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What is the approximate amount that should be invested today at an annual rate of 10% compounded semi-annually so that an amount of $1000 is received at the end of the first year?
- a)$830
- b)$890
- c)$910
- d)$970
- e)$990
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
What is the approximate amount that should be invested today at an ann...
Step 1: Question statement and Inferences
This question tests your proficiency in solving problems related to compound interest. Please note the formula used for compound interest problems:
Here A = $1000; r = 10% compounded semi-annually; t = compounding time intervals
Step 2: Finding required values
r is given as annual rate of interest. Since we know that the interest is compounded semi-annually. So the rate for calculation = r/2 = 10/2 = 5%. Also, the time intervals will be 2 since the compounding will be done 2 times till the end of first year. So t = 2.
Step 3: Calculating the final answer
Substituting these values we get:
Calculating and approximating (1.05)2, we get:
1000 = p × 1.1
Thus, p = 1000/1.1 = 909.09
Thus the closest answer is $910.
Answer: Option (C)
Most Upvoted Answer
What is the approximate amount that should be invested today at an ann...
Understanding the Problem
To determine how much needs to be invested today to receive $1000 at the end of one year, we need to account for the effect of compound interest. We are given:
- Future Value (FV) = $1000
- Annual Interest Rate (r) = 10%
- Compounding Frequency = Semi-Annually (2 times a year)
Calculating the Effective Interest Rate
The effective interest rate per compounding period can be calculated as:
- Effective Rate per period = r / number of periods
- Effective Rate per period = 10% / 2 = 5% or 0.05
Number of Compounding Periods
Since we are looking at one year with semi-annual compounding:
- Total Compounding Periods (n) = 2 (for one year)
Present Value Formula
To find the present value (PV), we can rearrange the future value formula:
- PV = FV / (1 + i)^n
- Where i = effective interest rate per period and n = total compounding periods.
Substituting the Values
Substituting the values we have:
- PV = 1000 / (1 + 0.05)^2
- PV = 1000 / (1.05)^2
- PV = 1000 / 1.1025
- PV ≈ 907.03
Conclusion
The closest amount to invest today, rounded to the nearest dollar, is approximately $910. Thus, the correct answer is option 'C'.
To determine how much needs to be invested today to receive $1000 at the end of one year, we need to account for the effect of compound interest. We are given:
- Future Value (FV) = $1000
- Annual Interest Rate (r) = 10%
- Compounding Frequency = Semi-Annually (2 times a year)
Calculating the Effective Interest Rate
The effective interest rate per compounding period can be calculated as:
- Effective Rate per period = r / number of periods
- Effective Rate per period = 10% / 2 = 5% or 0.05
Number of Compounding Periods
Since we are looking at one year with semi-annual compounding:
- Total Compounding Periods (n) = 2 (for one year)
Present Value Formula
To find the present value (PV), we can rearrange the future value formula:
- PV = FV / (1 + i)^n
- Where i = effective interest rate per period and n = total compounding periods.
Substituting the Values
Substituting the values we have:
- PV = 1000 / (1 + 0.05)^2
- PV = 1000 / (1.05)^2
- PV = 1000 / 1.1025
- PV ≈ 907.03
Conclusion
The closest amount to invest today, rounded to the nearest dollar, is approximately $910. Thus, the correct answer is option 'C'.
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