1.         Suppose that you put $100 in a bank.  For each of the choices below, what is the    Annual Percentage Rate (APR) for your account, and how much money would you have at the end of one year:

For each of the possibilities below, we need to remember that APR is by definition, the rate per period multiplied by the number of compounding periods within a year (which I refer to as m in the effective period rate formula). We will see that for each of the following, the APR is exactly 10%. Hence, the APR is the same for all of the examples, but we will end up with different amounts of money at the end of the year due to the compounding frequency. This gives us insights into the Effective Annual Rate that is earned due to compounding. The effective period rate can be calculated as below, where APR is the Annual Percentage rate, m is the number of compounding periods within a year, and y is the number of years in the period we are examining. Since we are examining only a single year in each of the following examples, y=1 throughout:

「 APR ]my

EPR = |L1+ m 」|   - 1

a.   The bank compounds once per year with a one-year rate of 10%

APR is rate per period x number of compounding periods in a year. Here we have a 10% rate compounded once per year (m=1).

The APR is thus 0.10 x 1 = 0.10 = 10%

Notice that with a single compounding period, the interest earned is ONLY on the original $100. Thus, we have 100 x 1.10 = $110 at the end of the year.

Notice that the effective annual rate is 10% (we started with $100 and ended

with $110). When rates are compounded annually, the APR and the EAR will be the same because m=1.

「 0.10 ]1x1

EPRAnnual = |L1+   1 」|   - 1 = 0.10

b.   The bank compounds twice per year with a six-month rate of 5%

APR is rate per period x number of compounding periods in a year. Here we have a 5% semi-annual rate compounded twice per year (m=2). The APR is thus 0.05 x 2 = 0.10 = 10%

As we see below, in the first compounding period, you will earn on your original $100 and thus have 100 x 1.05 = $105 at the end of six months. However, in the second six month period, you will earn interest not only on the original $100, but also on the $5 of interest you earned in the first six months. Thus you will end up with $105 x 1.05 = $110.25 at the end of the year. Notice that this is just $100 x (1.05)2 since you earn the 5% in two periods.

Notice that the effective annual rate is 10.25% (we started with $100 and ended with $110.25).

「 0.10 ]2x1

EPRAnnual = |L1+   2 」|    - 1 = 0.1025

When rates are compounded more than once per year (i.e. m>1), the EAR will be greater than the APR.

c.   The bank compounds four times per year with a three-month rate of 2.5%

APR is rate per period x number of compounding periods in a year. Here we have a 2.5% quarterly rate compounded four times per year (m=4). The APR is thus 0.025 x 4 = 0.10 = 10%

As we see below, in the first compounding period, you will earn on your original $100 and thus have 100 x 1.025 = $102.5 at the end of the first quarter. However, in the subsequent periods, you earn interest not only on the $100, but also on the interest earned in prior periods. As shown below,

you end up with $100 x (1.025)4 = $110.38.

Notice that the effective annual rate is 10.38% (we started with $100 and ended with $110.38).

「 0.10 ]4x1

EPRAnnual = |L1+   4 」|    - 1 = 0.1038

d.   The bank compounds 12 times per year with a monthly rate of 0.833%

Similar to the above, we can calculate APR as the rate per period multiplied by number of compounding periods in a year. Here we have a monthly rate of 0.833% with 12 compounding periods per year. The APR is thus 0.00833 x 12 = 0.10 = 10%.

You can imagine what the timeline would look like with twelve compounding periods. At the end of the year though, you would have $100 x (1.00833)12 = $110.47.

Notice that the effective annual rate is 10.47% (we started with $100 and ended with $110.47).

「 0.10 ]12x1

EPRAnnual = |L1+  12 」|     - 1 = 0.1047

2.         If the APR is 10%, what is the effective period rate for each of the following periods:

We know that the relationship between the effective period rate (EPR) and the annual percentage rate (APR) is:

「 APR ]my

EPR = |L1+ m 」|   - 1

Where m is the number of compounding periods per year (regardless of the period you are examining) and y is the number of years in the period you are examining. In the textbook (see pages 35-38), they are examining the effective annual rate. This is simply the above formula when y=1 because the period they are examining is a single year. For all of the following questions, APR will be 10%, it is a matter of changing m and sometimes y.

a.   One year when the rate is compounded annually

We are examining a period of one year so y=1, and the rate is compounded annually, so m=1 (1 time per year).

