Present Value of an Annuity Find the present value of the following ordinary ann

Question

Present Value of an Annuity

Find the present value of the following ordinary annuities. Round your answers to the nearest cent. (Notes: If you are using a financial calculator, you can enter the known values and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can "override" the variable that changes by simply entering a new value for it and then pressing the key for the unknown variable to obtain the second answer. This procedure can be used in many situations, to see how changes in input variables affect the output variable. Also, note that you can leave values in the TVM register, switch to Begin Mode, press FV, and find the FV of the annuity due.)

$600 per year for 10 years at 14%.
$   

$300 per year for 5 years at 7%.
$   

$600 per year for 5 years at 0%.
$  

Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

$600 per year for 10 years at 14%.
$   

$300 per year for 5 years at 7%.
$   

$600 per year for 5 years at 0%.
$  

Explanation / Answer

The present value is of each cash flow is determined by the present value from the future value formula and then added together for all the cash flows. The future value formula is:

FV = PV * ( 1 + i )^t

Therefore, the present value is:

PV = FV / ( 1 + i )^t

which can also be expressed as:

PV = FV * [ 1 / ( 1 + i ) ]^t

Therefore, the present value of all ten payments is:

PV = $200 / 1.08 + $200 / 1.08^2 + $200 / 1.08^3 + ... + $200 / 1.08^10

It's rather laborious to do all ten calculations but you'll notice that it's a geometric sequence. To make it fit in the summation of a finite geometric sequence formula, factor out $200 / 1.08 giving you:

PV = [ $200 / 1.08 ] * ( 1 + 1 / 1.08 + 1 / 1.08^2 + ... + 1 / 1.08^9 )

Apply the summation of a geometric sequence and you have:

PV = [ $200 / 1.08 ] * ( 1 - 1 / 1.08^10 ) / ( 1 - 1 / 1.08 )

This is usable as is, but if you simplify it further, it'll be more like the equations they expect you to memorize. You can simplify it to:

PV = [ $200 / 1.08 ] * ( 1 / 1.08^10 - 1 ) / [ 1 / 1.08 - 1 ]
.:
PV = $200 * ( 1 / 1.08^10 - 1 ) / [ 1 - 1.08 ]
PV = $1342.02

Then you just change the numbers for the second question giving you:

PV = $100 * ( 1 / 1.04^5 - 1 ) / ( 1 - 1.04 )
.:
PV = $445.18

The third is a trick question to see if you can see beyond just the formula ( or calculator ), cause you get a division by zero. But if you back up to the original formula where you are adding all the present values of the future value formula, you have:

PV = $200 / 1.00 + $200 / 1.00^2 + $200 / 1.00^3 + ... + $200 / 1.00^5

So you can see it's just:

PV = 5 * $200
.:
PV = $1,000

For the fourth, if you rework it so that the payments are at the beginning, you just don't have to factor out anything to meet the summation of a finite geometric sequence formula, therefore you have:

PV = $200 + $200 / 1.08 + $200 / 1.08^2 + ... + $200 / 1.08^9
.:
PV = $200 * ( 1 - 1 / 1.08^10 ) / ( 1 - 1 / 1.08 )
PV = $1,449.38

The fifth is then:

PV = $100 * ( 1 - 1 / 1.04^5 ) / ( 1 - 1 / 1.04 )
.:
PV = $462.99

The sixth is then:

PV = 5 * $200
.:
PV = $1,000

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.