a. Show (with a time line, for example) that the perpetuity in 4.3a. is exactly
ID: 1167714 • Letter: A
Question
a. Show (with a time line, for example) that the perpetuity in 4.3a. is exactly the same as the sum of the annuities and perpetuities in 4.3b. to 4.3d.
4.3 a. present value of a perpetuity (also called a perpetual annuity) of $50 received each year at the end of each year
4.3 b. present value of an annuity of $50 received at the end of each year for 5 years
4.3 c. present value of an annuity of $50 received at the end of each year for 10 years, with the first payment to be received at the end of the 6th year
4.3 d. present value of a perpetuity of $50, with the first payment received at the end of the 16th year
b. Show that their present values add up to the same amount.
Explanation / Answer
(a) If we assume that the discount rate is 8% the Present value will be $50 / 0.08 = $625
(b) $50 x 3.9927 (as per PVIFA table for 8% discount rate and 5 periods) = $199.64
50*((1-(1/(1+r)^t)/r) = 50(1+1/(1.08)5)/0.08
(c) PV of annuity at year-6
= $50 x present value interest factor of annuity
((PV interest factor of Annuity), 10 years or periods, discount rate of 8%) = $50 x 6.7101 7.1390
(from PVIFA table) = $335.51
Present Value of this amount at year 0 = $335.51 / (1.08)^5 = $211.43
(d) Present Value of perpetuity at year 16 = $50 / 0.08 = $625 Present Value of this amount discounted to year 0 = $625 / (1.08)^16 = $182.43
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