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You are set to receive an annual payment of $12,400 per year for the next 20 years. Assume the interest rate is 7.3 percent. How much more are the payments worth if they are received at the beginning of the year rather than the end of the year?

Question

You are set to receive an annual payment of $12,400 per year for the next 20 years. Assume the interest rate is 7.3 percent. How much more are the payments worth if they are received at the beginning of the year rather than the end of the year?

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Solution

To answer this question, we need to understand the concept of the present value of annuity due and ordinary annuity.

  1. Calculate the present value of an ordinary annuity: An ordinary annuity assumes payments are made at the end of each period. The formula for the present value of an ordinary annuity is:

    PVoa = PMT * [(1 - (1 + r)^-n) / r]

    Where: PVoa = Present Value of an ordinary annuity PMT = Periodic payment, which is $12,400 r = Interest rate per period, which is 7.3% or 0.073 n = Number of periods, which is 20 years

    Substituting the values into the formula, we get:

    PVoa = 12,400[(1(1+0.073)20)/0.073]PVoa=12,400 * [(1 - (1 + 0.073)^-20) / 0.073] PVoa = 12,400 * [9.81815] PVoa = $121,645.06

  2. Calculate the present value of an annuity due: An annuity due assumes payments are made at the beginning of each period. The present value of an annuity due is simply (1 + r) times the present value of an ordinary annuity.

    PVad = PVoa * (1 + r) PVad = 121,645.06(1+0.073)PVad=121,645.06 * (1 + 0.073) PVad = 130,593.91

  3. Calculate the difference: The difference between the present value of an annuity due and an ordinary annuity is the additional value of receiving payments at the beginning of the year rather than the end.

    Difference = PVad - PVoa Difference = 130,593.91130,593.91 - 121,645.06 Difference = $8,948.85

So, the payments are worth $8,948.85 more if they are received at the beginning of the year rather than the end of the year.

This problem has been solved

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