「 0.10 ]1*1

EPR = |L1+   1 」|   - 1 = 0.10

The effective yearly rate is 10%. This is the same as the APR, which is the quoted rate. This should make sense since the rate is compounded annually. If it is compounded annually, then it should equal the rate calculated AS IF the compounding were annual.

b.   One year when the rate is compounded monthly

We are still examining a period of one year so y=1, but now the rate is compounded monthly, so m=12 (12 compounding periods per year).

「 0.10 ]12*1

EPR = |L1+  12 」|    - 1 = 0.10471

The effective yearly rate is now 10.471%. Even though the quoted rate (APR) is 10% per year, because we are compounding within the year, the effective yearly rate is higher.

c.   One year when the rate is compounded weekly

We are still examining a period of one year so y=1, but now the rate is compounded weekly, so m=52.

「 0.10 ]52*1

EPR = |L1+  52 」|    - 1 = 0.10506

The effective yearly rate is now 10.506%. You can see that with the same quoted annual rate (10% APR), as we increase the frequency of compounding, we increase the effective annual rate.

d.   One year when the rate is compounded continuously

This gets us to the question of how big can we make m? We are still examining a period of one year so y=1, but now the rate is compounded continuously, so m is really infinite m=∞ .

「 0.10 ]w*1

EPR = |L1+  w 」|    - 1 = ???????

Before anyone panics about the infinity signs, it is helpful to know that people who are good at math have already been able to show that

x(l) = 1+ x = er

Where e is the exponential (2.718…) Thus for continuously compounded rates, the effective annual rate is simply eAPR – 1. With continuous compounding, the effective yearly rate when the APR is 10% is:

EPR = e0.10 - 1 = 0.10517

The effective yearly rate is now 10.517%. Again, with the same quoted annual rate (10% APR), as we increase the frequency of compounding, we increase the effective annual rate.

e.   One quarter when the rate is compounded monthly

Now we are examining a period which is NOT one year, but a fraction thereof. We want a quarterly rate so y=¼ (there are ¼ years in one quarter). The rate is compounded monthly, so m=12 (12 compounding periods per year).

「 0.10 ]12*( 14 )

EPR = |L1+  12 」| – 1 = 0.02521

The effective quarterly rate is 2.521%.

Recall from (b) that the effective annual rate with monthly compounding was 10.471%. Notice that the above quarterly rate is completely consistent with that. If you earned 2.521% each quarter, you would earn that 4 times over the course of a year. Hence your effective annual rate would be

[1+ 0.02521]4 – 1 = 0.10471

Which matches the answer in (b).

3.         Suppose your American Express (AMEX) card has a daily period rate of 0.0418%

a.   What is the Annual Percentage Rate that AMEX will report to you?  What annual rate are you effectively paying?  Use 365 days per year.

APR is rate per period x number of compounding periods in a year. Here we are told we have a daily rate of 0.0418% and there are 365 compounding periods within the year. The APR is 0.000418 * 365 = 0.1525 = 15.25%

The effective annual rate is 16.471% as shown below (m=365 and y=1).

「 0.1525 ]365*1

EPRAnnual = |L1+   365 」| – 1 = 0. 16471 = 16.471%

b.   What is the effective monthly rate you are paying on your AMEX card? Again use 365 days per year.

To get the effective monthly rate, we simply need to change y. Amex still compounds daily so m doesn’t change. It remains at 365. But now, we want one month so y = 1/12.

「 0.1525 ]365*( 112 )

EPRMonthly = |L1+   365 」|         - 1 = 0.01279 = 1.279%

The effective monthly rate is 1.279% per month.

4.         Western Sky Financial LLC offers a “Problem Solver” Loan where you can get up to  $10,000 in one day, without collateral.  The typical $10,000 loan has an upfront fee of $75 (thus you really only get $9,925) and is repaid in 84 monthly payments of             $743.49.  Assume that compounding is monthly.  What is the monthly rate for this      loan?  What is the APR for this loan? What is the effective annual rate for the loan?

In order to determine the monthly rate for this loan, we need to realize that the terms of the loan are such that you borrow $9,925 (PV) and make 84 (N) monthly payments of $743.49 (PMT). What we need to determine is the interest rate that makes those streams of cash flows equal. We can use our financial calculator.

