math compounded quarterly

Question

You deposit $ 10, 000 in an account that pays 5.5 % interest compounded quarterly.

A. Find the future value after one year.

B. Use the future value formula for simple interest to determine the effective annual yield.

Answer 1

You deposit $ 10, 000 in an account that pays 5.5 % interest compounded quarterly.

A. Find the future value after one year.

B. Use the future value formula for simple interest to determine the effective annual yield.

A. Find the Future Value after one year

1. First, we need the Future Value formula:
2. FV = PV (1 + r) ⁿ
3. FV = "future value"
4. PV = "present value" (this is often the principal (aka the amount invested), but it basically refers to whatever amount is currently in the bank account)
5. r = periodic interest rate (TAKE NOTE: this is periodic interest rate, not annual interest rate. In this problem, you are given the annual interest rate. In order to determine the amount of interest paid quarterly, we must divide the annual rate by the amount of periods (in this case, there are four quarterly periods in one year) EX: 5.5% = .055/4 = .01375 or 1.375%)
6. n = number of periods when interest was earned (if your interest compounds quarterly, this means after one year, you will have earned interest for 4 periods)
7. Next, we need to identify our variables. According to the definitions above and the information given by the problem:
8. FV = unknown (this is what we're trying to find out, the variable)
9. PV = 10,000 (because we have not gained any interest or added any other money, the principal is all we currently have in our account)
10. r = 5.5% / 4 = .055/4 = .01375 or 1.375% (don't forget to put this number in decimal form when using it in the equation!)
11. n = 4 (as discussed in the definition)
12. Now we can plug these variables into our future value equation:
13. FV = 10,000 ( 1 + .01375 ) ⁴
14. Finally, we simplify utilizing the order of operations (PEMDAS - parentheses, exponents, multiplication/division left to right, addition/subtraction left to right):
15. FV = 10,000 ( 1 + .01375 ) ⁴
16. FV = 10,000 ( 1.01375 ) ⁴ ... (parentheses are kept to clarify that 10,000 is being multiplied by 1.01375⁴)
17. FV = 10,000 ( 1.056144809...) ... (the decimal may continue on, but we don't need all the digits because we are working with a money problem, and will only include two decimal points)
18. FV = 10,561.44809... ...(now we round to the nearest cent)
19. FV = $10,561.45
20. The future value after one year of quarterly compounded interest on $10,000 at 5.5% is $10,561.45

B. Use the future value formula for simple interest to determine the effective annual yield.

1. First, we need the "Effective Yield" or "Annual Percentage Yield (APY)" Formula:
2. APY = [ 1 + ( r / n ) ] ⁿ - 1
3. APY = the Annual Percentage Yield (or Effective Yield) of the given account
4. r = a nominal rate (this just means the stated interest rate, it doesn't take inflation into account; we also need to divide this rate by the amount of periods, just like with the final value formula)
5. n = the number of payments received annually (because interest is compounded quarterly, we receive 4 payments annually, or during the year)
6. Next, we need to identify our variables. We've already done this for part (A), now we just need to apply them to this specific formula.
7. APY = unknown (the value we're trying to find, the variable)
8. r = 5.5% or .055 (remember to keep this in decimal form when simplifying the equation)
9. n = 4 (we receive payments quarterly, or 4 times per year, as discussed above)
10. Finally, we plug in our variables and simplify the equation.
11. APY = [ 1 + ( .055 / 4 ) ] ⁴ - 1
12. APY = [ 1 + ( .01375 ) ] ⁴ - 1
13. APY = [ 1.01375 ] ⁴ - 1
14. APY = [1.056144809...] - 1
15. APY = .056144809... ...(now convert this to a percentage and round to the nearest hundredth, or second decimal place)
16. APY = 5.61%
17. The effective annual yield for this account is 5.61%

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