Annuities
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
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