Suppose the interest rate is 7.7% APR with monthly compounding. What is the present value of an annuity that pays $110 every six months for five years?
EAR = [1 + (APR/m)]m - 1; m = no. of compounding periods in a year
= [1 + (0.077/12)]12 - 1 = 1.0798 - 1 = 0.0798, or 7.98%
Now, we need to find the APR with semi-annual compounding;
So,
APR = m * [(1 + EAR)1/m - 1]
= 2 * [(1 + 0.0798)1/2 - 1]
= 2 * [1.0391 - 1]
= 2 * 0.0391 = 0.0782, or 7.82%
PV of Annuity = Periodic Payment * [{1 - (1 + r)-n} / r]
= $115 * [{1 - (1 + 0.0782/2)-(5*2)} / (0.0782/2)]
= $115 * [0.3187 / 0.0391]
= $115 * 8.1464
= $936.84
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