© 2002 david a. stangeland 0 outline i.dealing with periods other than years ii.understanding...

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© 2002 David A. Stangeland

Outline

I. Dealing with periods other than years

II. Understanding interest rate quotes and conversions

III. Applications – mortgages, etc.

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I. Dealing with periods other than years

Definition: Effective interest rates are returns with interest compounded once over the period of quotation. Examples: 10% per year compounded yearly 0.5% per month compounded monthly

PV and FV Calculations for a single cash flow As long as you have an effective interest rate

there is only one thing to ensure: set the number of periods for PV or FV calculation in the same units as the effective rate’s period of quotation.

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Examples You expect to receive $50,000 in 90 days. What is the

PV if your relevant opportunity cost of capital is an effective rate of 6% per year? Note if the you are told it is an effective rate of 6%

per year, then this implies 6% per year compounded yearly.

You have just invested $100,000 and expect your return to be 4% per quarter compounded quarterly. How much do you expect to accumulate after 5 years?

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Annuities and perpetuities The annuity and perpetuity formulae require the rate

used to be an effective rate and, in particular, the effective rate must be quoted over the same time period as the time between cash flows. In effect: If cash flows are yearly, use an effective rate per

year If cash flows are monthly, use an effective rate per

month If cash flows are every 14 days, use an effective rate

per 14 days If cash flows are daily, use an effective rate per day If cash flows are every 5 years, use an effective rate

per 5 years. Etc.

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Example You are obtaining a car loan from your bank and the

loan will be repaid in 5 years of monthly payments beginning in one month. The amount borrowed is $20,000. Given the rate that the bank quoted, you have determined the effective monthly interest rate to be 0.75%. What are your monthly payments?

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II. Understanding Interest Rate Quotes and Conversions

The TVM formulae we have used all require rates that are effective. Unfortunately, rates are rarely quoted in a way that we can input, as is, into our TVM formulae or calculator functions.

Thus we must be competent in converting between the rates that are quoted to us and the equivalent rates that are necessary for our calculations.

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Interest Rate Conversions Step 1: finding the implied effective rate

Identify how the rate is quoted and, if not an effective rate, convert into the implied effective rate. Examples: 10% per year compounded yearly

This rate is already effective, so there is nothing to do for step 1.

60% per year compounded monthly This rate is not effective, but it implies – by definition –

an effective rate of 5% per month Note: the quoted rate of 60% per year with monthly

compounding is compounded 12 times per the quotation period of one year. Thus the implied effective rate is 60% ÷ 12 = 5% and this implied effective rate is over a period of one year ÷ 12 = one month.

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Step 1: finding the implied effective rate

In words, step 1 can be described as follows: Take both the quoted rate and its quotation

period and divide by the compounding frequency to get the implied effective rate and the implied effective rate’s quotation period.

The quoted rate of 60% per year with monthly compounding is compounded 12 times per the quotation period of one year. Thus the implied effective rate is 60% ÷ 12 = 5% and this implied effective rate is over a period of one year ÷ 12 = one month.

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Step 1: additional examples to find the implied effective rate

16% per year compounded quarterly

9% per year compounded semi-annually

11% per year compounded bi-yearly (every two years)

100% per decade compounded every 10 years

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Step 2: Converting to the desired effective rate

Example: if you are doing loan calculations with quarterly payments, then the annuity formula requires an effective rate per quarter.

Once we have done step 1, if our implied effective rate is not our desired effective rate, then we need to convert to our desired effective rate.

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Month:Quarter:

1 2 3 months1 quarter

$1 $1.05 $1.1025 $1.157625

Step 2: continuedConverting between equivalent effective rates

Use the example of 60% per year compounded monthly and the implied effective rate of 5% per month . . . we need an effective rate per quarter. Consider how $1 grows after 3 months . . .

x 1.05 x 1.05 x 1.05

x 1.157625

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Step 2: continuedEffective to effective conversion In the previous example, 5% per month is equivalent to

15.7625% per 3 months (or quarter year). This result is due to the fact that (1+.05)3=1.157625

As a formula this can be represented as

where rg is the given effective rate, rd is the desired effective rate.