Notice that we enter the payment as negative because the calculator uses sign to denote direction of payment (i.e. negative is money you pay and positive is money you get). On the BA=II+, you will notice that the rate is given by I/Y which means it can give us a yearly rate if we set the payments per year (P/Y) correctly. For now, let’s set P/Y=1 (even though it should be 12 because we are paying monthly). This will “trick” the calculator into giving us a monthly rate instead of converting it to an annual rate. Solving this we find that I/Y is 7.4735% or 0.074735 PER MONTH.

To convert it to an APR, we would need to multiply this monthly rate by the number of compounding periods per year which is 12. Thus the APR would be 0.074735 * 12 = 0.8968 = 89.68%

Notice that with the BAII+, if you set P/Y=12 and recomputed I/Y you will get 89.68% since you are basically having the calculator do the conversion to APR for you.

With monthly compounding, the effective annual rate is much higher than the 89.68% as shown below (m=12 and y=1).

「 0.8968 ]12*1

EPRAnnual = |L1+    12 」|    - 1 = 1.3747 = 137.47%

The effective annual rate is 137.47%. The loan may be called a “Problem Solver”, but it is a very expensive solution.

5.         After graduating from Georgetown and landing the job of your dreams, you decide to purchase a 4 bedroom, 2 ½ bath center hall colonial.  You are going to put $40,000 down and have just arranged for a  $300,000 mortgage to  finance the rest of the purchase  price.     The  mortgage  has   a   fixed-rate   6.00%  APR  with  monthly compounding and your first payment is due at the end of the month. The loan calls for monthly payments  computed over the next  30 years.   However, this is  a unique mortgage in that the loan has a ten year balloon payment, meaning that the loan must be completely paid off at the end of year 10.  This is known as a 30 due in 10 loan. You make regular monthly payments based on the terms of the 30 year mortgage, except that at the end of year 10 and then you must completely pay off the remaining balance of the loan.   Ignore any closing costs or escrow payments (i.e. taxes and insurance).

a)        What annual rate are you effectively paying on this loan?

To determine the effective annual rate, we use the following formula where APR is 6% and m is 12. Recall that m is the number of compounding periods within the year and y is the number of years in the period we are examining. Since we want an annual rate, the period we are examining is a single year. Thus y=1.

「 APR ]my

EPR = |L1 + m 」|   - 1

「 0.06 ]12(1)

EPR = |L1 +  12 」|     - 1 = 0.0616778

The effective annual rate for this loan is 6.16778%

b)        What is your monthly payment on this loan?

We know that we are making equal monthly payments on the loan so this suggests it is an annuity. We are borrowing $300,000 so this is the present value of the annuity. We want to solve for the annuity payment. Recall that the formula for present value of an annuity is:

C「 1     ]

PVAnnuity = r |L1 - (1 + r)t 」|

Because we are making monthly payments, we need to use an effective monthly rate.

「 APR ]my

EPR = |L1 + m 」|   - 1

Recall that m is still 12 because the loan is compounded monthly. Remember that you don’t get to choose m. We do get to choose y; however. Y is the number of years in the period we are examining. Since we are looking at a period of one month, y=1/12. There is

1/12 of a year in a single month

「 0.06 ]12( 112)

EPR = |L1 +  12 」|        - 1 = 0.005

Alternatively, because the period we are examining (one month) exactly corresponds to the frequency with which the rate is compounded (compounded monthly), we could also get the monthly rate by dividing the APR by 12. Thus, the monthly rate is 0.06/12 = 0.005. Remember, though that by using the EPR formula, you will always get the correct rate. On the other hand, blindly dividing the APR by number of periods can occasionally give you the wrong rate if the period you are examining is not the same as the frequency with which rates are compounded.

The last thing we need to know is what we use for t. The loan payment is calculated over a 30 year period even though the full amount will have to be paid off after 10 years. Thus, the appropriate t is 360 months (recall that C, r, and t all have to match).

C 「 1       ]

300,000 = 0.005 |L1 - (1.005)360」|

C = $1,798.65

The monthly payment on this loan is $1,798.65.

We could have also done this on the financial calculator. On the BA-II Plus, first clear TVM, then set P/Y=12, PV=300,000, N=360, I/Y=6, and compute PMT.