Introduction - University of Manitoba



Introduction

A holding period is the period over which interest is earned, e.g., one month, six months, or one year. Effective interest rates are rates that reflect the full interest earnings (or charges) over a holding period. Effective rates are compounded once per holding period.

Stated interest rates do not indicate how much interest is actually earned (or charged) over a holding period. The reason for this is that stated rates are usually compounded over some time interval different from the holding period; e.g., 15% per year compounded monthly — the holding period is one year, yet the interest is compounded monthly. Because of the compounding within the year, more than 15% interest is earned over the one-year holding period. The stated rate of 15% per year compounded monthly implies that 1.25% is effectively earned over every month or compounding period. Since this rate is compounded 12 times per holding period, we say it has a compounding frequency of 12.

In finance we usually calculate effective annual interest rates to give us a standard way of comparing returns. Effective rates give the true growth in value during a single holding period. They also show the equivalent growth in value over a holding period when multiple cash flows are received during the period. (See examples on pages 3 and 4.) Stated rates cannot be compared directly unless they are calculated over exactly the same holding period, in exactly the same manner. Even then they do not reflect the true cost of a loan or the true earnings of an investment.

A formula shown in most texts (p. 96, equation 4.5 in the 9.220 text) for converting stated rates to effective annual rates is:

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Where

reffective per year is the effective annual interest rate (return)

rstated per year is the stated annual interest rate (return)

m is the number of times per year the stated rate is being compounded — i.e., the compounding frequency (e.g., 12 for a rate of 15% per year compounded monthly)

Note: rstated per year (m is the stated annual rate divided by the number of compounding periods per year. Therefore rstated per year ( m is the effective rate over a holding period of one month.

One difficulty in using this formula in practice is that we often wish to have an effective interest rate over a holding period other than one year. For example, car loans and mortgages have monthly payments. In order to calculate the monthly payments, we need the effective monthly rate to use in the annuity formula. A more general way of calculating effective (and stated) rates is presented in the following section.

A General Method for Interest Rate Conversions

To convert from one rate to another we need to know the following information for the input rate (the rate we currently have that we want to convert into another rate) and the output rate (the rate into which we are trying to convert).

the types of rates we have: stated or effective

the compounding frequencies

the relevant effective rates and their compounding or holding periods

the relative lengths of the compounding periods

In this course, stated rates will be shown as “X% per holding period, compounded every compounding period”, e.g., 12% per year, compounded every quarter or as “X% per holding period with a compounding frequency of Y times per holding period, e.g. 12% per year, compounded four times per year. You can usually assume any rates not stated this way are effective rates over the holding period. In financial markets, however, several different conventions are in place. North American bonds and Canadian mortgages are quoted as stated rates per year compounded semi-annually. Most personal bank loans are quoted as stated rates per year compounded monthly; and most credit card rates are quoted as rates per year compounded daily.

There is a maximum of 3 steps necessary for interest rate conversions. These steps are illustrated with the following example: Convert a stated rate of 10% per year compounded semi-annually to Z% per year compounded monthly.

Step 1: Convert the stated input rate (10%) to its implied effective rate per compounding period. To do this, divide by the compounding frequency. (If the input rate is already an effective rate, then this step is not necessary as the compounding frequency is one.)

Input rate’s implied effective rate = 10% ( 2 = 5% per half-year = 0.05

Step 2: Convert the input rate’s implied effective rate (5% or 0.05) to the output rate’s implied effective rate. The output rate’s implied effective rate will be an effective rate with holding period equal to the output rate’s compounding period (i.e., in the example, we need an effective rate per month). To do the conversion we need to determine the relative lengths of the compounding periods. I.e., we need to determine how long is the output rate’s compounding period in terms of the input rate’s compounding period.

Output rate’s compounding period is one month = [pic]of the half-year compounding period of the input rate

The following formulas may be used to convert from one effective rate to another:

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Where L = the length of the output’s compounding period in terms of the input’s compounding period

So, in our example, we solve for routput’s implied effective = (1+0.05)1/6 - 1 = 0.0081864846 = 0.81864846% per month.

Step 3: Convert the output rate’s implied effective rate into the desired stated output rate. To do this, multiply by the output rate’s compounding frequency. (If the desired output rate is an effective rate than this step is not necessary as the compounding frequency is one.)

