ࡱ> 5@ bjbj22 XXyVVVjLDDD8D\En24FG"G: HHQVQ$QEnGnGnGnGnGnGnpRbrNGnQV1PNQVVGn   HHnXXXV" V HpHEnXVEnXbXUZhbiH(F miDVi1jn0nisWNs,i    si< R,9SXTT  R R RGnGn>DX"D An Effective Method for Interest Rate Conversions by David A. Stangeland* Charles E. Mossman** May, 2002 *Contact author; Associate Professor of Finance and Head, Department of Accounting and Finance, I.H. Asper School of Business, University of Manitoba, Winnipeg, MB, Canada. E-mail:  HYPERLINK mailto:D_Stangeland@UManitoba.Ca D_Stangeland@UManitoba.Ca Phone: (204) 474-6477 Fax: (204) 474-7545 **Associate Professor of Finance, Department of Accounting and Finance, I.H. Asper School of Business, University of Manitoba, Winnipeg, MB, Canada. E-mail: Mossman@ms.UManitoba.Ca (204) 474-9510 Fax: (204) 474-7545 An Effective Method for Teaching and Understanding Interest Rate Conversions Practitioners in the finance and legal professions are often required to use return or interest-rate quotes to determine interest charges, present values and future amounts. Unfortunately, due to government regulations, historical conventions, and many other arcane reasons, interest rates are rarely quoted in a form that can be used directly in basic time-value calculations. Even more distressing is that many finance and legal practitioners are unaware of the true meaning of the interest-rate quotes they are using thus resulting in incorrect calculations, an appearance of incompetence, and exposure to legal disputes. The purpose of this article is to provide precise explanations of interest-rate quotes and explain how they can be used correctly. We explain the different quotation methods and outline the circumstances when specific types of rates must be used in particular time-value calculations. We present a fail-safe method for converting any quoted interest rate into the correct rate to be used for time-value calculations. We also include discussion of Annual Percentage Rates (APRs) and real (inflation adjusted) versus nominal returns. Our presentation is intended for finance and legal professionals who require full understanding and correct methodology. This presentation is also useful for students and professors of finance who are usually only exposed to a cursory discussion of interest-rate converting presented in typical corporate finance textbooks. I. Methods of Quoting Interest Rates Several terms are used to describe the way an interest rate may be quoted. Real rates are returns in which interest is quoted in terms of purchasing power; nominal rates are those for which interest is quoted in terms of a currency (e.g., in terms of dollars). We will concentrate our discussion on nominal interest rates and return briefly to real interest rates at the end of the article. Probably the best way to quote an interest rate is as an effective rate. Effective interest rates are returns with interest compounded once over the period of quotation. For example, an effective annual rate is quoted over a one-year period. Since it is effective, it is compounded once per year. An effective monthly rate is quoted over a one-month period. Since it is effective, it is compounded once per month. To understand the true cost or return over a quotation period, it is most intuitive and accurate if all rates are expressed as effective rates. Following this, since most rates are quoted on an annual basis, we would expect effective annual rates to be the predominant quotation specification; unfortunately, effective annual interest rates are rare when examining the rate quotations for most financial contracts and securities. An interest rate may be quoted as a rate compounded more or less than once over the quotation period. This is usually due to the conventions or regulations related to the particular financial arrangement. For example, a loan quoted at 10% per year compounded monthly, has an interest rate quoted on a yearly basis but compounded 12 times over the quotation period. In this case the quoted rate is not a true-cost effective interest rate; in fact, since compounding occurs more frequently than once per quotation period, this quoted rate understates the true annual cost of the loan. Often the terms quoted, stated, or nominal are used to describe such rates. The term, nominal, in this last case often causes confusion with the previous definition of nominal so we will avoid such use. We will generally use the term quoted rates to refer to interest rate quotes and effective to refer specifically to rates compounded once per quotation period. The Truth in Lending and Truth in Saving acts passed in the United States add another dimension to interest rate quotations called the Annual Percentage Rate (APR). This rate is a quoted rate per year that is adjusted for the net cash flows related to a loan or investment rather than the stated principal and amortized payments. The net cash flows may differ from these stated amounts due to fees, compensating balances, points, etc. Following the general method for converting interest rates, we will explain the APR calculation and why it is usually not the same as an effective annual rate. II. Necessary rates for time-value calculations Most time-value calculations either involve a single cash flow or a series of cash flows (such as an annuity). Examples of present-value calculations are given in the two equations below:  (1)  (2) Equation 1 calculates the present value of a single cash flow, C, received in n periods, with slight manipulation, it can also be used to calculate the future value of an amount invested for n periods. Equation 2 calculates the present value of an annuity with equal cash flows, C, paid each period for n periods. With slight manipulating, equation 2 can also be used to determine annuity payments, C, for a loan amortization with principal amount equal to the PV. In both equations, r represents the interest rate used for discounting. For both of these types of calculations, r must be an effective rate. When dealing with a single cash flow as in equation 1, any equivalent effective interest rate may be used as long as n is adjusted accordingly. For example, if we wish to determine the present value of $100 received in 5 years, given an effective interest rate of 10.25% per year (i.e., a rate per year compounded once per year), the following calculation is correct:  An effective rate of 5% per six months (i.e., per six months compounded once every 6 months) is equivalent to an effective annual rate of 10.25%. We can use the effective rate of 5% per six months in our calculation as long as we adjust n, the number of periods to be stated in terms of the rates quotation period, i.e., using 5% per 6 months as the effective rate, we use n = 10 six-month periods.  