. Compute the amount of compound interest earned in 1 year for an investment of $1,000,000 with a nominal interest rate of 8% compounded quarterly.
Compute the Annual Percentage Yield (APY) for the investment in the previous question. (Round APY to the nearest hundredths of a percent.)
1. P = Po*(1+r)^n
r = (8%/4)/100% = 0.02 = Quarterly %
rate expressed as a decimal.
n = 4Comp /yr * 1yr = 4 Compounding
periods.
P = 1,000,000(1.02)^4 = 1,082,432.16
Int. = P-Po
2. APY = (I/Po)*100% =
To calculate the amount of compound interest earned in 1 year, we will use the formula:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual nominal interest rate (expressed as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, the principal amount (P) is $1,000,000, the annual nominal interest rate (r) is 8% (which is 0.08 as a decimal), the compounding period (n) is quarterly (which means it compounds 4 times a year), and the number of years (t) is 1.
Plugging these values into the formula:
A = $1,000,000 * (1 + 0.08/4)^(4*1)
= $1,000,000 * (1 + 0.02)^4
= $1,000,000 * (1.02)^4
≈ $1,082,432.64
To calculate the compound interest earned, we subtract the principal amount:
Compound Interest = A - P
= $1,082,432.64 - $1,000,000
≈ $82,432.64
Therefore, the amount of compound interest earned in 1 year for the $1,000,000 investment with a nominal interest rate of 8% compounded quarterly is approximately $82,432.64.
Now, let's calculate the Annual Percentage Yield (APY) for the investment. APY is a measure that takes into account the effects of compounding.
The formula to calculate APY is:
APY = (1 + r/n)^n - 1
where:
r = the annual nominal interest rate (as a decimal)
n = the number of times that interest is compounded per year
In this case, the annual nominal interest rate (r) is 8% (0.08 as a decimal) and the compounding period (n) is quarterly (4 times a year).
Plugging these values into the formula:
APY = (1 + 0.08/4)^4 - 1
= (1 + 0.02)^4 - 1
= (1.02)^4 - 1
≈ 0.082
≈ 8.2%
Therefore, the Annual Percentage Yield (APY) for the investment is approximately 8.2%.