Chapter 9: Mathematics of Finance

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Chapter 9 Mathematics of Finance
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9-1 Time Value of Money Problems

• Whenever we take out loans, make investments, or
  deal with money over time, common questions
  arise.
• How much will this be worth in 10 years?
• How will inflation eat into my retirement savings
  over the next 30 years?
• How much do I need to save each month to have
  1,000,000 at age 60?
• How much do I need to pay each month on a 30 year
  mortgage of 400,000 if the interest rate is
  5.25
• These are called Time Value of Money (TVM)
  problems.

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Common Notations

• P Present Value the current value of an
  investment or loan or sum of money.
• F Future Value The value of an investment,
  loan or sum of money in the future.
• r Annual Percentage Rate
• m Number of periods per year. Example If you
  make monthly loan payments, them m 12

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Common Notationscontd

• i Interest Rate per Period Example The annual
  interest rate is 6 and there are 12 payments per
  year. Then i .06/12 .005, or 0.5 per month.
• i r/m
• t Time, measured in years
• n Total number of periods. Example You have a
  10 year loan that is paid monthly. Then you have
  n 1012120 total periods. n mt
• R The amount of a Rent. This is the regular
  payment made on a loan or into an investment.

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Example

• Suppose you deposit 1000 into an account that
  compounds interest quarterly. The annual rate of
  interest is 2.3 and you are going to keep it in
  the account for 4.5 years. At the end of this
  time, the account will be worth 1108.72
• P 1000 F 1108.72 r .023
• m 4 i .023/4 .00575 t 4.5
• n 18

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9-2 Percent Increase/Decrease

• If a quantity increases by some percent, we can
  create a multiplier that helps us convert a
  beginning value to an ending value.
• To create the appropriate multiplier
• Percent Increase 1 i
• Percent Decrease 1 - i

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Example

• This year, the SCCC student population is 11,350.
• The administration estimates that will increase
  by 2 next year. How many students can we expect
  next year?
• The multiplier 10.02 1.02
• New student population 11,350(1.02) 11577

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Example

• The current balance of my retirement account is
  244,350.
• If the value of the account drops by 5.2 over
  the next year, what will be the new value?
• Multiplier 1 - .052 .948
• New Value 244,350(.948) 231,643

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9-3 Compound Interest

• When we invest money, interest may be applied to
  the account on a regular basis. For example, if
  we invest in an account that pays interest
  monthly, we say the interest compounds monthly.
  Anything in the account at the time of
  compounding gets interest added to it. In the
  monthly case, we have 12 compoundings per year,
  with each compounding representing 1/12 of the
  total annual interest rate.

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Example

• We invest 1000 in an account that compounds
  monthly. The annual interest rate is 3.6. If we
  keep it in the account for 5 years, adding or
  removing nothing, how much will be in the account
  at the end of 5 years?
• First, we need to note that the interest per
  period is i .036/12 .003.
• Lets begin by building a table

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Example
Period Previous Balance New Balance
0 0 1000
1 1000 1000(1.003)
2 1000(1.003) 1000(1.003)(1.003)
3 1000(1.003)(1.003) 1000(1.003) (1.003) (1.003)

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Example
Period New Balance
0 1000 1000(1.003)0
1 1000(1.003) 1000(1.003)1
2 1000(1.003)(1.003) 1000(1.003)2
3 1000(1.003) (1.003) (1.003) 1000(1.003)3

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Example
Period New Balance
3 1000(1.003) (1.003) (1.003) 1000(1.003)3
4 1000(1.003)4
5 1000(1.003)5
After 5 years, or 60 periods, we have After 5 years, or 60 periods, we have After 5 years, or 60 periods, we have
60 1000(1.003)60
After n total periods, we have After n total periods, we have After n total periods, we have
n 1000(1.003)n
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Compound Interest Formula

• If P dollars earn an annual interest rate of i
  per period for n periods, with no additional
  principal added or removed, then the future
  value (F) is given by
• F P(1i)n

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Example

• A bank account is opened with 4,000 in the
  account. It earns 6 annual interest. If it earns
  interest quarterly (four times per year), then
  what is in the account after 10 years?

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Example

• Suppose you invest 500 today at an annual rate
  of 1.5, compounded daily. How long before the
  balance doubles?

