# Lecture Compound Interest Formula by alicejenny

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```									                 Lecture 5: Compound Interest Formula

MA170: Introduction to Mathematics of Finance

January 18, 2011

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Outline

1 Equivalent Compound Interest Rates

2 Discounted Value

Text: Chapter 2, Sect. 2.2, 2.3

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Equivalent Compound Interest Rates

Annual Eﬀective Rate
Deﬁnition
For a given nominal rate jm compounded m time per year, we deﬁne the
corresponding annual eﬀective rate to be that rate j which if compounded
annually will produce the same amount of interest per year

Formula

• At the rate i = jm /m, \$1 will accumulate at the end of one year to \$(1 + i)m
• At the rate j, \$1 will accumulate at the end of one year to \$(1 + j)
• Therefore, 1 + j = (1 + i)m ⇒ j = (1 + i)m − 1 = (1 + jm /m)m − 1

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Equivalent Compound Interest Rates

Annual Eﬀective Rate: Example #1 (2.2.1a,b)
Determine the annual eﬀective rate (two decimals) equivalent to the following
rates: a) j2 = 7%; b) j4 = 16%

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Equivalent Compound Interest Rates

Annual Eﬀective Rate: Example #1 (2.2.7)
Application The eﬀective interest rate is useful for comparison of investments
that have interest rates with diﬀerent compounding periods
A trust company oﬀers guaranteed investment certiﬁcates paying j2 = 8.9% and
j1 = 9%. Which option yields the higher annual eﬀective rate of interest?

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Equivalent Compound Interest Rates

Equivalent Rates

• Recall that the rates that have the same eﬀect on money over the same
period of time are called equivalent rates

• Two nominal compound interest rates are equivalent if they yield the
same accumulated values at the end of one year, and hence at the end of any
number of years

• Let the nominal rate jm be compounded m times per year.
Let the nominal rate jn be compounded n times per year.
These rates are equivalent if
m                  n
jm                       jn
1+                   =   1+
m                        n

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Equivalent Compound Interest Rates

Annual Eﬀective Rate: Example #3 (2.2.2c,d)
Determine the nominal rate (two decimals) equivalent to the given annual
eﬀective rate: c) j = 10% determine j12 ; d) j = 17% determine j365

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Equivalent Compound Interest Rates

Annual Eﬀective Rate: Example #4 (2.2.4a,c)
Determine the nominal rate equivalent to the given nominal rate?
a) j2 = 8% determine j4 ; c) j12 = 18% determine j4

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Discounted Value

Discounted Value at Compound Interest

• The accumulated value of P after n period at compound interest rate i will be

S = P(1 + i)n

• Being given the accumulated value S, ﬁnd the principal P:

S
P=          = S(1 + i)−n
(1 + i)n

• P is called the present value of S or the discounted value of S.
• \$P invested at time 0 grows to \$S after n periods.

• The factor             1
(1+it)n     is called the discount factor.

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Discounted Value

Discounted Value: Example #5 (2.3.6)
What amount of money invested today will grow to \$1000 at the end of 5 years if
j4 = 8%?

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Discounted Value

Discounted Value: Example #5 (2.3.12)
A note dated October 1, 2007, calls for the payment of \$800 in 7 years. On
October 1, 2008, it is sold at a price that will yield the investor j4 = 12%. How
much is paid for the note?

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Discounted Value

Recommended Exercises

Section 2.2 Part A: 1(d,e), 2(a,b), 3(b,d), 4, 8a

Section 2.3 Part A: 7, 9, 11, 14

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