As long as equivalent effective rates are used and n is stated in terms of the effective rates quotation period, the choice of the particular effective rate does not matter. However, this result only holds when a single cash flow is analyzed and it does not hold true for multiple cash flows such as annuity, perpetuity, or amortization calculations. The derivations of annuity, perpetuity, and amortization time-value formulas require that the interest rate used be an effective rate and that the quotation period of the rate equal the time period between cash flow payments. For example, if an annuity has annual payments, then to correctly use formula 2, an effective annual interest rate must be used. If an annuity has monthly payments, then to correctly use formula 2, an effective monthly rate must be used. Consider a 20-year loan with an effective annual interest rate of 10.25%. If the loan had equal yearly repayments of $5,000, then the present value of the payments (or the principal value currently outstanding) would be calculated as follows.  If the loan had equal semiannual payments of $2,500, then we need to use the effective 6-month rate in the calculation and n = 40 semiannual payments.  Note: the two results are indeed different, due to the effect of making payments earlier, demonstrating that the earlier timing of the cash flows is important. The effective interest rates used (10.25% per year and 5% per six months) are equivalent. III. A fail-safe method for converting interest rates Since effective interest rates are usually not provided when interest rates are quoted but they are needed for time-value calculations, practitioners must understand how to convert between different types of rate quotations. The basic principle for conversion of rates is that the effective rate represents the growth in value of a principal amount during an entire period. With most quote conventions for securities or institutions, an effective rate for the quotation period is not stated; however an effective rate over the compounding period is implicit in the rate quote. For example, U.S. mortgages are usually expressed as quoted rates per year compounded monthly (e.g., 9% per year compounded monthly), Canadian mortgages are quoted as rates per year compounded semiannually (e.g., 9% per year compounded semiannually); most government and corporate bond returns (yields) are expressed as quoted rates per year compounded semiannually (e.g., 6% per year compounded semiannually). The effective rate implicit to the U.S. mortgage quote is actually 0.75% per month (compounded monthly). For the Canadian mortgage, it is actually 4.5% per six months (compounded semiannually) and the effective rate implicit to the bond quote is 3% per six months (compounded semiannually). Note: although the U.S. and Canadian mortgages appear to have the same rate (both have the same 9% amount quoted), the US mortgage is actually more costly because of the more frequent compounding. This highlights the importance in contracts to clearly specify not only the rate quotation, but also the terms for compounding. The following section explains how to handle rate conversions ranging from these simple cases to the most complicated ones. A. Conversions from a given quoted rate to a desired rate quoted with different terms To be complete and unambiguous, any interest rate quotation or contract must specify the following terms: the quoted amount the quotation period the number of times the rate is compounded during the quotation period (also called the compounding frequency of the quotation) For example, consider a rate of 20% per year, compounded quarterly. The quoted amount is 20%. The quotation period is one year. The interest is compounded every quarter of the one-year quotation period (so the compounding period is 0.25 years); therefore the compounding frequency is 4 (or 4 times per the one-year quotation period). To convert to an equivalent interest rate with different quotation terms, we must know the quotation period and compounding frequency of the desired rate, so that we can calculate the desired quoted amount. Below we explain the concept algebraically. The Appendix gives several examples that may be useful for practitioners new to this methodology. To continue our example, suppose we wish to convert our given quoted rate of 20% per year compounded quarterly into a desired quoted rate with a quotation period of 6 months and monthly compounding. For the purpose of a precise presentation, we must specify the following notation for the quotation terms of our given and desired interest rates: Given rate (subscript g): rg = the quoted amount of our given interest rate Lg = the quotation period from the given interest rate mg = the compounding frequency of our given interest rate Desired rate (subscript d) rd = the quoted amount we need to calculate for our desired interest rate Ld = the quotation period for the desired interest rate md = the compounding frequency of our desired interest rate So, for our example we have the following data for the given rate: rg = .20 = 20% Lg = 1 year mg = 4 and we know the following about our desired rate: Ld = 0.5 years (or 6 months) md = 6 (monthly compounding implies the desired rate is compounded 6 times in the half-year quotation period) Once we have completed the conversion process, we will have solved for the desired quoted rate, rd, which is currently unknown. Next, we introduce the concept of the implied effective rate. Given the quotation of any rate, we can divide the quoted amount, rg, by the compounding frequency, mg, to get the implied effective rate. The following notation is used for implied effective rates (subscripts ie). Given-rate implied data: rieg = the amount of the implied effective rate determined from the given interest rate Lieg = the quotation period for the given rates implied effective rate Desired-rate implied data: ried = the amount of the implied effective rate to be determined for the desired interest rate Lied = the quotation period for the desired rates implied effective rate By definition for effective rates, the compounding frequency for both rieg and ried must always be exactly 1 (hence notation is not required for these compounding frequencies). Expressing the process to determine ried in formula form, we divide the quoted amount by the compounding frequency as follows:  EMBED Equation.3  (3) We find the quotation period for an implied effective rate in a similar manner.  