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An annuity pays 50 in 1 yr, 55 in 2 yrs, 60 in 3 yrs, and so on until 150 in 21 yrs. Annual eff. int rate is 5%. Find PVo.
- PVo= (50+ (5/0.05)(1-(1.05)⁻²¹ / 0.05) - (5*21 / 0.05)(1.05)⁻²¹
- = (150)(12.8212) - (2,100)(0.35894)
- = 1,923.18 - 753.77896
- = 1,169.40
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A 10 yr annuity-immed. makes monthly pmts at a rate of $60 per yr in the 1st yr, $120 per yr in the 2nd yr, $180 per yr in the 3rd yr, and so on. The nominal annual int rate is 6% compounded monthly. Find AV10.
Because it is annuity-immed that makes monthly pmts throughout each year, you multiple the PIn by (i/i¹²):
- AV₁₀= (i/i¹²) [(P1 + I/i)*Sₙ̚ - (In/i)]
- = (0.06168 / 0.06)[(60 + (60/0.06168))*((1.06168)¹⁰ - 1) - (60*10 / 0.06168]
- = 1.028[(1,032.7626)*(13.28525) - (9,727.6265)]
- = 1.028[3,992.8828]
- = 4,104.68
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The present value of a 6 yr annuity is X. The annuity pays 5 at the end of the first month, 10 at the end of the second month, and for each month thereafter the pmt increases by 5. The annual nominal int rate is 7% compounded quarterly.
Calculate X
PVo= (P1 + I/i)aₙ̚ - In/i*vⁿ
- Note: i= (1+ 0.07/4)⁴/¹² - 1 = 0.0058
- n= 12 months * 6 yrs = 72 months
- = (5 + 5/0.0058)(1-(1.0058)⁻⁷² / 0.0058) - 5*72/0.0058(1.0058)⁻⁷²
- = 867.068965(58.720576) - 62,068.9655(0.65942065)
- = 50,914.78909 - 40,929.558
- = 9,985.23
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A 10 year increasing annuity makes cont. pmts at a rate of $50 per yr in the 1st yr, $100 per yr in 2nd yr, $150 per yr in the 3rd yr, and so on. No pmts are made 10 yrs have elapsed. The annual cont. compounded int rate is 7%.
Find PVo
- PVo= 50((1-e⁻ⁿᶦ / 1-e⁻ᶦ) - ne⁻ⁿᶦ) / i
- = 50((1-e⁻¹⁰*⁰.⁰⁷ / 1-e⁻⁰.⁰⁷) - 10e⁻¹⁰*⁰.⁰⁷) / 0.07
- = 50(7.44628 - 4.96585) / 0.07
- = 50(35.43467)
- = 1,771.73
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A perpetuity-due pays 1 per yr, and its present value is 26. An increasing perp-due pays X immed., 2x in 1 yr, 3x in 2 yrs, and so on. The PV of the increasing perp-due is 4,732. What is the amount of the 8th pmt made by the increasing perp-due?
- PV = 1/d
- 26 = 1/d
- d= 1/26 = 0.03846
- For increasing perp.-due:
- = 4,732 = X / (0.03846)²
- X= 6.999
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An annuity pays 50 in 1 yr, 45 in 2 yrs, 40 in 3 yrs, and so on until a final pmt of 5 in 10 yrs. The annual eff. int rate is 5%.
Calculate the PV of the annuity.
- PVo= (P1+I/i)äₙ̚ - In/i*vⁿ
- = (50 + -5/0.05)(1-(1.05)⁻¹⁰ / 0.05) - (-5*10/0.05)(1.05)⁻¹⁰
- = -50(7.72173) + 1000(0.61391)
- = -386.0865 + 613.91325
- = 227.83
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Payable continuously, but increasing once per unit of time.
Cont. pmts of $100 in 1st yr, $110 in 2nd yr, $120 in 3rd yr, and so on for 10 yrs. i¹²= 6%. Find PVo.
PVo= (I-1)ā₁₀̚ + (I(ä₁₀̚ - 10v¹⁰) / r)
- Note: ā= (1 - v¹⁰) / r
- r= 12ln(1.005) -> 0.05985 (1)
- i= (0.06/12) -> 0.005
- d= 1 - 1.005⁻¹² -> 0.058094 (2)
- ä_10= 1-v⁻¹⁰*¹² / d -> 1-(1.005)⁻¹⁰*¹² / d -> 7.7523 (3)
- PVo= 90(1-v¹⁰ / (1)) + 10((3) - 10(1.005)⁻¹²⁰) / (1))
- = 90(1-(1.005)⁻¹²⁰ / (1)) + 10((3) - 5.4963) / (1))
- = 90(7.5249) + 10(37.69423)
- = 677.241 + 376.9423
- = 1,054.1833
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If you are asked to find AV12 where i= 9%
AV12= PVo x (1.09)¹²
[Solve for PVo using PIn method then multiple by (1+i)ⁿ]
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PIn method formula for PVo and AVt
PVo= (P1 + I/i)(1 - (1+i)⁻ⁿ / i) - (In / i)(1+i)⁻ⁿ
- AVt= (P1 + I/i)(((1+i)ⁿ - 1) / i) - (In / i)
- OR
- AVt= PVo * (1+i)ⁿ
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PIn method for increasing annuities-due:
PVo(annuity-due)= (1+i) * PVo(annuity-immed.)
(1+i) = i/d
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PIn for increasing annuity-immed. payable monthly:
PVo(annuity-immed. payable monthly)= i/iᵐ * PVo(annuity-immed)
i= (1+ iᵐ/m)ᵐ - 1
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PIn for annuity-due payable monthly:
PVo(annuity-due payable monthly)= i/d ᵐ * PVo(annuity-immed.)
- i= (1+ iᵐ/m)ᵐ - 1
- dᵐ= m * (iᵐ/m) / (1 + iᵐ/m)
- Ex: Since its monthly, m=12. Lets say iᵐ= 6%.
- Therefore,
- i=(1 + (0.06/12))¹² - 1
- = 0.06168
- d= 12*(0.06/12) / (1+ 0.06/12)
- = 12(0.005) / 1.005
- = 0.05970
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PIn annuity payable continuously
PVo(annuity payable cont.)= i/r * PVo(annuity-immed.)
- i= (1+ iᵐ/m)ᵐ - 1
- r= ln(1+i)
- Ex: Compounded monthly. iᵐ= 6%. Pays for 10 years.
- Therefore,
- i= (1 + (0.06/12))¹² - 1
- = 0.06168
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PIn for Immed. Perpetuities:
PVo= (P1 / i) + (I / i²)
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PIn for increasing perpetuity-due:
PVo= Po + (P1 / i) + (I / i²)
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PIn method for decreasing annuities-due:
PVo= (1+i) * PVo(decreasing annuitiy-due)
[Same as increasing annuity-due. Find PVo using PIn method the multiple play (1+i)]
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