Question

5.23

Find the amount to which $400 will grow under each of these conditions:

1. 12% compounded annually for 8 years. Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

2. 12% compounded semiannually for 8 years. Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

3. 12% compounded quarterly for 8 years. Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

4. 12% compounded monthly for 8 years. Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

5. 12% compounded daily for 8 years. Assume 365-days in a year. Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

6. Why does the observed pattern of FVs occur?
   -Select-The future values increase because as compounding periods per year increase, interest is earned on interest less frequently.The future values decrease because as compounding periods per year increase, interest is earned on interest more frequently.The future values increase because as compounding periods per year increase, interest is earned on interest more frequently.The future values increase because as compounding periods per year decrease, interest is earned on interest more frequently.The future values decrease because as compounding periods per year decrease, interest is earned on interest more frequently.Item 6

Answer 1

This question talks about a single cash flow of $400 and hence requires application of basic time value of money function, which is mathematically represented as:

FV = PV * (1 + r)ⁿ

PV for each of the parts = $400

a) For this question, compounding is annual,

n = 8, r = 12%

FV = $400 * (1 + 12%)⁸ = $990.4

b) For this question, compounding is semi-annual,

n = 8 * 2 = 16 semi-annual periods , r = 12%/2 = 6% (semi-annually)

FV = $400 * (1 + 6%)¹⁶ = $1,016.1

c) For this question, compounding is quarterly,

n = 8 * 4 = 32 quarterly periods , r = 12%/4 = 3% (quarterly)

FV = $400 * (1 + 3%)³² = $1,030.0

d) For this question, compounding is monthly,

n = 8 * 12 = 96 monthly periods , r = 12%/12 = 1% (monthly)

FV = $400 * (1 + 1%)⁹⁶ = $1,039.7

e) For this question, compounding is daily,

n = 8 * 365 = 2920 days, r = 12%/365 = 0.0329% (daily)

FV = $400 * (1 + 0.0329%)²⁹²⁰ = $1,044.5

f) The future values increase because as compounding periods per year increase, interest is earned on interest more frequently.

Based on the answers above, it is clear that as the number of compounding period increases, the FV of the amount increases. However, the increase becomes smaller with each higher frequency (like change in FV from annual to semi-annual compounding is higher than change in FV from monthly to daily).