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.
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.