Question

Present Value of an Annuity Find the present value of the following ordinary annuities. Do not round intermediate calculations. 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 10%.

$ $300 per year for 5 years at 5%.

$ $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.

Present value of $600 per year for 10 years at 10%:

$ Present value of $300 per year for 5 years at 5%:

$ Present value of $600 per year for 5 years at 0%:

Answer 1

FV of Annuity Due:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time.
FV of Annuity = (1+r) * CF [ (1+r)^n - 1 ] / r
r - Int rate per period
n - No. of periods

CFs are made at the begining of the period

Part A:

Particulars	Amount
Cash Flow	$         600.00
Int Rate	10.000%
Periods	10

FV of Annuity Due = ( 1+ r) [ Cash Flow * [ [ ( 1 + r )^n ] - 1 ] /r ]
= ( 1 + 0.1 ) * [600 * [ [(1+0.1)^10] - 1 ] / 0.1 ]
= ( 1.1 ) * [600 * [ [( 1.1 ) ^ 10 ] - 1 ] / 0.1 ]
= ( 1.1 ) * [600 * [ [ 2.5937 ] - 1 ] / 0.1 ]
= ( 1.1 ) * [ $ 9562.45 ]
= $ 10518.7

Part B:

Particulars	Amount
Cash Flow	$         300.00
Int Rate	5.000%
Periods	5

FV of Annuity Due = ( 1+ r) [ Cash Flow * [ [ ( 1 + r )^n ] - 1 ] /r ]
= ( 1 + 0.05 ) * [300 * [ [(1+0.05)^5] - 1 ] / 0.05 ]
= ( 1.05 ) * [300 * [ [( 1.05 ) ^ 5 ] - 1 ] / 0.05 ]
= ( 1.05 ) * [300 * [ [ 1.2763 ] - 1 ] / 0.05 ]
= ( 1.05 ) * [ $ 1657.69 ]
= $ 1740.57

Part C:

If Int Rate is 0, FV of Annuity DUe is Normal Value

= $ 600 * 5

= $ 3000

PV of Annuity Due:
Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

PV of Annuity Due = Cash Flow + [ Cash Flow * [ 1 - [(1+r)^-(n-1)]] /r ]
r - Int rate per period
n - No. of periods

CFs are made at the begining of the period

Part A:

Particulars	Amount
Cash Flow	$               600.00
Int Rate	10.000%
Periods	10

PV of Annuity Due = [ Cash Flow + Cash Flow * [ 1 - [(1+r)^-(n-1)]] / r ]
= [ $ 600 + $ 600 * [ 1 - [(1+0.1)^-9] ] / 0.1 ]
= [ $ 600 + $ 600 * [ 1 - [(1.1)^-9] ] / 0.1 ]
= [ $ 600 + $ 600 * [ 1 - [0.4241] ] / 0.1 ]
= [ $ 600 + $ 600 * [0.5759] ] / 0.1 ]
= [ $ 600 + $ 3455.41 ]
= $ 4055.41

Part B:

Particulars	Amount
Cash Flow	$               300.00
Int Rate	5.000%
Periods	5

PV of Annuity Due = [ Cash Flow + Cash Flow * [ 1 - [(1+r)^-(n-1)]] / r ]
= [ $ 300 + $ 300 * [ 1 - [(1+0.05)^-4] ] / 0.05 ]
= [ $ 300 + $ 300 * [ 1 - [(1.05)^-4] ] / 0.05 ]
= [ $ 300 + $ 300 * [ 1 - [0.8227] ] / 0.05 ]
= [ $ 300 + $ 300 * [0.1773] ] / 0.05 ]
= [ $ 300 + $ 1063.79 ]
= $ 1363.79

Part C:

If disc Rate is 0%, PV is Sum of CFs.

= $ 600 * 5

= $ 3000