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9-4 Rule of 72

• Given some investment that grows at an annual
  interest rate, r, (not expressed as a decimal),
  then the amount of time in years it takes for the
  investment to double is approximately

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Example

• How long will it take for an investment to double
  if it earns 5 annual interest?
• Note that estimating the doubling time does not
  require that we know how much is originally
  invested!

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Example

• If you want your investment to double in 30
  months, what annual interest rate do you need to
  secure?

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9-5 Yield

• Because each compounding acts on the original
  balance and any interest that has been previously
  earned, the net interest earned will not be the
  same as the annual interest rate at the end of
  the investment. The true interest rate earned
  is called the Yield.

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Example

• Invest 100 for 1 years, compounded monthly at an
  annual rate of 12.
• F 100(1.01)12 112.68
• This represents a yield of 12.68, which is
  higher than the original 12 stated above. The
  yield is often called the Annual Percentage Yield
  (APY). Always ask what this is when you take out
  a loantime, compounding and bank fees can
  substantially increase your rate of interest and
  therefore your total payments due!
• APR 12
• APY 12.68

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Yield Formula

• The formula for yield in the t 1 year case is

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Where did the formula come from?
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Why is it important to know the APY?

• The APY, or yield, is helpful since it simplifies
  calculations. If we know the APY, then it does
  not matter how many times we compound per year
  because the APR will give us actual percentage
  increase at the end of a year.
• APR Annual Percentage Rate or nominal rate or
  rate
• APY Annual Percentage Yield or effective rate
  or yield

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Example

• If 375 is invested with an APY of 5.22 for 8
  years and 3 months, what is the Future value of
  the investment?

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9-6 Annuities

• When additional payments or deposits (rents) are
  made at regular intervals into an investment,
  then we call these annuities
• Ordinary Annuity Payment is due at the END of
  each period.
• Annuity Due Payment is due at the START of each
  period.
• This will complicate our Future Value
  calculations.

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Example

• We invest in an account that compounds annually.
  The annual interest rate is 3. If we add 800 at
  the end of each year (an ordinary annuity), how
  much will be in the account at the end of 5 years
  total?
• Lets look at a picture of what is going on here.

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Example
Year 1
Year 2
Year 3
Year 4
Year 5
Start

Investment Period
800
800
800
800
800
Each of these 800 investments earns interest for
a different period of time. Hence, the value of
each of these deposits is different at the end of
the 5-year period.
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Example
The End
End Year 1
End Year 2
End Year 3
End Year 4
End Year 5
Start

Investment Period
800
800
800
800
800
This one is worth 800(1.03)4 at the End 900.41
This one is worth 800(1.03)3 at the End 874.18
This one is worth 800 at the End 800
This one is worth 800(1.03)1 at the End 824
This one is worth 800(1.03)2 at the End 848.72
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Example

• We can add all of these up
• 800(1.03)4 800(1.03)3 800(1.03)2
  800(1.03)1 800
• 900.41 874.18 848.72 824 800
• 4247.31

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A General Formula

• Now imagine if the monthly payments were
  deposited and monthly interest credited. We would
  then have 512 60 different deposits to find
  the values for so we can add them up.
• To avoid this inefficiency, we instead use the
  following formula, which is equivalent to going
  through that process.

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A General Formula

• If R dollars are paid at the end of each period,
  with an interest rate of I per period, then the
  Future Value of the Annuity is

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Where did the formula come from?

• We can generalize the example before and think
  about adding R R(1i) R(1i)2 R(1i)3
  R(1i)n-1. Let us call this sum S.
• Hence, (1i)S - S R(1i)n - R
• S (1i - 1) R(1i)n - R
• Hence, the sum S is
• (R(1i)n - R)/i

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Check

• We invest in an account that compounds annually.
  The annual interest rate is 3. If we add 800 at
  the end of each year (an ordinary annuity), how
  much will be in the account at the end of 5 years
  total?