Lg is the quotation period of the given rate and Ld is the quotation period of the desired rate, thus Ld/Lg is the length of the desired quotation period in terms of the given quotation period.

1)r(1ror )r(1)r(1 g

d

g

d

L

L

gddL

L

g

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Step 2: additional examples to find the desired effective rate 9% per year compounded semi-annually; from step one

this gives us 4.5% per six months (effective rate). Suppose we desire an equivalent effective rate per

month

Suppose we desire an equivalent effective rate per year

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Step 3? For the purpose of doing TVM calculations,

generally we are ready after doing steps 1 and 2 as we have obtained our desired effective rate and can now use it in the TVM formulae.

Unfortunately, there are some circumstances when we desire a final rate quoted in a manner that is not effective – here a third step is necessary.

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Step 3: finding the final quoted rate Identify how the final rate is to be quoted and, if not an

effective rate, convert from the desired effective rate (determined in step 2) into the desired quoted rate. Examples: Desired rate is to be quoted as a rate per quarter

compounded quarterly This rate is already effective and was determined in step 2

(where, using a previous example, we calculated 15.7625% per quarter), so there is nothing to do for step 3.

Desired rate is to be quoted as a rate per year compounded quarterly This rate is not effective, but 15.7625% per quarter (from step

2) implies a desired quoted rate per year compounded quarterly of 63.05%

Note: the desired quoted rate is quoted per year with quarterly compounding; i.e., compounded 4 times per the quotation period of one year. Thus the desired quoted rate is 15.7625% per quarter x 4 = 63.05% quoted over one year (= 4 x one quarter of a year) compounded quarterly.

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Step 3: finding the final quoted rate

In words, step 3 can be described as follows: Take both the implied effective rate and its

quotation period and multiply by the compounding frequency of the desired final quoted rate. This results in the desired final quoted rate and its quotation period.

In our example, the desired quoted rate is a rate per year compounded quarterly. Therefore the compounding frequency is 4. We multiply 15.7625% per quarter by 4 to get 63.05% per year compounded quarterly.

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Step 3: additional examples Given an effective rate of 15.7625%

per quarter, find the following: The rate per six months compounded

quarterly The rate per 2 years compounded

quarterly The rate per month compounded

quarterly The rate per 1.5 months compounded

quarterly

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Interest rate conversions: additional examples Bank of Montreal is offering car loans at 8% per

year compounded monthly. You manage Catfish Credit Union where rates are quoted as “per year compounded semiannually”. What is the most you could quote to remain competitive with Bank of Montreal?

Step 1: Note: since your final quoted rate will be

compounded semiannually, you would like to (in step 2) convert the B of M rate into an effective rate per 6 months. So step 2 depends on the desired outcome in step 3!

Step 3

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Interest rate conversions – continuous compounding – self study

Consider steps 1 and 2 combined together in a formula to convert a quoted rate per period compounded m times into an effective rate over the same quotation period …

Note: this formula only handles steps 1 and 2 when the final effective rate has the same quotation period as the initial quoted rate. This formula is not recommended as it does NOT work in most situations and is only shown because of the derivation that follows.

effective

m

quoted rm

r

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Do not use this formula. Use the 3-step method shown in the prior slides as that method works generally and this formula only works in one special situation.

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Continuous Compounding – self study (continued)

Using the previous formula and mathematical limits …

)rln(1r

,r tor from

...direction other in theconvert To

interest of rate compoundedly continuous

thebe tosaid is r ,m As

e r1 ,m As

periodper effectivegcompoundin continuous with periodper

gcompoundin continuous with periodper periodper effective

quoted

reffective

quoted

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What is the effective annual rate, given a quoted rate of 20% per year with… Monthly compounding answer=21.939108% Daily compounding answer=22.133586% Compounding every hour answer=22.139997% Continuous compounding answer=22.140276%

What is the rate per year compounded continuously if the effective annual rate is … 10% answer=9.531018% 50% answer=40.54651% 100% answer=69.31472%

Continuous Compounding – self study – examples to try

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III. Applications of TVM

Quotations on mortgages Quotations on bonds Quotations on credit cards Quotations on personal loans and car

loans Mortgage and loan amortizations

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Canadian Mortgage Quotes Canadian mortgages are quoted as rates per year

compounded semiannually. In this course, unless otherwise noted, assume all mortgage quotes are quoted in the above manner. (Note, some of the text problems may not make this

assumption, but all class assignments and exams will make this assumption unless otherwise noted).