In our example the desired output rate is a rate per year compounded monthly. Since the compounding frequency is not one, this is not an effective rate. We multiply the output’s implied effective rate (from step 2) by the output rate’s compounding frequency, 12, to get Z = 12 ( 0.81864846% = 9.79781526% per year compounded monthly.

Note that not all steps are required for every rate conversion:

Step 1 converts a stated rate to its implied effective rate.

Step 2 converts an effective rate over one compounding period to an effective rate over a different compounding period.

Step 3 converts an effective rate to the stated rate that implies it.

To verify your conversion, check to see whether the change in magnitudes of the rates makes sense. When converting from an input rate with a shorter compounding period to an output rate with a longer compounding period, the output rate’s implied effective rate should be larger than the input rate’s implied effective rate (and vice versa). When both input and output rates have the same holding period, but the output rate has a longer compounding period (i.e., a smaller compounding frequency), the output rate should be larger than the input rate (and vice versa).

Note that the only time a stated rate will be the correct rate to use in time value calculations is when the compounding period equals the holding period, e.g. 10% per year, compounded annually. In this (trivial) case the stated rate equals the effective rate (the compounding frequency is exactly one).

Examples:

1. Find the effective annual rate for a stated rate of 7½% per year compounded quarterly.

Step 1 7.5 ( 4 = 1.875% effective per quarter

Step 2 (1 + 0.01875)4 - 1 = 0.07713587 = 7.713587% effective per year

2. Find the effective annual rate for a monthly (effective) rate of 1.75%.

Step 2 (1 + 0.0175)12 - 1 = 0.23143931 = 23.143931% effective per year

3. What is the stated rate per year, compounded semi-annually for an effective rate of 7.25% per half year?

Step 3 7.25% per half-year (effective) ( 2 = 14.5% per year compounded semi-annually

4. Given a 25-year mortgage for $70,000 at a rate of 8.00% per year compounded semi-annually,

(a) What is the effective annual interest rate?

(b) What rate do you need to calculate your monthly payments?

(c) What is the effective monthly rate?

(d) How much of the first four payments goes toward principal and interest?

(a) Step 1 8.00% per year compounded semi-annually ( 2 = 4.00% per half year (effective)

Step 2 (1 + 0.04)2 - 1 = 0.0816 = 8.16% effective per year

Thus, your effective interest cost per year is 8.16% per year.

(b) Given monthly payments, you need the effective rate per month for your annuity formula.

(c) Step 1 8.00% per year compounded semi-annually ( 2 = 4.00% per half year (effective)

Step 2 (1 + 0.04)1/6 - 1 = 0.00655819794 = 0.655819794% effective per month[1]

You would use a rate of 0.00655819694 (or 0. 655819694%) per month to calculate the mortgage payment using the annuity formula. If you had a $70,000 mortgage for 25 years (300 months) then you would then find your monthly payment by solving for C from the annuity equation:

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d) The interest charged each month is equal to the monthly interest rate multiplied by the principal outstanding at the beginning of the month. The principal reduction each month is the difference between the payment amount and the interest charge. The following table presents this data for the first five months. (Note: numbers shown are rounded to two decimal places; calculations are based on non-rounded numbers.)

|Month |A |B |C |D |E |

| |Principal outstanding at |Interest charged during the|Monthly payment |Principal reduction with |Principal outstanding at |

| |the beginning of the month |month | |monthly payment |the end of the month (after|

| | | | | |the payment) |

| | | | |=C-B |=A-D |

| | |=A•rateeffective per month | | | |

|1 |$70,000 |$459.07 |$534.25 |$75.18 |$69,924.82 |

|2 |$69,924.82 |$458.58 |$534.25 |$75.67 |$69,849.16 |

|3 |$69,849.16 |$458.08 |$534.25 |$76.16 |$69,772.99 |

|4 |$69,772.99 |$457.59 |$534.25 |$76.66 |$69,696.32 |

Note 1: As the principal declines, the monthly interest charge declines and thus the amount of the payment left for principal reduction increases each month.

Note 2: The principal outstanding immediately after a payment is simply the value (at that time) of all the payments remaining. E.g., the principal outstanding at the end of month 3 (or the beginning of month 4) is the value of an annuity of 297 remaining monthly payments discounted to time period 3. Try the present-value calculation to verify this for yourself.

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[1] Note: Canadian chartered banks use 10 decimal places for all calculations.

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