APY worksheet

1. Stated yearly rate, compounded annually: 7%

1. Write the APY as a decimal, so the formula can use it

   $\frac{\mathrm{7\%}}{100}$$=0.07$

2. Find the rate for one period that grows to the APY after 1 period. This undoes the compounding

   ${\left(1+0.07\right)}^{\frac{1}{1}}-1$$=0.07$

3. Multiply by 1 periods to get the stated yearly rate, which leaves compounding out

   $0.07\times 1$$=0.07$

4. Multiply by 100 to show it as a percent

   $0.07\times 100$$=\mathrm{7\%}$

2. Stated yearly rate, compounded daily: 5.8274%

1. Write the APY as a decimal, so the formula can use it

   $\frac{\mathrm{6\%}}{100}$$=0.06$

2. Find the rate for one period that grows to the APY after 365 periods. This undoes the compounding

   ${\left(1+0.06\right)}^{\frac{1}{365}}-1$$=0.00015965$

3. Multiply by 365 periods to get the stated yearly rate, which leaves compounding out

   $0.00015965\times 365$$=0.05827356$

4. Multiply by 100 to show it as a percent

   $0.05827356\times 100$$=\mathrm{5.827356\%}$

3. Annual percentage yield (APY): 4.0604%

1. Write the stated rate as a decimal, so 5% becomes 0.05

   $\frac{\mathrm{4\%}}{100}$$=0.04$

2. Divide by 4, because the bank adds a share of the yearly rate 4 times a year

   $\frac{0.04}{4}$$=0.01$

3. Grow by that rate 4 times in a row, then subtract 1 to leave just the interest earned

   ${\left(1+0.01\right)}^4-1$$=0.04060401$

4. Multiply by 100 to show it as a percent

   $0.04060401\times 100$$=\mathrm{4.060401\%}$

4. Annual percentage yield (APY): 3.0453%

1. Write the stated rate as a decimal, so 5% becomes 0.05

   $\frac{\mathrm{3\%}}{100}$$=0.03$

2. Divide by 365, because the bank adds a share of the yearly rate 365 times a year

   $\frac{0.03}{365}$$=0.00008219$

3. Grow by that rate 365 times in a row, then subtract 1 to leave just the interest earned

   ${\left(1+0.00008219\right)}^{365}-1$$=0.03045326$

4. Multiply by 100 to show it as a percent

   $0.03045326\times 100$$=\mathrm{3.045326\%}$

5. Stated yearly rate, compounded daily: 4.402%

1. Write the APY as a decimal, so the formula can use it

   $\frac{\mathrm{4.5\%}}{100}$$=0.045$

2. Find the rate for one period that grows to the APY after 365 periods. This undoes the compounding

   ${\left(1+0.045\right)}^{\frac{1}{365}}-1$$=0.0001206$

3. Multiply by 365 periods to get the stated yearly rate, which leaves compounding out

   $0.0001206\times 365$$=0.04401954$

4. Multiply by 100 to show it as a percent

   $0.04401954\times 100$$=\mathrm{4.401954\%}$

6. Stated yearly rate, compounded semiannually: 5.9126%

1. Write the APY as a decimal, so the formula can use it

   $\frac{\mathrm{6\%}}{100}$$=0.06$

2. Find the rate for one period that grows to the APY after 2 periods. This undoes the compounding

   ${\left(1+0.06\right)}^{\frac{1}{2}}-1$$=0.02956301$

3. Multiply by 2 periods to get the stated yearly rate, which leaves compounding out

   $0.02956301\times 2$$=0.05912603$

4. Multiply by 100 to show it as a percent

   $0.05912603\times 100$$=\mathrm{5.912603\%}$

7. Annual percentage yield (APY): 4.0808%

1. Write the stated rate as a decimal, so 5% becomes 0.05

   $\frac{\mathrm{4\%}}{100}$$=0.04$

2. Divide by 365, because the bank adds a share of the yearly rate 365 times a year

   $\frac{0.04}{365}$$=0.00010959$

3. Grow by that rate 365 times in a row, then subtract 1 to leave just the interest earned

   ${\left(1+0.00010959\right)}^{365}-1$$=0.04080849$

4. Multiply by 100 to show it as a percent

   $0.04080849\times 100$$=\mathrm{4.080849\%}$

8. Stated yearly rate, compounded continuously: 5.8269%

1. Write the APY as a decimal, so the formula can use it

   $\frac{\mathrm{6\%}}{100}$$=0.06$

2. For nonstop (continuous) compounding, the stated rate r is the natural log of 1 + APY, which undoes the growth

   $r$$=\ln \left(1+0.06\right)$$=0.05826891$

3. Multiply by 100 to show it as a percent

   $0.05826891\times 100$$=\mathrm{5.826891\%}$

9. Annual percentage yield (APY): 7.2458%

1. Write the stated rate as a decimal, so 5% becomes 0.05

   $\frac{\mathrm{7\%}}{100}$$=0.07$

2. Divide by 52, because the bank adds a share of the yearly rate 52 times a year

   $\frac{0.07}{52}$$=0.00134615$

3. Grow by that rate 52 times in a row, then subtract 1 to leave just the interest earned

   ${\left(1+0.00134615\right)}^{52}-1$$=0.0724577$

4. Multiply by 100 to show it as a percent

   $0.0724577\times 100$$=\mathrm{7.24577\%}$

10. Stated yearly rate, compounded continuously: 4.1622%

1. Write the APY as a decimal, so the formula can use it

   $\frac{\mathrm{4.25\%}}{100}$$=0.0425$

2. For nonstop (continuous) compounding, the stated rate r is the natural log of 1 + APY, which undoes the growth

   $r$$=\ln \left(1+0.0425\right)$$=0.04162167$

3. Multiply by 100 to show it as a percent

   $0.04162167\times 100$$=\mathrm{4.162167\%}$