Answer:
a. = $29,503.95
b. = $75,302.15
c. = $31,274.18
d. = $79,820.27
Explanation:
A financial product that gives an investor a fixed stream of payments over period of time is called an annuity.
The two types of annuity are in the question. The first is an ordinary annuity while second is annuity due.
An ordinary annuity gives investors payments at the end of each time period. The formula that is used to calculate the Present Value (PV) of ordinary annuity is:
PVo = P × [{1 - [1 ÷ (1+r)]^n} ÷ r] ....................................... (1)
Where
PVo = Present value of an ordinary annuity
P = periodical payment
r = interest rate
n = number of periods
An annuity due gives investors payments at the beginning of each time period. The formula is used to calculate the Present Value (PV) of annuity due is:
PVd = P × [{1 - [1 ÷ (1+r)]^n} ÷ r] × (1+r) .......................................... (2)
Where
PVd = Present value of an annuity due.
P, r and n are already described above.
Question "a"
This is an ordinary annual annuity, and equation (1) will be used to calculate the PV as follows:
PVo = P × [{1 - [1 ÷ (1+r)]^n} ÷ r]
Where,
P = yearly payment = $6,000
r = interest rate = 6% = 0.06
n = number of years = 6
PVo = $6,000 × [{1 - [1 ÷ (1+0.06)]^6} ÷ 0.06]
= $29,503.95
Question "b"
This is an ordinary quarterly annuity, and equation (1) will also be used to calculate the PV as follows:
PVo = P × [{1 - [1 ÷ (1+r)]^n} ÷ r]
Where,
P = quarterly payment = $6,000
r = interest rate = 6% = 0.06
n = number of quarters = 6 × 4 = 24
PVo = $6,000 × [{1 - [1 ÷ (1+0.06)]^24} ÷ 0.06]
= $75,302.15
Question "c"
This is an annual annuity due, and equation (2) will be used to calculate the PV as follows:
PVd = P × [{1 - [1 ÷ (1+r)]^n} ÷ r] × (1+r)
Where,
P = yearly payment = $6,000
r = interest rate = 6% = 0.06
n = number of years = 6
PVd = $6,000 × [{1 - [1 ÷ (1+0.06)]^6} ÷ 0.06] × (1+0.06)
= $31,274.18
Question "d"
This is a quarterly annuity due, and equation (2) will be used to calculate the PV as follows:
PVd = P × [{1 - [1 ÷ (1+r)]^n} ÷ r] × (1+r)
Where,
P = yearly payment = $6,000
r = interest rate = 6% = 0.06
n = number of years = 6 × 4 = 24
PVd = $6,000 × [{1 - [1 ÷ (1+0.06)]^24} ÷ 0.06] × (1+0.06)
= $79,820.27
All the best!
