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This document explains the concepts of simple and compound interest, including formulas for calculating interest earned on investments. It provides examples and problems related to investments with varying compounding frequencies and explores continuous compounding. Additionally, it discusses how to determine present value and effective interest rates for different investment options.

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5 views4 pages

PDF 3

This document explains the concepts of simple and compound interest, including formulas for calculating interest earned on investments. It provides examples and problems related to investments with varying compounding frequencies and explores continuous compounding. Additionally, it discusses how to determine present value and effective interest rates for different investment options.

Uploaded by

aditisharma072005
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Compound Interest

In this handout, we will use exponential and logarithmic functions to answer questions
about interest earned on investments (or charged when money is borrowed).

Simple Interest If a principal P is borrowed for a period of t years at a per


annum interest rate r, expressed as a decimal, the interest I charged is

I = P rt

Compound Interest: interest is earned (or charged) on a regular schedule (e.g. once a
year, every month, etc.); at the end of each payment period, interest is earned on
principal and on previously earned interest

Example. Gertrude invests $300 in a savings plan that earns 11% per annum compounded
quarterly. How much will be in Gertrude’s account after one year?

principal + interest

initial investment

end of 1st quarter

end of 2nd quarter

end of 3rd quarter

end of 4th quarter

Compound Interest r nt


A=P 1+
n

A = amount in account after t years


P = amount invested (or borrowed)
r = per annum interest rate (expressed as a decimal)
n = number of times interest is compounded per year
t = time (in years)

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Problem 1. Suppose $300 is invested in an account that pays 11% per annum.

(a) If interest is compounded quarterly, how much money will be in the account after 2
years? (This amount is called future value.)

(b) What happens to the amount in the account after 2 years as you compound more and
more times per year?
payment period # times compounded per yr. ≈ amount after 2 yrs.
annually
semiannually
quarterly
monthly
daily

continuously

Continuously Compounded Interest The amount A after t years due to


a principal P invested at an annual interest rate r compounded continuously
is
A = P ert

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Problem 2. A savings plan offers a rate of 8% compounded quarterly. How much should
be invested now in order to have $1000 after 5 years? (This amount is called present
value.)

Problem 3. Suppose $500 are invested at 9% per annum. If interest is compounded


continuously, how long will it take for $500 to double to $1000?

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Problem 4. If interest is compounded annually, what interest rate should you seek if you
want to triple your investment in 10 years?

Problem 5. Consider the following two investments:

Option 1 invest $1000 at a rate of 5%, compounded quarterly

Option 2 invest $1000 at rate R, compounded annually

What rate R should you seek in order to have the same amount in each account after 1
year? (This rate is called the effective interest rate or the annual percentage yield.)

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