Find the future values of the following ordinary annuities:
a. FV of $400 paid each 6 months for 5 years at a nominal rate of 16% compounded semiannually. Do not round
intermediate calculations. Round your answer to the nearest cent.
b. FV of $200 paid each 3 months for 5 years at a nominal rate of 16% compounded quarterly. Do not round intermediate calculations. Round your answer to the nearest cent.
C. These annuities receive the same amount of cash during the 5-year period and earn interest at the same nominal
rate, yet the annuity in part b ends up larger than the one in part a. Why does this occur?
Future Value of Ordinary Annuities
Requirement (a)-FV of $400 paid each 6 months for 5 years at a nominal rate of 16% compounded semiannually.
Interest Rate (r) = 8% [16% x ½]
Number of periods (n) = 10 Years [5 Years x 2]
Half yearly payment (P) = $400
Future Value of Annuity = P x [{(1+ r) n - 1} / r]
= $400 x [{(1 + 0.08) 10 – 1} / 0.08]
= $400 x [(2.158925 – 1) / 0.08]
= $400 x [1.158925 / 0.08]
= $400 x 14.486562
= $5,794.62
Requirement (b)-FV of $200 paid each 3 months for 5 years at a nominal rate of 16% compounded quarterly
Interest Rate (r) = 4% [16% x ¼]
Number of periods (n) = 20 Years [5 Years x 4]
Quarterly payment (P) = $200
Future Value of Annuity = P x [{(1+ r) n - 1} / r]
= $200 x [{(1 + 0.04)20 – 1} / 0.04]
= $200 x [(2.191123 – 1) / 0.04]
= $200 x [1.191123 / 0.04]
= $200 x 29.778079
= $5,955.62
= $4,030.56
Requirement - (c)
Yes. The future value in Part (b) is higher than the future value computed in Part (a). It’s because the frequency of number of compounding period is more in Part-B (20 periods) as compared to the period in part-A (10 periods). Therefore, more interest is earned under Part - B and thus the Future Value will be more.
FV of $400 paid each 6 months for 5 years at a nominal rate of 16% compounded semiannually
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