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Find the future values of the following ordinary annuities: FV of $200 paid each 6 months...

Find the future values of the following ordinary annuities:

  1. FV of $200 paid each 6 months for 5 years at a nominal rate of 5% compounded semiannually. Do not round intermediate calculations. Round your answer to the nearest cent.
    $  

  2. FV of $100 paid each 3 months for 5 years at a nominal rate of 5% compounded quarterly. Do not round intermediate calculations. Round your answer to the nearest cent.
    $  

  3. 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?

Solutions

Expert Solution

FV of Annuity :

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here deposits are made at the end of the period. FV of annuity is future value of cash flows deposited at regular intervals grown at specified int rate or Growth rate to future date.

FV of Annuity = CF [ (1+r)^n - 1 ] / r
r - Int rate per period
n - No. of periods

Part A:

Particulars Amount
Cash Flow $               200.00
Int Rate 2.500%
Periods 10

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 200 * [ [ ( 1 + 0.025 ) ^ 10 ] - 1 ] / 0.025
= $ 200 * [ [ ( 1.025 ) ^ 10 ] - 1 ] / 0.025
= $ 200 * [ [1.2801] - 1 ] / 0.025
= $ 200 * [0.2801] /0.025
= $ 2240.68

Part B:

Particulars Amount
Cash Flow $               100.00
Int Rate 1.250%
Periods 20

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 100 * [ [ ( 1 + 0.0125 ) ^ 20 ] - 1 ] / 0.0125
= $ 100 * [ [ ( 1.0125 ) ^ 20 ] - 1 ] / 0.0125
= $ 100 * [ [1.282] - 1 ] / 0.0125
= $ 100 * [0.282] /0.0125
= $ 2256.3

Part C:

For the difference between Part B & part A, there are two reasons.

1. In Part B, cash flows are deposited early.

2. As the frequency of compounding is increased, int on int is more compared toPart A.

In Part A, we have deposited $ 200 at theend of each 6 months.However inPart B we have deposited $ 100 at the end of each 3 Months. I.e $ 100 is deposited for a period of 3 Months extra compared to Part A.

Another thing, Amounts deposited are compunded twice per year in PartA and 4 times per year in Part B. As the frequency of compoundingis more in Part B, We will receive more Int on int in PartB compared to Part A.


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