Question

Lucy recieves a gift of a twenty-one year annuity on the day she was born. The annuity pays $500 on her odd birthday and $700 on even birthday. If the nominal rate of interest is 8% payable monthly, find the value of this annuity on the day she was born. Answer: $5,817.13

A Few things have me stuck. If the year of her birth time 0 is odd or even. Also, I understand I need to consider the interest for 2 years but how to get my monthly nominal to a two year rate. I just did 8% times 2.

Answer 1

Step-by-step explanation:

Let i represent the nominal annual interest rate (8%). Then the effective annual multiplier of the principal is ...

(1 +i/12)^12 = (1+.08/12)^12 ≈ 1.0829995

Then the 2-year multiplier is the square of this:

1+r = (1.0829995)² = 1.1728879

We can use the formula for the present value of an annuity to refer all of the odd-birthday payments to 1 year prior to birth, then adjust by the above annual multiplier. All the even-birthday payments will have the formula applied in the usual way.

PV = P(1 -(1+r)^-n)/r

where P is the payment amount, (1+r) is the periodic multiplier, and n is the number of payments.

For the odd-birthday payments, the PV (1 year early) is ...

PV = $500(1 -(1.1728879)^-11/0.1728879 = $2391.5704

so, the amount referred to the day of birth is ...

1.0829995 × $2391.5704 = $2590.07 . . . . . PV of odd-birthday payments

__

The value of even-birthday payments is ...

PV = $700(1 -(1.1728879)^-10/0.1728879 = $3227.06

And the total value of both sets of payments is ...

annuity value = $2590.07 +2337.06 = $5817.13

_____

Alternate solution

You could also work this using the PV formula on the sum of ten sets of two years' payments: (700 +500/1.083) and then add the PV of the first odd-year birthday payment: 500/1.083.

This gives ...

500/1.083 + (700 +500/1.083)(1 -(1.1729^-10))/0.1729 = 5817.13