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Example

• What is the future value if you invest 95 per
  month for 7 years at an annual rate of 3.75?
• R 95
• i .0375/12
• n 127 84

Note that I try to keep as many decimal places as
possible until the end
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FV for Annuities Due

• When payments or deposits are made at the
  beginning of a period (rather than at the end as
  in the previous examples), an adjustment is
  needed.
• We can view each payment as if it were made at
  the end of the preceding period. This would
  require one more payment (n1 total) than usual
  and would require that we subtract the last
  payment so we dont overpay.

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FV for Annuities Due
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Example

• If 300 payments are made at the beginning of the
  month for 18 years (a college savings fund), what
  is the Future Value if the annual interest rate
  is 5.5

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9-7 Future Value (FV) on Excel

• The FV command will do these computations for us
  automatically.
• Command Format
• FV(rate, nper, pmt, pv, type)

This is i, the rate per period
This is n, the total of periods
This is R, the amount of rent
This is P, the Present Value
Blank for ordinary annuity, 1 for annuity due
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Excel Examples

• If 300 payments are made at the beginning of the
  month for 18 years (a college savings fund), what
  is the Future Value if the annual interest rate
  is 5.5
• What is the future value if you invest 95 per
  month (paid at the end of the month) for 7 years
  at an annual rate of 3.75?

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9-8 PV of Annuities

• Suppose you have an ordinary 20-year annuity that
  you pay 500 into at the end of each quarter. The
  annual interest rate is 7.
• What is the lump-sum of money which should be
  deposited at the start of the annuity that would
  produce the exact same amount of money at the end
  of the period, without any additional payments?
• This is know as the Present Value of the Annuity

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Present Value of an Ordinary Annuity
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Example

• Suppose you set up an ordinary annuity account
  which is to last 10 years and earn 4 annual
  interest rate. If your rent payment is 150 per
  month, how much do you need to deposit as a lump
  sum up front to achieve the same end result
  without any regular payments?

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9-9 Present Value (PV) on Excel

• The PV command will do these computations for us
  automatically.
• Command Format
• PV(rate, nper, pmt, fv, type)

This is i, the rate per period
This is n, the total of periods
This is R, the amount of rent
This is the FV you want after the last
pmt Optional
Blank for ordinary annuity, 1 for annuity due
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• Can you figure out how this comes from the
  formula for the Future Value of an Ordinary
  Annuity?

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Loan Payment Formula
Start here with the original formula
Divide both sides by to get R alone
Here is the formula for the loan payment.
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Loan Payment Formula

• Using basic algebra, we can rewrite this as

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Example

• Suppose you want to buy a home and take out a
  30-year mortgage for 240,000. The annual
  interest rate is 5.75.
• What is the monthly payment?
• What total amount of money do you pay over the
  life of the loan (assuming all regular payments
  are made)?
• How much of your total payments is interest?

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Example (a)

• We use the formula to get 1400.57 per month .

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Example (b) and (c)

• The total amount of money we pay is
• 360x1400.57 504,205.20
• Hence, the amount of interest paid is
• 504,205.20 - 240,000 264,205.20

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69

• You borrowed 150,000, which you agree to pay
  back with monthly payments at the end of each
  month for the next 10 years. At 6.25 interest,
  how much is each payment?

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9-10 Loan Payments (PMT) on Excel

• The PMT command will do these computations for us
  automatically.
• Command Format
• PMT(rate, nper, pv, fv, type)

This is i, the rate per period
This is n, the total of periods
This is the present value of the loan
This is the FV you want after the last
pmt Default0
Blank for ordinary annuity, 1 for annuity due
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9-12 Adjusting for Inflation

• Inflation can seriously devalue a loan or asset
  over time.
• For example, if the average inflation rate is
  3.5, how much will 50 be worth in 5 years (in
  terms of todays dollars)? In other words, in
  five years how much can I buy with a 50 bill
  compared to what I can buy today?

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Example
Important Notice that we substituted 50 for F
since that is what we know we will have in the
future. We solve for P since we want to know what
the Future 50 is worth in Present dollars.

• F P(1i)n
• 50 P(1.035)5
• 50 P(1.187686306)
• 50/(1.187686306) P
• 42.10 P

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Terms

• Nominal Dollars are those that have not been
  adjusted for inflation.
• Real Dollars Present Dollars are those that
  have been adjusted for inflation and therefore
  reflect the spending power of some future amount
  of money in terms of todays dollars.

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• Well disregard amortization tables.