EMBED Equation.3  (4a) and  EMBED Equation.3  (4b) Taken together, equations (3) and (4a) give us the implied effective rate. This implied effective rate has a quotation period equal to the compounding period of our original given quoted rate. For example, with our given quoted rate of 20% per year compounded quarterly, we do the following calculation.  EMBED Equation.3  and  EMBED Equation.3  So our implied effective rate from the given rate is rieg = 5% per quarter (compounded once every quarter since the rate is an effective rate). Our next step is to convert this implied effective rate, rieg, into an equivalent effective rate with a new quotation period. We want the new quotation period to equal the compounding period of the desired quoted rate, ried. We have yet to determine how to calculate ried, but, from equation (4b), we know that the quotation period for the desired rates implied effective rate is 1 month.  EMBED Equation.3  Next is the process to convert between equivalent effective rates that have different quotation periods. The following illustration is useful for understanding the method. Suppose we have $1 invested at time 0 and we are given an effective rate of 1% per month (compounded once per month). If our goal is to convert the effective monthly interest rate to an effective 6-month interest rate, then we can compound the $1 investment 6 times to determine how much it has grown after 6 months. The result implies what must be the equivalent effective 6-month interest rate. Month 0 1 2 3 4 5 6 %%%%%%%<%%%%%%%<%%%%%%%<%%%%%%%<%%%%%%%<%%%%%%%$% $1 (1+.01) (1+.01) (1+.01) (1+.01) (1+.01) (1+.01) = $1(1+.01)6 The result of $1(1+.01)6 is an amount equal to exactly $1.0615201506 after 6 months. Therefore, the equivalent 6-month effective rate must be 6.15201506%. (I.e., $1 grows to the same amount, $1.0615201506, using 1% per month compounded monthly over six one-month periods or using the equivalent 6.15201506% per six months compounded once over the one six-month period.)  In the above illustration, we converted from the one effective rate to the second effective rate by compounding the given effective rate as many times as required to reach the quotation period of the desired effective rate. We can now return to our original example. Using our notation from above, we can convert our implied effective rate from the given quotation (an effective rate over a 3-month period) to the implied effective rate for the desired quotation (an effective rate over a 1-month period) as follows.  EMBED Equation.3  which implies that  EMBED Equation.3  (5) Note: care must be taken to ensure that both Lied and Lieg are expressed in terms of the same units. In our example, we can express them both in terms of months and do the following calculation.  EMBED Equation.3  To convert the implied effective rate for the desired quotation, ried, to the desired quote amount, rd, we use a variation of equation (3). Now we multiply the desired rates implied effective rate, ried, by the desired rates compounding frequency, md, to get the desired quoted rate.  EMBED Equation.3  (6) For our example, applying equation (6) results in the following.  EMBED Equation.3  The process illustrated here will produce accurate interest rate conversions in all cases that practitioners face. After the quotation terms of the given and desired rates are determined, the conversion process is, at most, three steps, illustrated further in the Appendix. Step 1: Determine the given rates implied effective rate.  EMBED Equation.3  Step 2: Convert the given rates implied effective rate to the desired rates implied effective rate.  EMBED Equation.3  Step 3: Convert the desired rates implied effective rate into the desired quoted rate.  EMBED Equation.3  In many cases, only one or two steps are necessary. If the given rate is an effective rate, then step 1 is unnecessary. If the given rate and the desired rate both have the same compounding period (i.e., their implied effective rates have the same quotation period), then step 2 is unnecessary. If the desired rate is an effective rate, then step 3 is unnecessary. Many textbooks only indicate the following formula for converting between interest rates.  EMBED Equation.3  (7) Unfortunately, this formula only accomplishes steps 1 and 2 from our method above and converts the given rate, rS, to an effective rate over the same quotation period as the originally given rate. Hence, while this formula is occasionally useful, it does not handle many of the required interest rate conversions necessary for input into time-value calculations. The three-step method shown above is recommended instead of equation 7 because the three-step method will always be able to get the necessary rate quotation desired. B. Conversions to APRs The Truth in Lending and Truth in Saving acts passed in the United States require the disclosure of Annual Percentage Rates (APR). The intention behind APRs is that the net interest cost is calculated after considering any fees, points, compensating balances, etc., that affect the cash flows of the loan or investment. In passing the legislation regarding APRs, legislators failed to require rates to be expressed as effective annual rates, so APRs are typically quoted on the same terms as is standard for the respective financial instrument. To do an APR calculation, the internal rate of return (IRR) of all net cash flows related to a loan or investment must be calculated. We illustrate the process with the following example. Suppose a $100,000 mortgage is required for a home purchase. A mortgage rate of 6% (per year compounded monthly) is