarter after death of the insured--the present value upon death is given by the formula $$a_{\overline{\infty}\rceil j} = \frac{1}{j}.$$

For an $n$-year annuity-immediate of $1$ paid quarterly, the present value is $$a_{\overline{4n}\rceil j} = \frac{1 - (1+j)^{-4n}}{j}.$$ Note the number of payments is $4n$ because there are $4$ payments per year for $n$ years.

For a lump sum of $1$ paid $n$ years after death, the present value is $$(1+j)^{-4n}.$$

For a lump sum of $1$ paid upon death, the present value is simply $1$.

Now, adjust each of the above by the amounts paid, and solve the resulting system. I'm not going to show you how to do it. You have to show your effort.

heropup
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