Normally a constant series of monthly payments is required to repay the mortgage. What interest rate is required to do TVM calculations for the mortgage if the quoted rate is 6%?

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Bond Yields A bond’s yield is essentially the IRR of the bond and is

quoted as a rate per year compounded semiannually. In this course, unless otherwise noted, assume bond yields are quoted in the above manner. (Note, some of the text problems may not make this assumption,

but all class assignments and exams will make this assumption unless otherwise noted).

Most corporate and government bonds have constant semiannual coupon payments and a lump sum terminal payment. What interest rate is required to do TVM calculations on the bond coupons if the yield is quoted as 8%?

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Credit Cards CIBC Visa quotes the annual interest rate of 19.50% and

the daily interest rate of 0.05342%. How are the two rates quoted? What is the effective rate per year charged by CIBC Visa?

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Personal Loans and Car Loans Most banks quote the interest rates on personal

loans and car loans as rates per year compounded monthly.

Since personal loans and car loans generally require equal monthly payments, what interest rate would be used in TVM formulae if the quoted rate was 12%?

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Mortgage and loan amortizations A mortgage contract specifies the quoted rate and the

amortization period for the payments. The amortization period is often longer than the duration of the contract. Thus we must determine the payments, the amount of interest and principal paid each month, and the outstanding principal at the end of the contract.

Example: You have just negotiated a 5 year mortgage on $100,000 amortized over 30 years at a rate of 8%.

What are the monthly payments? What are the principal and interest payments each month

for the first 3 months of payments? How much will be left at the end of the 5 year contract? If the mortgage terms do not change over then entire

amortization period, how much interest and principal reduction result from the 300th payment?

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Mortgage example First determine the relevant effective rate for TVM

calculations.

Next determine the monthly payment.

Now utilize the table on the next page to understand how a mortgage amortization schedule works.

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Mortgage amortization schedule

$99,721.71 $70.26 $724.71 $654.46 $99,791.97 4$99,791.97 $69.80 $724.71 $654.91 $99,861.77 3$99,861.77 $69.34 $724.71 $655.37 $99,931.11 2$99,931.11 $68.89 $724.71 $655.82 $100,000.00 1$100,000.00 0

= A - D= C - B= E0 ÷ anr%= A • r%

Principal outstanding at the end of the month (after the

payment)

Principal reduction with monthly payment

Monthly paymentInterest charged during the

monthPrincipal outstanding at the beginning

of the monthMonth

EDCBAColumn:

Note: as time goes by, the principal outstanding is reduced and therefore the interest charge per month drops. This results in more of the monthly payment going toward principal reduction as time elapses. The way the annuity payments are calculated, the last payment will have just enough principal reduction to repay the remaining principal owed and then the loan will be repaid.

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Mortgage continued How much will be left at the end of the 5-year contract?

After 5 years of payments (60 payments) there are 300 payments remaining in the amortization. The principal remaining outstanding is just the present value of the remaining payments.

How much interest and principal reduction result from the 300th payment? When the 299th payment is made, there are 61 payments

remaining. The PV of the remaining 61 payments is the principal outstanding at the beginning of the 300th period and this can be used to calculate the interest charge which can then be used to calculate the principal reduction.

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Summary and conclusions Cash flows that occur in different time periods cannot be added

together unless they are brought to one common time period. We usually use PV to do this and sometimes FV.

PV and FV calculations were done for single cash flows, constant annuities and perpetuities and growing annuities and perpetuities. In addition, we used the concepts of NPV and IRR.

For annuities and perpetuities, we must ensure the discount rate is effective and quoted over a period the same as the time period between cash flows.

TVM principles are useful for analyzing consumption and investment decisions. TVM principles are also useful for working with loan and mortgage amortizations.

If you understand TVM principles, you do not need to blindly rely on another party to determine value or interest costs. You know what factors affect these and you can determine reasonable numbers for yourself.

© 2002 david a. stangeland 0 outline i.dealing with periods other than years ii.understanding...

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