Annuity Equations

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Annuity-Immediate

Pays $5 at the end of each year
Annual eff. int rate of 4%
Pays for 8 years
Find PVo

PVo= 5(1-(1.04)⁻⁸) / 0.04)
Annuity-Immediate

Pays $5 at the end of each year
Annual eff. int rate of 4%
Pays for 8 years
Find AV8

AV8= 5((1.04)⁸ - 1) / 0.04)
Perpetuity-Immediate

Pay $5 at the end of each year
Annual eff. int rate of 6%
Pays forever
Find PVo

PVo= 5(1 / 0.06)
Annuity-Due

Pays $5 at the beginning of each year
Annual eff. int rate of 4%
Pays for 8 years
Find PVo

PVo= 5(1 - (1/1.04)⁻⁸) / (0.04 / 1.04)
Annuity-Due

Pays $5 at the beginning of each year
Annual eff. int rate of 4%
Pays for 8 years
Find AVt

Av8= 5((1.04)⁸ - 1 / 0.04)
Perpetuity-Due

Pays $5 at the beginning of each year
Annual eff. int rate of 6%
Pays forever
Find PVo

PVo= 5(1.06 / 0.06)
Deferred Annuity-Immediate

Pays $1000/yr
Annual eff. int rate is 6%
Pays for 15 yrs
Deferral period of 5 yrs (=k)
Find PVo

PVo= 1000(1.06)⁻⁵*(1 - (1.06)¹⁵ / 0.06)
Deferred Annuity-Immediate

Pays $100/month
Monthly eff. int rate is 9%
Pays for 10 years
Deferral period of 3 years (=k)
Find PVo

i¹²= 9% → 0.09/12 = 0.0075

PVo= 100(1.0075)⁻³ * (1 - (1.0075)¹⁰ / 0.0075)
Deferred Perpetuity-Immediate

Pays $100/month
Monthly eff. int rate is 9%
Pays forever
Deferral period of 3 yrs
Find PVo

i¹²= 9% → 0.09/12 = 0.0075

PVo= 100(1.0075)³⁶ * (1/0.0075)
Deferred Annuity-Due

Pays $300/quarter
Quarterly eff. int rate is 8%
Pays for 15 years
Deferral period of 5 years
Find PVo
- i⁴= 8% → 0.08/4= 0.02
- k= 5 yrs → 20 qrts and n= 15 yrs → 60 qrts
- d= i/(1+i) → 0.02/1.02= 0.01960
PVo= 300(1.02)⁻²⁰ * (1 - (1.02)⁶⁰ / 0.01960)
Annuity-Immediate-Paid in Monthly Increments

Pays $36/yr
Monthly eff. int rate is 12%
Paid in monthly increments for 10 years
Find PVo

PVo= 36(1 - (1.01)⁻¹⁰*¹² / 0.12)
Perpetuity-Immediate-Paid In Monthly Increments Forever

Pays $36/yr
Monthly eff. int rate is 12%
Paid in monthly increments for 10 years
Find PVo

PVo= 36(1/0.12)
Annuity-Due-Paid In Monthly Increments

Pays $36/yr
Monthly eff. int. rate of 12%
Paid in monthly increments for 10 years
Find AV10
- i¹²= 12% → 0.12/12= 0.01
- d¹²= 12[1-(1.01)⁻¹]= 0.1188
AV10= 36((1.01)¹⁰*¹² - 1) / 0.1188
Continuously Payable Annuity

Pays $36/yr monthly eff. int rate of 12%
Makes continuously pmts for 10 years
Find PVo
- i¹²= 12% → 0.12/12= 0.01
- r= 12ln(1.01)= 0.1194
PVo= 36(1 - (1.01)⁻¹⁰*¹² / 0.1194)

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Annuity Equations

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Updated

2024-10-02T14:13:07Z