Problem 6-53 Calculating Annuities Due [LO1] Suppose you are going to receive $1

Question

Problem 6-53 Calculating Annuities Due [LO1]

Suppose you are going to receive $12,500 per year for six years. The appropriate interest rate is 7.4 percent.

What is the present value of the payments if they are in the form of an ordinary annuity? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

What is the present value if the payments are an annuity due? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Suppose you plan to invest the payments for six years. What is the future value if the payments are an ordinary annuity? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Suppose you plan to invest the payments for six years. What is the future value if the payments are an annuity due? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Suppose you are going to receive $12,500 per year for six years. The appropriate interest rate is 7.4 percent.

Explanation / Answer

1. Present value of the payments in case of ordinary annuity

In the case of ordinary annuity receipt of payment is at the end of the year.

Using the formula

PVAn = R [(1- [1/(1+i)n]) /i ]

where PVAn = Present value of the annuity, R = Periodic Receipt amount, i = Rate of Interest, n = number of years

Putting the values we get

PVA6 = $12500 [ (1-[1/(1+0.074)6]) /0.074 ]

   = $12500 [ (1 - [1/1.53470775688623)]/ 0.074]

= $12500 [ 0.348410148112565 / 0.074 ]

= $12500 [4.70824524476439]

PVA6= $58853.07

2. Present value of the payments in case of annuity due

In case of annuity due the payments are received at the start of the year.

So present value of annuity in this case would be equal to present value of a 5 year ordinary annuity plus one non discounted periodic receipt. Hence-

PVA6 = R ( PVA5) + R

Now using the present value formula

PVA6 = $12500 [ (1-[1/(1+0.074)5]) /0.074 ] + $12500

= $12500 [ (1 - [1/1.42896439188662)]/ 0.074] + $12500

= $12500 [ 0.300192499072894 / 0.074 ] + $12500

= $12500 [ 4.05665539287695] + $12500

PVA6 = $50708.19 + $12500 = $63208.19

3. Future value of the payments in case of ordinary annuity

Using the future value formula for annuity

FVAn = R ( [ (1+i)n - 1] / i )

Putting the values in formula we get

FVA6 = $12500 ( [ (1+0.074)6 - 1] / 0.074 )

= $12500 ( [1.53470775688623 - 1] / 0.074 )

= $12500 ( 0.53470775688623/ 0.074)

= $12500 ( 7.22578049846263

FVA6= $90322.26

4. Future value of the payments in case of annuity due

In case of annuity due the future value of 6 year annuity would be equal to the future value of a comparable 6 year annuity compounded for one more year. Hence-

Future value = Future value in case of ordinary annuity x ( 1+i)

Taking value of FVA6 from the point 3 above

we get FVA (in case of annuity due) = $90322.26 x (1.074) = $97006.10

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