measurement of interest- slides

8/3/2019 Measurement of Interest- Slides

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Measurement of Interest - IMATH 384, Review lecture 1

Andrei Ordine

Measurement of Interest - I p.1/4


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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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Answer:

Table 3: Simple interest



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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?


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%?


E l P t V l


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Answer:

(6)

Discount factor is

The present value of

payable at the end of theyear is


E l P t V l


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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.


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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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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Answer: 2.797%Source: http://wwws.publicdebt.treas.gov/AI/OFNtebnd

Note: the solution can be found quickly using a financialcalculator.


F l d fi iti
http://wwws.publicdebt.treas.gov/AI/OFNtebndhttp://wwws.publicdebt.treas.gov/AI/OFNtebnd


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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.


R l ld


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Real world

1. Historical rates

2. What and who determines the rates


Historical Rates


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Historical Rates

Key rates of interest, last 10 years


Notes to the historical rates chart
http://www.bankrate.com/gookeyword/news/fed/fedchart.asphttp://www.bankrate.com/gookeyword/news/fed/fedchart.asp


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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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NBER a includes the average prime rate ofcommercial banks into the list of lagging indicatorsof the economic cycle.

aNational Bureau of Economic Research


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.


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


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.asp


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Sources - U.S. Rates

Selected interest rateshttp://www.federalreserve.gov/releases/h15/Current/


Sources - History
http://www.federalreserve.gov/releases/h15/Current/http://www.federalreserve.gov/releases/h15/Current/


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Sources History

10 year historical interest rateshttp://www.bankrate.com/gookeyword/news/fed/fedchart.asp


Sources - Fed Rates vs GDP
http://www.bankrate.com/gookeyword/news/fed/fedchart.asphttp://www.bankrate.com/gookeyword/news/fed/fedchart.asp


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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


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.txt


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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)

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.htm


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