available with a 1-point charge. The mortgage company also requires the borrower to pay a monthly mortgage insurance fee of $10.45. The mortgage is amortized over 30 years (360 months) but must be renegotiated in 5 years (60 months). The IRR should be calculated from the following set of cashflows. Month:012360Cash Flow:+ net amount borrowed- net amount repaid - net amount repaid- net amount repaid- net amount repaid and refinanced+$99,000-$610-$610-$610-$93,664.36 We will consider each cash flow in turn. The net amount borrowed is affected by the points that are charged. One point is 1% of the mortgage principal borrowed; in this case it results in a $1,000 charge paid by the borrower at the initiation of the mortgage. In effect, the borrower has a cash inflow of $100,000 (the amount borrowed that is paid to the vendor of the home) and a cash outflow of $1,000 (the one-point charge assessed with closing costs), so the borrowers net amount received is $99,000. The mortgage principal of $100,000 is used to calculate the amortization schedule. A 6% rate per year, as quoted above, (6% per year compounded monthly as is standard with US mortgages) results in an effective monthly rate of 0.5%. So the monthly mortgage payments (based on a 30-year amortization) are $599.55. Since the borrower is also required to pay monthly mortgage insurance premiums of $10.45, the net amount repaid each month (months 1 to 59) is $610. At the end of 5 years, the mortgage must be renegotiated. In effect, this can be represented by the remaining principal outstanding being repaid (in addition to the final payment). In reality, the principal outstanding would be refinanced. The principal outstanding immediately after the 60th payment will be $93,054.36. Thus, the net cash flow at month 60 (including the mortgage payment and the insurance) is -$93,664.36. Using a financial calculator or spreadsheet to determine the IRR of this stream of cash flows results in an IRR of 0.53090063% (effective monthly rate). Thus the APR would be 6.37080752% (i.e., 0.53090063 ( 12) as a quoted rate per year with monthly compounding. Conceptually, the APR calculation is relatively straightforward as long as a good financial calculator or spreadsheet program is available. In reality, the process is difficult because of ambiguous information regarding loan terms and prepayment or renegotiation options. Moreover, the APR is not the effective annual rate, as can be seen above. The effective monthly rate of 0.53090063% would compound to an effective annual rate of 6.5601639%. The effective annual rate is the true cost of each dollar of principal borrowed for one year. This is the intuition most people look for when they ask for an interest rate quotation. The effective annual rate has another advantage in that it can be compared to other interest rates or returns on a standard basis, unlike APRs that may be quoted with differing compounding assumptions. APRs cannot be compared directly to other APRs quoted with different compounding periods. For example, suppose an investor wondered whether to pay off a mortgage by cashing in a government bond. Converting both returns to effective annual rates after income tax (including the transaction costs specified in the APR approach) would provide the correct comparison. Simply comparing the APRs, though, would be misleading as the mortgage APR will have monthly compounding and the bond APR will have semiannual compounding. C. Nominal to Real Conversions After mastering effective and quoted rate conventions, practitioners often also need to consider real rates of return. As stated in section I, real rates are those where interest is quoted in terms of purchasing power, and nominal rates are those with interest quoted in terms of a currency (e.g., in terms of dollars). For example, if an investment earned 5% in real terms over the past year, then an investor is able to buy 5% more goods and services at the end of the investment, than was possible a year ago when the initial investment was made. If an investment earned 10% in nominal terms over the past year, then an investor receives 10% more currency (e.g., dollars) at the end of the investment, compared to the initial amount of currency invested one year ago. Practitioners use real rates for time value calculations involving real cash flows (cash flows that are expressed in todays purchasing power, i.e., the effects of inflation have been removed). For example a personal-injury lawsuit settlement may call for the defendant to purchase a real-return bond for the plaintiff. The real-return bond will generate annual income providing constant purchasing power to the defendant for the remainder of his/her life. If the constant purchasing power is expressed in terms of the amount that cash today can purchase, then the future annual income amounts are constant real cash flows and the effects of inflation have been factored out. By using equation 2 with both the cash flows and discount rate expressed as real amounts, the required investment in the real-return bond can easily be calculated as the present value of a series of constant real cash flows. In fact, in any situation where a recurring cost or revenue is expected to rise with inflation, it is simpler to determine what is the appropriate real rate and then apply time value formulas to the recurring constant real cash flows. The nominal rate of return is not usually equal to the real rate of return because inflation erodes the purchasing power of currencies. Consider our nominal rate of 10% and an initial investment of $10,000. Also suppose that inflation over the year was 4%. At the end of the year, the investment would have grown to $11,000. Suppose a basket of goods had a cost of $100 at the beginning of the year. Given 4% inflation, the same basket of goods now has a cost of $104 at the end of the year. Now consider how the investors purchasing power has changed. At the beginning of the year, the investor could have purchased 100 baskets of goods (calculated as $10,000 $100/basket). At the end of the year, the investor can purchase $11,000 $104/basket = 105.7692308 baskets. So, in terms of purchasing power, the investment caused the investor to forgo consuming 100 baskets of goods at the beginning of the year and now allows the investor to consume 105.7692308 baskets at the end of the year. Thus, the return in terms of purchasing power is 5.7692308%. We can illustrate the nominal and real return calculations on the following timelines. Let r represent the nominal rate of return, let i represent the inflation rate, and let R represent the real rate of return. Nominal Rate of Return Year 0 Year 1 %%%%%%%%%%%%%%%%%%%%%%%%%$% in dollars: $10,000 $10,000 (1+r)  $11,000 Real Rate of Return Year 0 Year 1 %%%%%%%%%%%%%%%%%%%%%%%%%$% in baskets of goods:  EMBED Equation.3   EMBED Equation.3   100 baskets  EMBED Equation.3   100(1+R) baskets  105.7692308 baskets Thus we get the following formula for converting between nominal and real rates of return.  