problem 37.13 in ASM MLC manual

[Thomas H]

The problem is:

An annuity-due of 600 per month is issued to (60) and (65). Payments are guaranteed for 10 years. After the gaurantee period, payments will be made as long as both are alive. If (60) has died and (65) is alive after the guarantee period, then each subsequent payment will be 450. If (65) has died and (60) is alive, then each subsequent payment will be 400.

Calculate the actuarial present value for this annuity.

Then there are choices given and it turns out the correct one is:

3000 a_{60:65:10} + 2400 a_{65:10} + 1800 a_{60:10}

where all annuities are annuities-due and are 10-year certain.

The solution simply says this has the correct pattern of payments.

How are these the correct payments?

[wat]

The problem is:

An annuity-due of 600 per month is issued to (60) and (65). Payments are guaranteed for 10 years. After the gaurantee period, payments will be made as long as both are alive. If (60) has died and (65) is alive after the guarantee period, then each subsequent payment will be 450. If (65) has died and (60) is alive, then each subsequent payment will be 400.

Calculate the actuarial present value for this annuity.

Then there are choices given and it turns out the correct one is:

3000 a_{60:65:10} + 2400 a_{65:10} + 1800 a_{60:10}

where all annuities are annuities-due and are 10-year certain.

The solution simply says this has the correct pattern of payments.

How are these the correct payments?

If you divide the numbers by 12, you'll get what each annuity is representing in monthly payments. The joint annuity represents $3,000/12 = $250 in monthly income, the (65) annuity represents $2,400/12 = $200 in annuity payments, and the (60) annuity represents $1,800/12 = $150 in annuity payments.

So, if both are alive, you get ($250 + $200 + $150) = $600/month.
If (65) is alive, you get ($250 + $200) = $450/month
If (60) is alive, you get ($250 + $150) = $400/month.

Also, you would need to assume that the joint annuity for $250 is a "last-to-die" annuity, where as long as one of them are alive, you'll be collecting $250, and an additional $150/$200/$350, depending on who's still alive after 10 years.

Final question - for the annuities you wrote down, they're supposed to be 10-year certain & life annuities, right?

[Thomas H]

Yes & life annuities. Also, they each have the superscript (12) for the monthly payments.

I'm not understanding why there isn't some kind of reversionary annuity in the solution.

[Abraham Weishaus]

Work out all 4 cases:
(1) In the first 10 years, 600 a month is paid. How much does (E) pay?

AFTER TEN YEARS:
(2) If both are alive, 600 a month is paid. How much does (E) pay?
(3) If (65) only is alive, 450 a month is paid. How much does (E) pay?
(4) If (60) only is alive, 400 a month is paid. How much does (E) pay?

If all 4 cases match (and they are exhaustive), they must be identical.