A bond purchased for $2100 earns interest of $115 every 6 months. The bond is sold at the end of the third year for $2000. What was the annual effective rate of return earned on this bond?
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Cash Flow TablePeriodCash Flow ($)0-21001115211531154115511562115
The next step is to calculate the net present value (NPV) of the cash flows at various interest rates to see which rate brings the NPV to zero. That rate is the effective rate for the cash flow. The formula for present value is:
P=F(1+i)nP=F(1+i)n where:
P is the present valueF is the future valuei is the raten is number of periods
A choice table for interest rates is created to help determine where the NPV of the cash flows is zero. A look at the approximate interest rate can be made by noting that every 6 months, an amount of $115 is paid on an invested sum of $2100. That gives us an approximation of around 5%. So the table only needs to calculate rates up to 5%. For example, the calculation for interest rate of 2% is:
NPV=−$2,100(1+2/100)0++$115(1+2/100)1++$115(1+2/100)2++$115(1+2/100)3++$115(1+2/100)4++$115(1+2/100)5++$2,115(1+2/100)6NPV=−$2,100(1+2/100)0++$115(1+2/100)1++$115(1+2/100)2++$115(1+2/100)3++$115(1+2/100)4++$115(1+2/100)5++$2,115(1+2/100)6
PeriodNPV0$590.000.5$516.541$446.111.5$378.582$313.832.5$251.733$192.183.5$135.064$80.264.5$27.704.6$17.454.7$7.284.77$0.214.772$0.014.8- $2.814.9- $12.815$22.73
The interest rate that brings the NPV to zero is 4.772%, but this is for 12 semiannual periods. Double that to get the annual interest rate of 9.54%.
(These calculations are made faster if Excel or other financial calculator is used. Use the Excel function called IRR, for example.)
The next step is to calculate the net present value (NPV) of the cash flows at various interest rates to see which rate brings the NPV to zero. That rate is the effective rate for the cash flow. The formula for present value is:
P=F(1+i)nP=F(1+i)n where:
P is the present valueF is the future valuei is the raten is number of periods
A choice table for interest rates is created to help determine where the NPV of the cash flows is zero. A look at the approximate interest rate can be made by noting that every 6 months, an amount of $115 is paid on an invested sum of $2100. That gives us an approximation of around 5%. So the table only needs to calculate rates up to 5%. For example, the calculation for interest rate of 2% is:
NPV=−$2,100(1+2/100)0++$115(1+2/100)1++$115(1+2/100)2++$115(1+2/100)3++$115(1+2/100)4++$115(1+2/100)5++$2,115(1+2/100)6NPV=−$2,100(1+2/100)0++$115(1+2/100)1++$115(1+2/100)2++$115(1+2/100)3++$115(1+2/100)4++$115(1+2/100)5++$2,115(1+2/100)6
PeriodNPV0$590.000.5$516.541$446.111.5$378.582$313.832.5$251.733$192.183.5$135.064$80.264.5$27.704.6$17.454.7$7.284.77$0.214.772$0.014.8- $2.814.9- $12.815$22.73
The interest rate that brings the NPV to zero is 4.772%, but this is for 12 semiannual periods. Double that to get the annual interest rate of 9.54%.
(These calculations are made faster if Excel or other financial calculator is used. Use the Excel function called IRR, for example.)
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