EMBED Equation.3 or R = (1+r)/(1+i) - 1 (8) where R is the real rate of interest, r is the nominal rate of interest, and i is the inflation rate. (In our example, R = (1+0.10) / (1+0.04) - 1 = 0.057692308) Care must be taken when using equation (7) in that all three rates must be expressed as effective rates and all three must have the same quotation period (e.g., all must be quoted over a 1-year period). If the rates to be used in equation (7) are not effective, they must first be converted to effective rates using the method outlined in section III.A. IV. Summary and Conclusion To produce accurate results, display competence, and avoid legal disputes, it is crucial that finance and legal practitioners understand interest rate terminology so that precise contracts can be written and correct time-value calculations can be performed. In this article, we explain the most common terminology and the quotation terms required for correct time-value calculations. We then demonstrate fail-safe methods for converting between various types of interest rates. Included are conversions from one form of quotation to another, conversions to Annual Percentage Rates (APRs), and conversions between nominal and real rates. Even with expertise in all these methods, often the terms and conditions affecting an interest rate are either not explicitly stated or are ambiguous; in some cases, the terms are even misleading. Only through a careful analysis of what the rates mean can practitioners use them correctly. Appendix Examples: 1. Find the effective annual rate for a stated rate of 7% 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 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:  EMBED Equation.2   EMBED Equation.2  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.) MonthA Principal outstanding at the beginning of the monthB Interest charged during the month =Arateeffective per monthC Monthly payment D Principal reduction with monthly payment =C-BE Principal outstanding at the end of the month (after the payment) =A-D1$70,000$459.07$534.25$75.18$69,924.822$69,924.82$458.58$534.25$75.67$69,849.163$69,849.16$458.08$534.25$76.16$69,772.994$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.  One rate is equivalent to another if it yields the same result from a time-value calculation (assuming proper use).  We will show how to calculate an equivalent rate later. For now, it is important to see that both rates, 10.25% effective per year and 5% effective per six months, result in the same present value amount.  The two rates are indeed equivalent as shown in the example with the single cash flow presented earlier. In this example, the calculated present values are different due to a different series of cash flows being analyzed.  For the US mortgage quote, the quotation period is one year and the compounding period (the length of time between compounding) is one month. For the Canadian mortgage quote and the bond quote, the quotation period is one year and the compounding period is six months. Note, though, that the Canadian mortgage has monthly payments but the bond has semiannual payments.  Once we use understand the method for converting interest rates, we may express both mortgage quotes on the same compounding terms. For example, restating the mortgage rates as effective annual rates (i.e., rates compounded once per year), the U.S. mortgage cost is 9.38068977% and the Canadian mortgage cost is 9.2025%  Often interest rate quotations are ambiguous because the terms may be stated in a misleading manner or are not stated explicitly. Compounding this problem is the fact that often the personnel providing the rate information are not informed regarding such matters. Knowledge of the conventions used for different types of rate quotes is essential when attempting to do time-value calculations, since effective rates are needed in actual time value calculations.  In the interest of accuracy, intermediate calculations should never be rounded.  The one exception is conversions to or from rates that are continuously compounded (note, for most practitioners these types of conversions are not encountered). To convert to a continuously compounded rate from an effective rate, take the natural log of one plus the effective rate, ln(1+rieg). This yields a continuously compounded rate with a quotation period equal to the implied effective rates quotation period. To convert from a continuously compounded rate to an implied effective rate, use the exponential function, exp(rg) = (1+ried). The quotation period of the desired effective rate is equal to the quotation period of the continuously compounded rate.  Note that if the borrower did not have the $1,000 for points in addition to the down payment and other closing costs, the borrower could choose to have a smaller down payment and a larger mortgage. In this example, suppose the borrowers total down payment plus points is $40,000 and the cost of the home is $140,000. Then the borrower could finance $101,010.10, pay 1% of this ($1,010.10) in points and have a down payment of $38,989.90.  This is calculated using equation (2) and solving for C. This results in  EMBED Equation.3 so we get  EMBED Equation.3   At the time of refinancing, the borrower may be subject to additional points or other fees. We leave these items as potential adjustments to the APR of the refinancing.  