Compare the results of the present value of a $6,000 ordinary annuity at 10 percent interest for ten years with the present value of a $6,000 annuity due at 10 percent interest for eleven years. Explain the difference.
To continue giving you solutions for these rather routine compound interest questions without any acknowledgement on your part that you are even learning anything from it, really serves no purpose.
I had asked you to show some effort on your part, but go no reply.
I had also explained to you in an earlier post how an "annuity due" differs from an "ordinary annuity".
Apply those concepts to each of the two parts of the above question.
To compare the results of the present value of these two annuities, we need to determine the present value of each annuity and then analyze the difference. The present value of an annuity represents the current value of a series of future cash flows, discounted at a specific interest rate.
To calculate the present value of an ordinary annuity, we can use the formula:
PV = PMT * (1 - (1 + r)^-n) / r
Where:
PV = Present value
PMT = Payment per period
r = Interest rate per period
n = Number of periods
For the first annuity, we have:
PMT = $6,000 (since it is an ordinary annuity, payments occur at the end of each period)
r = 10% per year
n = 10 years
Using these values in the formula, we can calculate the present value:
PV1 = $6,000 * (1 - (1 + 0.10)^-10) / 0.10
PV1 = $6,000 * (1 - 0.3855) / 0.10
PV1 ≈ $40,870.39
Now, let's calculate the present value of the annuity due. An annuity due implies that payments occur at the beginning of each period. As a result, the number of periods needs to be adjusted.
For the second annuity, the only difference is that the number of periods is 11 years because there is one additional payment. Using the same values as before, except for n = 11, we can calculate the present value:
PV2 = $6,000 * (1 - (1 + 0.10)^-11) / 0.10
PV2 ≈ $47,863.43
The difference between the two present values is:
PV2 - PV1 ≈ $47,863.43 - $40,870.39 ≈ $6,993.04
The difference of $6,993.04 represents the additional value in the annuity due compared to the ordinary annuity. This difference occurs because the annuity due has an extra payment at the beginning, which increases its present value.