In looking at an advertisement for a car you might see 2.5% APR financing on a $20,000 car. What does APR mean? What rate are they really charging you for the loan? Different banks may offer 8.1% annually, 8% compounded monthly or 7.9% compounded continuously. How much would you really be making if you put $100 in each bank? Which bank has the best deal?

### Nominal and Effective Rates of Interest

A** nominal interest rate** is an interest rate in name only since a method of compounding needs to be associated with it in order to get a true effective interest rate. APR rates are nominal. **APR** stands for **Annual Percentage Rate**. The terms are usually monthly, so \begin{align*}k=12\end{align*}

An **annual effective interest rate** is the true interest that is being charged or earned. APY rates are effective rates. **APY** stands for **Annual Percentage Yield**. It is a true rate that states exactly how much money will be earned as interest.

Banks, car dealerships and all companies will often advertise the interest rate that is most appealing to consumers who don’t know the difference between APR and APY. In places like loans where the interest rate is working against you, they advertise a nominal rate that is lower than the effective rate. On the other hand, banks want to advertise the highest rates possible on their savings accounts so that people believe they are earning more interest.

In order to calculate what you are truly being charged, or how much money an account is truly making, it is necessary to use what you have learned about compounding interest and continuous interest. Then, you can make an informed decision about what is best.

Take a credit card that advertises 19.9% APR (annual rate compounded monthly). Say you left $1000 unpaid, how much would you owe in a year?

First recognize that 19.9% APR is a nominal rate compounded monthly.

\begin{align*}FV =? \ PV=1000, \ i=.199, \ k=12, \ t=1\\\end{align*}

\begin{align*}FV = 1000 \left(1+\frac{0.199}{12}\right)^{12} \approx \$1,218.19\end{align*}

Notice that $1,218.19 is an increase of about 21.82% on the original $1,000. Many consumers expect to pay only $199 in interest because they misunderstood the term APR. The effective interest on this account is about 21.82%, which is more than advertised.

Another interesting note is that just like there are rounding conventions in this math text (4 significant digits or dollars and cents), there are legal conventions for rounding interest rate decimals. Many companies include an additional 0.0049% because it rounds down for advertising purposes, but adds additional cost when it is time to pay up. For the purposes of this concept, ignore this addition.

### Examples

#### Example 1

Earlier, you were asked about financing a car and the difference between APR and APY. A loan that offers 2.5% APR that compounds monthly is really charging slightly more.

\begin{align*}\left(1+\frac{0.025}{12}\right)^{12} \approx 1.025288\end{align*}

They are really charging about 2.529%.

The table below shows the APY calculations for three different banks offering 8.1% annually, 8% compounded monthly and 7.9% compounded continuously.

Bank A |
Bank B |
Bank C |

\begin{align*}FV &= PV(1+i)^t\\ FV &= 100(1+0.081)\\ FV &= \$ 108.1\end{align*} \begin{align*}APY=8.1 \ \%\end{align*} |
\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ FV &= 100 \left(1+\frac{0.08}{12}\right)^{12}\\ FV &\approx 108.299\end{align*} \begin{align*}APY \approx 8.300 \%\end{align*} |
\begin{align*}FV &= PV \cdot e^{rt}\\ FV &= 100e^{0.079}\\ FV & \approx 108.22\end{align*} \begin{align*}APY \approx 8.22 \%\end{align*} |

Even though Bank B does not seem to offer the best interest rate, or the most advantageous compounding strategy, it still offers the highest yield to the consumer.

#### Example 2

Three banks offer three slightly different savings accounts. Calculate the Annual Percentage Yield for each bank and choose which bank would be best to invest in.

Bank A offers 7.1% annual interest.

Bank B offers 7.0% annual interest compounded monthly.

Bank C offers 6.98% annual interest compounded continuously.

Since no initial amount is given, choose a \begin{align*}PV\end{align*} that is easy to work with like $1 or $100 and test just one year so \begin{align*}t=1\end{align*}. Once you have the future value for 1 year, you can look at the percentage increase from the present value to determine the APY.

