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Savings Accounts and Annual Percentage Yield
1. Suppose that you deposit $500 in a bank that offers an annual percentage rate of 6.0% compounded annually.
a. What is your account balance after one year?
$500⋅(1.06)=$530.00
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b. What is your account balance after 10 years?
$500⋅(1.06)^10=$895.42
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c. What is the annual percentage yield of this account? (Recall that the annual percentage yield is the percentage change in the account for one year.)
6% in this case
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2. How long will it take $1000 to triple at an annual
percentage rate of 8% compounded annually?
Approximately 15 years (14.27 years to be more exact). You need to solve 3000 = 1000⋅(1.08)^x or make a table in Excel.
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3. Suppose that you deposit $500 in a bank that offers an annual percentage rate of 6.0% compounded monthly.
a. What is your account balance after one year?
$500⋅(1+0.06/12)^12≈$530.84
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b.What is your account balance after 10 years?
$500⋅(1+0.06/12)^120≈$909.70
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c. What is the annual percentage yield for this account? (Recall that the annual percentage yield is the percentage change in the account for one year, in this case in 12 months. It is usually rounded to decimal places.)
(1+0.06/12)^12-1≈6.17%
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4. Compare the accumulated balance in three accounts that all start with an initial deposit of $1000. All three accounts have an annual percentage rate of 5.5%, but the first account compounds interest annually, the second account compounds interest quarterly while the third account compounds interest monthly. Make a table that shows the accumulated balance in all three accounts for the first 3 years. Your first table will 4 rows; your second table will have 13 rows; your third table will have 37 rows.
a. Paste the first 5 rows of each table your Word document.
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b. Write a short paragraph discussing the differences between the accounts indicating which account is the largest and which is the smallest in one year. You will be looking year 1 in the first table, quarter 4 in the second table, and month 12 in the third table.
The more often the compounding, the higher the final amount (1055, 1056.14, 1056.41). Another thing you might observe is that you don't get to much more going from quarterly to monthly. The positive effect of more often compounding seems to diminish as you compound more. In fact, it is a litte surprising that compounded every day(for example) is not too much better than compounding every month.
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c. Calculate the annual percentage yield for each account. As usual please round to two decimal places. Do you want an account that compounds interest annually, quarterly or monthly (given that the annual interest rate is the same)?
The APY for the annual account is 5.5% of course. The APY for the quarterly is (1+0.055/4)^4-1≈5.61%. THe APY for the monthly is
(1+0.055/12)^12-1≈5.64%. All things being, we would prefer the account that compounds more often.
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