Annuities

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Basic Formulas:
Present value interest factor of an annuity,
PVIFA(k,n) = [ 1 – ( 1 + k )-n ] / k
Present value interest factor of a perpetuity,

PVIFA(k; ∞) = 1 / k

Future value interest factor of an annuity,
FVIFA(k,n) = [ ( 1 + k )n –1 ] / k
Annuities Due, payments at start of period,
PVIFADue(k,n) = PVIFA(k,n) * ( 1 + k )
FVIFADue(k,n) = FVIFA(k,n) * ( 1 + k )
Where: k is the effective discount rate per payment period and n is the number of payments. Note: to use these formulas, the payments must be equal and at regular intervals and k cannot vary.

Finding Effective rates:
As mentioned in last week’s tutorial, find the future value of $1 over the given period and subtract off that $1 principal to find out how much interest accumulated over the period. Example:

Convert 8% with semi-annual compounding to an effective monthly rate. Note that the stated rate really means 4% per six months, so a monthly rate is 1/6 of a period. k1/12 = ( 1 + k1/2 ) 1/6 – 1 = 0.6558% per month Quick Derivation, in case you forget your formula sheet.

At a fixed interest rate, if you withdraw all of the interest every period leaving the principal untouched, you can withdraw the same amount per period forever. This is perpetuity. Therefore the present value of a perpetuity of $1 per period, is the amount that you must deposit to generate $1 of return per period.

PVIFA(k; ∞) * k = 1

PVIFA(k; ∞) = 1 / k

An annuity can be viewed as gaining a perpetuity and losing it after the end of the annuity payments. PVIFA(k;n) = PVIFA(k; ∞) – PV(n)[ PVIFA(k; ∞)] PVIFA(k;n) = 1 / k – [ 1 / k * (1 + k)-n ]

PVIFA(k;n) = [ 1 - (1 + k)-n ] / k
Note that the PVIFA(k;n) cannot exceed 1 / k
To find the future value of an annuity, simply find the present value and future value the result. FVIFA(k;n) = ([ 1 - (1 + k)-n ] / k) * ( 1+ k )n
FVIFA(k;n) = [( 1+ k )n – 1 ] / k
Sample Problem 1.
Bob wants to be a millionaire. If Bob has just turned 35 how much would he have to invest at the end of each year in order to have $1 million on his 65th birthday, assuming that he is able to get a return on investment of 10% per year? How much would he have to invest if he made the investment at the end of each month instead? Future Value = $1,000,000k = .10n = 30

FV = CF x FVIFA(0.10, 30)
CF = FV / FVIFA(0.10, 30)
CF = 1,000,000 / 164.5 = $6079.25
Therefore he would have to invest $6,079.25 per year to end up with one million dollars on his 65th birthday.

With monthly investment:
Future Value = $1,000,000n = 30x12 = 360
k = (1 + .10)1/12 –1 = 0.007974
CF = FV / FVIFA(0. 007974, 360)
CF = 484.77
Therefore he would have to invest $484.77 at the end of every month to end up with one million dollars on his 65th birthday. Sample Problem 2.
If Bob gets his $1 million, how much could he afford to withdraw at the end of each month through his 90th birthday that would leave him with a balance of zero at that time?

Present Value = $1 million
k = 0.007974
n = 25x12= 300
CF = PV / PVIFA(0. 007974, 300)
CF = $1,000,000 / 113.8
CF = $8.784.96
Therefore Bob could withdraw $8.784.96 at the end of each month through his 90th birthday. Sample Problem 3.
How expensive a house can you afford to purchase if you have $23,000 for a down payment and you can afford to pay $1,800 per month on a mortgage, if the mortgage rate is 9% per annum with semi-annual compounding and a 20 year amortization period? If you can afford $900 every two weeks instead, what is the maximum you can pay for a house? What you can afford = down payment + PVmortgage.

CF = 1,800n = 20x12 = 240 months
k = (1 + 0.045)1/6 – 1 = 0.0073631 per month
PV = CF x PVIFA(0.0073631, 240)
PV = $202,432
Therefore you can...
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