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
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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
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<em>Alternate solution</em>
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