Bank A |
Bank B |
Bank C |

\begin{align*}FV &= PV(1+i)^t\\ FV &= 100(1+0.071)\\ FV &= \$ 107.1\end{align*} \begin{align*}APY=7.1 \%\end{align*} |
\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\
FV &= 100 \left(1+\frac{0.07}{12}\right)^{12}\\
FV &\approx 107.229\end{align*}
\begin{align*}APY \approx 7.2290 \%\end{align*} |
\begin{align*}FV &= PV \cdot e^{rt}\\ FV &= 100e^{.0698}\\ FV &\approx 107.2294\end{align*} \begin{align*}APY=7.2294 \%\end{align*} |

Bank A compounded only once per year so the APY was exactly the starting interest rate. However, for both Bank B and Bank C, the APY was higher than the original interest rates. While the APY’s are very close, Bank C offers a slightly more favorable interest rate to an investor.

#### Example 3

The APY for two banks are the same. What nominal interest rate would a monthly compounding bank need to offer to match another bank offering 4% compounding continuously?

Solve for APY for the bank where all information is given, the continuously compounding bank.

\begin{align*}FV=PV \cdot e^{rt}=100 \cdot e^{0.04} \approx 104.08\end{align*}

The APY is about 4.08%. Now you will set up an equation where you use the 104.08 you just calculated, but with the other banks interest rate.

\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ 104.08 &= 100 \left(1+\frac{i}{12}\right)^{12}\\ i &= 12 \left[ \left( \frac{104.08}{100}\right)^{\frac{1}{12}}-1\right]=0.0400667\end{align*}

The second bank will need to offer slightly more than 4% to match the first bank.

#### Example 4

Which bank offers the best deal to someone wishing to deposit money?

- Bank A, offering 4.5% annually compounded
- Bank B, offering 4.4% compounded quarterly
- Bank C, offering 4.3% compounding continuously

The following table shows the APY calculations for the three banks.

Bank A |
Bank B |
Bank C |

\begin{align*}FV &= PV(1+i)^t\\ FV &= 100(1+0.045)\end{align*} \begin{align*}APY=4.5\%\end{align*} |
\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ FV &= 100 \left(1+\frac{0.044}{4}\right)^4\end{align*} \begin{align*}APY \approx 4.473\%\end{align*} |
\begin{align*}FV &= PV \cdot e^{rt}\\ FV &= 100e^{0.043}\end{align*} \begin{align*}APY = 4.394\%\end{align*} |

Bank B offers the best interest rate.

#### Example 5

What is the effective rate of a credit card interest charge of 34.99% APR compounded monthly?

\begin{align*}\left(1+\frac{.3499}{12}\right)^{12}=1.4118\end{align*} or a 41.18% effective interest rate.

### Review

For problems 1-4, find the APY for each of the following bank accounts.

1. Bank A, offering 3.5% annually compounded.

2. Bank B, offering 3.4% compounded quarterly.

3. Bank C, offering 3.3% compounded monthly.

4. Bank D, offering 3.3% compounding continuously.

5. What is the effective rate of a credit card interest charge of 21.99% APR compounded monthly?

6. What is the effective rate of a credit card interest charge of 16.89% APR compounded monthly?

7. What is the effective rate of a credit card interest charge of 18.49% APR compounded monthly?

8. The APY for two banks are the same. What nominal interest rate would a monthly compounding bank need to offer to match another bank offering 3% compounding continuously?

9. The APY for two banks are the same. What nominal interest rate would a quarterly compounding bank need to offer to match another bank offering 1.5% compounding continuously?

10. The APY for two banks are the same. What nominal interest rate would a daily compounding bank need to offer to match another bank offering 2% compounding monthly?

11. Explain the difference between APR and APY.

12. Give an example of a situation where the APY is higher than the APR. Explain why the APY is higher.

13. Give an example of a situation where the APY is the same as the APR. Explain why the APY is the same.

14. Give an example of a situation where you would be looking for the highest possible APY.

15. Give an example of a situation where you would be looking for the lowest possible APY.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 13.5.