This can be calculated, using equation (2) as the present value of the remaining 300 monthly payments of $599.55 discounted at the 0.5% effective monthly rate. I.e.,  EMBED Equation.3   Calculated as (1+0.53090063)12 - 1 = 0.0637080752  For this section, it is assumed that all nominal and real rates are expressed as effective annual rates. If they are not, then the methods from the previous sections should be used to first convert to effective annual rates.  The consumers price index CPI measure "<=>nqvw/ @ A m n o V o x +5||-1 2= !ֵǧ֝hS6CJ] hCJ hS6CJh)PB*CJphhS0J>*B*CJph"jhSB*CJUphjhSB*CJUphhSB*CJph h)PCJ htSCJ h)P5CJ hS5CJ hSCJ0  =>ABWXmnxyz{|}~$a$<V X Y jk$a$  ?@ I"K"L"M"N"##%$a$()  kl ! H!I!!!/"0"I"J"""$$$%'%7%]%s%w&x&&&''(((L())*-*5*<*`* h)PCJhS@mH sH jhSUmHnHujhS0JCJUhS6CJ] hS6CJhS5CJ\ hSCJjhSCJUmHnHuB%%w&y&z&{&|&'''''((L(M(0)1)//r/s///00073 & F$ !*$`*r*++-,.,..//0|1}11834444444444555j5k5m5n5o5555555o6p6q6~666666666͹hS@CJ]mH sH hS6@CJH*mH sH hS6@CJmH sH hS6@CJ]mH sH hS@CJmH sH hSmH sH hS@mH sH h)PhSjhS0JCJU h)PCJ hSCJhS6CJ]2738344445Q5R5m555+6,6o6~666666O7P7772888 0*$^`0$*$*$666667778T8U8v8w888999\9]9`9999: :#:::::B;E;;;;;;;;;;phS5@CJ\mH sH #jhS@CJEHUmH sH jr!M= hSUVjhS@CJUmH sH hS6@CJ]mH sH hS@CJH*mH sH hS6@CJH*]mH sH hS6@CJmH sH hS@CJmH sH hS6@CJH*mH sH )889\9999:j:k:;;;;;; < <<<=== !*$ $ !*$$ !*$ !*$ !*$$ !*$$*$*$; < < << <!<"<,<0<b<c<v<w<x<y<|<<==========Ͻꤞ~saVj/M= hSUV#j\hS@CJEHUmH sH j"M= hSUVhS5CJ\j.hSCJEHUjl.M= hSUV hSCJjhSCJUhS5@CJ\mH sH #j hS@CJEHUmH sH j_= hSUVjhS@CJUmH sH hS@CJmH sH hS@mH sH ===2>3>6>T>X>>>>>j?k?n????@@@*@+@,@-@A ABCCDD0D2DEEĵѱњ|ѱrgrgr[hS0J@mH sH hS@H*mH sH hS@mH sH hN@CJmH sH #j hS@CJEHUmH sH j5S? hSUV hS6H* hS6hShS6@CJH*mH sH hS6@CJmH sH hS@CJmH sH jhS@CJUmH sH #j hS@CJEHUmH sH #==>>@@.@/@@@BBCCDDEErF $ 4X  $ 4X $ $ 4X $ ~ 4X  !~ 4X *$$ !*$EEEGGGGGGGGGGGGGGHHHHHHHHHHHپ١ـsdsdYG#jghS@CJEHUmH sH jqN= hSUVhS6@CJH*mH sH hS6@CJmH sH hS5@CJ\mH sH #jhS@CJEHUmH sH jKoN= hSUV#jhS@CJEHUmH sH jnN= hSUVjhS@CJUmH sH hS@CJmH sH hS@mH sH jhS0J@UmH sH rFsFGGGGGGHHHHIIII@JAJYJZJJJpKqKK !"*$^`" 00*$^`0 !HHII'I(IIIIIIIIIIIIIAJBJUJVJWJXJJJpKqKKKKKȶ㜊xk`jr!M= hSUVhS5@CJmH sH #jhS0J@CJUmH sH #jPhS@CJEHUmH sH jU? hSUVhS5@CJ\mH sH #jAhS@CJEHUmH sH jqN= hSUVhS6@CJH*]mH sH hS@CJmH sH jhS@CJUmH sH KKK,L-L@LALBLCLLLLLLLLNNNNNNNNNNOƴѩэrbUOI hNCJ hSCJhS5@\mH sH j!hS@EHUmH sH j$Y? hSCJUVjhS@UmH sH hS@mH sH #jrhS@CJEHUmH sH jqN= hSUV#jhS@CJEHUmH sH jKoN= hSUVhS@CJmH sH jhS@CJUmH sH #jhS@CJEHUmH sH KDLELLL%N&NNNNNPPPPRRTUUU\U^U_UaUbU $$Ifa$ !*$ !"*$^`" !*$ !"^`"OOOOTO\OjOPPPPPPUUmUUVVHVGXHXYY[[G[I[e[f[\\^^bbbb-c:chereee.k/kmmnnooPoRooojppppp۪ۥۗjhSUmHnHu hS6hS@CJmH sH hS6CJ] jhSCJ hSCJH*jhS0JCJUhS hS6CJ hSCJ hNCJhhH* hCJ;bUdUeUgUhUiUlUmUnUoUpUqUrUsUtUuUvUwUxUyUzU{U|U}U~UUUFf*$IfFfJ% $$Ifa$UUUUUUUUUUUUUUUUUUUUUUUUUUUU $$Ifa$Ff61$IfUUUUVVV$V%V+V,V2V3V9V:V;VGVHVIVtVuVKXLX Z!Z[Ffv9 $$Ifa$$IfFf5 $$Ifa$[[\\^^+b,bLbMbUeVeiimmnnoooojppppp !*$$  ! *$ppppqqqqqqqqqqrrrrrrr(r)rr?r@rArBrPrQr[r\r]rrrvlg]jhSCJU hS6j3@hSEHU%jU? hS@CJUVmH sH jhSUmHnHujhSCJUmHnHuj=hSEHU!jlY= hS@UVmH sH j;hSEHU!jdY= hS@UVmH sH jhSUhS@CJmH sH  hS>*hS hSCJ$pqqrrr@rBr[r]rxryrrrs sssuu,u-uwwxxx$a$ !$$  ! *$rrrrrssss/s0sVsWsxxxxxCyPyQyyyyzzzQzzzz{Z|||||}}}~7~u jh)P@CJmH sH h)P@CJH*mH sH  jh)P@CJmH sH h)P@CJmH sH h)P5@CJmH sH  h)P5CJ hS5CJ hS6CJhS5CJ\jhSCJUjBhSCJEHUj4 Y= hSCJUV hSCJ)xxxCyDysytyyyzzPzQzzz{{s{{{ | 0*$^`0gd)P 0*$@&^`0gd)P 0*$@&gd)P *$gd)P *$@&gd)P |Y|Z|||2}3}}}}}9~:~Nfg!')]_ $$Ifa$gd $Ifgd & Fgd)P$a$gd)Pgd)P 0*$^`0gd)P *$gd)P7~8~NObcdegh{|}~!vwGH)*˺˕ujcYUYUYUNUY hS6]hSjhS0JU h)P5CJh)P@H*mH sH h)P@mH sH h)P#jDh)P@CJEHUmH sH  jA8 h)PUVmHnHu&ja6 h)P@CJEHUmH sH  ja6 h)PUVmHnHu h)PCJjh)P@CJUmH sH h)P@CJmH sH #jh)P0J@CJUmH sH _܁݁&+ $$Ifa$gd $Ifgd+,.6>FM>55555 $IfgdkdH$$Iflֈ$~ &#Z^^^ t0#4 laMXY[fnv5kdH$$Iflֈ$~ &#Z^^^ t0#4 la $Ifgdv}5kdI$$Iflֈ$~ &#Z^^^ t0#4 la $IfgdƂ5kdJ$$Iflֈ$~ &#Z^^^ t0#4 la $IfgdƂ΂ւ݂5kdK$$Iflֈ$~ &#Z^^^ t0#4 la $IfgduvFG()dW$a$gd)P*158,/ (+deĐŐƐǐѐҐ?@STUVWXuw hSCJ hSH*jQhSEHUj_= hSUVjOhSEHUj%_= hSUVj}LhSEHUj_= hSUVjhSU hS6 hSH*jhS0JUh)PhhS8qr;<=CDEFGHNOQRSTVWjklmopԿ޷xjXYhSEHUjL= hSUVjVhSEHUj-VN= hSUVjThSEHUjUN= hSUVjhSUhO/_0JmHnHuhtS htS0JjhtS0JUhh)Pjh)P0JUUjhS0JUhS.pq;<EFGSTUVnoh]hgdtS &`#$gdtSgd)Ps the relative price level of a representative basket of goods purchased by the average consumer. Changes in the CPI are often used to calculate the rate of inflation in an economy.  Calculated as 105.7692308/100.00 - 1 = 0.057692308  Note: Canadian chartered banks use 10 decimal places for all calculations. PAGE  PAGE 15  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  $a$ ޾ޭ h)P5CJjtahSEHUjzݨdu23o>)j勐) ^"}$&^\c< n_i}VwM\guУR5C$](y^sG}*VҺ_K`W-v19#]WQ0J) Uigyk_Q+؏ ko5JG.k^=ͫOL_j2?kIS[8g#<T7@l-˛$% Ò>˻gGUZ\ވ`!xm|`@ PFxڍR=KA}3w]./#(+m;EL ^& ) ; ; =wg+Dc0ޛ%TgЫvuD!βL24#O_7BEs1"M~Vѽw>)a?n b`M^uQ\">::Tѭ~p~\F:I^_FݤRBPX ӵ_iX60_ed7ʙkaJoX&awĹҪ̰sWFYX}asvOM/srkrbj={ zs Po0|X`!_'\M XA%@YkxڕRMK@mSV_ JQT*xV*ꉋd3kX63ë~_σf x]WoJtJ:is1DfyyS/q?!YTu[4r$^.G-:MBWS=Ҭ1WGgl=뮤wBg8i L]}#*EX;~ w43_`!:!gpAE-{  `\xڕJAggA U(jP""I6a/$Ry7 >?ݬA qonfvf`c q4K2&- (6#-d;8C|lf! H v-d#p,$V9Q PE>Iڒmlܣ!v?U<(tOU> ]u#KP-Sa6Wb?iCb/FlFNqM~hT7Y-Qw$=-փJ Jur2om~=21 Mmb_5=7M[@ =sEqpB⣭ ڌ4%ɝeR To`!|8 qp H(`\xڕ+Q=wkkwM"lkBbSPb @ɓRbPbsm⅙{ιsw. f #db cqɱ!sE62Vau4 %%֑)Q&D`jigX*"[*\m1,T} Xŵri;3]VK2xB6'$%Hɗ .!͟q0+iݝ1.'QzU+z&ӌ;]èƛjywTc&u4{;7 P)3_UFOZq~ZK*u-:r77\0| սWi'#z qcTcT1 efC S$%͒cٹN S-.DyK D_Stangeland@UManitoba.CayK Bmailto:D_Stangeland@UManitoba.Ca"Dd B  S A? 23df$6{S!( {Aa^d y׃uυ׻y3'-xp8^r˗'NbxEHo4ҭ8+zONbFyI!Dd B  S A? 2 J$<~9gQ`!_ J$<~9@8 (+-xڍQ=KAݻxgT ADA@*? ? aO''X&E*wٻfg"`dE|dFs9-&Z4ݸh'^mpRWFObi WW*quZgI?;ҏ%]~17lUV"a(Ѩxi_ 7T~lI;h~Og}aNUkfPDEL `gyF.Dd 8B  S A? 2^xEixܿvtr`!l^xEixܿv^`v P:xڕQJP=3II BTtJԅXiBpĝ E?Ekܹ]ԍ07 gfN2H.DL⪪4Ifr:Ob>G`kL#Ae JiSwỲO4t˜oВⴲ!?>#iue7q/o_/ a^;Q?׌ 5yYvFYO޽ojrU%?df$"?흃'`~3ҽ_NO%䀘MF]9eoqbn4 ˁYw~YV5SDd  lB  S A? 2SVzT^U-%9`!SVzT^U-%9`z_xڅRAKAޛt[ :PD/BCz/V%&;tWt+:t~C?  f.f7o9aƚ D2C̔iGf"vc]8v<':{qDN֨k[%lɶtO?I+TUl5mNޠm?KtBPpDϕ,[A?t]T$)~u5+%hy*jc#M͸06 #6Qsmbb nLt]ۊ)Ԅ޹P/qt&2h!KL]͞EAzR3 r =mCZ+~0Yx?e)Dd lP  S A? "2Chewta `!Chewt@}"~xڍR=KADs xh6!-?F-LD=HW} KXX X+ٽ=b*7ޛE]X^sđ)6! R22S#LyҺ}墈ֲ 1x$gFeGChNN3{/R ^Y}MJ"zܭs"~9fz7pOC+g5V.J|YC#t;(j!+"? WsӞ-:O{ ,#LuG{hm`|jnYg:*zFf* v#+ =M@i/g6SwRicRJރ%ߌl5iÜD/!Q0۵!y^;kDd pB  S A? 29.<+!`rGb`!9.<+!`rGb. @2wxڝRJ@m&m1ҋQDś >`{fŠ[\={#x |qwvhE n0o Q`ԲIE) 3HQqf 9S&4  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz|}~3Root Entry F׽iBData {cWordDocumentObjectPool"ڠi׽i_1028465010[F`i`iOle CompObjfObjInfo !