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8/3/2019 Measurement of Interest- Slides
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Measurement of Interest - IMATH 384, Review lecture 1
Andrei Ordine
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In this lecture
Compound interest, simple interest, timemeasurement conventions, present value, discount
factor.Applications: T-bills.
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Interest
Interest = time value of money
Virtually all financial transactions involve interest.
Objective of this course is to provide precisemathematical descriptions of financial transactions.
Terminology, formulas, real-world examples.
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Example: Compound interest
Problem 1 Suppose that $100 is deposited into a bankaccount which earns interest at the effective annual rate
of 10%. What is the balance after 3 years?
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Example: Compound interest
Answer:
Table 1: Compound interest
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Example: varying rates
Problem 2 A Bank of Montreal mutual fund has thefollowing effective rates of interest in years 1998 - 2003:
Year Rate of interest
Table 2: Investment ratesMeasurement of Interest - I p.6/4
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Example: varying rates
Answer:
(1)
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Compound interest: usage
Compound interest is almost exclusively used for trans-
actions spanning more than 1 year, and for many shorter-
term transactions as well.
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Definitions
Consider an investment of an amount of money atinterest.
Principal: amount initially investedAccumulated value: amount after a period of time
Amount of interest: difference between theaccumulated value and the principal
Measurement period: a unit of time, in most cases ayear, sometimes a month, a day, etc.
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Definitions II
Amount function
: accumulated value at time
of an original investment. For compound interest,
we have
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Definitions
Accumulation function
: amount function for anoriginal investment of $1.
, where
is the amount of theoriginal investment
.
These definitions assume that the fund is only
growing through interest, ie., no principal is addedor withdrawn.
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Formal definitions
Effective rate of interest during the first period
(2)
During the th period:
(3)
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Varying rates of interest
Generalizing our last example, we have
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Example: Simple interest
Problem 3 Suppose that a principal of
is investedat simple interest of 10% per year. What is the
accumulated amount at the end of 3 years?
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Example: Simple interest
Answer:
Table 3: Simple interest
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Example: Simple interest
Problem 4 A Canadian T-Bill with face value $100 is asecurity which is exchangeable for $100 CAD on the
maturity date.Suppose that a T-Bill with face value $100 is issued on2005.09.08 and matures on 2005.12.15 (there are 98days between the dates). Given that the price of theT-Bill is $99.27076, find the annual rate of interest.Assume simple interest.
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Simple interest: usage
Simple interest is occasionally used in short-termtransactions.
Simple interest can be used to approximatecompound interest.
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Simple v Compound interest
and
100
0
100
200
300
400
500
A
4 2 4 6 8 10 14t
100
0
100
200
300
400
500
A
4 2 4 6 8 10 14t
Figure 1: Simple interest and compound interest,
,
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Assumption
Amount functions for simple and compound interestare continuous, not stepwise.
For example, if an investment of $100 earns simpleinterest of 10% per annum, paid at the end of eachyear, then the amount function at time
is $105,
not $100.
50
100
150
200
250
300
A
50
100
150
200
250
300
A
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actual/actual
365 days in a year
Actual number of days between dates
Simple interest calculated on this basis is calledexact simple interest, denoted actual/actual.
This method is used for Canadian Treasury Bills.
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30/360
360 days in a year
Number of days between two dates is
(5)
where is the first date, is
the second date.Simple interest calculated on this basis is calledordinary simple interest, denoted 30/360.
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actual/360
360 days in a year
Actual number of days between dates
Simple interest calculated on this basis is calledBankers Rule, denoted actual/360.
This method is used for U.S. Treasury bills
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Present value
Present value: two equivalent formulations.
How much is a promise of a future payment
worth now?How much should I invest now to have at acertain point in the future?
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E l P t V l
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Example: Present Value
How much do I need to invest at the beginning of theyear to have $100 at the end of the year, if the annualeffective rate of interest is 10%?
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E l P t V l
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Example: Present Value
Answer:
(6)
Discount factor is
The present value of
payable at the end of theyear is
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E l P t V l
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Example: Present Value
Problem 5 A Canadian T-Bill with face value $100 is asecurity which is exchangeable for $100 CAD on thematurity date.Suppose that a T-Bill with face value $100 is issued on2005.09.08 and matures on 2006.03.09 (there are 182days between the dates).Assuming simple interest, given that the annual rate ofinterest is 2.810%, find the price of the security.
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E l P t l
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Example: Present value
Answer: 98.61821
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Example: U S Treasury Notes
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Example: U.S. Treasury Notes
Problem 6 Suppose that a 2-year Treasury note withface value $1,000, coupon rate 2.75% is sold for$999.09. Find the effective annual rate of interest. Roundto 4 digits.
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Example: U S Treasury Notes
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Example: U.S. Treasury Notes
Solution plan
1. Calculate the two payments (coupon and
coupon+face value)2. Write the present value of each payment
3. The sum of the two present values is the purchase
price
4. Solve the equation to find and then calculate
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Example: U S Treasury Notes
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Example: U.S. Treasury Notes
Answer: 2.797%Source: http://wwws.publicdebt.treas.gov/AI/OFNtebnd
Note: the solution can be found quickly using a financialcalculator.
