Mathematics of Finance

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Mathematics of Finance
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We can use our knowledge of exponential functions
and logarithms to see how interest works. When
customers put money into a savings account, the
money is a loan to the bank and the bank pays
interest to the customer for the use of their
money. If money is borrowed the customer will
pay interest to the bank for the use of the banks
money.
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Suppose that a principal (beginning) amount P
dollars in invested in an account earning 3
annual interest. How much money would be there
at the end of n years if no money is added or
taken away? (Let t time and A(t) amount in
account after time t)
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Suppose that a principal (beginning) amount P
dollars in invested in an account earning 3
annual interest. How much money would be there
at the end of n years if no money is added or
taken away?
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By extending this pattern we find that, where P
is the principal and r is the constant interest
rate expressed as a decimal. This is the
compound interest formula where interest is
compounded annually.
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Example. Joe invests 500 in a savings account
earning 2 annual interest compounded annually.
How much will be in his account after 5 years?
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What happens when interest is compounded more
than one time a year?
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Let P principal, rannual interest rate,
knumber of times the account is compounded per
year, and ttime in years. Thus, r/kinterest
rate per compounding period, and ktthe number of
compounding periods. The amount A in the account
after t years is
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Let P principal, rannual interest rate,
knumber of times the account is compounded per
year, and ttime in years. Thus, r/kinterest
rate per compounding period, and ktthe number of
compounding periods. The amount A in the account
after t years is
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Example. Suppose P1500, r7, t5, k4
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Example. Finding time. If John invests 2300
in a savings account with 9 interest rate
compounded quarterly, how long will it take until
Johns account has a balance of 4150?
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Example. Finding time. If John invests 2300
in a savings account with 9 interest rate
compounded quarterly, how long will it take until
Johns account has a balance of 4150?
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Compounding Continuously

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Example. Suppose P3350, r6.2, t8 yrs
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It can be very difficult to compare investment
with all the different compounding options. A
common basis for comparing investments is the
annual percentage yield (APY) the percentage
that, compounded annually, would yield the same
return as the given interest rate with the given
compounding period.
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Example. What is the APY for a 8000 investment
at 5.75 compounded daily?
Let xAPY. Therefore, the value of the
investment using the APY after 1 year is
A8000(1x). So,
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Example. Which investment is more attractive, 6
compounded monthly, or 6.1 compounded
semiannually?
Let
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The 6.1 compounded semiannually is more
attractive because the APY6.19 compared with
APY6.17 for the 6 compounded monthly.
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Guided Practice
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• 1. Jean deposits 3000 into a savings account
  earning 3 annual interest compounded
  semiannually. How much will be in the account in
  6 years?

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• 2. Becky Jo deposits 10,000 into an account
  earning 2 annual interest compounded
  continuously. How much will be in the account in
  7 years?

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• Which investment is more attractive, 4
  compounded daily, or 4.1 compounded
  semiannually?

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Annuities Future Value
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So far we have only discussed when the investor
has made a single lump-sum deposit. But what if
the investor makes regular deposits monthly,
quarterly, yearly the same amount each time.
This is an annuity
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Annuity
a sequence of equal periodic payments
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We will be studying ordinary annuities deposits
are made at the end of each period at the same
time the interest is posted in the account.
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Suppose Jill makes quarterly 200 payments at the
end of each quarter into a retirement account
that pays 6 interest compounded quarterly. How
much will be in Jills account after 1 year?
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Since interest is compounded quarterly, Jill will
not earn the full 6 each quarter.
She will earn 6/41.5 each quarter. Following
is the growth pattern of Jills account
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End of quarter 1 200
End of quarter 2 200 200(10.015)403
End of quarter 3 200 200(1.015)
200(1.015)2609.05
End of the year 200200(1.015)200(1.015)22
00(1.015)3 818.19
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This is called future value. It includes all
periodic payments and the interest earned. It is
called future value because it is projecting the
value of the annuity into the future.
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Future Value (FV) of an Annuity
where Rpayments, knumber of times compounded
per year, rannual interest rate, and tyears of
investment.
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Example. Matthew contributes 50 per month into
the Hoffbrau Fund that earns 15.5 annual
interest. What is the value of Matthews
investment after 20 years?
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Example. Matthew contributes 50 per month into
the Hoffbrau Fund that earns 15.5 annual
interest. What is the value of Matthews
investment after 20 years?
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Loans and Mortgages Present Value
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Present Value
the net amount of money put into an annuity
This is how a bank determines the amount of the
periodic payments of a loan/mortgage.
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Present value (PV) of an annuity
Note that the annual interest rate charged on
consumer loans is the annual percentage rate
(APR).
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Example. Calculating a Car Loan Payment What is
Kims monthly payment for a 4-year 9000 car loan
with an APR of 7.95 from Century Bank?
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Kims monthly payment will pay 219.51 for 48
months.
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Review
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• When would you use the compound interest or
  continuous interest formulas?

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• When would you use the compound interest or
  continuous interest formulas?
• When the investor makes a single lump-sum deposit
  and allows the money to grow in an account
  without depositing more or withdrawing any for a
  specified amount of time.

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• What are the similarities and differences Future
  Value and Present Value?

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• What are the similarities and differences Future
  Value and Present Value?
• Both future value and present value require that
  a deposit or payment be made regularly (monthly,
  quarterly, yearly, ) and the amount deposited or
  the payment made must be the same each time.

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• What are the similarities and differences Future
  Value and Present Value?
• Future Value is used when you want to know what
  the value of an investment will be in the future.
  The future value is used most often to find the
  value of an annuity after a specified period of
  time.

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• What are the similarities and differences Future
  Value and Present Value?
• Present Value is most commonly used when you want
  to determine what the periodic payments should be
  on a loan or mortgage. Present value is the
  amount of money put into the annuity (or how much
  the loan is for) before interest is added.

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Can you determine what you would use to answer
the following?

• 1. Sally purchases a 1000 certificate of
  deposit (CD) earning 5.6 annual interest
  compounded quarterly. How much will it be worth
  in 5 years?

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• 2. Luke contributes 200 a month into a
  retirement account that earns 10 annual
  interest. How long will it take the account to
  grow to 1,000,000?

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• 3. Gina is planning on purchasing a home. She
  will need to apply for a mortgage and can only
  afford to make 1000 monthly payments. The
  current 30-yr mortgage rate is 5.8 (APR). How
  much can she afford to spend on a home?