$%&),-.1458;<=>?BEHKLMNORUVY\_bcfijmpsuvwxyz{|}~ FMicrosoft Equation 3.0 DS Equation Equation.39qTtmIpyI r ieg =r g m g FMicrosoft Equation 3.0 DS EqEquation Native p_102967395916 Fڠi`iOle CompObj fuation Equation.39qYTII L ieg =L g m g FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native  p_1028468332qF`i`iOle  CompObj fObjInfoEquation Native t_1028465292F`i`iXII  L ied =L d m d FMicrosoft Equation 3.0 DS Equation Equation.39qdĀII r ieg =0.204=0.05=5%Ole CompObjfObjInfoEquation Native _1028468505F`i`iOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qߜĀII L ieg =1 year4=0.25 years=3 monthsEquation Native _1062417839lF`i`iOle CompObj f FMicrosoft Equation 3.0 DS Equation Equation.39qά܀II L ied =6 months6=1 month = 112 yearObjInfo!"Equation Native #_1028550326 $F`i`iOle 'CompObj#%(fObjInfo&*Equation Native +_1028550475)F`i`i FMicrosoft Equation 3.0 DS Equation Equation.39qߨtmIpyI (1+r ieg ) L ied L ieg =(1+r ied )Ole /CompObj(*0fObjInfo+2Equation Native 3 FMicrosoft Equation 3.0 DS Equation Equation.39qߠ I`uI r ied =(1+r ieg ) L ied L ieg "1 FMicrosoft Equation 3.0 DS Eq_1028551154O.F`i`iOle 6CompObj-/7fObjInfo09uation Equation.39qDĀII r ied =(1+.05) 13 "1=.0163963568=1.63963568% (effective rate per one month)Equation Native :`_1028551134'@3F`i`iOle @CompObj24Af FMicrosoft Equation 3.0 DS Equation Equation.39qPI̎I r d =r ied m d{ FMicrosoft Equation 3.0 DS EqObjInfo5CEquation Native Dl_1062591218g;8F`i`iOle FCompObj79GfObjInfo:IEquation Native JP_1062853412J=F`i`iuation Equation.39q4IyI r d =.01639635686=.0983781409=9.83781409% per 6 months compounded monthlyOle PCompObj<>QfObjInfo?SEquation Native T FMicrosoft Equation 3.0 DS Equation Equation.39ql܀II r E =1+r S m() m "1_1029250916,EBF`i`iOle WCompObjACXfObjInfoDZ FMicrosoft Equation 3.0 DS Equation Equation.39qJ\ĀII $10,000$100/basketI FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native [x_1029250924GF`i`iOle ]CompObjFH^fObjInfoI`Equation Native a_1062593188LF`i`iOle dJȢII $10,000(1+r)$100(1+i)/basket FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjKMefObjInfoNgEquation Native h_1029245748QF`i`ilIyI 100(1+r)(1+i) baskets FMicrosoft Equation 3.0 DS Equation Equation.39qJTtmIpyI (1+R)=Ole kCompObjPRlfObjInfoSnEquation Native op(1+r)(1+i)IL`6[  .1  `&  & MathType -]]]]@ -J-_915104006VFPiPiOle qPIC UXrLMETA t(]] Times New RomanR- 2 <PV 2 C 2 r~ 2 = r~ 2 C 2 BPV 2 _r~ 2 CGr~Times New Roman - 2 Wg n 2 qnSymbol- 2 = 2 - 2  + 2 0 2 H0 2 (0 2 W  2 HW  2 (W  2  \J 2 = 2 k 2 - 2 C"+ 2 : 2 : 2 h: 2 a 2 a 2 haTimes New RomanR- 2 1 2  1 2 ,1 2 1 2 1 2 C61 2 (~ 2  )~ 2 C(~ 2 C)~ &  "System- FMicrosoft Equation 2.0 DS Equation Equation.2GRQQ PV=Cr1-1(1+r) n []\C=PVr1-1(1+r) n []-1CompObjWYZObjInfoZEquation Native _943786007T{]FPiPiOle PIC \_LMETA CompObj^`fLM.M@$M.M6[ Q .1  *&)K & MathType -=="-#Symbol- 2 `\J 2 `= 2 l* 2 s- 2  2 ( 2  2 " 2 (" 2 " 2 `#=Times New RomanR-2 `v Monthly MoUkk`U2 ` rtgage Pay~k` 2 `Nment+k 2 l"4 2 4 2 l$50 2 lM,` 2 l.` 2 (~ 2 .` 2 L )~ 2 `%%$381 2 `(.` 2 l000 2 l02 l+ 0065581979 2 1 2 S1 2 12  0065581979 2 `W(61Times New Roman - 2 7 300 &  "System- FMicrosoft Equation 3.0 DS Equation Equation.39qdt 4"Monthly Mortgage Payment=$70,000" 0.006558196941"1(1.00655819694) 300 []=$534.25ObjInfoaEquation Native _1029674151dFPiPiOle  FMicrosoft Equation 3.0 DS Equation Equation.39qYJdI  e@$100,000=  e@C  e@.005  e@1"  e@1  e@1.005  e360 []CompObjcefObjInfofEquation Native _1029674277biFPiPiOle CompObjhjfObjInfokEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qY\JI  e@C=  e@$100,000.005  e@  e@1"  e@1  e@1.005  e360 [] e@=$599.95. FMicrosoft Equation 3.0 DS Equation Equation.39qYJ|I  e@PV=  e@$599.55  e@.005  e@1"  e@1  e@1.005  e30_1029674385nFPiPiOle CompObjmofObjInfopEquation Native  _1028543988sFPiPiOle CompObjrtf0 [] e@=$93,054.36. FMicrosoft Equation 3.0 DS Equation Equation.39qTĀII PV=C n (1+r) nObjInfouEquation Native p_1028544045xFPiPiOle  FMicrosoft Equation 3.0 DS Equation Equation.39qlġInI PV=Cr1"1(1+r) n []CompObjwyfObjInfozEquation Native _1028454023}FPiPiOle CompObj|~fObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qĀII PV=$2,500.051"1(1+.05) 40 []=$42,897.72_1028453799FPiPiOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qĀII PV=$5,000.10251"1(1+.1025) 20 []=$41,851.43Equation Native _1028544070FPiPiOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39qxpII PV=$100(1+.05) 10 =$61.39 FMicrosoft Equation 3.0 DS EqObjInfoEquation Native _1028544060vFPiPiOle CompObjfObjInfoEquation Native 1Tablesuation Equation.39q|xII PV=$100(1+.1025) 5 =$61.39Oh+'08DT p|    uyT0-!REJ*;5jwZ;>@GyOJ|*jYj`Rь53[o~/vhhd݊"/'-p9-K L`yQ'8"ClVꋃhFJK,pbAxtJxwTyV>w_h.ufVܩ?|Zπrd+J2 o' w̚cbt!'J>~b]jX_Dd B   S A ? 2i#WD}h+L`!i#WD}h+ _@2kxڍRJ@m UЃ(R=*`=>-PzEͳ7R`FۋݰaLP/X}f: _BR$ETmtFyA&lega>W|<A;k :!@G̖sJXF>OMkkޠL7ZQ ~p؎,hXPxn :ⷜ)v2ߟɄ"qTGաf2]0%#~#D2{EZX`j>j4}>ZlbuW3nFSZo%WM)r^dCTt{a0١D'hDd (0B   S A ? 2DC3f )pgu `!C3f )pgu@0JkxڍK@߽hր_WbQAQuqz҂iJn΂8/8;/"]I4%\w1Hh/zZś@Zċʑ #9d }Mm>!X;K&0?( 0 1``jM 'ޅXSDo:/=gfsl̏]4UI;|tLDvR7x!%ʄj#Qʳk {NHQ+:W )0wwS> t'$)g ´!U&p=w -. |>?ʼnSc,'Z+x T,B)it2=vTW3{c6qH:Q[OʕѕLK:_es*qsbv/cHV=AYiOup.3ƴ"ܟ. =׋}+zb1!<^ݳJ bԊc-\<'eJ=p|ޏ<"fKanj *zd0NV^ߴ`_p1rv"Dd B  S A?  23df$6{S!( {Aa^d y׃uυ׻y3'-xp8^r˗'NbxEHo4ҭ8+zONbFyI_Dd B  S A ?  2i#WD}h+W`!i#WD}h+ _@2kxڍRJ@m UЃ(R=*`=>-PzEͳ7R`FۋݰaLP/X}f: _BR$ETmtFyA&lega>W|<A;k :!@G̖sJXF>OMkkޠL7ZQ ~p؎,hXPxn :ⷜ)v2ߟɄ"qTGաf2]0%#~#D2{EZX`j>j4}>ZlbuW3nFSZo%WM)r^dCTt{a0١D'hDd hB  S A?  2yԢҁ7"xU`!MԢҁ7"x@e |xڍQAJAMv Oo ^=' _ !<{Nl0]=U]ݽ;N5$K⪪4sH]16ot>g]oa;rѓ;tvbjk.*m$ F>Z3/z澸 "疻:I;i?Df _kIX468|)<8钝S5x3OA6xY{%=r$O^/)h_IYeDd J   C A ? "2Y#}#V(">}n!`!Y#}#V(">}n F(+ixڍ;KAg x )l$B,1&eϨt Yx3YB*ؽwPpvQ 0<ՋĖ:! Re22S#4ꀆJtIe&`2{eݪ^6hJ7s;U$biщ֔roJ_Də lQRފ;up Pcx%|j32uA9o$|Lo17 םDZ6^hXO6KSJUWaϬZk~z/6޶˼`W7˛25WDF7D19O'`b$$If!v h55"55N55N55N5 5 5 #v#v"#v#vN#v#vN#v#vN#v #v :V l4 t055"55N55N55N5 5 / 4f48kd#$$Ifl4 >* x d,""NNN t0,,,,4 laf48$$If!vh55555'5'55'5 '5 5 '5 '5 5 555H5H#v#v#v#v'#v#v '#v #v '#v #vH:V l t05555'55 '5 5 '5 5H/ / / / / /  / / /  / / /  / / /  / / / /  /  / / / /  / / / / / / /  4<kd'$$Ifl֐>* Q x d,"''''''HH t0HHHH4 la8$$If!vh55555'5'55'5 '5 5 '5 '5 5 555H5H#v#v#v#v'#v#v '#v #v '#v #vH:V l t05555'55 '5 5 '5 5H/ / / / / / / / / / / / / / / / / / /  /  / / / / / / / / / / /  4<kd-$$Ifl֐>* Q x d,"''''''HH t0HHHH4 lab$$If!v h55"55N55N55N5 5 5 #v#v"#v#vN#v#vN#v#vN#v #v :V l4 t055"55N55N55N5 5 / 4f48kdt4$$Ifl4 >* x d,""NNN t0,,,,4 laf4b$$If!v h55"55N55N55N5 5 5 #v#v"#v#vN#v#vN#v#vN#v #v :V l4 t055"55N55N55N5 5 / 4f48kd8$$Ifl4 >* x d,""NNN t0,,,,4 laf4 Dd <B   S A ? 