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F l d fi iti
http://wwws.publicdebt.treas.gov/AI/OFNtebndhttp://wwws.publicdebt.treas.gov/AI/OFNtebnd8/3/2019 Measurement of Interest- Slides
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Formal definitions
Discount factor:
Recall:
is the accumulated amount of an
investment of
at time 0.Present value of $1 payable at time
is
For simple interest:
For compound interest:
For the Treasury note, the price is the sum of presentvalues of the payments from the security.
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R l ld
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Real world
1. Historical rates
2. What and who determines the rates
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Historical Rates
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Historical Rates
Key rates of interest, last 10 years
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Notes to the historical rates chart
http://www.bankrate.com/gookeyword/news/fed/fedchart.asphttp://www.bankrate.com/gookeyword/news/fed/fedchart.asp8/3/2019 Measurement of Interest- Slides
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Notes to the historical rates chart
Fed funds rate is the rate at which banks lend theirfederal reserve deposits overnight.
Prime is one of several base rates used by banks toprice short-term business loans.
Rates of interest depend on the market and timing of
the transaction.
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Notes to the historical rates chart
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Notes to the historical rates chart
NBER a includes the average prime rate ofcommercial banks into the list of lagging indicatorsof the economic cycle.
aNational Bureau of Economic Research
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What and who determines the rates
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What and who determines the rates
Rates depend on the economic climate
Rates are a lagging indicator of economic cycle
Central banks are major players in the moneymarkets. They promote monetary policies usingregulation and open-market operations.
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Sources Canadian rates
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Sources - Canadian rates
BankRate.comhttps://www.bankrate.com/can/rate/mtg_home.asp
Weekly financial report - Bank of Canadahttp://www.bankofcanada.ca/en/wfsgen.htm
Bank of Canada rates page
http://www.bankofcanada.ca/en/rates.htmBank of Canada securities auctions:http://www.bankofcanada.ca/en/cars/cars.html
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Sources - U S Rates
https://www.bankrate.com/can/rate/mtg_home.asphttp://www.bankofcanada.ca/en/wfsgen.htmhttp://www.bankofcanada.ca/en/rates.htmhttp://www.bankofcanada.ca/en/cars/cars.htmlhttp://www.bankofcanada.ca/en/cars/cars.htmlhttp://www.bankofcanada.ca/en/rates.htmhttp://www.bankofcanada.ca/en/wfsgen.htmhttps://www.bankrate.com/can/rate/mtg_home.asp8/3/2019 Measurement of Interest- Slides
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Sources - U.S. Rates
Selected interest rateshttp://www.federalreserve.gov/releases/h15/Current/
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Sources - History
http://www.federalreserve.gov/releases/h15/Current/http://www.federalreserve.gov/releases/h15/Current/8/3/2019 Measurement of Interest- Slides
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Sources History
10 year historical interest rateshttp://www.bankrate.com/gookeyword/news/fed/fedchart.asp
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Sources - Fed Rates vs GDP
http://www.bankrate.com/gookeyword/news/fed/fedchart.asphttp://www.bankrate.com/gookeyword/news/fed/fedchart.asp8/3/2019 Measurement of Interest- Slides
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Sources - Fed Rates vs GDP
http://www.federalreserve.gov/releases/h15/data/m/fedfund.txt
http://www.bea.doc.gov/bea/dn/nipaweb/SelectTable.asp
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Sources - FED policy tools
http://www.federalreserve.gov/releases/h15/data/m/fedfund.txthttp://www.bea.doc.gov/bea/dn/nipaweb/SelectTable.asphttp://www.bea.doc.gov/bea/dn/nipaweb/SelectTable.asphttp://www.federalreserve.gov/releases/h15/data/m/fedfund.txt8/3/2019 Measurement of Interest- Slides
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S p y
Open Market Operationshttp://www.federalreserve.gov/fomc/fundsrate.htm
Federal Reserve Banks discount ratehttp://www.federalreserve.gov/monetarypolicy/discountrate.htm
Reserve requirements
http://www.federalreserve.gov/monetarypolicy/reservereq.htm
How the Federal Reserve conducts its policyThe Economist: Monopoly power over money (Nov 18th 1999)
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http://www.federalreserve.gov/fomc/fundsrate.htmhttp://www.federalreserve.gov/monetarypolicy/discountrate.htmhttp://www.federalreserve.gov/monetarypolicy/reservereq.htmhttp://www.economist.com/displaystory.cfm?story_id=260590http://www.economist.com/displaystory.cfm?story_id=260590http://www.federalreserve.gov/monetarypolicy/reservereq.htmhttp://www.federalreserve.gov/monetarypolicy/discountrate.htmhttp://www.federalreserve.gov/fomc/fundsrate.htm8/3/2019 Measurement of Interest- Slides
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