You are given an annuity-immediate paying 10 for 10 years, then decreasing by one per year for nine years and paying one per year thereafter, forever. The annual effective rate of interest is 4%. Calculate the present value of this annuity.

124.17

Explanation:

since the first payment is immediate, then this is an annuity due:

we must divide this annuity into 3 separate parts:

1) today plus 9 years = PV = 10 x 8.43533 (PV annuity due, 4%, 10 periods) = 84.3533

2) the second group of years where annuity decreases by $1

PV year 10 = 9/1.04¹⁰ = 6.08

PV year 11 = 8/1.04¹¹ = 5.20

PV year 12 = 7/1.04¹² = 4.37

PV year 13 = 6/1.04¹³ = 3.60

PV year 14 = 5/1.04¹⁴ = 2.89

PV year 15 = 4/1.04¹⁵ = 2.22

PV year 16 = 3/1.04¹⁶ = 1.60

PV year 17 = 2/1.04¹⁷ = 1.03

sum of PVs = 26.99

3) terminal value at year 17 = 1/0.04 = 25

PV of terminal value = 25/1.04¹⁷ = 12.83

now we add the three parts = 84.3533 + 26.99 + 12.83 = 124.17

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