2w❕Ed\S;`!K❕Ed\` @2x=PJP=374 UA\"|~?`Z\!H4]tƯ ?O\`{pܙ!t =k]VH0e Yf1/谳}4ZXD3'5^EVߍ'e|j/B~<}熶CwӿVN3#yiB=\tfC^>zPINLS,#+JT%ъ2ǒϩumcvJ̃IU&oq#9vDd B   S A ? 2cUC~l*>`!cUC~l* dxڅR=KA$D[KS (z,m$F_`&(x-Gn̛7PP`x.[ބ((ij&1J<*CoR[ U[OzzK$vF$r/MqT\X}ĚV~7Pz&a.ՂҮ w wj5ۇAI|_Xo ^2f\5Rx-W*Qf'XმUTh;YEM+FQbƊ{DR37W 39j^[sL%Kz? ݰIzd^rL]6\Nh?l\a?䝸䊓|(zׇDz B3Y`PDd P  S A? "2.,(%c3kuZHw@`!.,(%c3kuZHP@ hdNx]N@ ?;%B (ꀊ 0>@[)E vf&`HN?ǟ e|sfX\&Rq٥54\Ot!~3\kuFHE'C!ńx3œV!}M~ޒ̈4">pmVݫt.udayI;}[_* > 4 ^U9CiZ? 峂i{V_v\z]2DqX!tьEO/kǏmTtc:HW3B;&jkdIIDd B  S A? 2e;$IB`!e;$Iɢ `dUxڅRJ@$m jA<" U>m #"yEٟwEЇ<)gg%ŅI\uPaqIi6 !)IlXڄ){f~UjnViV QN3~e%t*ǽijo4%Kt(mWC{R3Dᨱ^4:q?Q*ۊ"]Rmjc|c1>M,1d;: gT60,>3dKhaV=˫Afw2@GKxTCGDd gB  S A? 2F2q9 Oϋ~E`!F2q9 Oϋ~t@*~A SxkAfflhclcmJ &Lnm E x06-4bBbEHU<&xГwO(S0Λk|{3%(@o-BJd\ٍ ڍ+ dt2G7H 3>Zj-XaцA=4Zcz3?Z@{cJ nv+]"k o9ﵝVքͯ2:&癵n:v4VBgŻ쁃|~}]w88G8fc2SȮroy͏K;{)*Od>y#9i~$8+Or:xˆgB'q] q.: zĐ^{ vn 6:N%fo%Ew]4wprPi\mmؙQegp^Td+SbY`!xT޿o)Ōgn0W)Efj@@O NǸx]Mv8 W_;xV(Qh }dm5GiyU˥R0*]z2[ m%z+v&uUQpb818a?VLݽPC[Cm:Lt-nAUZ|"OyDd P B  S A? 20HEЯPXFt XO`!HEЯPXFt ~ xڍ;KAggo%AJD(&  F#*, װX),FGacݝу湻 \nPEe9r!czaEzgeN`.wHk =)'𥾑֥\RJ4m ,n-kZSFK}}:](PVv&hfh=$S5#;D+p37+ku#+x6paw>piEVD!!L _#ѪCgUvʦSUUY(,A9-pE[#iv迳$ )TG}Zs~VYZ_lN"p;GOvC֍`S?TLIq*+Frӳz<˓~=d$|3oLևVe2ľRms+Ļ%lDd 8B  S A? 2"0RǓ3~P\.- FR`!0RǓ3~P\.- F@ D#xڍKA߼1mW헔D-CХKPDF  2D?nx :UE̼٭20x/y@Eč CU&UfV1KRLtR"DxFkS۬r-AWFc(qhDD 7a9o@/?eksVkqzR9x gwXehB.`5gq|mGW^I)`F?19^_+F~;6JTqsCsJ98bXq¥bhi@5W1FyV5[T/V JbȓlHBR|CsvLS'~ď}Xϳ2}[2i76AwkALѭzbT2@w\{EBDdJ  C A? "2mC%p`!xm|`@ PFxڍR=KA}3w]./#(+m;EL ^& ) ; ; =wg+Dc0ޛ%TgЫvuD!βL24#O_7BEs1"M~Vѽw>)a?n b`M^uQ\">::Tѭ~p~\F:I^_FݤRBPX ӵ_iX60_ed7ʙkaJoX&awĹҪ̰sWFYX}asvOM/srkrbj={ zs Po0|XDdpJ  C A? "2:!gpAE-{'p`!:!gpAE-{  `\xڕJAggA U(jP""I6a/$Ry7 >?ݬA qonfvf`c q4K2&- (6#-d;8C|lf! H v-d#p,$V9Q PE>Iڒmlܣ!v?U<(tOU> ]u#KP-Sa6Wb?iCb/FlFNqM~hT7Y-Qw$=-փJ Jur2om~=21 Mmb_5=7M[@ =sEqpB⣭ ڌ4%ɝeR ToDdTJ  C A? "24|8 qpp`!|8 qp H(`\xڕ+Q=wkkwM"lkBbSPb @ɓRbPbsm⅙{ιsw. f #db cqɱ!sE62Vau4 %%֑)Q&D`jigX*"[*\m1,T} Xŵri;3]VK2xB6'$%Hɗ .!͟q0+iݝ1.'QzU+z&ӌ;]èƛjywTc&u4{;7 P)3_UFOZq~ZK*u-:r77\0| սWi'#z qcTcT1 efC S$%͒cٹN S-.Dd0J  C A? "2Cg-XDw^j);?Zp`!g-XDw^j);?*`\xڕKA߼h~5DĢݴ^*Dr3B)(Jտ?@(H[Xz9 xoDH.M!e3~NA w #dYb crarf=3.20.՜N kn < \{"MB$a|xW"b/CAόc*8b5&h8wNCw*>zݨdu23o>)j勐) ^"}$&^\c< n_i}VwM\guУR5C$](y^sG}*VҺ_K`W-v19#]WQ0J) Uigyk_Q+؏ ko5JG.k^=ͫOL_j2?kIS[8g#<T7@l-˛$% Ò>˻gGUZ\ވiDd J  C A? "2yS}ϥ]u/p`!yS}ϥ]u@mxڕRJA=nM D|`H"*LaAōBJ?A N\l{kι3 4|0YI3,Vl&Rq)Ѥz*LQvu9LG5c 3XfD99l7; >V*ꉋd3kX63ë~_σf x]WoJtJ:is1DfyyS/q?!YTu[4r$^.G-:MBWS=Ҭ1WGgl=뮤wBg8i L]}#*EX;~ w43_gDd J  C A? "2_'\M XA%%2p`!_'\M XA%@YkxڕRMK@mSV_ JQT*xyA MAn Effective Method for Teaching and Understanding Interest Rate Conversions Title 8@ _PID_HLINKSA|k!mailto:D_Stangeland@UManitoba.Ca      !"#$%&'()*+,-./0128@8 Normal_HmH sH tH T@T Heading 1$ !*$@&@CJmH sH uF@F Heading 2$@&>*CJmH sH uDA@D Default Paragraph FontVi@V  Table Normal :V 44 la (k(No List 0U@0 Hyperlink>*B*6@6  Footnote Text@&@@ Footnote ReferenceH*2B@"2 Body TextCJFV@1F FollowedHyperlink >*B* ph4 @B4 tSFooter  !.)@Q. tS Page NumberH -$&'<AGOPReRUY.bd7s yJ,  g Z tS2345678Qj2345678Qj    =>ABWXmnxyz{|}~XYj k   ?@IKLMNwyz{|  L M 0!1!''r's'''(((7+8+,,,,-Q-R-m---+.,.o.~......O/P///200001\11112j2k2333333 4 44455556688.8/888l:m:::;;<<r=s=>>>>>>????@@@@@AAAYAZAAApBqBBDCECCC%E&EEEEEGGGGIITLUL\L^L_LaLbLdLeLgLhLiLlLmLnLoLpLqLrLsLtLuLvLwLxLyLzL{L|L}L~LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMMM$M%M+M,M2M3M9M:M;MGMHMIMtMuMKOLO Q!QRRSSUU+Y,YLYMYU\V\``ddOePeeeee5fZf\fkffffggg@gBg[g]gxgygggh hhhjj,j-jllmmmmmCnDnsntnnnooPoQoooppsppp qYqZqqq2r3rrrrr9s:sNtftgtt!v'v)v]v_vvvvvvvvvvvvv&w+w,w.w6w>wFwMwXwYw[wfwnwvw}wwwwwwwwwwwwwwwwwwxxyyuzvzF{G{(|)|}}~~dWpqOPԉՉމ߉ !"#$%&'?@XYqr0000000000000p00000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0 0 0 0 0 0 0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00606060606060600m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:0m:00e005f0000f00000g00Bg00]g0]g0]g0]g0]g0]g0]g0]g0]g0]g0]g0]g0]g0]g0]g0]g00m00m0m0m0m0m00n0n0n00Qo0Qo0Qo0Qo0Qo0Qo0Qo0Qo0Qo0Qo0Qo0Qo0Qo0Qo0Qo0Qo 0Qo 0Qo0Qo0Qo0Qo0Qo0Qo 0Qo0Po 0Po0Po0Po0Po0Po0Po0Po0Po0Po0Po0Po0Po0Po0Po0Po0Po0Po@0Po0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00Po0Po0Po0Po@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0p0S(@0@0My00@0@0My00H@c0lb0000000000000000000 `*6;=EHKOpr7~*JOQSUVXZ[]bdgopt%738=rFKbUUU[px |_+MvƂKMNPRTWY\^_`acefhijklmnqsL@n333 44!4b4v4x45555558*8,8>>>>>>???@@@AAUAWABBB,C@CBCCCCEEEffffgg(gggggNtbtdtgt{t}tX::::::::::::::::::::: !! B V X :::-/8LNQegj~::::::t)+42$g-XDw^j);?*2$mI2$_'\M XA%$2$yS}ϥ]un2$:!gpAE-{2$|8 qp@ *(  \  S A ?"  s A ?^ ` !F V v/B =BR!RRRRF!F"\  S A  ?" \  S A  ?"  s A  ?^ ` !F V v/B =BR!RRRRF!F"  s A  ?^ ` !F V v/B =BR!RRRRF!F"\B & S D"\B ' S D"\B ( S D" \B ) S D"\B * S D" B S  ?IwZfgg@g[gTx(p@X -T  T@|t@&@t)Yt' Y t(t*t OLE_LINK2 OLE_LINK1 OLE_LINK4 OLE_LINK3UV=<<><?_@gAtlBCd* DlE cF=^GdlHgIJKTL<jM@55>>>>>>??@@BBCCEEyՉ333>Bn8'( 4455>>>>??@@BBCCEEULLMIM,YMYmm!v'vvvvv&w+w,wwy9WPՉyՉDave StangelandDave StangelandDave StangelandDave StangelandDave StangelandDave StangelandDave StangelandDave StangelandDave StangelandDavid A. Stangeland#J0#A>*)k <>Og-LMWE즺G)R즺bZW{*^`OJPJQJ^Jo(- ^`OJQJo(o pp^p`OJQJo( @ @ ^@ `OJQJo( ^`OJQJo(o ^`OJQJo( ^`OJQJo( ^`OJQJo(o PP^P`OJQJo(^`.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.0^`0o(()h   ^ `OJQJo(h ^`OJQJo(oh ^`OJQJo(h | | ^| `OJQJo(h LL^L`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh ^`OJQJo(h^`.h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L.h ^`OJQJo(h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L.^`OJPJQJ^Jo(- ^`OJQJo(o pp^p`OJQJo( @ @ ^@ `OJQJo( ^`OJQJo(o ^`OJQJo( ^`OJQJo( ^`OJQJo(o PP^P`OJQJo(k Og->MWEG)RbZW#J @hh^h`CJOJQJo( )PO/_tSNSA 455>>EEITLUL\L^L_LaLbLdLeLgLhLiLlLmLnLoLpLqLrLsLtLuLvLwLxLyLzL{L|L}L~LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMMM$M%M+M,M2M3M9M:M;MGMHMqqgtt!v'v]vvvv+w,w.w6w>wFwMwXwYw[wfwnwvw}wwwwwwwwwwwwwwwwwyy9ԉy0a0M@L:;eg@@@B@@@n@@@@UnknownGz Times New Roman5Symbol3& z Arial?5 z Courier New;Wingdings"hYf8Le3Le5g>5g>#24dyy 3QH(?LAn Effective Method for Teaching and Understanding Interest Rate Conversions3David A. Stangeland, Ph.D. & Charles Mossman, Ph.D.David A. Stangeland,          FMicrosoft Word Document MSWordDocWord.Document.89q