02b Interest Rate

8/19/2019 02b Interest Rate

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syamsul irham, TIME VALUE OF MONEY

The terms ‘nominal’ and ‘effective’ enter into consideration when thecompounding period (i.e. interest period) is less than one year .

A nominal interest rate(r) is obtained by multiplying an interest rate that is expressed over a short time period by the number of interest periods in a longer time period:

nominal rate r ! interest rate per period x no. of periods

"or example if i ! #$ per month the nominal rate per year r is (#)(#%) ! #%$&yr

'ince nominal rates are essentially simple interest rates they cannot be used

in any of the interest formulas (effective rates must be used as calculated below).

Effective rates can be obtained from nominal rates via the following formula:

i ! (# r &m )m # where m ! no times interest is comp’d

"or example if i ! #$ per month effective i&yr ! (# *.#%%)#% # ! #%.+,$


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There are - general ways to express interest rates as shown below:

When no compounding period is

given, rate is effective

nterest /ate 'tatement 0omment

i = 12% per month

i = 12% per year

1!

When compounding period is given

and it is not the same as period of

interest rate, it is nominal

i = 1"% per year, comp#d semiannually

i= $% per uarter, comp#d monthly2!

i = effective 1"%&yr, comp#d semiannually

i = effective '% per uarter, comp#d monthly$!

When compounding period is given

and rate is specified as effective,

rate is effective over stated period


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1ominal rates can be converted into effective rates via the following e2uation:

i = 1 ( r & m!m ) 1

Where : i = effective interest rate for any time period

r = nominal rate for same time period as i m = no. times interest is comp#d in period specified for i

3xample: "or an interest rate of #.%$ per month determine the nominal

and effective rates (a) per 2uarter and (b) per year

a! *ominal r & uarter = 1.2!$! = $.+% per uarter

ffective i & uarter = 1 ( "."$+& $!$ ) 1 = $.+'% per uarter

b! *ominal i &yr = 1.2!12! = 1'.'% per year

ffective i & yr = 1 ( ".1'' & 12!12 ) 1 = 1-.$% per year

/olution:


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"or problems involving single amounts the payment period(44) is usually

longer than the comp’d period(04). "or these problems there are an infinitenumber of i and n combinations that can be used with only two restrictions:

(#) The i must be an effective interest rate and

(%) The time units on n must be the same as those of i ( if i is a rate

per 2uarter then n must be the no. of 2uarters between 4 and ")

3xample: 5ow much money will be in an account in 6 years if 7#**** is

deposited now at an interest rate of #$ per month8 9se three different rates:

(a) monthly, (b) 2uarterly and (c) yearly.

a! 0or monthly rate, 1% is effective:

0 = 1","""0&,1%,+"! = 13,1+4

b! 0or a uarterly rate, effective i&uarter = 1 ( "."$&$!$ )1 = $."$%

0 = 1","""0&,$."$%,2"! = 13,1+4

c! 0or an annual rate, effective i&yr = 1 ( ".12&12!12 )1 = 12.+3%0 = 1","""0&,12.+3%,-! = 13,1+-


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"or series cash flows first step is to determine relationship between 44 and 04

hen 44;!04 the only procedure(% steps) that can be used is as follows:

1! 0irst, find effective i per e5: if is uarters, must find effective i/quarter

2! 6hen, determine n, 7here n is eual to the no. of A values involved e5: uarterly

payments for si5 years yields n = 2'!

3xample: 5ow much money will be accumulated in #* years from a deposit

of 76** every + months if the interest rate is #$ per month8

/olution: /ince 8C, first step is to find effective i per si5 months!:i &+ mos. = 1 ( "."+ &+!+ ) 1 = +.1-%

*e5t step is to determine n:n = 1"2! = 2"

*o7, 0 = -""0&A,+.1-%,2"!

= 13,+2 5cel!


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hen 44


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hen the interest period is infinitely small interest is comp’d continuously

"or continuous compounding e2uation is: i ! er #

3xample: f a person deposits 76** into an account every - months for

five years at an interest rate of +$ per year compounded continuously

how much will be in the account at the end of that time8

/olution: *ominal rate,r, per three months is 1.-%. 6herefore,

ffective i& $ months = e"."1- ) 1 = 1.-1%

0 = -""0&A,1.-1%,2"!= 11,-4$

44;04


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hen interest rates vary over time use the interest rates associated with their

respective time periods to find 4.

3xample: "ind the present worth of a uniform series of 7%6** deposits in years

# thru , if the interest rate is =$ for the first five years and #*$ per year thereafter.

/olution: = 2,-""&A,4%,-! ( 2,-""&A,1"%,$!&0,4%,-!

= 1',+3$

An e2uivalent A value can be obtained by replacing each cash flow amount

with ‘A’ and setting the e2uation e2ual to the calculated 4 value as follows:

1',+3$ = A&A,4%,-! ( A&A,1"%,$!&0,4%,-!

A = 2-""

02b Interest Rate

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